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PARTIAL DIFFERENTIAL EQUATION APPROACH TO
MATHEMATICAL MODELING OF IMAGE AND BIOLOGICAL
SURFACES

presented by

Yuhui Sun

has been accepted towards fulfillment
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PARTIAL DIFFERENTIAL EQUATION APPROACHES
TO MATHEMATICAL MODELING OF IMAGE AND
BIOLOGICAL SURFACES

By
YUIIUI SUN

A D I S S F. R TAI I ( )N
Submitted to

Michigan State Ifiiiwrsity
in partial fulfillment of the I‘vqnirmnmits

for the (lvgl‘ri*(‘ ()I'

li)(i)("l'()l{ OF PHILOSOPHY

I)(‘[):II'IIII(‘III of lel'limlml'it's

ZIIIIT

ABSTRACT

PARTIAL DIFFERENTIAL EQUATION APPROACHES TO MATHEMATICAL
l\l(")DEI.ING OF IMAGE AND BIOLOGICAL SURFACES
By

Yuhui Sun

Partial differential equations (PDEs) are widely used in the theoretical modeling
of scientific and engineering problems. Recently, they have been introduced to
the mathenrat-ical modeling of geometric design and image analysis. The present.
dissertation is dedicated to PDE approaches to a class of important problems in
image surfaces and biological surfaces. To this end. \r'arious geometric PDEs are
[)I‘()[)(IS(‘('l and their pert})rmances are analyzed.

We first deyeloped an evolution operator based single time step method for image
surface processii’ig. The key conmonent of the proposed method is a local spectral
eyolution kernel (LSEK) that analytically integrates a class of 1')ara.bolic partial
differential equz-ttions. From the point of yiew PDEs. the LSEK provides the ana-
lytical solution in a single time step. it is of spectral accuracy. and free of instalnlity
constraint. Frmn the point. of image/signal processing. the LSEK gives rise to a.
family of low-pass Iilters. These filters contain emrtrollable time (IOlt’l‘V’ and ampli-
tude sealing. The new evolution t‘)1’)(~>rator based method is ecmstrticted by pointwise
adaptat ion of anisotropy to the coeIIicients of the LSHK. The Perona-Alalik type of
anisot’rt’ipic dill'usion scheme is im-orpm‘at ed in the LSEK for image denoising. A
forward-I)ackwartl diil'usion process is adopted to the LSEK for image del:>lurring
or sharpening. A coupled PDE system is modified for image edge detection. The
resulting image edge is utilized for image enhancement. Extensiye computer experi-
ments are carried out to demonstrate the performance of the proposed method. The

major z-idyantages of the proposed method are its single step solution and readiness

for multidimensitmal data analysis.

We then proposed a new ccmcept. molecular multiresolution surfaces. for biolog-
ical surfaces modeling. L\Iolecular nniltiresolution surfaces contain not only a family
of molecular surfaces. which are. ccn‘respoiidii1g to different probe radius, but also
the solvent accessible surface and Van der \Vaals surface as extreme cases. All the
proposed surfaces are generated by the diffusion map of continuum solvent over the
van der Waals surface of a molecule. The LSEK is used for the numerical integra—
tion of the diffusion equatitm in a single time step. \Ve compared our method with
MSMS, one of the most. widely used molecular yisualizz‘ition tool. and found that our
results agree well with those obtained by the .\IS.\IS. Applications of our method to
the electrostatic analysis are considered for a set of twenty three proteins.

It is noticed that the molecular surfaces in earlier models suffer from geometric
singularities. such as cusps and self-intersecting surfaces which are not only unphys-
ical. but also deyastate iu‘nuerical siirnilations. \\'e therefore introduce a noyel con-
cept. the constant. mean—curyature molecular surface (CHRIS). which encompasses
a family of l’iiomolectilar surfaces for the theoretical modeling of biomolecules. The
C\I.\IS unilies an earlier minimal molecular surface as well as the popular molecu-
lar surface. Ilowtwer. unlike the molecular surface. the CMMS is typically free of
geometric singularities. The CMka is formulated based on the theory of differen-
tial geometry. A computalional scheme is proposed for practical control of linean
curyature eyolution of a hy])ersurface. which leads to the CMMS after taking an ap-
propriate level set. Extensiye numerical ex1')eri1nents are carried out to demonstrate

Ifillt‘ proposed (fulltft-‘Ifl and algorithm.

Acknowledgments

This dissertation is the result of five—year work that I have accomplished with the.
support of many people. I am very pleased to have this opportunity to express my
gratitude to all of them.

First of all. I would like to thank my supervisor. Professor Guowei \V'ei. for giving
me the opportunity to work in his research group. It. is such a great experience that.
will be the most valuable part of my life. I feel fortunate to have met: him and have
him as my supervisor. ,I appreciate all of his support. guidance. patience. continual
understanding and encotiragement throughout my PhD study.

I am very grateful to all the members of my advisory committee who moni-
tored my work and took the effort to read my thesis and provide me with valuable
comments and insightful feedback: Professor Tien-Yien Li. Professor Chichia Chin.
Professor Andrew Christlieb and Professor Moxun Tang“. I thank all of them yery
much.

I would also like to acknowledge the support. of some very special individuals who
helped me immensely by giving me enctiuragement. numerous fruitful discussion and
friendship: Sining Yu. \Veihua (Ieng. '/.han (.‘hen. Shan 7.hao. Duan Chen. Li Yang.
\Venmei Huang and Maureen .\Iorton. I am really glad to have gotten to know all
of them in my life.

Sincere thanks are given to Ms. Barbara .\Iiller. (Tiradnate Secretary in the

Department of Mathematics. for her generous help throughout my graduate study

at Michigan State University.

I would love to express my utmost ap];)reciz-ttion to my husband. Yongcheng Zhou.
who is my everlz-rsting strength and support. My life. is beautiful due to him. I am
grateful for the extraordinary chance of having him as my husband. best. friend and
companion.

Finally. I would like to thank my parents. my brother and my sister for their love
and eneotu'agement. Thanks go to them for always l’)('>1ieviiig in me and supporting

me in whz—‘ttever I do.

Table of Contents

Chapter 1 Introduction to PDE—Based Image Processing

1.1
.1.)

.a~

Linear and Nonlinear Diffusion Processes in Image Processing
Numerical Schemes for Diffusion Equattions

Chapter 2 Theory and Algorithm: Evolution Operator Based Single
Time Step Method

2.1
2.2
2.2.1
2.4

Local spectral evolution kernels .
Filter properties

Numerical test

Adaptation of anisotropy

Chapter 3 Applications of Single Time Step IVIetliod in Image Process-

ing
3.1

31.21

11111g11 de 11'11111111g . . . . . . . . . . . .

3.1.1 ’rief int11'1du1ti1111 to image 111111111ing .

3.1.2 (1e 111,1al 111111111 .

1.1.31 Results 11l.'111_1111111'1‘i11g

Image denoising . . . . . . . .

31.2.1. l1rief introduction to image denoising

31.2.2 General models and results .

Image 11111111111111111111 . . . . . . . . . . .

1 '11 I11ief introduction of image 111ge dete1tio11

313’ 1. ()111 te1l111111ues

11.1.1 Results of image edge 1,11,1tecti1111

Digital image 11111111111'13111er1t

3.1.4.1 Brief introduction to image enhancement

3.1.11.2 Our techniques and results of image enhancement
Summary 11f .\lathe111atical Modeling of Image Surfaces

Chapter 4 1\v'loloc11lar 1\'Iultircsolution Surfaces

1.1

111111'111111111111 . . . . . .
4.1.1 Smtac es in 11111111111111 1111 1s
1.1.2 Objective of this chapter
.\l(.'l‘111111s .

Results and Discussion

1.3.] Singularity ’1‘11st

1.3.2 General test

1.31.51 (‘avity and pocket test

vi

001—s|-‘

4t)
11)
41
'11
111'
116
17
4.8

63
(1'11
61
(511'
(17
7:1
7:1

711

1.3.1 Large 1111‘1lecule test ........................... 77

4.51.5 Applications ............................... 78

11.11 (i‘1’1111'111sio11s .................................... 831
Chapter 5 Singularity-free lVIolecular Surfaces 85
5.1 Introduction ................................... 86
5.2 Tl1e1.)retical 11111111111111,T .............................. 87
5.2.1 H11'1'11'1'surface and its mean curvature ................. 87

5.3 Results ...................................... 93
5.3.1 Validation ................................ 93

5.11.2 Cavities ................................. 94

5.3.31 Si11g11larities ............................... 95

5.4 Applications ................................... 95
5.5 Conclusions .................................... 99
Chapter 6 Thesis Contributions and Hiture Work 106
6.1 'I‘liesis c.011tril1uti1'111s ............................... 106
(1.2 Future \Vork ................................... 108
References 110

List of Tables

;1

Errors of the solutit‘ms for the 1—D diffusion equation.

Errors of t he solutions and the CPU time [or the 2-1) dilhision equation.

Main operations for one step of the m—diinensionzil Perona hlzilik,
A08. /\— resolution, and the LSEK. (M/D: l’l’lllltlpllCdlilUIIS and di—
visions: A/S: addition and subtractions: N: the number of pixels;
m: the dimensions of the [)l’()l)l(‘lll eonsidered; H": the hull of the
eomputntionnl lmndwidth.) .

("Idl-"l‘iiiie eoinpzu'ison between the forwnrt'l Euler selien'ie and the
l..SlCl\'. Running the linear diffusion equation with :l : l to time
i :2 3t).

(_‘oiiipzii‘isoiis of total inoleeulzn' surl'eu'e urea and eleetrostntie solva—
tion energy.

Comparison of total molerulzu‘ surlzu-e urea and eleetrostzitie solmtion

(-‘nergy.

Error analysis in surl'aee en'eu mleulntions for diatomie system with
(Jay. 3) : ('—.'il.l.").(7).()\). (3.13.0.0) and I'Hny : ‘2A .

viii

List of Figures

V

V‘v
>—--5

\A.
W

s.»
to—

:Hi

The LSEK in the coordinate domain with xlU) : BU) : CU) = t). . .

The fretpienev responses of' the LSEK with BU) 2 CU) = 0.1)] = ($4

and ill), 2 88. The solid lines from outside to inner are for A; :

Ulluolagltolngruulolng ........................

The frequeney response of the. LSEK with AU) = BU) 2 CU) =

‘ .

0.1)] = 6 and .sll), : 88 ...........................

The frmptetreV responses of the LSEK with ill : (ti-l. AI], 2 88, x'lU) 2

(no:0mmmmmmIfitospzuoanpzauqupzyr.

The frequeney responses of the LSEK with .-\l : (i-l. ill), : \8. AU) 1—-
lj’U) : t) and different ("f/r (a)("';, = ‘2: (l))(';, : —2 ..........

The numerit-al solutions otthe l—l) (litlusioii equation. rl‘he (‘enterlines
in the deseending order are at time t : 1.0. $311.51). 7.0 and 9.0. .

l‘)ifl'usion (-oeflieient .‘l(:s) plotted as a hint-tion of the smoothed gra-

dient. magnitude. ....... . . . . . .................

l_)el,)lurring proeess of the parrot. image. ta) ()riginal parrot image:
(h) Gaussian liltll‘red image: (e) l’arrot image after delilurring.

Zoomed parrot images for del>lurring proeess. (a) Original: (h) Blurred:
(e) li’roeessed. ......... . . . . . .................

l.)el)lurrin” )roeess of the eolumliia image. a Gaussian blurred im—
l“) m

age: (l)) (‘()lllllll)lél image alter (’lel.)llll‘l'i11g. . . .............

Denoising pi'oeess [or the lena image. (a) Noisy Lena. image. with

PSNiltz'ZUH: (h) Restored Lena image with l’SNT{:2tl.t$l .....

Denoisinu iroeess for the lirain imaee. (a) Noisx' lirain image with
(-5 F3 . D

l’SNRzlfiTt): (Ii) Restored ln‘ain image with l’SNR:‘25.lt~T. . . . . . .

"‘t) plotted as a tum-tion of. the smoothed gl‘zulient. magnitude with
E: tl.t).t)t)2.t).t)l.tl.l (front top to liottom): irlfi: 2‘ L2 Wt).

18

19

.37

37

4 . :3
4 . :1

gig) plotted as a function of the smoothed gradient magnitude at. the
following times t:0.()(ll. t=0.()l. t:0.1, and t:1t)t) (from bottom to
top): 3216; (- 20.01.

Time (.iependent (jlenoising proeess. (b) lit-istored Lena image with
PSNR.:29.41. (b) Restored brain image. with PSNR=25.22.

Edges of the Barbara image deteeted by different edge detectors. (a)
The original Barbara. image: (b) edges detected by the Sobel detector;
(e) Edges detected by the Prewitt detector; ((11) Edges detected by
the Canny deteetor: (e) Edge deteeted by the anisotrt‘ipie diffusion
scheme; (f) Edges detected by the LSEK-based edge detector.

l\lo(’leratel‘\' high texture images. .

The edges of 1'n('_)(,leratel_\' high texture images.
High texture images.

The edges of high texture images.

Lungs and Abdomen CT images.

The edges of Lungs and. abdomen CT images.
The manimograplis used for image enhaneeinent.
Enhaneed mammograplis.

The CT images of mouse used for image enhaneement.
Enhaneed CT images of mouse.

Illustration of various surfaees of l)ioim)leeules.

.\lultis(-;Ide surfaees marked up by flow eontours. (1a) The diffusion
map of a fractal (Courtesy: The. .\Iandelbrot Explorer Gallery); (b)
The nuiltiresolution surfaees of a water moleeule.

Illustration of multirestiilution molecular surfaees.

The aetix'e grid points (squares) in the proeess of moleeular surface
generation. Dashed line: Solvent aeeessible surfaee. The diffusion
t-oetfieients D : (Mil at aetive grid points and D : t) at all other grid
points.

39

39

\l
[0

r“:
Cf!

1:1 . 6

\l

1.9

1.10

CIT
p-a

Left column: .\lolecular surfaces of two-. three- and f(.1ur-at.om sys-
tems by MSMS: Middle column: Molecular surfaces of two.. three-
and four-atom svstems bv diffusion; Right column: Corresponding
comparisons of cross sections :1" = () (Circles: MSMS with rp = 1.4
A: Solid line: our molecular surface with 1'], = 1.4 A). For the di-
atomic system. (.13.;1/12) = (— ‘2. 3. (1.0) and (2 3.0.0). For the three-
atom s)-'st.(‘111. ($.11. :) : (— ‘2. 3 t) 0). (‘2. 3 (1. 0) and (0.3.9810). For
the four-atom system. (17.3}. 2)— — (— 3. t). 0). (0, —‘2.(1'.()). (30,0). and
(0. 2.6. 0).

Molecular surfaces of the buckball and their comparison with those
obtained by the .\~ISMS (Solid lines: present; Circles: MSLVIS). (an
der \Yaals r 2: 1.5 A; Probe radius 1‘1) 2 1.4 A. (a) The n1(,)lecular

surface: (11) The cross section at .1' : t):

l\lolecular surfaces of the open l'1uckb1-1ll and their comparison with
those obtained by the MSMS (Solid lines: [‘nfesent: Circles: MSMS).
Van der \Vaals 1" : 1.5 A; Probe radius 1'], = 1.0 A. (1-1) The molecular
surface: (b) The cross section at .1: =- 0: (c) The cross section at 1/ : 0:
(d) The. cross sect ion at Z : t).

'.\lolecul1~1r mulliresolut-ion surfaces of cvclohexane and their compar-
ison with those obtained bv the 1\lS.\lS (Solid lines: pre sent; Circles:
.\lS.\lS). (a) The solvent excluded surfa(ce: ((b 1) The cross section at
.1' : "(1.9: (c) The cross section at y: ).(i: (d) The cross section at
.: :—. ~05.

.\lole(,-11lar multiresolution surfaces of the cell division protein. (a)
The van der \\'aals surfaces: (b) The solvent excluded surface.

An illustration of the continuum dielectric model in bionu1lecular sim-
"1,,llati11ns.

Graphic cmnparisons of solvation energies and total 1111,1lecul1’1r surface
areas for ‘235 molecules. Solid litres are perfect matches 1/ = .1'. (a) Plot

of solvat ion energies for our molecular surface vs. .\lS.\lS: (b) Plot of

molecular surface areas for our molecular surface vs. MSMS.

Surface. electrostatic potentials of (:(1/1 ,1) factor (1151,51) ID: 11163) pro-

jected onto its molecular surface. if..eft: Generated with our molecular

surface: Right: Generated with the MSMS. Colors are. chosen accord-

ing to the magnitude of the potential as indicated with the color bar.

1111111 1] condit 11111.1( 1)lllust1at 11111, of initial maps of 8(1' 1/. :) at the cross
section 2' : ()2 1' : "MW and 1' : "1-1/H'+T1 (b)5(.1._1/. .: : 0) shows
11 familv of level surfaces.

xi

p,—

78

TO

80

83

81

91

5.2

2;!
C1»:

(.51
L;—

(J!
of!

in
*1

The constant 111ean curvature surf1-1ces of 11 diatomic molecule plotted
at H“ = (1.133.(1.(1T4. (1.000 and ~(.1.(133 in (a). (b). (c) and ((1). Their
comparison with the corres11onding molecular surface (circles) with
the probe radius 1'], = 1.4A is given at the cross section 2 = 0 in
(e). (f). (g) and (l1). The coordinates of the. diatomic. molecule are
(1./z) = (_—‘2.4. (1. (1) and (2.4. (1. (1). The van der \Yaals radius is set

it) I‘m/(1' : 2.0.‘A. . . . ..........................

The constant mean curvature surfaces of a seven-atom system and

their comparison with the corresponding molecular surface with Tim/(1’:

1.(1.(1A. (a) H”— —— (1.(1T4j.(l1) HO— — (1: (c ).The comparison of cross section
2 = (1 (Cirles: .\IS with r—p -— 1.4A: Solidi ine: H” 0.074; Dashed

line: HO : (1). The, coordinates are generated by % rotations of a

‘2.4A diatom. ...............................

