Some filtering techniques for digital image processing

Some Filtering Techniques for Digital Image

Processing

by

Chi-M ing Leung

l.l.S c . M cM aster University, 1975 M. A. U niversity o f B ritish Colum bia, 1971 B .S c. T h e C hinese U niversity of H cng K ong, 1969 A D issertation S u b m itted in P artial Fulfillm ent of the

R equirem ents for th e D egree of D O C T O R O F PH IL O SO PH Y

in. th e D epartm ent of E lectrical and C om puter Engineering

W e accept this thesis as conform ing to the required standard

Dr. W -S T.ii Srmp.rvisnr fO ept. of Elec. & Com p. Eng.)

Dr. P. A V ^hok lis, D ep artm en tal M em ber (D ept, o f Elec. & C om p. E n g.) Dr. R. L_K iU in. D eoartm eh tal M em ber (D ept, of Elec. & Com p. E ng.)

Dr. A. R. Sourour, O utside M ember (D ep t, of M ath em atics) Dr.

<

P. van den D riessche, Orrtsidh M em ber (D ept, of M ath em atics) Dr. F.H T'T'eet, ExternaT Exam iner (Forestry Canada)

© C h i-M in g Leung, 1994 U n iversity o f V ictoria

All rights reserved. D issertation m ay not be reproduced in whole or in part, by photocop yin g or other m eans, w ithout th e perm ission o f th e author.

Supervisor: Dr. W.-S. Lu.

Abstract

This thesis is primarily concerned with the theory and implen:cntation of

digital filtering methods as applied to restoration and edge detect.ion of noise

con-taminated images that are further degraded due to either linea.r rnotio11 of t;he obj<•d.

or defocusing of the camera.

is known tha.t restoring

severely blurred but

noise-free image is relatively easy, but restoring

blurred pidure that; ha.s been con·tq·,t;(•d

by noise is not. This is especially the case when the noise is widc-ba.11d, since

conventional linear filtering techniques using lowpa.ss, highpa.ss, bandpass, or their

combinations will in general not work. A fundamental question that; a.rises from the

above observation is · is there a filtering approach to this image restora.tio11 problcrn

that is significantly better than conventional techniques ment;ioued a.bovc

with the

filtering mechanism remaining linear? This thesis presents

systematic study of

this question and gives a possible answer by proposing a class of linear, recursive

regularization filters (R-filters). These R-fiJters extend the one-dimensional recursive

R-filters proposed by M. Unser and his co-workers in 1991 for noise removal

the

two-dimensional case. The R-filters derived ar:c image independent;, therefore they

can be designed off-line and stored in the computer. Furthermore, the convent;ional

single-parameter regularization methods are extended to a multiplc-para.rnctcr

set-ting, rendering a better balance of fidelity and smoothness of restored images. In

Ill

addition, other image processing tasks such

edge detection and enhancement can

a.lso be performed within this 11-filter framework.

Filtering techniques other than R-filters may also be useful in various image

processing tasks. In particular, a modified Wiener filter for the restoration of blurred

images is presented, showi::i.v; how low-order one-dimensional and two-dimegsional

linear finite impulse response filters can be used for detecting edges in noisy images.

Conventional smoothing filters always tend to blur the images, so for noise

re-rnova.I tasks, the filter should have the ability to preserve features in an image while

reducing the noise. Two approaches are taken here.

In

the first, the Lee filter is

extended to two nonlinear filters utilizing local statistics of the degraded image.

The nonlinear filters so designed have the advantage that they reduce noise without

blurring the details of the image. The second apprnach proposed for noise removal is

a two-step regularization algorithm.

In the first step of the algorithm, a smoothing

filter is applied to reduce the noise level of the image. Of course, the sharp features

that; occur in the image are then unavoidably blurred. A critical observation ma.de

at this point is that the blurring mechanism is precisely known.

In the second step

of the algorithm, an R-filtcr is applied to restore the image

the

smooth-ing filter

i11

the first step as a degradation operator. As is expected, this two-step

approach permits smoothing without introducing undesirable blurring.

distri-bution of relative minima. of the power spectra. of subimag;<·s f.ha.t n.rl' ,)bl.a.ined fr,)111 the given degraded image. This algorithm is efficient. for the ide11t.ifica.t.io11 of t.hc linear motion blur, defocused blur, and exponential blur. As the second meU1od for blur identification, a two-phase blind restoration algorithm is proposed. Jn the first phase of the algorithm, the blur function is identified by the prccedin~ blur identification approach. The power spectra of the f-ubimagcs are then estimated. In the second phase, Wiener filtering is employed to obtain a restored iniagc using the estimated power spectra obt;i.ined in phase one

*

---Dr. W .-S. Lu, Supervisor (D ept, of Elec. & Com p. Eng.)