The CAIMS of the buckyball Cm with van der \Vaals radius and
scaled atomic radius. (a) Van der \Vaals radius. I‘m/11' = ‘20 A: (c)
Atomic radii. ('1-1111' 2 1.7 A. (e) Scaled atomic radii. I‘m/n" = 1.4
A: (g) Scaled atomic radii. I'm/H' : (1.8 A (l1). (d). and (f) show.
respectively. the C‘.\l.\IS of the same buckyball as in (a). (c). and (c).

with half of the data. of .S(.1'. ,1/. .3) removed. . . . . .......... 100

The C.".\I.\IS of the open buckyball (‘51) with van der \Vaals radius
11nd scaled atomic radius. (a) Van der \Vaals radius. I'm/(1' 2: '20 A;
(c) Atomic radii, 1'Hm' :- 1.7 A: (e) Scaled atomic radii. I‘m/(1' : 1.4
A: (g) Scaled atomic radii. l’H/H' : (1.13 A (I11. ((1). and (f) show.
re spe c tiyely. the C .\I\IS of (hes same buckyball as in (a). (c). and (e).

with I111 If of the data of .5(.1‘. ,1/. :) removed. . . . . .......... 1(11

1\Iolecular surfaces of two-. three- and four-atom systems and their
correspontling constant mean—curvature surfaces. For the diatomic
system. (121/. 3) : (—1'5._ll:3.(1.(1) and (3.15. (1. (11:1,.(m' = ‘2 A For the
tln‘ee—atom system (.1' .1/. :i) : (—2.3.(1.t1 ). (2. 5. (1(1). and ((1. 3. 981. (1):
1'1“)” : ISA. 1'], : 1.? 1A. -- (1.1)A. 1111 the foul—atom system.
(.1.1/. r.) : (—~3_5.(1.(1). ((.1 —"2.(1()).(3.(1.t1). and ((12.6.0): I'm/Hr

1.5 A. [p : l.r).\. T : (1.94%. ....................... 102

The comparison of the C.'1\l.\lS (a) and the .\IS (I1) of the hemoglobin

(1)1113 111: 1 115.1111. ............................. 111:3

Comparison of the electrostatic solvation free energies AC of twenty
three proteins listed in 'l‘able 5.1. (a) Plot of solv1;1tion free energies
for CLAIMS vs. .\IS.\IS: Solid lines are ,1/ :: .1': (b) iclativc ('litf'erences

of solvation free energies. . . . . . . . . . . ...... . ....... 1114

xii

5.1) Surface electroste-ttic potentials of (1012'. [1 factor (P151) 11): 12163) pro-
jected onto its molecular surface. Left: Generated with the CMMS:
Right: Generated with the. MSMS. Colors are chosen according to
the magnitude of the potential as indicated with the color bar ..... 1053

xiii

Chapter 1

Introduction to PDE-Based Image

Processing

1.1 Linear and Nonlinear Diffusion Processes in
Image Processing

Image denoising. restorz-ttion. edge detection and enl’i-(‘uicement are fundamental
problems in image processing. computer vision and artificial intelligence (11.7]. The.
solution to these problems is crucial to automatic control. robotics. imaging. tar-
get tracking. and tt1lecomnnmication. .A variety of 111ethods have been proposed
to tackle these problems in the past few decades. Among these 111ethods. partial
differential equation (1)1115) 1:1ased nietlmds have recently received much attention.
\Vitliin (17(1) introduced a parabolic evolution equation (1.11.. the heat ecpiation) for
image denoising. The basic idea behind \Vitltin's approach is to evolve an original
image under a diffusion process so as to effectively remove noise in the image surface.
Such process is formally equivalent to the standard Gaussian tiller. Ilis al.111111ach

has been intensively studied in terms of the scale—space process ‘2 111. 1(171. Based

on a discrete version of the diffusion equation

5+1 s . .' s .
u‘). : u.)- + ”(”3—1 — 219+ (137%) (1.1)
(If; : 21.j(t1) j: 1......=\7. and s 20.1....‘3,
Where
RZbAf. .S'=(f,2—f1)/Af, (1.2)

\Vei and Zhao showed that. [lb-l] the effect of the diffusion after S 'terations can be

expressed as a (‘c’involtttitnl
uV : Z ll'(f.'..5')tq.(t1l, (1.3)

where the filter lit/125') can be explicitly given by

at}: ' 2 »
‘5 A)’ {If/u". .5. 2/1.). V .5 — ff “"011

 

no. 5) :- t‘f’f” ,, (1,1)
\‘(n5"" lli+l)/2 I“ gv .)} 1 V SY _ I]. 11
24/):1 {If .1.- I ..__ l k “(a (-
and
. Stir-9' "' '1 —— 21?“
t/(lx‘. S. It) : Q's—Iv}: (‘ S'm/A Jr) . (1.5)
(fi—)l(‘—+_}——)llzl

'l‘herefore the result. of the multiple-step time evolutitm described in (‘liscrete t'lif‘fusion
liq. (1.1) can be exactlv obtained via the single—step convolution (1.3). It can be

\'t"*1'ifi(‘t’l that

1+8
Z tl'(l.-..S"):l. (1.0)
t- a; -- s
and
ll'tjwv/gS)xiii/.28). w.- _:r ...... s. (1.7)

A two-dinn—nrsional (2D,) generalization of Eq. (1.3) is straightforward. \Vei and
Zhao also showed that. ll'(/;. 5) is a generalizaticni of the I’lanning filter [66] and
applied it to chaos synchronization (16.1] and trend estinmtion [177].

Like the Ge-‘tussian filter. the diffusion process not only removes noise. but. also
smears image edges and leads to poor Visual effects. To address such an issue.
Perona and .\fa.lik proposed anisotropic diffusion process [118] in which the diffusion

coefficient is replaced by a function of position gradient

'9 . i .
—( (If): f) = V - (tl(l'\7'tt(r.t)|)Vt/(r. H].
c. .
u(r.tl) : [(1‘). (1.8)

Where. [(r') is the original image and (1(IVHI) is a generalized diffusion coefficient
which is so designed that its values are very small at the edge of an image. Pcrona

‘)
and Malik suggested the (.lanssian t/(J‘) : exp [—35] and the Lorentz (1(1) 2

(l + (fit? l where (T is a constt-nrt to be tuned for a particular application. The
Perona—.\lalik equation has recently stimulated much interest in image processing
and applied mathenmtical communities [10. 225. 88. 1.12, 120. 135. 150. 1.51. 166.
167. 171]. It is conunonly believed that the l’erona—Malik equation f’t'icilitates a, new
and potentially more effective algcn'itlnn for noise removing. image restorzjttion. edge
detection. and image enhancernent.

(.‘atte rat: til. 23] pointed out. the possibility of generating additicmal maximum

.

and minimum which do not belong to the initial image data. 1 he problem was arra—
lyzed in detail by Kichernrssamy (88] further studies (25 1 12. 167. 171] showed that
the anisotropic diffusion process may break down when the grz-tdient generated by
noise is comparable to image edges and features. A 1n‘e—convoltttion with a smooth-

ing‘ function. such as a (iaussian filter was proposed as a regular'ixt-ttitm procedtu'e

to alleviate the instability. Such a preprocess in fact degrades image quality. ()ne

of the present authors 11:38] proposed a statistical method which discriminates noise.
from image edges. The idea is to choose (I in the Gaussian or Lorenz as a local

statistical variance of V11

 

(7(1‘.f):qu.-— W12. (1.9)

where if?) denotes the local avert-ige of X(r) centered at point r. The area of
the local average can be selected in particular a}_>plication. This simple approach
works well in restoration of noisy images. More sophisticated approaches. including
Variational methods. curvature and active contours, have been extensively explored
in the literature {17. 2'2 115. 116 128.129]

Another problem with the original Perona-Rfalik equation concerns its efficiency
in noise removing and image enhancelnent. This problem was addressed by \Vei
(158} by introducing high order t1t'lgt1-cc.111trolled diffusion operators. A fourth order

variation—based I’DE was proposed by Chan (:1‘ r11. (27 ] fo1 image restoration. You

r
f

and Iiaveh (17 ‘2] introduced a fourth order curvature diminishing diffusion equation.

The high order l’erona—Xfalik equation (1718] takes the form

(it/(1'. f)
T 22V [(/q( “(1‘ f (Vt/(1‘. f )()VV)(IU(.I' H]
+1- ((11 r t (Vut (I t) ).|) (1.10)
Solutesohent dielectric interfaces he11e (1,](11.|Vul) are edge sensitive diffusion func—

tions and ct at r. t). )tht( r t )l) is an edge sensitive (kinetic) production term. Nu-

111erical experiments were carried out in Ref. (1581 for a simplified version of Eq.

(1.111)

Outfit)

71 = V - [111(11(r t)Vu(r. t)f)V11(r.t)]
(

+v . [112((2111- f).|V1/(.r 1))1vV2-utr.t)l

+c('tl(l‘.t).(Vu(r.t)l). (1.11)

A linear and discrete version of the fourth order diffusion operat-(n‘ was tested for
time series analysis (1611]. Clearly. the r‘rrotivatiorrs litrehind the generalized Perona-
.\Ialik Eq. (1.11) include high order plrerrorrrerrological kinetic equations for pattern
forrrratiorr in alloys. glasses. polymer. eonrlmstion and biological systems. and fry—
perdiffusion used to stabilize the direct numerical simulation of turbulence flow by
damping small scale oscillations around the Nycnristt frequency. Image sharpening
can be achieved by a balance between forward diffusion and. backward diffusion in
liq. (1.11) via apprt1priately choosing the signs of (11 and (1'2. Bertozzi and Greer
1H. 71. 72) have carried out rrrathcr‘rrzittical analysis of the generalized l’erona—Malik
1'31). (1.11). The latter was found to be efficient for treating rrredical magnetic res-
onance images (11155). (filboa cl (1.]. ((12) constructed an efficient image sharpening
scheme. by using Eq. (1.1l) with a triple—well potential for the. kinetic production
((111121). )Vrr(r‘.1‘)|). \\itesl ki and Bowen [1111) proposed alternating direction im—
plicit (ADI) schemes for the integration of Eq. (1.10).

The other problem in the l’ercnra—Malik ecpration is its inefficit'1rrcy for image
edge detection. particularly for edge (iletetirtion of images with a. large. arrrount of
texture. This is due to the fact that edge detection is naturally a high-pass filterng
operation. while the diffusion process is inherently a low-pass filtering process. \Vei

and .1111 [162) address this problem by introducing a pair of weakly coupled nonlinear

L"!

equations

111 : F1(11..V’11.V2u.---)+3.1(11—11). (1.12)

1', : F.1(11.V1.'.V2e.-~) + 132(11. — 1). (1.13)

where 51 and :2 are the coupling strengths. Here. F1 and F3 are nonlinear functions

and can both be chosen as the Perona-Malik operator

1"}, = V - [11,,(IV11|)V11]. 11:1.‘2. (1.14)

but evolve at entirely different time scales. For complicated 2-11)])li("ttti()ll. the gerr—
eralixed Pt1rona-.\lalik operator (1.11) can be used. This approach was motivated
by the synchronization analysis of two nonlinear chaotic systems. Image edges are
detected by the so—called syncln'onization residual at an appropriate time

Ir’tr./) :11t'r.l)—1'(r.t) (1.15)

‘1

provided that the common initial value is a digital image field 11(I‘.f)) I 1'(r.0) =
[(1‘). This 1711111111111 I’DF. apprm-ich has been shown [1621 to be very robust. and
efficient. and it provides superit'1r results in image edge detection compared to those
1:1btaint‘1d by using other existing 1~1p];1roacln~1s. such as the Sobel. I’rewitt and Canny
<_1per;_1tors. and by anisotropic diffusion.

Furthermore. the tinre in the evolution equation is an extra parznneter for images
and often does not have much justification for its necessity. In particular. there is
no rigorous and common rule to determine the period of the time evolution. An

antomatically controlled generalized I"erona—;\Ialik operator was proposed 1158] by

6

introducing a time dependent. factor

(0

_______ 1.16~
f +-fo ( )

where 1?” and 1‘” are positive constants. Since each term in the generalized I’erona-
Malik equation may have a different. time dependence. this actually turns the time
into an additional dimension for optimizing image properties. Such an idea was
adopted and the related technique was further explored by Gill’1oa ct 1.11. [63]. They

proposed two time—detj11-1n1lent diffusion 1_:1;1eflicients for Eq. (1.8) in the. form of

 

[Vt/.1 2
l V1 .t : 1 .
((1 111 1 + W)
. . . 1
with 1111') : ——+—; and
f (11
112f
1/(1V11|.1‘): 1+ ELM—l

where 111. 113 are para-uneters for controlling the diffusicm rate. 6 is a constant. and 11'
is the gradient. tlrresh1'1ld. C'tmtrolled evoluti1i1n is an i111111ort1'mt concept for I’erona—
.\falik type of evolution eqm-rtions. 1n fact. two types of controls have. been intro-
duced in the lit.1f1rature. One 1_-1,1nt'rol. originally proposed by \\'ei [1.58]. is to 1;'<_1nt.rol
the, magnitude of an individual 1.1p1'1ratuor as an explicit function of time. such as Eq.
(1.1(1'). so that its effect 1111 the image can be optimized. The other is to control the
exit. time 11f the time evolution 1'1as111‘1 on the characteristic of the 1.1v171lving image.
The second type 11f control was prop1'1sed by \Vei and .11a 1162] to stop the time evo—
lution 11f their couple edge detection mutations based on the (11111111111111 in variance
of two evolving images. (’ertz—tinly. more r11s1'11‘n‘clr work on the automatic control of

time evolution is requiret'l in order to make I’DIC based image processing a really

\l

competitive approach e-igainst a host of other 111ethods. such as wavelets. \Viener

filter. neural 111'1tworks. etc.

1.2 Numerical Schemes for Diffusion Equattions

One advantage of PDE based image processing methcxls is their elegant and rigor-
ous nrzgithematical f1f1nndation. Another advantage is their readiness for systematic
generalization to the. pro1'11i1ssing of three (iiimensional (3D) and higher dimensional
data. (11111111111111? selectivity and 1‘11'11'ameter c1:1ntr1f1llability also endow these methods
great flexibility and potential for a variety of applications in medical imaging. as-
tronomic 11111111111114. pattern 1'121c1'1gnition and computer vision. Ntwertl’ieless. there are
111.1stacles which prevent I’lJE based image processing 111ethods from being used in
prz‘rctical applications. 1.11.. industrial co1’les. First 11f all. nniltiple iterations involved
in solving nonlinear I’Dl‘ls are in general much more expensive than the single step
operation of most classic image processing 111ethods. This lack of efficiency is very
severe for the processing 11f large 31) medical image data. .\1oreo\1'er. the numerical
solution 11f l’erona—Xlalik type of nonlinear I’I)l£s is nontrivial. Numerical instabil-
ity exists when the time step is too large in explicit. schemes. For implicit schemes.
stability may not be a problem. but other undesir1-1d features might be introduced
to the solution.

Many efficient solution techniques were pr1'11'1ose1l (151. S114. 111). 175] for the linear
heat 111'luation. Ilowever. when the emphasis of the diffusion process is changed from
the linear filtering to the nonlinear one. poor conquitational efficiency 1’1ecomes a
main obstruction in the practical applicz‘rtions of the nonlinear diffusion 1'1qt'iat'ion.
In time. explicit schemes like forward f‘luler (first order). multi-level explicit finite
difference schemes are most widely used. However. these explicit schemes require

very small time steps in order to be stable. It is doomed that: the whole diffusion

procedure is rather time consuming. C(msequently. accelerated ex1_)licit. schemcs
are (,lesirable A fast. explicit numerical scheme (A—resolution) was presented to
approximate the solution of the linear diffusion filtering [153]. More sol_)l'1isticated
and absolutely stable approaclu—s are semi—implicit schemes [23. 166]. \Veickert (it (if.
[160] proposed an additive operator splitting (AOS) scheme for nonlinear diffusion
process. In cc‘mtrast to traditional mtiltiplicative splitting such as the ADI and
locally one dimensional splitting. all axes are treated in the same manner in AOS
scheme. However this scheme needs to compute matrix inversions which is usually
a source of computational errors and requires much computer memory. In space.
since the pixel structure of digital inmg‘es provides a natural discretizz—rtion on a
fixed rectangular grid. it is not. surprising that finite difference based 111ethods. such
as second order. fourth order. and sixth order central finite difference schemes. are
commonly used for the task of (liscretization. Alternatives to the finite difference

1

methods include finite element methods [9. 5'2. 8, ] finite volume schemes [91. 107].

i5‘)l. wavelets f-ST. St). 158]. multigrid methods [1], fast.

L ‘ l
. J L

pseudospectral approaches
level—scar 111ethods [133]. high-order EN() [110]. lattice 3oltzmann methods [81] and
stmhastic simulations [120].

1 Recently. a local spectral (LS) metlmd. also called the discrete singular convo—
lution (l)S(_‘) [157}. was proposed as an efficient numerical tool for solving PDEs.
The mathematical foumlations of the DSC algorithm are the theory of distribution
and wavelet amilysis. The DSC algorithm has found to be useful in a number of
scientific and engineering applicz-itions [10. 76. 1.315. 1.40. 1:39. 101). 1.65. 176, 179, 180].
More recently. a family of local. spectral evolution kernel (LSElx's) f173] have been

u

constructcd for analytically integrating a class of evolution 1’1)le

l

a , a~ a) . . . _
_—f./(.i'.t) : (do) +BU):——+('t/))flint). (Ht)
0/ (11'

.' ‘)
0.1“-

9

where. A. B and C" are functions of time. and Retrl) 2 0. The proposed local spectral
method has the same level of accuracy as that of spectral methods in space. and
is analvtic in time. The local spectral evolution kernels are constructed by using
Hermite functions. However, unlike the. standard global Hermite spectral methods
[9‘2]. the LSEIis adopt uniform grids and have banded iruitrices. just like other DSC
kernels. In fact. the length of the computational stencil. and thus the accuracy of
the LSEK. can be controlled according to the needs of the problem of interest. which
is another feature of the DSC inherited from wavelet analysis. l.\*l(‘)reover. the local
propertiv endmvs the LSEK method with sufficient flexibility to handle moderately
complex geometry and complex boundary conditions. In principle. Fourier spec-
tral methods can also solve many evolution equations analytically. However, thev
are restricted to periodic lamndarv conditions and lead to distortion near inurge
boundaries.

The objective of the present work is to introduce an evolution operator based sin—
gle step image processing methml. rl‘he LSEK svstemat icallv inct‘rrporates anisotropic
diffusion coefficients at each pixel. The new method arrives at a desired evolution
time in a single step. Moreover. the proposed method is free of instabilitv constraint.
and is of spectral accuraev for integrating evolution PDEs (1.17). Furtherniore. by
an appropriate design of coefficient ("(1). one can achieve the automatic termination
of the time integration. In this work. we explore the use of the prcmosed single
step method for several importz-urt tasks. including image deblurring, denoising.
edge detection and enhancement. Image denoising is accomplislred by incorporat-
ing anisotropic diffusion tvpe of coeflicients in the LSICK. While image deblurring is
realized bv an appropriate forward—lrackxvard diffusion process. The coupled I’DE
svstem (1.12) is simplifiet'l and utilized as a mode for constructing new schemes for
image edge detect ion. and the resulting image edge is emploved for image enhance-

ment. It. is believed that the proposed evolution operator based method is some of

10

the fastest schemes for image processing. In particular, the prt‘)p<:)sed I’lel based
method has great. potmitial for processing large scale multidimensionz-rl data and
for data mining. Some theoretic work and application results have already been
rmblished. the interested reader may turn to [1:15] for references.

In the following cl'iapters. the theorv of local spectral evolution kernel (LSEK)
is briefly reviewed. The frequency response of the LSEK is studied. The efficiency
of the LSEK is examined by 1D and 2D diffusion processes that are exactly solv-
able. Numerical results are. compared with those obtained bv using other standard
111ethods. A LSEK based qriasi—mmlinear method is proposed for image processing.
Chapter Three of the thesis is devoted to the applicz-‘itions of the proposed method.
Comrmter experiments are corulucted to demonstrate the performance and cfliciencv
of the proposed single step method for a number of image processing tasks. At the

end of ('hapter Three. a conclusion of the proposed 111ethod is given.

11

Chapter 2

Theory and Algorithm: Evolution

Operator Based Single Time Step
Method

In this chapter. we give a brief review to the local spectral method and its evolution
kernels. Discrete Fourier analysis is carried out to study the filter properties of the
lb‘lilx’. .‘\ccurac\' of the the kernel and efficiency of t he scheme are examined. Anew

erolution operator based method is introduced for image processing.

2.1 Local spectral evolution kernels

The discrete singular convolution (NSC) is a. general framework for constructing
local spectral methods. It is an ellicicnt approach for the numerical realization of

singular convolution

”c
I‘ll) -_— (T * alt/l : / TU .r)//(.r)(/.r. (2.1)

. -*x

where 0(1)) is an element of the. space of test. functions, and TH —.r) a singular kernel.
Interesting examples include singular kernels of delta. type and Hilbert. (Able) type.
The latter plavs an important role in the theorv of analytic functions. processing of
analvtical signals. theory of linear responses and Radon transform. The. delta type

kernels are of the form

To) = (it"i’o-‘i. n = 0.1.2..-- . (2.2)

where Mr) is the delta distribution. \Ve focus on these. singular kernels since they are
key elements in the approximation theory and the munerical solution of differential
equatit'ms. Betause of their singular nature. their approxirmittion is necessary for
numerical computatioiis. A variety of ('alltlltlz;l.t-t"s are available for the approximation
in the literature. Allltfllg these examples. Shannon's delta kernel is of particular
interest. It essentially gives rise to classic l‘euricr spectral methods. Some of local

srwctral kernels HST] are constructed bv the regularized Shannon kernel

 

, .l .' Tr . . 9
Jr , (I h]ll-(.I 11‘.) —(.r—-.r. - .,‘
(All, :7(.I — IL.) I ——-/- T? 1’ ("XP ————)—A—)— . - (2.13)
‘ ('/.1' EU -- .I'i.) 20-

where f) is the grid spacing. The delta. distriluttion can also be approximated by
clz-issical pt.)l_\,'nomials. I{(’)l'(‘(‘\'t-tl‘ [91)] pmposed a Hermite. function approximation
to the delta distribution. IndepemlcntIv. IIoll'man ct (1.1. [74] derived the same.

approximatitin and an expression for its arbitrru'v (‘lerivi‘ttives

 

‘l/l . . . _ (.1)’, Y‘MII 1 II 1 .I'—-.‘l'].
(ii/Lnkl _ 'I [1) — 2/2 7; L4,):T.” (-.-T) \E:;]’I/1277+] \l,-‘§{7 ‘

l3

where the Hermite function (2,, is defined as
.)
li,,(.r) : exp(—:1#“)H,,(.r). (2.5)

Here H,,(.r) is the classic Hermite polynomial.
In the DSC algoritlnn. a. function and its derivatives can be a1)proxinrated by

.U

' st!)
We): Z (>,m(.1-—.z‘A.);,(.1-(.) 1:0.1.2..--. (2.6)
k2~AI

where 2.11 + l is the computational bandwidth. or effective kernel support. This
approach has been extensively vz-ilidated by solving nonlinear PDEs [159. 176]. vi-
bration analysis [160. 107)]. Maxwell's equations (It). 130]. incompressil)le and com—
(n'essible Navier Stokes equations (17!). 180]. and im-age edge ("letection [70']. It is
also used in this work for coiinrmting derivatives.

\\'hat is crucial for the present work is an analytical integrator of the I’DEs given

in Eq. (1.17"). Their formal solution can be expicssed as

fo. r) : /(1.,-’1\'(.ir. .1". /. t/)f(\.r’. x’). (2.7)
where Iv(.r. .1". I. 1’] is an evolution operator. defined as
-f.
/\'(.r. .r’. t. (I) -— .r.exp /t’ [.(H .rI . (2.8)

Here the standard inner product (fly) is defined as (fig) : If"(.r)_q(.r)r'l.r. The
operator can be converted into an algel'u‘aic exl'u‘ession by using a Fourier projection
operator and then carrying out the integration over the waveuumber. The. resulting

kernel is diagonal in the position representation l\'(.r. .2". I. 1’)

ll

= I\'(.r »— .r’. t. f’) and can be cast. in a local spectral representatiou

11',,fl(.1-.t. t") = A / I\'(.r — .1". t. t’himl (.I'lldl'l.

he

\
~Ir—-

One therefore obtains (173]

.ll) (2 212+1 f

h ('t I, 1 n 1 O .1.. + B I
K .r. f. t’) z: —c f’ E —— —— —+. 1,.) .____I_ a 2'9
hfl< (I He‘ll 4 mill (7;; -n fin], ( )

where A], : .fz.’ .\ (sh/s. for .I : .1. B and ( . Here. a], is given by

f . ‘
at, : V n? + 2.11,. (2.10)

Expression (2.9) is called a local spectral evolution kernel (l.SlCI\”). The case of .‘l : (1
can be recovered by setting a], r: (7. Another simplification occurs when .113 and
('i' are independent of time. i.e.. .\'(f) : X. Then. X], '-——- -\'(t — (I) : KAI. and
Mmtr. r. r’) :: lx',,.(,(.r./ —— r’) : lx',,.(,(.z-. Ax).

I’or an initial function ,/'(.r. (I). the solution ((.1: t) of liq. (1.17) at an arl.)itrary

time 1‘ can be analytically given by
.V
(“(.1-.1) : Z It',,fl(.i-~1)..t.r’),/'(.r(..r’). (2.11)
1-11

or for a grid point (1')

[v
H
Ix.)
V

.1/
fl.1'_/-./) : Z Khfltlrli.(.1/Mt”—~l.'//./’). (‘. '
1.':~_ll

This implementation is the same as that for other local spectral kernels. given in Eq.
(2.0). liq. (2.12). together with liq. (”2.9). implies that the evolution operation has

been converted into an algebraic operation and the local spectral solution (2.12) to

PI)Es given in liq. (1.17) is analytical. This e-‘ipproach is therefore free of stability
constraint and is in principle of spectral accuracy when the support of the stencil is
sufficiently large.

For image. and data analysis. it. is necessary to deal with higher dimensions. If

the. operator in Eq. (1.17) is of q-dimensit)nal.

() .' . _ -"‘1 ., ‘0“ . 4L
meIImH ..1/.~- ..Iq./) — Huff1 [.l,(f)m+B,(fl(-)J.i +C1U1] (913)
>-’./'(J'1..--- ..-1:i.--- (JV-(la
its solution on an arbitrary grid point (.1111. - -- ..ITfjj,"' ..'qu-qf) can easily be

constructed by tensor product

. ll-
- . .- . U1 ~ - . I
j (.1 1J1. ° ' ' ..f (.1). ' ° ' ..f (llq. I) : [IIZLII :41‘1'2—_‘\[’[\III'.KT"(/‘.I'fl’/I'f'f )
><f(.'ll.l‘] — A'.t‘lhl' -'° ..'IIJ',‘ — 1".1'1'1’13'”

/
(Tl/(l '— It.‘)'(_l/)(l.f )1

where definitions of quantities with subscripts are self—evident.

The local spectral method also provides a freedom to choose a desired level
of accuracy so that the computing time is reduced. This can be achieved by an
zqipropriate choice of .l/),. a and .1]. Both forward and backward time evolution
can be resolved by using the LSIEK as long as “WWI/l .1: t) for a given f. A detailed
analysis and validation of the LSEK. including an efficient algorithm for solving
complex nonlinear l’l)le via time splitting formulation are out of the scope of the

present work and have been accounted elsewhere [173.1].

2.2 Filter properties

To achieve a better understanding of the lb'l'llx'. we study its liltcr properties by

discrete Fourier analysis. Such a study allows us to optimally choose the USC

16

parameters. i.e.. the computational bandwidth M], and the scaled Gaussian window
size 1' :— (I/h. The Fourier analysis is a. classicz-il technique for characterizing the
Fourier resolution of an interpolation or differentiation scheme applied to a, class of
compactly supported periodic functions and for examining the frequency response

of a. filter.

Figure 2.1: The LSElx’ in the ("-oordinate domain with AU) 2 BU) = C(t) : (l.

 

 

Amplitude

0.5» i

 

 

 

1r Frequency

Figure 2.2: The frequency responses of the LSEK with BU) = C(f) = 0. AI = 64

and M}, = 88. The solid lines from outside to inner are for xii, = 0.000102. 0.0102

and 0.102.

Note that if the function ./'(.r./‘) in Eq. (2.1l) is regarded as a two—parameter

. . . . . - . /. . . . . .
family of signal. the convolution kernel I\/,fl(\.r .r/‘u I./ It is a linear time lllWll‘lileli.

17

 

Amplitude

 

 

 

Tr Frequency n

Figure 2.3: The frequency response of the LSEK with AU) 2 BU) 2 CU.) =
0.1)] = 6 and All, 2: 88.

system and is practically used as a finite impulse response (FIR) filter. This filter is
stable and inevitable for (H' <:.‘ X. ()n the one hand. when B : ( : (l and AU.) is a
constant. liq. (l .17) reduce to the heat. equation. As the LESK provides analytical
solution to the heat equation. one might assume that the analytic integrator is a
Gaussian filter as the action of heat mpiation is known to be equivalent to the
Gaussian filter [170]. However. this is not true. In fact. it looks like a wavelet filter
in the ('-oor(lin;-ite domain. see Fig. 2.1. Mathematically. it. is called a kernel of
Dirichlet type in contrast to the kernel of [)Lh‘ll-lYC type. eg” a Gaussian kernel. ()n
the other hand. when B : U. the kernel Khflfrx LI") is obviously symmetric and
its Fourier” response is of linear phase and real. since it contains only even terms
of the Hermite. ft’uictions. As it is also the evolution kernel for the parabolic I’DE
(when .-l at U). it must be a. low—pass filter in general for t :> I" as shown Fig. 2.2.
It. is noted that the LSEK low—pass is almost ideal for small 2‘. and appropriate 3/},
and r values. In general. large .l/l, and I' values lead to a better a])proximation to
the ideal low-pass filter. Nevertheless. since they are constructed from polynornial

approximation. they are ("hebyshev type of filters as illustrated with appropriate

18

.‘l/ and r values. see Fig. 2.3. ;\loreover. for a Niven set of other )au'ameters. when
fl h r)
t - f’ is getting larger i.e.. long time evolution. the LSEK filter response becomes

more and more focused in the low frequency part. corresptniding to large amount. of

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

diffusion.
1
1 1
:3 is“, as 0
_ an
o: E
0 0 -1
TE 0) 1t 7'3 (1) 7t 1t (1) 1t
(8)
1 1 *
1
__ :8: E
’37 >50 1 x 0
V a) v
E. o: E
0 4 -1 i
TE 0) 7t T! (1) TE 1t (0 it
(b)
1 1
1
:- 3 E
El as o s, 0
m
E. {I §
0 —1 —1
TC (1) TE 7: (1) TC Tl: (1) TC
(V)

Figure 2.4: The frequency responses of the LSEK with 2‘] : (54.510, 2 88. AU)
CU) : 0 and different B;,. (a.)B;, : f); (b) B:, : h: (t) B;, : 2]).

I

In eeneral. the iarameter B U'ives rise to a translation in the com'dinate. which
’.’) C”)

leads to an extra phase factor in the frequency response. see fig. 2.1. ()bviously.

1 S)

 

10

 

 

r

— Original function
- - Evolved function with C(t)=2

 

 

 

 

 

 

~10 1 1 .
0 10 20 30
x
(at)
-——- Original function
1, - - Evolved function with C(t)=-2 .
0.5»
O f \ I \ \ I \ \ / I \ ’T
-O.5 .
-1 -
O 10 20 30
x
(b)

Figure 2.5: The frequency responses of the LSEK with 1)] = 64. .11}, = 88. AU)

BU) : U and different. (1,. (MCI, 2; (MCI, = —2.

in signal analysis. it is the time shift fan-tor that is useful for the circuit design. In
other words. the drz-ifting term [ft/)fi); in Eq. (fl?) is a. time—shift. operator from
the point view of signal processing. Since. BU) is a function of time. it, in fact can

be used for nonlinear time shifting. As shown in Fig. 2.5. the parameter (" provides

'20

an overall scaling factor (either exponential growth and decay) to the amplitude of

the signal. which can be useful in practical t—mplications.

2.3 Numerical test

\Ve illustrate the proposed local spectral method for the diffusion equation in this
se(.:tion. The diffusion equation is a special case of the class of PDEs of (1.17)
(B : (.1 and C :: ll). The most important fact is that the diffusion equation has
extensive applications in image processing. The accurz‘icy of the proposed local
spectral method is obtained by cmnparing its results with exact. solutions. Two
advantages of the present local spectral approach (accurate and fast.) have been fully
('lt‘IIlOIlHlll‘tlft‘tl. Since both forward and bz-rckward time evolution can be resolved by
using the LSEK as long as lie(r7:,) Z t) for a given t. \Vhen 51:, is positive in Eq.
(2.11)). it is obviously a forward 1')ri.)p;-igation process. In this case. the numerical
solution of the forward diffusion equation at any time can be obtained by Eq. (2.12)
in only one time step. However. when time :1:, is negative. the problem becomes a
backward propagating process. and we still can use Eq. (2.12) to get the solution
as long as we ensure lie(r7;,) >3 (1. However. the backward diffusion is an unsteady
problem. In order to obtain. a reliable numeriml solution. we have. to terminate the
backward diffusion in a limited time. As was mentioned earlier, the accuracy of the
proposed local spi-‘ctrz—il method is cmrtrollable by adjusting its support parameter
.1] and the lflermite parameters .11], and (T. In this numerical example. we choose the
ll’lermite par'z-uneters and the computational bandwidth as M}, : 128. (r : 37$ and
.1! : 15 to illustrate the spectral order of accuracy of the method. Unless stated
otherwise. the above set of parameters is used in this section. In EthIllldtf 2 (2f)
diffusion process). we will compare the performance of the present local spectral

method with that of liftinu'ier spectral method which is realized by using the fast

Fourier translorimit ion (l’FT).

Example 1. \Ve consider a one-dimensional linear diffusion process given by

(f)f(.r.f) ( ()2j'(.r.t)
__ _._. 7(_______

0t 0.1‘2 , (214)

where o is the (.lifl'usion coefficient. \Vith an initial delta function distribution lo—

calized at .r”. the analytic solution is

 

f(.I.‘.l‘) : ’ ___1_____(.J—(.r---.2'0)2/[1o(t:-—t())]I (2.15)
4(i7rU —- f“)

The computation is conducted using a sufficiently large interval [—10.10] of coor—
dinate space to ensure that boundary reflection is negligible. Fifty grid points are.
used for this interval. The diffusion coefficient a is set to be (1.5 in all cmnputations.
Since it. is difficult to accurately represent a delta. function on a grid as the initial
ccmdition. we choose the initial value as the Gaussian of Eq. (2.15) at f. : 1.1) for
the forward diffusion process and the (i1aussian of liq. (2.15) at t" 2 2.11 as the ini-
tial condition for the backward diffusion process. The numerical solutions at time
t 2: 1.1). 3.1). 5.11. 7.1) and 0.0 are given in Fig. 2.6. ’lwvo kinds of numerical error mea—
sures. i.c. L.K and 1.2 are used to evaluate the. quality of the. present local spectral
method. The computational errors are listed in 51311110 2.1. [t can be observed from
Table 2.1 that the present local spectral method gives highly accurate munerical
results. On the other hand. it can be seen clearly that. the results of the backward
diffusion are not as good as those of the forward diffusion. The loss of accul':€t(_‘y is
due to the fact that the lau'kwa.1‘(l diffusion is itself an unsteady [in‘ocess .\l(n'eover.
since no time evolution scheme is nemled in the present methoi‘l. there is no accumu-
lation of time discretimtion error. This property guarantei‘s the accuracy and the
speed of the present method and shows great advantz-ige over many other munerk‘al

algorithms.

[\3
1' Q

 

0.5

f(x,t)

 

 

 

 

Figure 2.6: The numerical solutions of the 1-D diffusion equation. The centerlirres
in the descending order are at tinre t = 1.0. 3.0. 5.0. 7.0 and 9.0.

Table 2.1: Errors of the solutions for the 1-D diffusion equation.

 

 

Backward Diffusion

Forward Diffusion

 

 

 

Tirrre Lg 1.x Tirrre L2 Lx
2.0 71713—17 2.221‘3-10 1.8 82513-13 65015-13
3.0 37213-17 813313-17 1.6 2.32E—11 1.01E-10
4.0 3.80E—17 133913-16 1.4 1.19E-09 42813—00

5.0 05315—16 (5.011347 1.2 1.74E-07 SHOE—()7

 

Example 2. \\"e consider a two-tiinrerrsional linear diffusion cqrration

my; 22113111 W , (2.16)
()r‘ ().I'- On“

It is a good test problerrr because it adrrrits an exact solution of the forrrr

' , ., 0.. -)
l (1‘50"), hrt (

(,
‘)
1(1~7tt.

[v
p—a
‘l
V

 

ltr. .11. I) :

where. u is a positive parameter and we will use a = 0.7 in this test. To compare
with the 15FT method. we irrrpose periodic boundary conditions in both .‘I.’ and y
directions. The mnrputation domain is taken to be. [0. 10] X i010] with equal spatial
discretixzirtion. Although the grid can be set to be any integer values for the LSEK
method. for the purpose of comparing with the FFT method in which the grid is
required to be powers of 2 we choose NJ. : A}, = 32 in our computation. The

forward diffusion is initialized to be a Gaussian wave at t = 0.1

 

1 '. . 2 , 2 , 2
10613.11) : .1 ()hfi‘f‘fi'ffll +(f/"f/U) l/“l” t_ (2.18)

where the point. tr”. gm) is the center of the 21.) square donrairr. The detailed LSEK
method used for the 2D problem are given in Eq. (2.1 1). Our interest is to explore
the efficiency of the present LS method. Therefore. we will not only conmare the
rnrrrrerical errors of the LSICK method and the FFT method but also the computing
time of both methods. They are giyen in Table 2.2. It is clear from Table. 2.2 that
the present LSlCK method is as accurate and fast. as the l’l’T method when they
are solving the diffusion equations.

Table 2.2: Errors of the solutions and the CPU time for the 2-D diffusion equation.

 

 

 

 

 

FFT LSEK
Tirrre L3 [.1 CPU Tirrre L2 Lac CPU Time
0.5 1.31E-T 2.16E—T 0.0156 10813-7 1.70E-T 0.0156
1.0 51013—7 16013-7 0.0156 7.08E-7 4.6015}? 0.0156
1.5 28413-5 2.20E—5 0.0156 3.81E-5 2.20E-5 0.0156
2.0 20813-1 1.38E-1 0.0.156 27013-4 13813—1- 0.0156

 

 

By comparing the present l.ocal spectral method and the Fourier spectral method.

we reacfr conclusions which are summarized as follows:

0 The accuracy of the proposed LSlCli method is controlled by its computation

bandwidth .1]. llermite parameters 1]}, and (7. f'nder' choice of .1! = 15.

21

41]}? 2 128. and r7 : 3.7;\. the LS rrretlrod can provide. the sarrre level of

accuracy as that of Fourier spectral method.

0 The global nature of the Fourier spectral rrretlrod makes it difficult for practi-
cal applications to 1’)roblerns of complicated boundary condition and complex
geometry. However. the present LSEK is a local rrretlrod which can be used

for treating complicated bmurdary conditions and geometries.

o For the Fourier spectral method. the number of grid points is required to be

the power of 2. but there is no such a. linritatiorr for our LS method.

0 The present LSEK rrretlrod transforms the diffusion equation into algebraic
calculations. ”'f‘his irrrplics that. the LSEK rrretlrod can speed up the cornl'm—
tatiori dramatically and thereby reduce the cornputational cost. Spcwifically.
the C‘l’U cost of the LSFK scales as (,)(.‘ll-\"1_. where .1/ and .‘\-" are half of
ctnrrputnational bandwidth and the nurrrber of grid points to deal. llmvever.
asynrptotically. the LSEK rrretlrod requires less CPU time than the Fourier

1)serrdospectral method does as the latter scales as Otr\71.og.\7).

In. practical applications. especially in imz-rge 1')r'o<.‘°essirig. the aforernentioned preci—
sion is entirely unnecessary. A three percent error does not produce rrrucl‘r difference
in visual perception. Therefore. in the following image experiments. we choose
.11}, : .36. a : 1.7.1 and .11 : 8 In fact. even less costing parameters can be used
for certain tasks in image processing. such as image deblurring and denoising where

the image quality is very low to start with.

2.4 Adaptation of anisotropy

Obviously. the l’erorra—Malik type of anisotropic diffusion equat ions cannot be solved

directly with the LSICK in a single. step. albeit the local spectral method given in

liq. (2.6) can be used to solve these ecpiation iteratively. Fortunately. a close
exz—nnination on the solution scheme. Eq. (2.12) or Eq. (2.14). and the LSEK. Eq.
(2.9). one. realizes that the solution scheme is of collocation type. and the kernel

z o v'sz ocafle< or," c; o o' )e i 'e . .4. a it, f. i.e..
ill w 11 117 lm llfl it] n f(( ff1c1 nts l B I f(

Asmwm.3smwm.eamwm.

This flexibility of selecting local coefficients endows us a new single step evolution
operator based method for image processing and data analysis.

In l’eromr-Malik liq. (1.8). the break up of the. operator leads to a gradient
cont rolled diffusion term and a gradient, controlled drafting term. The. local strength
of the both terms can be easily computed and l]l(‘(,)1'p()l‘tlt(‘(fl in the LSEK expression.

Since the diffusion coefficients computed from Penina—.\f;-11il< type of ecntations
may vary from point to point. it implies possible N?“ sets of LSEK weights to be
computed. This requires much computation if .\7 is about 512. On the other hand.
since image resolut ion is normally limited to 8 bits. it. is obviously umiecessary to use
more than 2:30 sets of Lb‘f‘lfx' weights. \‘Ve tl'1erefore classify the diffusion coefficients
into a. total of NF groups. For each group. we. calculate a set of LSEK weights
according to the mean value of the groul’). in most practical applications. taking
.\>',. m 121) is encmgh and it costs less than 1 second to compute these w-I'eights. In the
above argument. we have assumed that the drafting coefficient behaves similar to the.
diffusion ccmfficicnt at each points. If this is not the case. the discussed classification
method can be easily modified. Thus. the basic procedure for the proposed evolution

operator |')ased image processing method is the follows:

0 ("onipute local coefficients .~l(|\7/|). BthII) and ('(HVI!) from an image. func—

tion [(.r. y) and classify the weights into a total of -\",. groups.