Dr. P. A^hthoklis, D epartm ental M em ber (D ept, of Elec. & Com p. E ng.)

— --- — --- 7 “ *---

---Dr. R. L. K irlin, D epartm ental M em ber (D ept, of Elec. & C om p. Eng.)

— ---7---

---Dr. A. R. Sourour, O utside M em ber (D ep t, of M ath em atics)

Dr. P. van den Driessche, O utside M em ber (D ept, o f M ath em atics)

A b stract

ii

C on ten ts

vi

L ist o f Tables

x v

L ist o f Figures

x x iv

L ist o f A b b reviation s

xxv

A ck now ledgem ents

x x vii

D ed ica tio n

xxviii

1 In trod u ction

1

C O N T E N T S vii

1.2 Problem s E n c o u n t e r e d ... 3

1.2.1 Im age R estoration as an Ill-Posed P r o b le m ... 3

1.2.2 E dge D e t e c t i o n ...10

1.2.3 Identification o f Im age M o d e l ...1 1 1.3 C o n t r ib u t io n s ... 14

1.4 O rganization o f th e T h e s i s ... 15

2 O ne-D im ensional and T w o-D im ensional R -F ilters

19

2. L I Introduction... 19

2.1.2 T h e M ethod o f U nser, Aldroubi, and E d e n ... 21

2.1.3 G eneralized 1-D R - F i l t e r s ... 23

2.1.4 A D esign I s s u e ... 30

2.1.5 R ecursive R egularization F ilters w ith A p proxim ate Linear P h ase 31 2.1.6 R ealization ... 32

2.1.7 C om putational C om parison o f Spatial and Frequency D om ain I m p le m e n ta t io n ... 33

2.1.9 C o n c lu s io n s ...

2.2 T w o-D im ension al R - F ilt e r s ...

2.2.1 Introduction ...

2.2.2 2-D R - F i l t e r s ...

2.2.3 A n Iterative Im p lem en tation of 2-D R -Filters ...

2.2.4 A n A p proxim ate Factorization for th e Im plem entation of 2-D R - f i l t e r s ...

2.2.5 E xp erim ental R e s u l t s ...

2.2.6 C o n c l u s i o n s ...

3 O ptim al D eterm in a tio n o f R egularization Param eters and the

S tabilizin g O perator

3.1 A n L-Curve Approach to O ptim al D eterm ination o f Regularization P aram eters ...

3.1.1 I n tr o d u c tio n ...

3.1.2 M ore A b out T ikhonov R e g u la r iz a tio n ...

3.1.3 D eterm in a tio n o f th e Regularization Param eters via the L- Curve A p p r o a c h ...

C O N T E N T S ix

3.1.4 An A d aptive R estoration M ethod for Linear M otion Blurred

I m a g e s ...65

3.1.5 C o n c l u s i o n s ...6 8 3.2 O ther O ptim al Approaches to the D eterm ination of th e R egulariza­ tion P a r a m e t e r s ... 70

3.2.1 in t r o d u c t io n ...70

3.2.2 O ptim ization F o r m u la tio n s... 71

3.2.3 T h e C onstrained Least-Squares M ethod ... 73

3.2.4 T h e G eneralized Cross-V alidation M e t h o d ... . 74

3.2.5 T h e M SE and IS N R M e t h o d s ...76

3.2.6 E xp erim en tal R e s u l t s ...77

3.2.7 C o n c l u s i o n s ...78

3.3 A C om m ent on th e Stabilizing O perator G enerated by D iscrete Lapla-cian O p e r a t o r ... 79

3.4 Sim ultaneous O p tim al D eterm ination of th e Stabilizing O perator and R egularization P a r a m e t e r s ...8 6 3.5 A M ultiple-P aram eter G eneralization of the T ikhonov R egularization M ethod ... 90