0 For each given group of diffusion coefficients. the 5.3;i'oup's mean value as the

26

diffusion coefficient. labeled as A‘- and 8.». Then compute a set of .ll+1 LSEK
coefficients. Here we assume a uniform grid in both .r— and ,tj-directions and
a common stencil size. of 2.1] + 1. Due, to the symmetry of the LSEK. only

1‘] + l coefficients are indepeinlent.

o For a pixel I(-(.r,-. 11,-. f). whose diffusion coefficients are labeled as Ac and Br.

its value at arbitrmw time t can be updated as

Al Al

[cf-7')- .f/i- f) : Z Z 1"II..0..4(_r.B(.(Air/b t) X

[xi/,flv..1(,.B(,(Icy/2.. f) X 1(.rj — A‘_,~h. y,- — Icy/1). (2.19)

where [(1,fixgulggfljrh. /) is obtained from Eq. (2.11) by setting I’ = f). A :

.'l,-f and Hi, : [3,}.

Although the proposed method has its root in nonlinear I’DifTs. it does not. exactly
solve the anisotropic diffusion equation. Instead. it is constructed by the pointwise
adaptation of anisotropy to the evolution kernel for the isotropic diffusion equation.
ft's major advantages are its single step operation. free of munerical instability.
readiness for multi<limensional problems. .\f(n‘e<.)ver. by appropriately choosing B
and C. systematic time shift and amplitudescaling can be easily acconn')lisln>d in a

single step.

Chapter 3

Applications of Single Time Step

Method in Image Processing

In this chapter. we apply the evolution operator based single step approach for
image debfurring. dermising. image edge (.lt.‘t(‘('fft)ll. and image enhancement. The
performaiu-e of the proposed method is illustrated by a large number of standard

test images and is compared with that of other existing methods.

3.1 Image deblurring

3.1.1 Brief introduction to image debluring

Image blur refers to serious degradat ion to image qualities by either the environment

of an imaging object. or the process of imaging. or the processing of an image

(117]. Backgrounds of imaging objects. such as fog. air pollution. texture to a finger

print residual and illumination. are common e1wirtunnental stun‘ces of image blur.

Notions of the imaeinU ob'ect and/or camera. or out—of—focus lens Uenerate blur in
h (”'3 _ , ("W

the )rocess of imagine. ban0 and \\'an” '82. Nil discussed image blurring due to
i. h (‘7 ('73 . J :3 h

reccmstruction techniques. Very often. image processing may lead to blur during a

variety of mathematical o[,.)erations. such as Gaussian convt)lution. interpolation or
image registration. Deblurring is a basic image processing task and the subject is
so extensively studied that it is virtually impossible to give a cornprehensive review
to include all inmortant ccmtribution and development in the literature.
Essentially. deblurring methods can be classified as linear and nonlinear. global
or local. time deperufent or time independent. a prim")? knmvlcxlge based or blind.
etc. Linear filters [18. 40. (it). 80]. do not make use of local information of an image
under processing. and are usually simple. and low cost. Many linear filers. such as
\Viener filter. Hanning filter and Fourier filtering. work effectively when there is a
prim? knowledge about the source of the blur. Mormver, linear filters are often
utilized to obtain initial inftn'mation for other nonlinear filters. Nonlinear filters
make use of the local information of an image to gain a. better effect of debfurring.
Some nonlinear filters 2188111110 partial priori knowledge of an image. such as Iiafman
filter and Lucy—Richardson algorithm. The former makes optimal use of imprecise
data on a linear (or nearly linear) system with (It-mssian errors to continuously up—
date the best estimate of systems current state. ‘While the latter maximizes the
likelihood that the resulting image is an instzmce of the blurred image. assuming
the Poisson noise statistics. .\Iany blind debhn'ring algorithms can be used effec-
tively when no information about the source or status of the distortion (blurring
and noise) is known. They utilize the local infornration or Fourier spectrum of
an image intensively and achieve the goal of deblurring by iterations. (i)bviousfy.
Iii-’erona—.\fafik type of 1’1)le based image deblurring algorithms are time-tfepclufent.
nonlinear. blind filters. \Yang and his coworkers proposed an efficient blind deblur-
ring algorithm for spiral computed tonmgraphy ((ii'T) images [8-1]. \Vhile. wavelet.
transform is a linear algorithm in nature jltll]. its application to image ju'ocessing
can be made nonlinear by a])propriately iiiccu'ptu'ating image iiiftu‘mation into the

select ion of truncation parameters 18 '29. 18. l 13]. Recently. some titne—imh‘pendent.

total variation (TV) based regularization functioin-ils have attracted much attention
because of their ability of recovering image edges during (.leblurring. see Rudin (if,
(if. [126]. Li and Santosa [9:3]. Dobson and Vogel [47]. and Chan et. (If. [26]. These.
metlmds are often f(n'mulated in terms of Euler-Iagrange equations and are asso-
ciated with ill—posed inverse problems. Ideally. TV based methods should work as
blind algorithms. Nevertheless. the main difficulty associated with TV methods. as
indicated by many researchers. is the determiiration of the unknown regularization
parameter(s), which lmlances the fidelity and smoothness of a TV solution. Con-
sequently. these methods are. most. efficient when some ‘1)"1‘fO‘I‘f knowledge about the
image is given.

A common used degradation process is often j’nfesented in terms of a. blur oper-

ation and additive noise.

I” _—: If] 4* 7]. (3.1)

where 1/ stands for a white additive (iaussian noise and where If is a linear operator
representing the blur (usually a. convolution). Given degradation version of image
I”. the restoration problem is then to obtain 1 knowing (3.1.). In this sttufy, we will
concern ourselves only with two zrtspects of image registoration which are deblurring

and (.letniising.

3.1.2 General model

\Ve also make use of the sharpening property of the bat'kward diffusion to restore a
blurry image. The effect of noise amplification (an inherent ffivprculuct of backward
diffusion process) will be minimized by terminating the diffusion process in a limited

time or bv choosing an appropriate diffusion coefficient. \\’c consider the following

30

anisotropic diffusion equation

0., * V1|.t)V2[.

 

s = 4t
.’ .
111:0 = 10~ (32)

UNIIJ‘EOSZ : 0~

as a model. Here n is a unit. vector. outward normal to the boundary 852. The
diffusion coefficient. A is a function of the smoothed gradient magnitude s 2 Ga *VI
evaluated at t. = f). \Ve further create the LSEK analogy of Eq. (3.2). which is given
by Eq. (2.19) with B : C = (I. Here. we design a. smoothed gradient. magnitude

depemfent diffusion coefficient for (“leblurring

 

j I ‘ __ /_fl.l8 .‘ 522.
\f/ l + (5/1". )2 VII 1+ ( 5/ A ) 2

q .
__1__ _ ——(l']—L—.=_—:. ot herwise.

\’ I +121 A ’l

(3.3)

 

V

 

l/ ‘ ’1' , \
h \fI'tfz/tl

where c is a small strictly positive real number for avoiding zero deiiominator and
I“ is a threshold parameter like that in the I’eronal—Malik model. Here It can be
a constant threshold or be adjusted locally by gradient. In the latter case. the
parzuneter It varies gradually with the image. and edge shz-n‘pening is acconiplished
by (.fiff'ereut thresholds in different locations. As an illustration. we present. in Fig.
3.1. a plot of the suggested diffusion coefficient :1 with 1c : 80. It is seen that
the diffusion coefficient considered in this work is strictly i1'r("1‘(>asiiig and negative.
Negative diffusion coefficient is for edge sharpening and increasing tendency ensures

less backward diffusion for lam“ .‘s‘l‘adient magnitudes.

3.1.3 Results of deblurring

\Ve use the Parrot image for our demonstration. The blurry l’arrot image is obtained

by convolving the. initial f’arrot image with a Gaussian kernel ((7 :: ‘2). As for

31

 

-005

-015

 

 

 

 

50 100 150 200 250 300
S

Figure 3.1: Diffusion coefficient A(s) plotted as a. function of the smoothed gradient.
magnitude.
another Gaussian kernel used for the calculation of the smoothed gradit-ent G0 >:< VI
of original image. we set (7 :_- .3. \Ve stop the diffusion process at time t. = 1.0. Fig.
3.2 shows the results of deblurring. For a more clear observation. we zoomed the
Fig. 3.2 out near one parrots head and the momed inmges are shown in Fig. 3.3.
We found that edges at different locations have been improved. For example. the.
edges near parrots beak and those around the eve are well sharpened. Fig. 3.1
shows the deblurring process for a lutilding. The blurring and deblurring processes
are similar to those described in the Parrot. image. In this case. we are interested in
the features that characterim‘ the architect.ure. such as edges and lines.

In fact. one can intru‘pret the above deblurring process as an adaptive resolution
of the linear diffusion equation which is conceptuallv different from the cmiventim1al

ntmlinear diffusion filtering in which the nonlinearitv is spatial.

   

(a)

 

Figure 3.2: Deblurring process of the parrot image. (a) Original parrot image; (b)
Gaussian blurred image: (c) Parrot image after deblurring.

3.2 Image denoising

3.2.1 Brief introduction to image denoising

Noise is one of the most common sources of image degradation 1100']. It is often
referred to the rapid changes over spatial extension in image intensities or color
plan. Either imaging acquisition process. or naturally occurring phent‘nnena can
lead to noise ctmtaminated. Typically. noise is assumed to be additive and satisfies
a mathematical model. such as the Poisson. or the laussian. or Laplacian. Some
noise is multiplicative. e.g.. the speckle noise. The image denoising problem refers

to the process of recovering an image contaminated by noise 33] and is one of the

33

 

Figure 3.3: Zoomed parrot images for deblurring process. (a) Original: (b) Blurred:
(c) Processed.
most. elementarv operations in image processing.

The earlier algorithms include (linear) spatial filtering approach [80] and fre-
quencv cut—off filters in the Fourier domain. which are first motivated bv the prim—
itive ideas from signal processing. These filters reduce the noise by removing the
high frequent-v components. However. they also blur images. Stochastic modeling
[18. #10. 60] originated from estimation theorv has also been introduced for noise re—
moving. In the past two decades. a number of new techniques have been introduced
to improve the qualitv of image denoising. Statistical approaches such as maximum-
likelihood estimation and adaptive (\Viener) filters are developed. Local adaptive
filters are combined with impulse removal filters in the trz-msform domain to remove

not only white and mixed noise. bttt also their mixtures [-30]. “7avelet theorv [104]

34

 

 

(1))

Figure 3.4: Deblurring process of the columbia image. (a) Gaussian blurred image;
(1)) Columbia image after deblurring.
has been used for denoising by many researchers [28. 3:3. «18. 113. 138. 155. 156]. The
essential idea in wavelet schemes is to decompose the noisy image into a number
of l'reoucncy bands. and remove the noise in appropriate frequency bands. \Vavelet
approaches usually perform well with a viable st rategy to discriminate noise from
image in each wavelet subdoniain. PDl‘: based denoising methods were proposed to
Preserve image edges drain}: the brocess of noise redaction as briefly reviewed in the
lntroduct ion.

11' impressive results are the main reasons lor using nonlinear dillusion iiltering in

image processing. poor ciliciencv is the main reason tor less widely spread application

35

of nonlinear diffusion filtering in industrial codes. Finite diffm'ence schemes are
extensively used to discretize nonlinear anisotropic diffusion equations and they
make the whole filtering procedure quite time-consuming. “7e resolve this problem
by using the LSEK. The efficiency of the proposed LSEK for diffusion equations has

already validated in Chapter 2.

3.2.2 General models and results

In the present work, we explore the performance of the LSEK image. denoising. We
take two different LSEK apI'n‘oaches.
In our first denoising approach, we choose a time-independent diffusion coefficient

of the Gaussian suggested by I’erona and Malik

lVU‘2

[V =Q.’ ———.—'
((l (1|) exp 202

, (3.4)

where the. \s'ariance a is based on a local statistical estimate of Eq. (1.9). By using
this variance. no additional kernel smoothing is needed. The success of this apprtmch
has been fully demonstrated in [158] In this study, we. used the proposed evolution
operator based single—step 111etlmd to obtain a fast denoising process. A Gaussiz’m
noise. contaniinated Lena image has been used to test: the 1;)e1'f(.)r1nan(.-e. It is found
that the PSNR of the noisy Lena image is 21.03db. After denoising, the qtmlity of
the Lena image has been improved and the resulting PSNR is 29.81db. Both noisy
and dermised Lena images were presented in Fig. 55.5. We also examine our scheme
for an MR] image ofbrain. An improvement of 6.3tldb in the I’SNR is obtained for
this medical image. see fig. 3.6.

In the second approach. we use Eq. (2.19) directly for image denoising in a
single step. I‘lrom qus. (2.10) and (2.0). it is seen that the diffusion coefficient

jl:, : ll; rlf‘Tlt/T (outrols the local diffusion. Ihcrefore. one can tailor :l:, for a

36

      

(: {i I) f b)

Figure 3.5: Denoising process for the lena image. (a) Noisy Lena image with
PSNR:21.U3: (bl Restored Lena image with I’SNH:29.81.

 

(all (bl

Figure 3.6: Denoising process for the brain image. (a) Noisy brain image with
PSNRleTO: (b) Restored brain image with PSNHr'ZSlS.

gircn ['iiii'pose, In this work. we propose the following gradient dependent sill]

, If) , . ,
.lfdxii 'ejrflan‘ I sit vi i/lj' 'ltlll lbs)}. (3.")

 

where s — |(}0*Vl| is the smoothed gradient magnitude. if and 6 are two parameters
controlling the tendency of .46. \Ve fix i)’ = 10 and vary (,- between 0 and 0.1.. Fig.
3.7 shows the change of .46 with sets of e = 0. 0.002. 0.01 and 0.1 at the time t = 100.
Obviously. c = 0 almost corresponds to the linear diffusion process. Therefore. we
set ( to be larger than zero in order to preserve edges of images. In our test. we take
6 = 0.01. \Vith ,j and 6 fixed. we plot the relation l‘)etween 46 and the time t in
Fig. 3.8. One can (‘il‘iserve that .48 basically does not change any more as the time
t. is large enough. This implies that the denoising process is stable with respect. to
the stopping time. Thus. the stopping time can be chosen to be quite large without

worrying possible over smootllillé’é-

 

 

 

 

 

 

1 _ -i
.. 0
<1:
05* .
O 1 1
O 100 200 300

Figure 3.7: ['46 plotted as a function of the smoothed gradient magnitude with
F : 0. 0.002.001.01 (from top to bottom): {3:10; 2‘ = 100.

\Ve classify .46 by equally dividing the values of 4") into A}. : ‘20 groups. \Ve
therefore generate ‘20 sets of LSEK coefficients accordingly. \Ve still use the last.
two noisy images for our denoising rlemonstration. see fig. 3.0. The I’SNH for the
restored Lena image is '20. 1 [db and that for the restored brain image is 25.22db. It is

can be seen that the performance of this denoising procedure with a fIl]l(‘-(lt‘])(‘ll(lt‘lll

38

 

0.5 - .

 

 

 

Figure 3.8: .48 plotted as a function of the smoothed gradient magnitude at the.
following times t=0.001. t=0.01. t=0.1. and t:100 (from bottom to top): .3216:
( : 0.0].

diffusion coefficient is ahnost the same as that of the timc—indepeiident one. As for
the stopping time. we can set it to be 5. 10 or a larger number. It is not very

sensitive.

   

(b)

Figure 3.9: Time dependent denoising process. (b) Restored Lena image with
PSNRsz). 11. (b) Restored brain image with PSN’RzBSZ‘Z.

39

()lwiously. more efficient diffusion coefficients can be designed to remove blur
and noise from images. but they are beyond the scope. of the present work. What
we want to convey to the reader is just an efficient single-step procedure for image

deblurring and denoising.

3.3 Image edge detection

3.3.1 Brief introduction of image edge detection

Edges clraracterixe object: boundaries and are therefore crucial for image segmenta-
tion and registration. and for the l(lt“llflfi(‘21fl()ll of (.il'rjects in scerws. In recent years.
considerable research has been carried out in the (.level(_>[.)ment and application of
efficient algorithms for the detection of the image edges. A well-known approach in
the edge detection is to associate edges with zero crossing of second-order deriva-
tive. The resulting operator in use is called il.aplacian of Gaussian (LoG) operator
[103]. An alternative approach is the method of curve fitting [73. 100]. in which
an apprt)ximz-ited function is built. based on the least square method. rI‘herefore. a
derivative on the fitting curve yields the approximate differentiation for each pixel
directly. Compared with the finite difference method. curve fitting approach results
in better performz-tnce in a noisy enviromnent. They are. htm'ever. cmnputationally
more expensive. Recently. Canny l‘21] has fortnulated edge detection as an optimiza-
tion problem and defined an optimal filter which can be efficiently approximated by
the first derivative of the Gaussian function. \Vith a recursive procedure [4'2]. the
Canny detector provided an efficient way for image noise filtering and edge detection.
()ther edges detection methods include differentiation-based 111ethods that use the.
logarithmic image processing (LIP) models If] contrast based methods [8'3] and
relaxation labeling techniques f78]. In fact. these delicate techniques can achieve

better performance in one way or another for connnon tasks. flmvcver. for images

40

with large amount of texture. these traditional edge detection techniques are no
longer reliable. because of severe edge distortion. The latter is due to the wrong pre—
diction of high frequency edge components by low accurat‘ry methods. Most recently.
\Vei and J ia [162] proposed a. novel sy1rchronization-based realistic edge detection
scheme that has demonstrated a success in detecting edges of high texture images.
In this scheme. the residual of syncln'onization. defined by the difference in the syn-
cln'onization of two (.iyrn-unic systems. gives rise to image edges as if obtained by a
high-pass filter. However, solving two dynamic systems makes it computationally

expensive.

3.3.2 Our techniques

In this work. we propose a simplified version of this synclu'onixat ion scheme. The
essential idea of this new edge ('fetvector is quite simple: A snmothed version (SV) of
the original image I is obtained by modifying the forward nonlinear diffusion opera.-
tor. f7.q. (3.21). In such a nonlinear operator. the diffusion coefficient at. image edges
are significantly large. contrast to the original. use of nonlinmr diffusion operators.
As a result. image values near the edges are effectively suppressed. This smoothing
operation. from the point of view of image processing. is a low—pass filter. \threas.
image edge detection should be a l‘iigh-pa‘tss filtering process. Therefore. edge de-
tection version (EDV) denoted by .Igl_)tr(.r.y) is then ext'n‘essed as the difference

between the original image and the smoothed image:

1].;jpt'fjf. y) ;: (111(11‘. y) — (321314.72 y). (3.6)

ft is noted that image edges are effcctively cmplntsized in I[;~Dy('.‘l'.}/) due to the
edge suppressive nonlinear diffusion operator. flcre ("'1 and (‘2 are two constants.

either l or - l. \\'e emphasize that the sign of ("1 and (7-3 must be same. If the edge

41

of the object. has a white color on a black lmckground. we take. C1 : (7'2 2 1. ()n
the cmrtrarv. if the edge of the object has a black color on a white l.)ackground. we
choose C1 = C2 = — 1. For instance. the tiger image is composed of the black object
(a tiger) on the white backgrouml. (7'1 and ('3 are. therefore. set. to be —1. \Vhile
the Pine cone image is composed of the white object on the black l.)ackground. we.
take ('1 = ('3 = 1.

Con‘iputational cost is very low in the present approach since only a. single step
is needed in rumring the nonlinear diffusion equation with our LSEK. Nevertheless.
one can further speed up the edge detection by using a linear diffusion operator.
e.g.. running the forward (-liffusion liq. (33.2,) till 7‘ = 3 by our LSEK. with a etmstant
diffusion coefficient .41 : 1. In order to assess the computational complexity of the
LSEK method. we compared in Table 3.3.2 the number of basic operations for four
different methmls in a single time step. 'l‘hese methods include the adaptive resolu-
tion scheme [1.33]. the l’eroua—Xlalilx uonline:u' diffusion ecpiation 118] resolved by
the (j-onventioinil explicit scheme. the A08 scheme [166] without the step of regular~
ixation. the LSFZK for a linear diffusion process. Table. 3.3.2 provides a quantitative
analysis of corn])utational cmnplexity for these schemes. .\lulti]’)licz~1tion or division
and the addition or subtraction are calculated se1,)arately. as given in [153] and {166}.
In the Table 3.3.2. .\"' denotes the number of pixels to treat. m is the dimensions of
the. problem considered. and H" is the half of the computational bandwidth. Note
that the first two methods requires multiple time steps. while the last. two methods
can be integrated in a single time step. llmvever. the reliability of the implicit AOS
scheme for a large. time step needs to be validated.

We next discuss an efficient edge extraction scheme. As it is well-loiown. thresh-

olding is one of the most common approaches. Usually. thresholdiug may be viewt-éd

as an (7)1)(Jl'?litl(_)ll that tests a hint-tion '1’ of the form

T : Tl.1'.,1/.1)(1'.t /-l 1L1)\'( 11.1)] (3.7)

where p(.r,y) denotes some loeal property of the point. Fortunately, (?X[")(Tl‘l(‘Il(‘(‘,
has shown that. the threshold T is eonneeted to the estimated varianee whieh is a

measure of eontrast. The ehoiee of yarianee (an be

i+l\ 1+1?

1 - 2
= (I —I~ . 3.8
2L+1 “1+1 :2] le tDIi .1) Am”) ( )
\j’:_} J

[\D

 

’) . é .
where 0“ denotes the loeal variant-e. Here. 151”". 7. represents loeal average and 1s

defined as

 

l 1 Hair j+L
— . . I I. " fi’ (
[Ling]- ‘21. +1 ,1\ +1 L2] ZLIEDi (I a.) )- (3-3)
\ .’/ ”“1

where It and l. are the support of subin’iage in .r and y—direetion.

The, edge deteetor is susceptible to streaking if it, uses only a single threshold.
Streaking is the breaking up of an edge contour caused by fluctuation around the
value of t he threshold. 'l‘heret'ore. a more elli-(-ti\'e threshold seheme uses two thresh—
olds Thigh and TIM. with Thigh it 2'00”” 111 this ease, streaking oeeurs only when

lluetuates above. the high threshold and below the low threshold. Tlu‘l't‘fUl'O. the
probability of strez-iking is greatly redueed. The probability of isolated false edge
points is also redur-ed beeause the strength of sueh points must. be above the high
threshold. \Ve note that if the threshold TIN”. is too small. there will be some false
edges. Similarly. it NW], is too large. some portion of real eontour may be missing.

hi our seheme. the ratio of the yarianee oi [Igni- to that of original image is used

43

to determine T101,” from which Thiqh (:an be fixed. i.e..

0' w

Time : %%0E01e

(3.10)
ling}; 2’ 2Thin?"

where a EDI and 01 represent; the variance of the edge detection image and that of

the original image respectively. In addition. rum—maximum suppressicm is performed

to thin the edges in the process of edge extraction. For more detailed (,lescription.

the reader is referred to WI].

L

3.3.3 Results of image edge detection

To (:lemtmstrate the caps-ibilities of the edge detector described above. we carry out
experiments on yarious synthetic and real images. and compare the results with the
output of four other detectors: Sobel detector. li’rewitt detector. Canny detector
and anisotropic diffusion scheme (l’erona-Afalili model).

A yariety of ‘iiii;~iges are employed for the edge detection. \Ve first consider
the Barbara image. which is a challenging test. Figs. 3.10 (a). (b). (c). (d). (e),
and (f) show the Barbara image and edges of the Barbara, image ('letected by the
Sobel detector. the li’rewitt. detector. the. Ginny detector. the anisotropic diffusion
scheme. and the present [EEK edge detection method. It. is seen that the. present.
LSEK edge detection method is the best. Facial edges are accurately extracted and
image texture is clearly detected. lloweyer. none of the other four detectors gives
correct edge response to the texturml part of the image. The regularity of the thin
lines in the original image has been entirely distorted.

We next consider a few standard modem/(alt; hit/l) litifizi/‘(t‘ images (.-"\irpl:me.
floats. Pillow. Mailbox. heather. and Turtle images) and hilt/l) f(f’J'fi/x/‘(i illir1_(]t.‘.s (Ba—

boon. ‘Tig’er. (.loldhill. firidge. Boats and l’inecone). see l’ig. 3.11 and Fig. 3.13. hi

general. the processing of texture images is a difficult task for most existing edge
detection methods because the involvement of high-frequency components. For in-
stance. air1_>lane image has a lot of small details such as small characters “01568." and
“US. AIR F(.)ltCE" on the body of the airplane. The ability of resolving such de-
tails is also crucial to edge detection and pattern recognition. Most existing methmls
are also not able to distinguish Babtmn‘s bristles from the background. l\"Ioreover.
it. is a challenge to fully detect the thin ropes on the Beats by many other existing
methods. Similar fine texture. such as that in the grass and trees can be found in
other four images. The. ability of resolving such details is crucial to edge detectit‘m
and pattern remgnition. It is seen from Fig. 3.12 and Fig. 3.14 that the proposed
LSEK does an excellent job in the edge detection of these moderately high and high
texture images. The detected image edges provide even (i-learer features than do
the original images. For example. the [‘iattern on the house roof and window in
Goldhill image is not very clear in the original images. This feature is visible from
the detected edges.

Having validated the proposed method for the edge (iletection of texture images.
we finally apply it to a group of medical images as shown in fig. 3.15. The edge
detection of X—ray images can be difficult because of" small gradient in image resulted
from small variation in X—ray attmmation. see Lungtll. Lungtl'Z and LungU-l. huge.
amount of \-'<_~ssels can lead to high texture in‘iz‘iges as shown in Lungtl3. Fig. 3.16
indicates that the proposed method is able to reveal a variety of features in these
images.

Based on the above discussion. it is evident that the present LSEK edge detection
method is one of the best edge detectors in 1)t’irforinance. .\lorei_)ver. the. present
method requires only a. single time step operation. It: is these two advzmtages.
high efficiency and excellent pt‘i‘forn'iancc. that qualifies the proposed LSEK edge

detection method as one of best 2*t])]‘)1'()zl(}ll(‘5 for image edge detection.

3.4 Digital image enhancement

3.4.1 Brief introduction to image enhancement

In general. image enhairccment is a process designed to increz—rse the usefuhnss of the
image. \Vhen images are enhanced for human \-'i('*we1‘s. the objective is to improve
perceptual aspects. including image quality. intelligibility, or visual appearance. One
important. class of image enhancement. is to modifv the contrast. and/or dynamic
range of the image. Another class of image enlrancement is to reduce the degradation
of the image. In this work. we focus on the former class of issues. i.e.. to improve
the contrast. of mammograms.

rat lzrrigt' llllllll)(‘l‘ til' iri\'<wstiig;;iti()11>; ll&l\'(‘ lititiri Illil(ltT tr) (‘IllittIl('(? Illtlllllll()g§rtl})ll}’
hwuiuc “inh‘reducunriunse. (hnwhaiileiuwd an adapthxriufighborhood hnage
I)Y()('(¥SHi11§1 f(‘(illli(111(‘ l() (‘Illltlll(7(‘ tllt‘ (1)1117Y2181’ ()f il'ttt1111‘>$ 11‘l(‘\}111l1 tti Iiiriiiiiiitiggrzrl)11§2
lh3w1w112 thisiinwluid enlnurced the ccnniwed (a inarnincgnmnihw'features as “Idlzus
nome. anuuurrt at ill l6]ckwchunwlzuiadapthtrnendnnnhooddxnmd unage
processing technicpie hillowing Gorden‘s work. 'I‘hey utilized low-level analysis and
l§11<)V\l1><lg;t‘iir (lt‘HllT‘tl iiizitiiixj (ltrsiggii tt) i1111)r()\«) tllt‘ (1)11tfttSif ()f s[)<‘(ili(: fixatiiixis.

}7iltx>ri11ig ins ziri ()ltlt‘Sl t(3(illli(|11(‘ lizis 211st) l)(wfiir (f}{i(‘Iltii\T)l}’ 11st>tl iii 111(JCliCYil irrizigg—
iiigg. I:()Y (‘Fittlll[)l(‘. 'lirlicitwrs; (it (11. [l'lts] (lrrxrfilt>[)(itl 21 Illt‘il1()(l tiii' tll(¥ (‘11l12111c131116?11t; c>f
(lustzuullnema nuhognuduesln'an aunnuaficrqnuhd hhefing. hithen'“rwk.a
hrnairtaindiuiathin in aircirnrnral hruage aini twrirniniotlnnl nintgcs vvas usemt 'Tina
LHT)CPSS \vas C(Hlu)h‘lvtll))'tlIlUIdi1HV1Y ctnrtrast stretclning.

'lelt‘(lliit’11‘ll(1‘ iirreigg' is it (xiiirriitiiili' 11st‘rl (dlllilllt1‘lllt‘lli f(‘(illll(111(‘ttll(l is tlrca l)EiSiES
()T iritirst (1);istiir;; (“t)lll])11i(‘t-tli(lt?(l (ii21§;11()5l>% si'sti*1iis. 'Tfliii (iliit‘lt‘llt1‘ iiirzig§(* is ljhfllztll)’
(daahnwlln'subtnntingziruuantsmwliunnn‘than Huicannnaltnnato HHHUVPldK¥

backgrouml i23l.

46

3.4.2 Our techniques and results of image enhancement

As we have discussed in the last subsection. the edge of an image is obtained based
on the difference between the original and the smoothed image. A similar idea is
extended to image enlraneement. by increasing the contrast of the image. Specifically.
the enhanced version ( EV) of the image. is produced by adding edge detected version

(EDV) to the original version

1Ei'(-’II« .0.) = [(13.11) + IEDV(-'T- .111
z [(.1". y) + C'3[I(.r. y) — [Stir- 31)].

(3.11)

in which the parameter (‘3 is used to further increase the contrast. between neigh—
boring edges. There are a couple of criteria constraining the choice of the parameter
("3. ()n one hand. ('3 must be larger than or equal to l. ()n the other hand. if ('3 is
too large. the results may exceed the maximum and minimum yefiilnes in the original

yersion. 'l‘hus we give mmther (expression of the enhanced version as

[Hi-13.91 : (hit-I'- y) + (311(111/1 — ISM-11.1111

= ("51(17- .9) — Falsiftr- 11).

(3.12)

where the parameter ("'5 is used to control the Inaxima in scale-space and C3 is a
inagnil‘ication Inmnneter. Basically. ('5 is an element of (12) and C3 is an element.
of 1 ‘2). In the aboye equation. the smmn’hed yersion of the image is still produced
by solving the linear forward diffusion Eq. (3.2) which can be rapidly acconi]>lished
by the LSlili. The diffusion coefficient is also fixed to be :1 = l and the whole
dillnsion process is terminated at time I : ll).
\Ve first consider {our mannnography images collected from the FTP site

fl;).}»"//n'i/w.cuss-(21‘.ac.'z//.',,//'/)(l,/pInf/m.ins. The original images from the FTP site all

haye the size of 11121 x IUZJ pixels. l’hnyeyer. the images used in our study are

47

cropped to the useful portion. as shown in. Fig. 3.17. The above image enhancement
scheme employs two important parameters C5 and (‘3. which are set to 1.95 and
1.05. respectively. Fig. 3.18 shows good results for alnmst all test images. i.e..
abnorinalities on all mamniography images can be seen more clearly in the enhanced
images than in their original ones.

\Ve next consider four CT images of a mouse as shown in Fig. 3.19. The
same. parameters used in the last. example are utilized for these images. Results
shown in Fig. 3.20 indicate that the proposed method is very efficient for medical
image enhaiicement. Nevertheless. it is worth mentioning that more attr-(‘rctiyx'e image
enhancernent results may be obtained if one chooses the diffusion equation coefficient.

adaptively.

3.5 Summary of Mathematical Modeling of Im-
age Surfaces

l’artial differential equation (1)1.)1‘1) based approaches have great potential for the
processing of images and lllllli-ltlllllt.‘llh’it.)lléll data because they can be made, to be
systematic and automatic for large—scale high dimensional image data. Nevertheless.
research in this direction is often hindered by problems in pare-unetcr (..)ptimizat'ion
and extra computing time. The present work proposes an (,‘volution operator based
single time step method for image and multit'limensional data processing. By uti-
lizing a local spectral evolution kernel (LSICK) that anz’ilytically integrates a class
l’lllis. we show that a number of image processing operations. such as image de—
blurring. denoising. edge detection and e1thancement. can be effectively carried out
in a single step of time integration. Filter properties of the LSfilx' is studied by using
the Fourier analysis. Local information about the image function and its gradient

is analyzed and incorporated in the time evolution of the anisotropic type equatitm.

418

111 image denoising. a local variance based method is employed and is compared to
the regularizatitm type of apprt’iaches. Automatic exit schemes are proposed to stop
image evolution in the denoising. \Ve have achieved about 8 dB in the peak signal
to noise improvement. In image edge. detection. the proposed method utilized a
simplified version of the previous coupled f‘DlC algorithm. The present method is
compared with a number of standard approaches. including Sobel. Prewitt. Canny
detectors and tuiisotropic diffusion operators. High texture images and X-ray lung
images are employed to demonstrate the proposed method. It is found that the
prtmosed scheme provides some of the, best performance for the edge detectitm of
high texture images. which is a challenge to other existing methods. Image enhance-
ment is carriml out by the combination of the image edges and the original image.

Significant enhancement is kl(‘llit,‘\"t1’(-l due to reliable image textures.

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Table 3.2: CPU-Time comparison. between the forward Euler scheme and the
LSEK. Running the linear diffusion equation with A = 1 to time t = 20.

 

 

 

 

Scheme 7' CPU time(s) Rel. [12 error
Forward Euler 0.2 5.01 0.78%
2nd-order Runge Kutta 0.2 11.2 0.76%
LSEK ‘20 0.88 0.15%

 

 

Figure 3.10: Edges of the Barbara image detected by different edge detectors.
(a) The original Barbara image: (b) edges detected by the Sobel detector: (c) Edges
detected by the Prewitt detector: ((1) Edges detected by the Canny detector: (0) Edge
detected by the anisotropic diffusion scheme: (f) Edges detected by the LSEK-based
edge detector.

 

(a) Airplane (b) Boats

 

(c) Pillow (d) Mailbox

 

(e) Feat her (f) Turtle

Figure 3.11: Moderately high texture images.

53

 

 

ats

(b) B0

 

 

(a) Airplane

 

(d) Mailbox

(c) Pillow

 

 

 

 

               

(f) Turtle

) Feather

(0

Figure 3.12: The edges of moderately high texture images.

54

 

 

(cl Boats (1) Pinecone

Figure 3.13: High texture images.

 

 

 

 

 

 

ICE.

:‘="
'Tvr,‘

'1;

 

 

(a) LungOl tb) Lung02

 

(c) Lung03 (d) Lung01

 

(er Abdomenfll fl”) A bdomentl‘l

Figure 3.15: Lungs and Abdomen CT images.

“I

(:1

 

 

(a) LungOl

 

 

(c) Lung03

 

(c) AbdomenUl

 

 

 

 

 

 

 

 

 

 

 

(f) AbdoincnU‘Z

Figure 3.16: The edges of Lungs and alxloinen CT images.

 

(e) mdblfil (d) mde 70

Fi ure 3.17: The n’ianinioura )hs used for imavc enhancement.
h h

 

(c) indbltil (d) mdbl70

Figure 3.18: Enhanced Inaminographs.

60

   

m C "r001 (b) (“"1‘002

   

(c) e'r003 ((1) ("rear

Figure 3.19: The CT images of mouse used for image enhancement.

61

 

(a) C’l‘OOl (b) CT002

 

(c) C1003 (d) CT004

Figure 3.20: Enhanced CT images of mouse.

Chapter 4

Molecular Multiresolution

Surfaces

In this chapter. we introduce molecular illllllfil‘t‘solllfioll surfaces as a new para-
digm of multiscale biologicz-il modeling. .\lol(_'cular mtiltiresolution surfaces contain
not only a family of molecular surfaces. which are corresponding to different probe
radii. but also the solvent accessible surface and van der \Vaals surface as extreme
cases. A novel approach. the diffusion map of continuum solvent over the van dcr
\Vaals surface of a. n'iolecule. is proposed to generate molecular s1_11'fa.(‘-es. The local
spectral evolution kernel (LSEK) for the munerical integration of the diffusion equa—
tion protmsed in the previews chapters will be used to obtain this niultiresolution
surface. \Ve compare our method with one of the most widely used molecular visu-
alization tool (MSXIS‘). It. is found that our results agree with those of MSMS very
well. \\'e also calculate the solvat ion free energies of ‘23 proteins by using our molec—
ular surface and .\IS.\IS as the dielectric boundaries between the solvent and solute.
A linear regression result. demonstrates the accuracy of our method in molecular
surface generalization. Some results of the proposed approach and their extensions

are zu'ailablc in :1 ll. lfifl].

(i3

4.1 Introduction

4.1.1 Surfaces in biomolecules

Continuum dielectric descriptions of solvent environments [6. 54] have become in-
creasingly DOIHIlPtl‘ as a less costly alternative to explicit representat.ions of the envi-
ronment, especially in simulaticms of biological macromolecules. Such models have
been used successfully to understand fundamental physical principles related to the,
stability and solubility of macromolecules. such as proteins. DNAs. RNAS and nu-
cleic acids. in the aqueous solvent and dielectric environments [75]. Important appli-
cations of molecular surfaces. as well as solvent. accessible surfaces [1‘22] and van der
\V’aals surfaces. are flaring in protein folding [09. 142]. prott—éi11—1)r(_)t(_‘i1'i interaction
[38]. oral drug classification [13]. DNA binding and bending [~19], parameterization
of heat ('i-apacity changes [0%)]. macrmnolt‘cnlar docking [16]. enzyme, catalysis [79].
calculation of solvation free energies fl‘21f. molecular dynamics [39]. and implicit.
solvent models f7. 94. 125].

A crucial aspect of such models is the definition of the (‘lielectric interface between
an explicitly modeled molecule of interest- and the surrounding e1n-‘iromnent. There
are three commmr definitions of surface of a linolecule. Fig. 4.1 shows an illustration
of these definitions in 2D. The simplest is the \an der \Vaals (:vd\\') surface. which
is merely the surface of interlocking atomic spheres. The solvt*nt—accessible surface
(SAS) is formed by rolling a solvent sphere. or a probe sphere over the van der
\Vaals surface. The trajectory of the center of the. solvent sphere. defines the solvent-
accessible surface. ft, is (*‘SStfllflzrtllfi" a vd\\' surface using extended radii. \V'herez-is.
the st)l\-'triit.—ex<_-ltitletl surface (SES). or molecular surface (MS) :3le is the boundary
of the region imiccessible to a solvent sphere as it. rolled around the set, of van der
\Vaals spheres. The molecular surface is most often applied in order to reflect the

act nal solvent" accessibility in a given molecular conft)rrm-ition. This surface envelopes

64

the joint set of atomic spheres with different atomic radii corresponding to their
respective Van der Vt'aals interaction potential. The MS is therefore composed of
two distinct parts. The first part is the contact surface. which is the vdW surface
of the molecule that is direct contact with the solvent probe. The second part
is the re—cntrant surface. which consists of the interior facing portions of the probe
sphere that touches two or more. atoms simultaneously. In addition. cavities that are
inaccessible to a spherical probe mimicking a solvent molecule are carved out. These
cavities have many important functions. For example. the buried water molecules in
the internal cavities may contribute to protein stability [149]. \Vater—filled cavities
also play a role of modulating pKa values of acidic and basic residues surrounding

the cavities [97].

   

solvent accessible.’
surface 1
\

T~—-—'

Figure 4.1: Illustration of various surfaces of biomolecules.