3.5.2 A G eneralization o f the Tikhonov Regularization Met hod . . . 93

3.5.3 C hoice of M ultiple Regularization P a r a m e t e r s ...96

3.5.4 E xp erim ental r e s u l t s ...99

3.5.5 M ultiple-P aram eter 2-D R-.fi..lters... ... 99

3.5.6 C o n c l u s i o n s ...101

4 A M odified W ien er F ilter for th e R estoration of Blurred Im ages 103

4 .2 W iener F ilterin g and R egularization M e t h o d s ... 106

4 .3 A M odified W iener F i l t e r ...108

4 .4 E xp erim ental R e s u l t s ... 11.0 4 .5 C o n c l u s i o n s ... 112

5 N o ise R em oval

117

5.2 C onventional Sm ooth ing F i lt e r i n g ...119

5.2.1 Average F i l t e r i n g ... 1.19 5.2.2 G aussian F ilterin g ... 123

C O N T E N T S xi

5.4 Noise Rem oval by a T w o-S tep R egularization Approach ... 127

5.5 N onlinear F ilters for N oise R e m o v a l...129

5.5.1 N oise Rem oval by Lee F i l t e r s ...129

0 .5 T N oise R em oval by Q uadratic F ilters ... 130

5.5.3 N oise R em oval by C ubic F i l t e r s ... 132

5.5.4 Sim ulation R e s u l t s ...133

5.6 Perform ance E valuation of th e Sm oothing F i l t e r s ... 133

5.7 Deblurring W ith a P re-Sm oothing S t e p ...136

5.8 C o n c l u s i o n s ...137

6

D e te c tio n o f E dges in N oisy Im ages by 1-D and 2-D Linear FIR

D igital F ilters

147

6.2 1-D First-O rder F IR Edge D etectors ... 148

6.3 2-D First-O rder F IR E dge D etectors ... 152

6 .4 2-D Second-O rder F IR E dge D e t e c t o r s ... 154

6.5 C o n c l u s i o n s ... 158

7.1 I n t r o d u c tio n ... 161

7.2 A Posteriori Blur Identification in th e Spectral D o m a i n ...162

7.2.1 Linear M otion B l u r ... 162

7.2.2 D efocusing B l u r ... 167

7.2.3 Blur w ith C ircularly f ym m etric P S F s ... 169

7.3 A B lin d R estoration A l g o r i t h m ...169

7.4 C o n c l u s i o n s ...172

8

C on clu sion s and Future Research Considerations

174

8.2 Suggestion s for Future R e s e a r c h ...177

xm

L ist o f T ab les

2.1 N um ber o f operations for obtain ing f k \ through base-2 D F T algo­ rithm . F ( C ) represents th e D F T o f C ... 35

2.2 N um ber o f operations for obtain ing f k \ through sp atial R -filtering approach. T h e blurring distance is L sam pling u n its...36

2.3 Q u a n tita tiv e com parison betw een R -filtering and W iener filtering of the degraded im age o f Fig. 2.2. T h e filter th at produces sm aller M SE and N M SE , or larger ISN R is considered to be a b etter one. . . . 40

2.4 Q u a n tita tiv e com parison betw een R -filtering and W iener filtering of th e degraded im age Lena on Fig. 2.3. T h e filter th a t produces sm aller M SE and N M SE , or larger IS N R is considered to be a better one. . . 40

2.5 Q u a n tita tiv e com parison betw een th e R-filter and W iener filter (re­ lated im ages are show n in F ig. 2 .5 )... 55

2.6 Q u a n tita tiv e com parison betw een the R-filter and W iener filter (re­ lated im ages are shown in Fig. 2 .6 )... 55

3.1 R estoration results for exponentially blurred Airplane im age with 30 dB n o ise... 82

3.2 R estoration results for linear m otion blurred Lena im age with 20 dB n o ise...82

3.3 R estoration results for defocused (r = 7) Lena, im age w ith 28 dB noise. 90

3 .4 Q u a n tita tiv e com parison betw een single-param eter and m ulti pie-param eter regularized im age restoration. T h e sam ple im age is a linear m otion blur degraded Lena im a g e... 98

3.5 Q u a n tita tiv e com parison betw een single-param eter and m ultiple-param eter regularized im age restoration. T he sam ple im age is a defocused and noise-con tam in ated Lena im a g e... 98

3.6 Q u a n tita tiv e com parison betw een single-param eter arid m ultiple-param eter regularized im age restoration. T h e sam ple im age is a defocused and n oise-con tam in ated T ext im a g e... 102

4.1 Q u a n tita tiv e com parison betw een th e W iener filter and modified W iener filters. R elated im ages are shown in Fig. 4 .2 ...11.6

L I S T O F T A B L E S x v

4 .2 Q ua n tita tiv e com parison betw een th e W iener filter and m odified W iener filters. R elated im ages are shown in Fig. 4 .3 ... 116

5.1 Q ua n tita tiv e com parison of sm oothing capacity of different filtering techniques. (T h e sam ple im age is Peppers.) ... 139

5.2 Q u an tita tiv e com parison o f sm oothing capacity o f different filtering techniques. (C orresponding sm ooth ed im ages are show n in Fig. 5.8 .) 142

5.3 Q u a n tita tiv e com parison o f sm oothing capacity o f different filtering techniques. (C orresponding sm oothed im ages are shown in Fig. 5.9.) 145