The calculation and analysis of solvent accessible surfaces and molecular surfaces
have attracted much attention in the past three decades after the first calculation
of molecular surfaces by Greer and Bush [70]. A variety of methods have been pro—
posed. including the (1anss—l3onnet theorem BO. 123]. closed—form z’uialytical method
[til]. space transformation '35]. alpha shape theory :90]. triangulation [37"].C‘vartcsian

grid based methods [1213]. contour-buildup algorithm [132]. binary tree [1‘27]. vari-

65

able probe radius [15] and parallel methods [154]. \V'hile most methods represent the
resulting molecular surface by triangularion [37, 127. 17-1]. a Cartesian grid based
method was preposed by Rocchia et al. [125]. Despit of much success. in general.
calculation of molecular surfaces is both topologically and coniput—ationally challeng—
ing due to the multiple overlap of atomic spheres. overall irregular structure of a
macromolecule. and self-intersecting singularities [64. 127]. The situation where the
probe sphere is simultaneously tangent to four atoms poses another challenge [51].
Molecular surface calculations a re commonly the bottleneck in the molecular dynam-
ics of macromoleciiles with implicit. solvent method. where the molecular surfaces
need to be recomrnited at each time step during the course of a simulation. More-
over. as biological phenomena occur over a wide range of length scales. molecular
multiresolution surfaces ought to provide the scientific community with a nmltiscale
framework to reveal complex l’iiologim’d processes across scales. ll11ftn‘tunately, such

a nniltiscale framework does not yet exist.

4.1.2 Objective of this chapter

The objective of this chapter is to introduce the concept of molecular multiresolu—
tion surfaces and to propose a novel dilfusion process for analyzing these. surfaces.
The IllUlt‘t‘lllzvll' nniltiresolutioii surfaces provide. a mnltiscale framework for l')i(,)logical
modeling and a. unified description of the vziin der \V'aals Slll'feu'tf‘. solvent. accessible
surface and nmlecular surface. The essential it‘lea is similar to the nniltiresohition
profiles of a. fractal. which is generated by restricted diffusion. see Fig. 4.2 (a). Use
is made to continuum solvent and its curyatiire—controlleil diffusion. The matihmnat-
ical description of the solvent diffusion process is formulated by reaction—diffusion
equations. All geometric fmtnres of a molecular surface. such as convex spherical
patches. saddlc—shaped pieces of tori. and concave reentrant faces. are naturally

2’~l("('()llllt(‘(‘l by the geometric property of partial differential equations with i-‘ippropri—

($6

 

(a) i (b)

Figure 4.2: Multiseale surfaces marked up by flow contours. (a) The (llffusion map
of a fractal (Courtesy: The Mandelbrot. Explorer Gallery); (b) T he Inultiresolution
surfaces of a water molecule.

ate initiiiH)oumlziry eontlitioiis. A family of moleeular multiresolutioii surfaces are
found to (‘()1‘1‘t‘s[)(illtl to (lilierent probe l'rttlll. and are generated in the continuous
distribution of solvent (lensity. .\loreo\'er. the vein (ler \Vazils surfaee is a speeial
(use in \\'lll(‘ll the probe rmlius vanishes. Finally. a fast eoinputz-itionzil algorithm

is utilized to generate moleeulur nuiltyiresolution surfuees in at single, step of time

evolution.

4.2 Methods

\Vithout the loss of generality. we assume that the fluid flow of eontinuum solvent
has a, density distribution /)('r.f‘) at a spuee—tinie lot-ation (Int). where I‘ 6 RB and
t E [0. DC). In an arbitrarily small domain $2. the motion of the solvent density is
elmrneterimwl by rlensity tlux Jtr. I). In this Work We assume thut the tlux is given

by H generalized l‘lieli's lmr

J

1‘. Ii) :7 — Z Dqt/{t/I). I‘, fllvvgqptr. fl- (4.1)
(1

67

where Dq(I\'(/)). r. I) is the diffusion coefficient depending on the Gaussian curvature
[\f of solvent density isosurfaces. and high order terms (i.e... q > 1) characterize the
lllllt'nlitigt‘llt‘lU’ of the solvent density and flux—flux (i-(u'relation in the density flux.
In [)Ell't-l('tllzll‘, the term VV2 can be regarded as an energy flux operator. A similar
fourth order flux term was employed in the Calm-Hilliard equation [20] to describe
the evolution of a conserved concentration field during phase separation. The total
mass of the solvent in the domain Q is given by j” [)(r. t)(1r. The rate of change of
the solvent mass in $2, % [3, p(r. t)(lr. is balanced by the solvent flux on the boundary

032. — jg)” J-n(r)(/S. and solvent production. [$2 FUN/2), p, r. t)dr

(I

/ 7””-’l"r:- J'ntrMSJr / 1)<Ix'<p).p.r..t)dr (4.2)
I Q

- (2 (I: - HQ

— _‘

\vhere ntr) is the unit outer normal direction at r. Bv using the divergence theorem.

we convert the surface integral into a volume integral and obtain

(7/)(1‘. /)

")1‘ '—'—' V-Dthvt/i). r. f)VV2(1/i(r. I) + I’ll/((7)). p, r. f). (1.3)
(I

‘l
The diffusion coefficients [)(1(I\'(p).r. ii) are position dependent because of the pos-
sible presence of inht)mogeneitv. such. as multivalent ions [I‘ll]. and their curvz‘tture-
dependence is designed for the geometric control of solvent. density isosurfaces in
the final diffusion map. For example, the solvent density distribution near the

molecular surfi-ice can be made to preserve the curvature of the probe molecule
(]'\"'I\'[‘))2

2H}?

) ,z \' 5' ( I r. ‘. ‘ N eX) —~ 'i‘c '7, 's , e “1;, s2 ‘f "t—
Iv t ll'llL’) D,(/\(/) 1 t) \\l(1 I\, 1 th C ut s1 in (IIHI
ture of the probe and m] is constant. The solvent prtuluction can be either neg—
ative or positive. and is designed for further geometric control of the molecular

surlace. It can also be used to add or remove atoms at specific locatituis. High or—

der gradient-controlled diffusion equations were proposed bv \Vei [138] for efficient

68

image processing. .~\nisotropic reaction-diffusion equations have been introduced for
shock-capturing in the context of computational fluid dynamics [161].

In this work. we define the problem of finding molecular multiresolution surfaces
as an initit'll—bountlary value problem governed by the new curvattire-controlled dif-
fusion equation (1.3) in a. domain S2 that is large enough to include all the surfaces
of a molecule. The Dirichlet boundary condition is employed at domain boundary
092. i.e.. [)(r. f) : /)(). r E Oil. Initially. there is no presence of solvent in the domain.
i.e.. [)(r. fl) :2 t). r 6 f2. \Ye set qulx'fp). r. t‘) 2 () indiscriminately in all the atomic
volumes with the radius r,- of it'll atom of the molecule. \Ve choose the evolution
time sufficiently long so that the solvent can reach the center of the computational
domain. The diffusion of the solvent toward the domain center leads to a family of
profiles of solvent density distributitm. which in turn gives rise to a family of mole-
cular multiresolution surfaces. correspomling to different solvent probe radii 7'1, and
different resolutions. Note that the van der \Vaals surface. 03211139- corresponds to
r], : f). and is obtained at the level set p :- 0. Here we define the midway molecular
surface as the mult iresolution molecular surface whose convex surfaces are located
between the van der \Vaals surface and the solvent accessible surface. Such a surface
should be useful as it is unclear whether the solvent accessible surface or the solvent
excluded surface better ('lescribes hydration effects [75)]. In general. the. nonlinear
terms in liq. ("1.3) can be utilized to generate large gradient. near the desirable
molecular surface. This approach directly results in a Cartesian grid representation
of the molecular surface. Such a representation is important for many applications.
particularly for implicit solvent modeling of electrostatics of inatrtuuolecules [1‘25].

In general. it might not be convenient to solve the nonlinear reaction—diffnsitui
equation (1.5%). Thus some simplifications are desirable. for homogeneous solvent.
it is sufficient to set [)1 a piecewise constant [)1 : Utr). and [)(l r: t) if q ?4 ‘1. For

simplicity. we also drop the production term in the rest of discussion. 'l’licrefore. we

fit)

have a much simpler diffusion problem

Upfr. if)
(N
[)(I‘. t) : I)”. I' 6 0f).

:VJXHVMnfl.rEfZ H4)

pfl‘. 0) = f). r E 52
D(r) =1. 1‘ E SZ/l'i'u'D,

Dfr) = 0, I‘ E l’ryu'D.

where l‘l'H'D is the molecular van der \V'aals domain given by

line = {1‘ I ll‘ - ril S 7'x-i=1

lv
0»)
W4

and 1', is the atomic center for the 1"“ atom of the molecule. This diffusion ecpiation
does not yield a closed-form solution due to the irregular atomic surface infornn-rtion.
’l‘herefore. finding the solution of Eq. (4.1) numerically is quite robust. For example.
both implicit and explicit approaches. as well as various spatial discretizations ‘an
be used to generate the solvent density profiles near the van der \Vaals surface.

The van der \Vaals surface generated by this approach is exact in the sense that
its resolution is only limited by the finite mesh size. l’lowever. the molecular surface.
or the solvent excluded surfzfrce computed in this manner could be distorted because
of the different diffusion for different parts of a molecule. To eliminate this error of
molecular anisotropy. we set the bounden'y of the computational domain to be the
solvent accessible surface (Of? : Hill-53h") defining st)lvent-accessil)le radius of the
if“ atom as the sum of its atomic radius and the probe radius. riff/f : I',j + rp. The
domain enclosed by the solvent—accessilile surface is denoted by l'S‘f‘g:

V&W:{Vh~nHM”WJ:rzam}

I

TO

\\'e initialize the solvent density on the desirable. solvent accessil.)le surface by setting
[)(I‘.fl) : 0. VT 6 VS‘LS' and pfr.0) = [)0 in rest of the domain. A large probe
radius r!) can be chosen to ensure a. variety of molecular multiresolution surfaces.
Note that solvent accessible surface marked out by the solvent density {)(r. O) is also
exact

The diffusion equation of (4.4) can be solved with standard central finite dif—
ference and Euler time evolution schemes. To speed up the solution of Eq. (4.4),
we will apply the fast local spectral evolution kernel (L‘SEK) method described
in previous chapters. The LSEK is able to analytically solve a class of reaction-

. . ‘)

diff11sio11-ctu'1vection ecplations ~52”) = D(1)5% + C(' )5); + Pff ) p in a single time.
step. it is exact in time and is of controllable accuracy in space. Its complexity at
a. fixed level of accuracy scales is ()(\). where N is the number of grid points. The
solution of Eq. (If) at an arbitrary point of space—time. (.17j.a‘/J' 3). t)1 is obtained

in a finite (.lif'fercuce 111211111er

.‘l1)"[?/.“[:

WWI-.1: Z Z A1110!” II, ti)><

[.1'7.llr](/:—l[1/]::~—'\/;
[\lf.I/~(Ty(-,.Uhy‘ t)[\-h:‘0: (13,12. t) X

/1(.r,j — {Jr/Ila}; —Z,/hy.3 —f II~ (I). (4.5)

where 2-)!” + I are the stencil width for (it = .r. _I_/. 3. II“, is the grid spacing. and

11';,fi(.17.t) is the. [SICK [I73]
..\ [I] 1'2 . ~ ) . 1
II “a l n l (T “"T .I‘
1.0 . . (7 9:7, 4 V27?!” (71' ..H \r/Zflf ( )

lIere l23,,(.r) is the Hermite function defined by the Hermite polynomial [12”. i.e..

. 1 ‘). ~ » . . . .
[23,,(J') :— (‘pr‘~"".l.""l]'l~_),,f.l'). .l/h 1s the h1ghest degree of the used Hermite polyno—

-f
. . . ‘) ‘) , . , ,_ ‘
mm]. (7 1s window parameter and a,“ z (7“ + ‘2. / [)(s)ds. \\e set M}, : 1:). Ma 2 ‘21
. (l
and (7a : 0.111“ in above diffusion evolution process.

 

Figure 4.3: Illustration of multirest)lution molecular surfaces.

As a proof of principle. we consider a model molecule consisting of four atoms
to demonstrate that the proposed approach provides a familv of multiresolution
molecular surfaces. including the van der \Vaals surface and the solvent. accessible
surface. A cross section of these surfaces. generated from a single computation. is
shown in Fig. fit. ()bviouslv. each surface corresponses to a different probe radius.

Due to the extensive application of the nmlecular surfz—u'e. we. present. a series of
controls to get an apprt)priate isosurfzu-e which exactlv corresponds to the molecular

’ atom has the radius I',: and is

surface. \\'e consider a protein with H atoms. The I”
centered at 1‘). I : l. II. \Ve choose the domain enclosed by the st)lvent-accessil’de

surface to be

We consider initial value [)fl‘.fl) to be a piecewise constant function which is 1000

‘s' .- l 5'

outside the domain l." and ptr.t)) :: () elsewhere. We choose an iso—surface

/)(r. t) -: (HI/1H to be solvent accessible surface. Then all intersection points r3. 3 _—_-—_

‘1
[Q

 

 

 

 

 

Figure 4.4: The. active grid points (squares) in the process of molecular surface
generation. Dashed line: Solvent accessible surface. The diffusion coefficients D =
0.01 at active. grid points and D = 0 at all other grid points.

1.2. of SAS with the mesh lines can be efficiently calculated by the marching
cube. rrretlrod [101]. Now. we consider a domain 11' : {r : |r — r3] g I"). s z 1. ‘2. ...}
and set. an active region as 11' W 17539 where the diffusion coefficients D 2 (1.01
and D : f) for elsewhere. With the initial p“ and given diffusion coefficients D.
the solution of Eq. (4.4) can be obtained by the. LSEK expansion (4.5) in single
time step. See Fig. 1.4 for the application of this procedure to a two dimensional

problem.

4.3 Results and Discussion

4.3. 1 Singularity Test

Bion‘iolecular surfaces in earlier models. such as the 'an der \Vaals surface and
the solvent accessible surface. are non—smooth. Although molecular surface was
intrmluced to create smooth surfaces by smoothly joining atomic surfaces with the.

probe surface. the molecular surface is not smooth everywhere (it is not ('1). There

73

are self-intersecting surfaces. cusps. and other singularities in the molecular surface
definition. In this study, we shall carefully cxan‘iine whether similar singularities
occur in our method by a systematical ccn‘nparison of our molecular surface and
those generated by using the MSMS.

Fig. 4.5 illustrates ottr molecular surfaces and the MSMS for two-. three—. and
four-atom topologies. Our molecular surfaces are (lt’*[)iti-t;.e(l in the second column.
.\-'Iolecular surface by the .\IS.\IS are sl’iown in the first column. It is seen that
cusps occur when the probe radius is relatively small. Overlapping of two probes
results in a. self—intersecting surface. see Fig. 4.5 (h). The singularity is found at the
et‘lge of such a self—intersecting surface. This surface. will appear if the probe sitting
above the three atoms intercepts with the probe below. The self—intersecting stu'face
singularities also occur in the four-atom system. see Fig. 4.5(k). A con‘iparison is
made between our method and the MSMS by present ing their cross se.ctions (.r : 0).

It is observed from the figures that their results are quite consistent.

4.3.2 General test

The second test case. cyclohexz-tne ((701112). reconnnended by Sanner (if a]. [127].
is considered to validate the proposed 111ethod for calculating the solvent (‘rxcluded
surface. The van der \Vaals radii of the carbcn'i and hydrogen atoms are taken as 1.7
A and 1.2 A. respectively. \Ve choose. a. grid of 200 points in each dimension and set
[)0 : 1000. r1, = 1.5 A. The solvent excluded surface is extracted at. [)(r. '10) = 0.6.
see Fig. 4.8 (a). \Ve also conniute the solvent. excluded surface of cyclohexane by
using the. MSMS algtn'ithm [127]. The. conmarison of three cross sections generated
from the present z—tmn'oach and those from the .\lS.\lS is depicted in Fig. 4.8. There

is an excellent agreement among these results.

 

('95) (h) (i)

   

(,1) (k)

Figure 4.5: Loft (011111111: Molcculzu' surfaces of two—. t111‘cc- 11.1111 fuur-atom systems
by MSMS: Middle ("0111111112 310101111111“ surfaces of two: thrcc— and four—atom systems
by diffusion: Right c011111111: C‘orrcs11011111113r ("(111111211‘is011s of cross scctions :1: = 0
(Circles: MSMS with 7") = 1.4 A: 8111111 11119: our 111010c111ar surface with TI) = 1.4
A). F111 11111 (liutouuc system. (1.y.:) : ('2.3.().()) 111111 (2.3.0.0). For the three-
;1111111 systmu. (1.1;. 3) = (—2.3.().()). (2.25.0.0) 111111 (0.3.9810). For 1110 fo1,11‘—ato111
systcul. (12.11:) : ( 73.0.0). ((1. —‘2.(i.()‘). (35.0.1)). and (0.2.6.04).

 

Figure 4.6: Molecular surfaces of the buckball and their comparison with those
obtained by the MSMS (Solid lines: present: Circles: MSlVIS). Van der Waals 7‘ = 1.5
A: Probe radius rp = 1.4 A. (a) The molecular surface; (b) The cross section at
.l‘ = 0:

4.3.3 Cavity and pocket test

\Ve proceed to consider the ability of the proposed 111ethod to capture internal cav—
ities and open civilities (pockets) which are comnmnly encountered in macromole-
cules. It is thus interesting and unportant to explore the representation of these
cavities in the proposed algorithm. A buckyl’n-ill molecule (bucknii11—sterfullere11e
Cm) is chosen as an example. This buckyball has 12 pentagons which do not share
any edge. To study the. performance of 011' algorithm for open cavities. we create
a C51 ‘1nolecule' by removing six carbon atoms. i.e., one hexagon being excluded;
the coordinates of the remained titty four atoms are kept the. same. For both study
cases. we set r], : 1.0 A. The results of the buckyball are. shown in Fig. 4.6 with 21
Comparison of the cross sections to those obtained by the MSMS. Apparently, the
internal cavity of the buckyball has been resolved very well by our approach. while
the RISK-'18 fails in the inner cavity generation. The configuration of the open cavity
in the fictitious molecule (.‘51 is presented in Fig. 4.7. where we observe a very nice

consistence between our results and those of MSMS.

76

 

 

 

 

 

 

 

 

 

 

 

—4 o 4 —4 o 4

Figure 4.7: Molecular surfaces of the open buckball and their comparison with
those obtained by the. MSMS (Solid lines: present; Circles: MSMS). Van der VVaals
r = 1.5 A: Probe radius rp : 1.0 A. (a) The molecular surface; (1)) The cross section
at .r = 0: (c) The cross section at. y = 0; (d) The cross section at z = 0.

4.3.4 Large molecule test

We will further evaluate our molecular surface generator on large biomolecules 111
particular. A cell wall biosvnthesis protein [31] is considered. This protein has 92-15
atoms in eight polymer chains and 1328 residues (PDB ID: 1NOE). We choose the
same set of computational parameters as those described 111 the preceding paragraph.

The van der \Vaals surface and the solvent excluded surface of the cell division

77

 

 

 

 

 

 

 

 

 

 

 

Figure 4.8: Molecular multiresolution surfaces of cyclohexane and their comparison
with those obtained by the l\IS.\lS (Solid lines: present; Circles: MSMS). (a) The
solvent. excluded surface: (b) The cross section at. .r 2 ~09; (c) The cross section
at y = 0.6: (d) The cross section at z : —0.5.

protein are depicted in Fig. 4.9. It is seen that the proposed method works well for

a large biomolecule.

4.3.5 Applications

Finally. we consider the z-ipplications of our molecular surface to the electrostatic

analysis. The solvation free energy is calculated from numerical solution of the

78

 

(a)

 

Figure 4.9: l\rlolecular multiresolution surfaces of the cell division protein. (a) The
van der VVaals surfaces: (b) The solvent excluded surface.

Poisson—Bollzinann (PB) equation in a continuum dielectric model:

 

., 1717.7), , , _.
r—V . (t(l‘lV1f)(I‘)\) + H'OU‘) 7: ,- ,1. E (1,-a(_r-— rll) (4x)
~n .
I

where (15(r) is the electrostatic potential. ((1‘) is the spatial dependent dielectric
constant. a is the Debye screening function describing ion strength, 8“ is the electron
charge. K B is the Boltzmann constant, qi is the unit(proton) charge located at
position rig, and T is the absolute temperature. A dielectric constant of 6p 2 1
is used for the solute interior, while the external dielectric constant, 6w, of 80 is
applied for the exterior aqueous medium. Here. p and w are subscripts representing
protein and solvent. See Fig. 4.10 for an illustration of this continuum dielectric
model. The PB results strongly depend upon the definition of the dielectric surface.

In many biomolecular modeling and simulations, the molecular surface is used as

the dielectric boundary between high and low dielectric constants.

 

Figure 4.10: An illustration of the continuum dielectric model in biomolecular
simulations.

The implementation of a well—defined molecular surface in the solution of PB
equation depends on the underlying numerical techniques. In finite difference based
PB solvers such as Delphi. MEAD and PBEQ, the molecular surface is only used
to define. the map of dielectric function. but the position of the molecular surface is
not explicitly computed. A recent proposed matched interface and boundary (MIB)

method [181] utilizes an explicitly defined molecular surface on which the following

80

two interface conditions are enforced.

()1) : (Dil'. (I) :

—. 4.
(9/2. F“ 0n. ( 8)

The first condition describes the continuity of the electrostatic potential across the
interface. while the second cliaracterizes the. l‘)alance of the derivative of the electro—
static potential field at the interface. \Ve therefore use the .\‘IIB method to solve the
PB equation with our molecular surface and the MSMS to compare their electrosta-
tic solvation energies. Since the l\IlB 111ethod has incorporated accurate intersection
points of the molecular surface with mesh lines and associated normal directions,
the obtained electrostatic solvation energies from different surfaces will be different.
Due to the iso—surface nature. our molecular surface can be easily implemented with
the MIR method. The intersectitui points of molecular surfeurc with the mesh lines
and the correspomling normal directions can be readily calculated by marching cube
method [1111]. The. solvation energy calculations are conducted on a mesh with grid
spacing I: of 0.13 A. In order to involve less surface error. MSMS surface with a den-
sity of 11) is used and the molecular iso~surface by the ('liffusion process is generated
with a resolution of 0.25 A.

'f‘wenty three proteins. most of t hem are adopted from a test set used in previous
studies (533178]. are employed to validate our approach. For all structures. extra
water molecules are removed and hydrogen atoms are added to obtain full all—atom
models. l’artial charges at atomic sites and atomic van der Waals radii are taken
from the (‘i'll.-‘\li.\f.\f'22 force field [801.

The numerical results of elect rostatic free energies of solvat ion (AC) are listed in
Table 1.1. For twenty three proteins. the current results based on the our generated
surfaces are in excellent agreement with those reported in the literaturcs [313.178].

This validates our computational prm-cdure.

81

Table 4.1: Comparisons of total molecular surface area and electrostatic solvation

 

 

 

 

energy.
318318 MS
PDB code No. of Atoms Area AG Area AC
lajj .519 2166 4136.9 2136 4126.3
lbbl 576 2604 -994.2 2552 -999.6
Ibol‘ 832 2898 -854. 1 2841 —847.1
lbpi 898 3233 -1304.0 3177 -1309.9
lcbn 648 2366 —305.4 2328 «299.5
1fca 729 2547 -1201.3 2521 -1204.7
1frd 1478 4367 -2852.2 4277 -2858.9
1fxd 824 2922 -3299.8 2966 -3308.7
llipt 858 3263 -811.6 3194 -811.2
liiibg 903 3071 -1350.9 3002 -1358.9
1neq 1187 4716 —1730.7 4590 -1736.1
1ptq 795 2897 871.8 2839 -870.5
1169 997 3054 -1088.9 2991 —1089.3
lshl 702 2744 -754.6 2679 -738.9
lsyl‘ 1435 4644 ~1713.5 4564 ~1717.1
lttxc 809 2835 ~1141.6 2.774 ~1150.6
lvii 596 2476 —902.5 2397 —903.7
2er1 573 2318 -950.2 2280 -955.5
2pde 667 2716 —819.5 2662 -794.8
451(' 1216 4159 —1025.0 4046 —1013.8
1a2s 1272 4437 —1912.8 4324 —1915.3
1a63 2065 697 1 —2375.5 6871 —2377.7
IaTm 2809 7733 —2158.0 7011 -2143.3

 

 

Table 4.1 illustrates the total surface areas for these twenty three protein mole,-
cules. The surface area is obtained by summing the areas of all the triangular
elements of the surface .4 : Z 4,». where :1,- is the area of the 1"” triangle. It can
be concluded from the table 4.1 that our molecular surface usually 1‘)1'o(_lu<;‘es a little
smaller area comparing with the values generated by the 318318. This is explained
by the fact that the chosen iso—surface strongly depends on the grid resolution. The.
graphic comparisons of the solvation free energies of twenty three proteins and total
molecular surface areas are shown in l7ig. 1.11. The solid lines in two figures per-

fectly matches .1/ r: .l' with slope 1. Linear regression of twenty three scatter points

82

will lead to a slope. of 1.0047 for (a) and a slope of 0.9732 for (1)).
Fig. 4.12 shows the electrostatic potential projected at. the molecular surface
computed with the MSMS and our generated surface at It : 0.5 A (PDB ID: 1:163).

It can be. seen that there is a very good agreement between two potentials.

 

 

 

 

 

 

 

 

A 0 . . ' 8000 . . -
6 g C
E 9 J.
g 00 1‘"
V —1 0 h ,
>‘ 8 6000
9 “g
“c’ a)
L” -2ooo~ i ‘3 ”V
,‘g: a 4000' v”
‘65 2
> O
53) 2
5 -3000- “ g _ ,
O . . . 2000’ . ' . . *
-3000 -2000 -1000 O 2000 4000 6000 8000
MSMS Solvation Enerov (kcal/mol) MSMS Areas
(a) (b)

Figure 4.11: Graphic comparisons of solvation. energies anti total molecular surface
areas for 23 molecules. Solid lines are perfect. matches y = .r. (3.) Plot of solvation
energies for our molecular surface vs. MSMS: (1)) Plot of molecular surface areas
for our molecular surface vs. MSMS.

4.4 Conclusions

In summary. we have introduced a new concept. molecular inultiresolutitm surfaces.
for the multiscale modeling of biomolecules. The proposed concept provides a uni-
fied description of the van der \Vaals surface. solvent accessible surface and solvent.
excluded surface. .»\ novel approach based on the ('()ll11'()ll("(l diffusion of continuum
solvent density is proposed to generate the i'nultirestilu’tion surfaces. The fast, local
spectral evolution kernel is implemented to integrate the diffusion equation in a sin-
gle time step. The proposed method does not. need to discriminate buried atomic

surfaces from unburied ones during the computat ion. Thus. it bypassts the difficulty

83

of singularities in other existing methods. Ntunerical experiments are carried out
to validate and ('lOIHOIlStl'fth’ the proposed approach. The formulation and examples
given here hopefully will spur the. interest of other investigators in this and related

fields.

 

Figure 4.12: Surface electrostatic potentials of colt p factor (PBD ID: 1&163) pro—
jected onto its molecular surface. Left: Generated with our molecular surface; Right:
Generated with the MSMS. Colors are. chosen according to the magnitude of the
potential as indicated with the color bar.

84

Chapter 5

Singularity-free Molecular

Surfaces

In this chapter. we introduce a novel concept. the. constant. mean-cur\.'ature moltcular
surface (CHRIS). which encompasses a familv of hioniolet-ular surfaces for theoretical
modeling of biomolecules. The (‘7l\l.\lS is formulated l‘msed on the thcorv of (‘litferen-
tial geometry unifving earlier minimal molecular surface and the popular molecular
surface. and is typically free of geometric singularities which occur frequently on
the nmlecular surfaces. A conmutational scheme is proposed for pre-urtical control
of mean curvature evolution of a hypersiu'face. which leads to the C.\I.\IS when an
approprizrte level set is chosen. Extensive numerical experiments are carried out
to demonstrate the proposed concept. and algorithm. (i‘cnnrmrisons are, conducted
between the C'.\I.\IS and the popula‘u‘ molecular surface. .i\p1,)lications of the CMMS

to the electrostatic analysis are considered for a set. of twenty three proteins.

5.1 Introduction

The multiresolution surfaces and the molecular surfaces suffer from geometric sin—
gularities. such as cusps and self-intersecting surfaces [37, 51, 64. 127]. which not
only are nnphysical, but also devastate I'nu‘nerical simulations. Smootth varying
functions were proposed to avoid this problem [68. 77, 102]. Nonetheless, it has
been shown that some atomic centered dielectric functions n‘iay lead to urn.)hysical
interstitial high dielectric regions in implicit solvent models [147]. It is necessary
to develop a new generation of biomolecular surfaces which are free of geometric
singularities. while possess the solvent exclusion property of the original MS.

Recently. Bates et al. proposed a new concept. the minimal molecular surface
(l\l.\lS) ill]. for l)iomolccular surface modelings and developed a new algoritlnn for
the practicz—il generation of the .\[i\ISs under biological ctmstraints. The, i\l.\[S is
not only free of geometric singularities in general. but. also consistent with surface
free energy minimization. The theoretical underpinning of the. minimal surfaces is
differential geometry [(39)]. l‘ixtcnsivc studies on this topic have been carried out
in past decades [3. l. 119]. [n the middle of nineteenth century. Belgian physicist
Jmeph l’lz-rteau challenged niathematicians to give a general description of the area-
minimizing surface (minimal surfz-ice) by working extensively with soap bubbles. As
a ct)Ilsequein‘e. the problem of determining the surface of least. area. ccmstrained by a
given boundary is known as 'Plateauis problem'. Leonhard Euler and Joseph Louis
Lagrange had already begun the. study of the minimal surfaces almost a century
earlier than Plateau.

The mathematical necessity to resolve many of the. conjectures and the problems
of Platez-m had not been developml until the twentieth century. Indeed. the study of
minimal surfz-ices remains an active area of research today. its various :«rpplications

have been ex )lored in )hvsical and bioltwical sciences. flit). I52. 53. 8f). 13%;) . and also
. h t.

86

in molecular engineering [32] and material science [132]. The generation of minimal
surfaces with given boundary constraints can be accomplished by using the existing
software like .\Iatlab or l\latlit-innittica [69]. Evoluticm equation approaches to the
minimal surface generalization were also available in [‘24. 33]. However. the method
of Bates rt al. [12] is the first algorithm that. generates minimal surfaces constrained
by obstacles. such as arbitrarily distrilmted atoms in biomolectiles.

The objective of this study is to generalize the. concept. of the .\IMS by introduc-
ing a family of mean—curvature controlled molecular surfaces. the constant mean—
curvature molecular surfz-ui-e. It is interesting that the C.\I.\IS includes the original
.\l.\[S as a special case. and can be it'lentical to the .\IS for certain I'nolccular biology
because the latter is made of facets of [)iCCPVVlHO constant mean curvatures. However.
unlike the M5. the proposed (‘.\l.\lS is typically free of geometric singularities. \Vc

will discuss its theoretical modeling in the following sections.

5.2 Theoretical modeling

5.2.1 Hypersurface and its mean curvature

., . ,i) . . . . ._ , _ . .
(, onsidcr a (“ immersion f : I —’ 113;]. where I. c IRS3 is an open set. Here
f(u) —- (fltu). jgtu). j’gtu). j'lt'ufl is a hypersurface element (or a position vector).
and 11:: (1/|.112.u;;) E U.
y . . . . . (if . ‘. .
Tangent vectors (or directional vectors) of j are .\,- : (hr' 1 he .lacobl matrix of
' I
the mapping f is given by Uf : (X1. X2. X3). The first frmdarm‘ntal form off is a
symmetric. positive definite metric tensor I of 1‘ given by I I: (Vii) : (Df)1 (DI).
The entries of this matrix can be expressed as 9,-1- : <.\',-..\'.j). where <.) is the

1‘:ll(’ll(l(‘illl inner product in lfiil. I.) : l.‘2.3.

(“J
*1

Lvt 1/(11) be thu unit normal vet-tor of f giwn by the Gauss 111211) I/ : U —> 5'5.

 

 

I/(l11.112.113) I: .\'1 X 3'2 X i\';;/, i\’_1 X Al} X A73” 6 _Luf. (5.1)

\\'il(‘l'(‘. the (Toss product. in R1 is a gOHOI‘t’liiZflt.i011 of that in R3. HON) _l_uf is the
IlUI'llliLl spate of f at point p : f(u). T110 ywtor I/ is pm‘pmnlicnlar to the tangent
hy1wrplano Tuf at 1). Nntv that Tuf E .Luf : 'Tf1u1R3, the tangent space at 1). By
11mins of the normal vector u and tangent vector X1. the second fnndmnontal form

of f is giyvn by

r , / ()I/ , \ r 9
Ilixl‘\/):(hI/):(\—)—“\j/) (0....)
( (1,1
The 111mm ('in'yntin'v (inn b0 ("‘rlh‘lli‘divti fimn
i l‘,‘ ,. ,
If T: "h (1' . ().t;)
3 .I.
1.1 -y1.1':' 1; . .' -\ .' . 1 U— "’1
\\ n It. m HM t n Jinsti 1n mnnnmtlnn (mm 1111011. .111( g _ 1].].

[mt ( tlw (‘iuh’lll‘v of L' C 1R". bv mnipnvt with boundary 0U. Let f : (1' -—> R1
bv a family of i[Y1)(‘I'Slll'i‘éu‘OS indvxml by 3 L?» U. nbtainml by (infiii‘iiiilig / in thv

nm'nml (lirm-tinn zu-(km'ding m the 111mm ourmtni‘o. Explicitly. “1‘ set

flier-U- -:) ::f(.1-.;/..:) +511”! - [foil/(11.04). (5.4)

\yhvrv g : I)('t(1_(/,~IE). \Vv wish to itvmtv this to obtain a 113'])(‘I‘Slll'hK‘t‘ of (,‘(_)111\Ii‘(lllt.
moan (“lll‘\’&'li'lll'(‘. that is I] : H11 in all ()f I". vxwpt possibly Wil(‘l‘(‘ l'nn‘i‘iors (atomic
(‘()IlHH‘HiIIiS) m'v (‘]l('()lllli("]'('('i.

“'0 ("hump f(u) : (1:11;. :. S) \yhm‘u 5(1'. y. 3) is :1 function (if illtl‘l'PSi. \V'v haw

(x:

tht‘ first hnidannmital form:

1 + S?
(fl/J) = 5,61,
SI.

whose inverse matrix of ((1,1)) is giyvi’i by

5.1-5},

1 + 5;} s1, 5';

CH
V

Sys; 1+—s§

— ‘51,. ‘S"(/

1 + 8}}. + 55

“‘ 15:1] ‘8':

—‘ L91 S:
_SU.S'3

.t)

1+£+$

,‘) 3') .9 . — . _
\yhvrv f] : 1 + .81; + 517 + .8; is tho Gram (lvtvrnnnnnt. Tho nornn—rl yvrtor can be

('()Ill])ili(‘(i from H1. (15.1)

i.(*.. thv Hossinn matrix of S.

“1* ("onsnlvr zii fzn’nily I : (.r. y.

* Sg- ). whvrv

WI

51r41:)E—fvfi(HVslflfl.

#1

a
A i

A
$1

'I‘hv (*xphrit form for thv 111mm mn'yntnrv is \\'1'iH(‘ll us

5'
n : 1v. 1::
c; \r/‘I-(l

89

1 .

J

Thus. we obtain the final evolution scheme

-, ,1 1 VS
.H5(.1;. y. z) = 5(1'. y. 3) + EV/Q —V-

3 fl

 

— HU . (5.11)

The minimal molecular surface can be obtained by setting H1) 2 0.

\Ve iterate (5.11) so that S1:(.r.y.z) ——> S(.r,y,:) and H —> H1). For a given
set. of atomic coordimrtes, we initialize S ($.31, 2) by the same way as we did for
the initial solyent density p in Chz-ipter 4. \Ve set. 5(3'. 31.2) as a piecewise constant
functirm. That is. 5(1). y. 2') : S1) for (.r. y, ;) inside a domain enclosed by the solvent
accessible surface V'SAS and S(.r. y. 3) = () otherwise. The van der \Vaals radius
rmmr and the probe radius 7' are prescribed to achieye desired elTect in this work.
By iterating liq. (5.11). the mine of .S'(.1'.g.:) is updated in each iteration except
for the constrained yan dcr \Vaals regions.

The aboye hy1)crsurface e\-'(_)lution 1')rocess can be fornn’ilated as a constant. mean

curyature (geometric) flow

US
9—- : 3 ,i/F— H — H
_ VS
: V/f/V‘ 7; ’ 3\/.‘_/Hd (5.12)
v.

The iteration of the process (5.11) is e(,1ui\'alent. to solying EL1.(5.1‘2) to the steady
state. In this study. the discrete approximation. of this equation is achiewd by
using second order central finite difference scheme for the spatial discretization and
explicit Euler scheme for the time eyolution. The reader is referred to the literatures
[50. (.35. 108. ll 1. 130. .131. 131.. ill) for some of Yi-ll‘lélllf-S of liq. (5.12).

‘- 7.") ,‘) 4‘) . . .
Since (1 : l + .8; + .81; + b; at the hypersurface lmundary is dominated by

{)0

 

(a)

Figure 5.1: Initial condition.(a)Illustrati0n of initial maps of S(:r, y, 2) at the cross
section 3 = 0; r = r'Wm' and r’ = rwmr + T: (b) 5(1‘. y. z = 0) shows a family of
level surfaces.

IIVSHQ : 5% + 5;: + 5'3 Eq. (5.12) can be approximated by the equation

L): : [IVS'HV (—VS > ~3||VSHH11 (5.13)
()f “VSH

if the Grain determinant g is replaced by )IVS'H'). Some other approximations of the
constant mean curvature will also work.

The CMMS can be constructed in a procedure similar to techniques used for the
generation of the HHS [1‘2]. The hypersurfzu-e S(.r. I/- 2) obtained via the constant
mean curvature flow is not the ("HHS that W(‘ are looking for. It gives rise to a
family of iso—siirfaccs instead. which include the desired (‘.\I.\IS. The level set or an
isosui'face of the hypersurface function S gives rise to the desirable CMMS. Nuiiieri-
cally. isosurface extraction can be done with existing soft wares. All Visualizations of
molecular surfaces in current work are done by .\Iatlab or \-"'.\ID. \Vhen applying the
C.\I.\IS to the electrostatic analysis by solving the PB equation with the matched
interface method (NIB). the marching cubes algorithm {101] is used to find the iii—

tersectioii points of the iso~surface and grid lines. .\Iaiiy Work for lille‘lllIig cubes

91

 

Figure 5.2: The constant mean curvature surfaces of a diatomic molecule plotted at
HO = 0.133. 0.074. 0.000 and 70.033 in (a). (b). (c) and ((1). Their comparison with
the corresponding molecular surface (circles) with the probe radius r], = 1.4A is
given at the cross section 2 = 0 in (e). (f). (g) and (h). The coordinates of the
diatomic molecule are. (.r.y. :) : (—2.4.0.0) and (2.4.0.0). The. van der Waals
radius is set to I'mmr = 20A.

algorithm and its improvement [19. 31. 100. 137. 130. 108] can be utilized for more

efficient applications.

92

 

Figure 5.3: The constant mean curvature surfaces of a seven—atom system and
their comparison with the corresponding molecular surface with rum: 2 1.0A. (a)
H0 2 0.074: (b) H0 2 0; (c) The comparison of cross section .r = 0 (Cirles: MS
with r], : 1.4A: Solid line: H0 2 0.074: Dashed line: H0 2 0). The coordinates are
generated by 1;; rotations of a 2.4A diatom.

5.3 Results

5.3.1 Validation
We will solve liq. (5.12) to get a hyrwrsurface function 5'. while protecting the
molecular van der \Vaals surfaces. As discussed earlier. we 1,)rescribe a piecewise

constant initial function for .S'(.r y. 2). i.e.. a non—xero ('ronstant S” inside the domain

V.S'.~’1.S'

V8315 : {(11.1}. 2) : [(111]. L) — rjl _<_' l',‘ + 717: 1.2.3..}

93

and 5(17. y. s) = 0 in the rest of the domain. Here. It,- and r.,- are the atomic radius
and the atomic center of the it!" atom: T is the probe radius. Fig. 5.1 (a) gives an
illustration of the initial value setting of S (.r. y. z). A cross section of the graph of
S is depicted in Fig. 5.1. (b). In Fig. 5.1 (a). one. has I” : I'me + T.

\Ve use a diatomic molecule to illustrate the impact of HO on the topology of
molecule surface. For this diatomic system, various values of the mean curvature
H” are prescribed. Fig. 5.2 shows the. CMMS results of the diatomic system. Note.
that the signs of the (:turvatures take into account the orientation of the surface.
That is. the surface lmlges out if H“ > 0 and in if [[0 < 0 when they are compared
with the minimal molecular surfi‘rce in which we. have HO 2 0. Indeed. we can
exactly reproduce the molecular Slll‘fEICO of the diattjnnic molecule at an appropriate
1]“. A one-to-one correspond between the probe. radius T of the molecular surface
and the three—dimensit)nal constant" mean curvature 1]” can be empirically fitted as
H” : 0.1777'152 — (tltifll'lj'l + 0.316. The ability of the proposed CMMS to capture
the solvent excluding feature of the molecular Sill'ltit't" is further demonstrated by a

Ht‘Vt‘ll—HitHll system in Fig. 5.35.

5.3.2 Cavities

It is interesting to examine the ability of the biomolecular surface modeling to
create the embedded cavities that are. larger than the solvent molecular volume.
\Vhen polar water molect’iles reside inside a protein molecule. their enviromnent is
less polar or apolar. 'l‘herefm‘e. the surface area of the water molecules tends to
be minimized, subject. to the geometric constraint. of the solvent accessible surface
which is mapped out by rolling a probe with radius T on the van der \Yaals surface
of the lll(_)l(-‘(f'tll(‘. Thus. we set Stray. :) : 0 on the domain outside the solvent
accessible surface during the cmnputation. To verify the proposed approach. we,

consider a buclcvl‘mll molecule (l)ucltminsterfullermre Cm) as that in Chapter 1. By

9.1

an apm‘opriate clmice of the solvent. excluding radius 7' = (ISA. a cavity is produced
for the lmckyl.>all as shown in Fig. 5.4 (b) (d) (f). .\Ioreover. by choosing relz-rtively
smaller I‘m/Hf and T. a tormlogically interesting CMMS for the buckyball molecule is
created in Fig. 5.—l(e). The same test. has been done. for the C54 for the open cavity

captures. The results for this case are given in Fig. 5.5.

5.3.3 Singularities

There are self—intersecting surfaces. cusps. and other singulan'ities in the molecular
surface definition. It is interesting to \-'erifv that. the proposed CKIMS is free of
these geometric singularities. To this end. we consider geometric parameters where
molecular surface singularities occur. Fig. 5.0 depicts three difl‘erent molecular
systems which admit cusps and/or self-intersecting surfaces. The diatomic system
has a pair of cusps when the atomic distance is continuously enlarged. The three.-
atom system has a sharp self—intersecting surface. The four—atmn system admits both
cusps and self—intt‘rsecting surfaces. Unlike the MS. the proposed CMMSs are very
smooth at these geometric parameters. Indeed. by systematically varying atomic

parameters. the singiilarity—free pi'ol')e1't-_y of the proposed ("AIMS is confirmed.

5.4 Applications

We now cmisider a slightly more complicated application of the CMMS. \Ve will
generate the (‘31le of a, complex biomolecule of henniglobin which is an important
metalloprotein in red blood cells with 1010 atoms (PDB ID: .lhga). \Ve choose the
same set of computational [‘mralneters as those (’lescribed in the preceding paragraph.
but with a grid size I) : t).3;\ for a fast computation. The CHRIS of lhga. and

its corresponding MSMS are depicted in Figs. 5.7 (a) and (M respectively. The

a
1

molecular surface is geiwratml by MSMS with 1'], : 1.5.—-\ . It is noted that the (_'.\l.\lS

emphasizes the skeleton of protein structure. and the. C.\I.\IS is much smoother than
the MS.

We now compare the CMka with the MSMS from a quantitative point of view.
The a[,)plic2—1tion of the C.\ll\lS to the electrostatic analysis is considered. By (.lefining
the CMMS as the dielectric interface. the electrostatic potentials of proteins are. at-
tained via the numerical solution of the Poisson—Boltzmann equation. Twenty three.
proteins in Chapter .1 are used to examine the CIR-IRIS. All other techniques used
here are exactly the same as those described in Chapter 4. By setting the CMMS
and the MS as the dielectric bmmdaries. electrostatic free energies of solvation AG
are computed by using the MIB method with mesh sizes 12.. 2 0.5.4. A probe radius
of rpzl.4.¥\. was used for the MS. The CRIMS is generated with the probe radius
of T‘: t).b"5.\ to enforce the cavity constraints considering the choice of T = 0.9
corresponding l'p :: 1.5 .-'\ in singularities free tests. The half of the grid spacing is
used in solving the Poisson—Boltzrnann equation to ensure the accuracy

The numerical results of electrostatic free energies of solvation are listed in Table
5.1.. The graphic comparisons of the solvation free energies of twenty three proteins
and relative differences of these solvat ion free energies with two different surfaces
(AG'C-vflym — Act/Sill.S'l/CAGJISA/S' are shown in Fig. 5.8. It is observed from
'Iiiible 5.1 that results of the CMle are in good consistence with those of the MSMS.
The solid red line in [fig 5.8(a) perfectly matches the line y 2 .r with a slope 1.0.
Linear regression of twenty three scatter points will lead to a. straight line with a
slope of 1.009 which agrees very well with the desired slope 1.0. Furtherniore. Fig.
5.8 (b) confirms that the deviations of the results computed with two surfz-ices are.
very small (the relative differences of AC are between —0.0‘2 and 0.01). For reference.
we also list the solvat ion free energies associated with the minimal molecular surface
I13] in Table 5.1. .\lthough there exist some dilferences among the results. we found

that the results of the HHS are consistent with those of the other two. It is noted

96

that this consistence depends on the given probe radii for the MS. the (731318 and
the .\f‘.\_fS. Inconsistent1e occurs when any one of these radii is dramatically changed.
Nevertheless. for a given MS probe radius. we can find an CMNIS or a. 1\1.\IS probe
radius such that their electrostatic potentials have a good agreement for a large set

of proteins.

Table 5.1: Comparison of total molecular surface area and electrostatic solvation

 

 

 

 

 

energy.
MS MS M MS CM 1\ IS
PUB code No. of Atoms. Area AG Area AG Area AG
lajj 519 2166 -1136.9 2117 -1150.4 2063 -1106.4
lbbl 576 2604 —994.2 2638 -1043.5 2596 —999.9
1bol‘ 832 2898 -854.1 2859 —875.0 2810 -836.0
lbpi 898 3233 -1304.0 3208 ~1358.7 3150 -1301.2
lcbn 648 2366 -305.4 2322 —315.9 2280 ~285.0
1fca 729 2547 -1201.3 2629 —1245.5 2679 —1205.3
1 frcl 1478 4367 -2852.2 4361 -2944.8 4283 ~2859.2
1fxd 824 2922 -3299.8 2971 -3356.0 2919 —3311.0
1hpt 858 3263 -811.6 3200 -857.5 3144 -799.0
1111bg 903 3071 91350.9 3036 -1407.9 2981 —1362.7
111cc; '1 187 4716 -1730.7 4743 -1833.0 4666 —1757.9
1ptq 795 2897 —871.8 2855 ~902.45 2805 —852.9
1169 997 3054 -1088.9 2997 -1131.6 2943 41067.5
1shl 702 2744 —754.6 2715 -779.7 2668 —728.8
lsvl‘ 1435 4644 -1713.5 4660 -1786.8 4581 -1703.2
111xc 809 2835 -11-11.6 L773 -1206.8 2726 -1156.0
1\'ii 596 2476 —902.5 2410 —949.5 2370 —915.8
2crl 57L 2318 —950.2 2283 -986.1 2244 -952.1
2pde 667 2716 —819.5 2682 —833.3 2638 -767.7
451c 1216 4159 ~1025.0 4080 ~1061.9 4004 —992.3
1112s 1272 4437 -1912.8 4367 -1973.6 4288 —1906.6
1116.3 2065 6974 -2375.5 7056 -2486.4 6939 —2367.8
1217111 2809 7733 —2158.0 7758 —2233.6 7624 —2126.9

 

 

Table 5.1 also illustrates the total surface areas for twenty three proteins. The
surface area is obtained by summing the areas of all the triangular elements of
the surface .-1 : 2: .41,‘. where .1, is the area of the iff’ triangles. It can be seen

from the table 5.1 that the ('.\f.\18 has produced smaller areas then the X18318.

97

This can he explained by two fat-ts: (1) The ohtained CMle is of an iso-surfaee
in nature. its resolution greatly depends on the grid size It; (2) The C';\'[l\IS is
of free of singularities. Moreover. it: is worth mentioning that the, triangulation
of the iso—surfaee in this dissertation is generated by Math-if). which appears not.
sutlieieutlv uniform. The ohtaiued triangulation is not. uniform enough. \Ye take a
simplest. example to demonstrate relative approxinration errors of the surface areas.
A diatomic system with (1.31. s) = (—3.15.0.0). (3.15.0.0) and rmmr = 2.0 A are
eonsit‘lered. The analvtieal value of surface area for this two-separate—‘itoni system
is 2 x ‘lT’i—itm“ The approximate surfaee areas of the CMle with two resolutions
h : 0.25.151 and I) : 0.125A are given in Table 5.2. It. is seen that the relative error
is tip to 1.7% for the (‘liatomie svstem when it : 0.25A. This error may explain the
differet'es among the ahove three surfaee areas. Thus the sulfate area given in Table
5.1 is a quantitative eomparisou for a referenee onlv.

Table 5.2: Error analysis in surfaee area ealeulations for diatomic svstem with
(111]. 3) : (—3.15.0. 0) (3.15 0. fl) and irt‘dll' = 2A.

 

 

 

I) Actual Surfaee. Area Calculated Surfaee. Area Rel. Error
0.25 100.5309 98.8440 1.7%
0.125 100.5300 09.6542 0.9%

 

 

Fig. 5.0 shows the eleetrostz—rtie potential projeeted at the moleeular surfaee
eomputed with the MSle and our (“RIMS at. h : 0.5A (PDB ID: latiii). It ran
he seen that there is a good agreement between two potentials. Ne\t'ertheless. some.
different-es do exist. and their impaets on some aspeets like elet-trostat ie ft)l'('(‘8 remain

to he further analyzed in the near future.

5.5 Conclusions

In summary. we have introduced a novel concept. the constant mean-eurvature. mole-
cular surface (CMRIS) in this chapter. for the modeling of biomt)lecules. based on
the theory of differential geometry. The ]_)roposed CMMS encompasses a family of
biomolecular surfaces. and unifies the minimal molecular surface (HMS) [11] as well
as the popular molecular surface [122]. The most important. feature of the pro-
posed C.\l.\lS is that it. is typically free of geometric singularities, which devastate
many applications of implicit. solvent models. The CMMS is generated via a novel
hypersurtate for a given set of molecular coordinates. Numerical experiments are
carried out to demonstrate the proposed method. Twenty three proteins are used
to illustrate the electrostatic analysis using the proposed CRIMS. It. is believed that
the proposed (.'.\l.\lS provides a new paradigm for the studies of surface biology.
chemistry and physics. in particular. for the analysis of stability. solubility. solva-
tion energy. and interaction of macrtanolecules. such as proteins. membranm. DNAs
and RNAs. It might have potential applicaticms not only in biological science. but.

also in industrial engineering. such as vehicle design and packaging problems.

00

 

(a) (b)

 

Figure 5.4: The CHRIS of the buckyball C60 with van der \Vaals radius and scaled
atomic radius. (a) Van der \Vaals radius. I‘Hny = 2.0 A: (c) Atomic radii. ’"mlH' = 1.7
A: (e) Scaled atomic radii. ’"mIH' = 1.4 A: (g) Scaled atomic radii. "mitt" = 0.8 A.
(b). (d). and (f) show. respectively. the CMMS of the. same buckyball as in (a). (c).
and (e). with half of the data of 5(1'. y. .3) removed.

100

 

Figure 5.5: The CMMS of the open buckyball C50 with van der VVaals radius
and scaled atomic radius. (a) Van der VVaals radius. rmm- = 2.0 A; (c) Atomic
radii. "mIH' : 1.7 A; (e) Scaled atomic radii. 1}.de = 1.4 A: (g) Scaled atomic radii,
rmmv = 0.8 A. (b). (d). and (f) show. respectively. the CMMS of the same buckyball
as in (a). (c). and (t) with half of the data of S(.r. y. :) removed.

101

 

(e) (0

Figure 5.6: :\Iolecular surfaces of two—. three- and four—atom systems and their cor—
responding constant mean-curvature surfaces. For the diatomic system. (1.1/,2) =
(—3.15.0. 0) and (3.15.0.0); 7‘udW : 2A. For the. three—atom system. (r.y.z) =
(—2.3.0.0). (2.3.0.0). and (0.3.9840): rmm : 1.5A. rp 2 1.5.4. 7 = 0.911. For
the four-atom system. (1111.2) 2 (—3.0.0). (0.—2.6.0). (3.0.0). and (0. 2.6.0);
I‘l‘dli' I 1.5A. 1‘1): 1.5A. T = 09A.

102

 

Figure 5.7: The comparison of the CMMS (a) and the MS (b) of the hemoglobin
(PDB ID: lhga),

103

n
U

 

-500

-1000

-1500'

-2000 '

-2500 '

-3000 *

CMMS Salvation Energy (kcal/mol)

-3500 *

 

 

-400 ‘ - -
—4(000 —3000 -2000 -1000 0
MSMS Solvation Energy (kcal/mol)

(a)

 

0.02

 

0.01:

Relative Difference
O

-0.01-

 

 

 

'O‘G‘o 5 1‘0 1‘5 2‘0 25
(b)
Figure 5.8: Comparison of the electrostatic solvation free energies AC of twenty

three proteins listed in Table 5.1. (a) Plot. of solvation free. energies for CMMS Vs.
MSMS: Solid lines are y = .r: (b) Relative differences of solvation free energies.

104

 

Figure 5.9: Surface electrostatic potentials of cult p factor (PBD ID: 18.63) pro—
jected onto its molecular surface. Left: Generated with the CMMS; Right: Gener—
ated with the .\IS.\IS. Colors are chosen according to the magnitude of the potential
as indicated with the color bar.

Chapter 6

Thesis Contributions and Future

Work

The objective of this dissertation is to develop partial differential (.‘quation (PDE)
approzuxhes to the mathematical modeling of image. surfaces and biological surfaces
of molecules. This chapter concludes the contril)utions of this dissertation and

proposes future related work.

6.1 Thesis contributions

In this dissertation. we proposed a. single time step method for a class of parabolic
partial differential ecpiations. The local spectral evolution kernel (LSEK) analyti—
cally integrates the partial differential equations to provide the analytic solution in
a single time step. It is of spectral accru'acy. of free of instability constraint. From
the point of view of image procc—‘ssing. the I..Sl€l\' gives rise to a. family of low—l1)ass
filters. The filtering properties of LSICK are studied, by using the I’tnn'ier analy—
sis. 'l‘he LSI‘ili is incorptirated into the I’e1.'<ma—.\lalik type of mrisotropic diffusion

scheme for imagt'r denoising. Automatic exit schemes are pmposed to stop image

1 06

evolution in the image denoising. The peak signal to noise improvement of 8(‘113 lras
been achieved. A ftirward—backward diffusion process is developed and is combined
with the LSEK for image blurring. A coupled PDE system is modified for the im-
age edge detection. The present rrretlrod is compared with many previous standard
approaches. including Sobel. Prewitt. Canny image edge detectors and anisotropic
diffusion operators. Extensive numerical experiments of this rrrethod have been con-
ducted on high texture images and X-ray lung images. It. is found that our method
provides some of the best performance for the edge detection of high texture images.
which is a challenge to other existing methods. The. results of nrammography images
and CT images of a mouse indicate that the proposed method is very efficient.

In biomt)lecular surfaces simulations. we proposed a new concept. multiresolution
surfaces. for the nnrltiscale modeling of biomolecules. The pr.‘<‘)1‘.)osed concept contains
not only a family of molecular surfaces. correspontling to different probe radius. but
also the solvent. accessible surfzwe and Van de \Vaals surface as extreme cases. A
novel approach based on the controlled diffusion of continuum solvent density is
proposed to generate the multiresolution surfaces. The LSEK is applied to integrate
the diffusion equation in a single time step. The proposed method does not need to
discriminz-ite buried atomic surfaces from unburied ones during the comrmtation. It
bypasses the (‘lifficulty from singularities in other existing methods. We compared
our rrretlrod to one of the urost widely used molecular surface generators. MSMS.
A very good agreement. was achieved. \Ve also conducted the solvation energy
calculations of 23 proteins. A linear regression result shows the accuracy of the
proposed rrretlrod in molecular surface gt‘neralization.

Since the molecular surface suffers from unphysical geometric singularities which
may devastate munerical simulations. we introduced a concept of the constant: mean
curvature molecular surface (CMMS). The, CMMS is formulated based on the theory

of differential geometry. It encompasses a family of bimolecular surfaces for the

107

theoretical modeling of biomolecules and unifies the minimal molecular surface and
the molecular surface. l-lowever, unlike, the molecular surface. the proposed C".\I.\IS
is typically free of geometric singularities. A computational scheme is proposed for
practical control of mean curvature evolution of a hypersurface. which leads to the
CMMS after taking an appropriate level set. The. validity of the mean curvature
concept, the efficiency and controllability of the algorithm are. demonstrated by

extensive munerical experiments.

6.2 Future Work

The. I’DE—based modeling of image surfaces and bioinolecular surfaces discussed in
this dissertation are proved to be very promising. as illustrated by analysis and
numerical simulations. Nevertheless. there are many ways to extend this thesis
work.

Firstly. we have explored the simplified version (linear diffusion) of coupled PDE
algorithm for the image edge detection and the image. enhancernent. The. obtained
numerical experimental results indicate that our method is very efficient for the
images with a large amount. of texture and medical image enhancement. However.
it is worth testing the coupled nonlinear diffusion equation. More attractive results
might be found by choosing the diffusion coefficient adaptively.

Secondly. we have used the ('liffusion map of coutinmnn solvent to generate mole—
cular surfaces. There are several applications associated with this method. since it.
does not. need to discriminate buried atomic surfaces from unburied ones during
the computation and bypasses the difficulty of singularities. In addition. because
of the iso—surface nature of our method. it. can be easily implemented into the NIB
method. 'l‘herefore. the applications of this molecular surface to the large l,)iotiiole—

cular simulations will be promising.

108

Thirt’lly. we have, used the constant met-m-cnrvatnre flow equation to generate
singularity free molecular surface. It. is believed that. the CMMS provides a new
1,)a.radig1n for the studies of surface biology. chemistry and [,iliysies, in particular, for
the analysis of stability. solubility, solyz—ititm energy. and interaction of macromole-
cules. The advantages of CMMS need to be further explored in the future.

F(")urthly. the CMMS can be computed Via standard program based on the march-
ing cubes triangulation, and the computational expense of the CMMS depends on
the desired level of resolutions. In the eurrtuit study. the generalization of the CMMS
usually uses more CPU time than that of the molccular surface by the MSMS and
our approach in Chapter 4. Issues of efficient generation of the CMMS will be
Considered in the. future.

Finally. the generz-itions of the molecular surfaces in this dissertation are I’DE-
based approaches. \Iit'll‘lUllS geometric equations have been used. Their efficiency
in modeling has been fully denimistrmed by our extensive numerical experiments.
In the future. we expect to ('lm'elop other efficient FDIC-based apin‘oaehes for the
surface modeling. .\latl’ie1natical anz—ilysis of the {noposed PDEs will be provided to

support the modeling.

1 05.)

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