L ist o f F ig u res

1 . 1 A n im a g e form ation m o d e l... 3

2.1 A m p litu d e response o f S \ ( z ) w ith A = 0.2 and L = 1.4... 29

2 . 2 R -filtered and W iener filtered im ages, (a) Original im age, (b) Lin­ early blurred im age w ith L — 14 units and pseudo w h ite G aussian noise at SN R = 30 dB level, (c) Restored im age by W iener filter, (d) R estored im age by 1-D R-filter w ith A = 0 . 2... 38

2 .3 R -filtered and W iener filtered im ages, (a) T h e original im age Lena. (b) L inearly blurred im age w ith L = 14 units and pseudo w h ite G aus­ sian n o ise at S N R = 25 dB level, (c) R estored im age by W iener filter. (d) R estored im age by 1-D R-filter with A = 0 .3 ...39

2 .4 (a) A m p litu d e response o f th e discrete L aplacian operator (2.54). (b) T he corresponding contour plot o f (a )...45

L I S T O F F I G U R E S xvii

2.5 Com parison betw een R-filtered and W iener filtered im ages, (a) T h e Lena im age, (b) D efocusing blurred Lena im age w ith r = 7 units and pseudo w h ite G aussian noise at SN R = 28 dB level, (c) R estored im age by W iener filter, (d) R estored im age by R-filter w ith A = 0.005. 53

2.6 Com parison betw een R-filtered and W iener filtered im ages, (a) T h e im age Text, (b) D efocusing blurred Text im age w ith r = 5 units and pseudo w h ite G aussian noise at SN R = 33 dB level, (c) R estored im age by W iener filter, (d) R estored im age by R -filter w ith A = 0.00006. 54

2.7 Im age restoration by R-filtering. (a) Linearly blurred Lena im age with L = 14 u n its and pseudo w hite G aussian noise at S N R = 25 dB level, (b) R estored im age by R -filtering w ith A = 0 .006... 56

2.8 Im age restoration by approxim ate-factorization R -filtering. (a) T h e im age A irplane, (b) D efocused Airplane im age w ith r — 3 units and pseudo w h ite G aussian noise at S N R = 30 dB level, (c) R estored im age by approxim ate-factorization R -filtering w ith A = 0.0006. . . . 57

3.1 An L -curve... 63

3.2 (a) Linear m otion blurred Lena im age w ith L = 14 units and pseudo w h ite G aussian noise at S N R = 30 dB level, (b) R estored Lena im age w ith A = 0 . 0 1 obtain ed by L-curve approach, IS N R = 1 0 .5...6 6

3.3 (a) D efocused Lena im age w ith r = 7 units and pseudo w hite G aus­ sian n oise at S N R = 28 d ti level, (b) R estored Lena im age with A = 0.005 obtain ed by L-curve approach, ISN R = 11.0...

3.4 (a) T h e Tiffany im age w ith a syn th etic line segm ent, (b) Linear m o tio n blurred Tiffany im age w ith L = 9 units and pseudo white G aussian noise at SN R = 40 dB level, (c) Restored Tiffany im age us­ ing ad ap tive approach, (d) R estored Tiffany im age using nonadaptive approach...

3.5 D eterm in ation o f th e regularization param eter, (a) Original im age A irplane, (b) E xp on en tial blur (r = 5 and a 2 = 1) with noise at S N R = 30 dB level, (c) R estored im age by CLS m eth od (A = 0.012). T h e im age is overly sm oothed, (d) Restored im age by GCV m ethod (A = 0.0005). (e) R estored im age by M SE or IS N R m ethod (A =

0 .0 0 0 8 ) .

3.6 D eterm in a tio n o f th e regularization param eter, (a) T h e Lena im ­ age. (b) Linear m otion blur (L = 1 1) with noise at S N R = 20 dB level, (c) R estored im age by CLS m ethod (A = 0 .088). T h e im age is oversm oothed, (d ) R estored im age by G C V m ethod (A = 0.02). (e) R estored im age by M SE or IS N R m ethod (A = 0 .0 2 5 )...

L I S T O F F I G U R E S x ix

3.7 (a) A m p litu d e response of th e discrete Laplacian operator L ' . (b) T he corresponding contour plot o f (a ) ... 85

3.8 (a) D efocused (r = 7) Lena im age w ith noise added at SN R - 28 dB level. R estored im age by (b) CLS m ethod, (c) M SE m ethod and (d) ISN R m eth o d ...91

3.9 M ultiple-param eter regularized im age restoration, (a) Linearly blurred Lena im a g e w ith L = 14 units and pseudo w hite G aussian noise at SN R = 30 dB level, (b) R estored im age by m ultiple-param eter restoration...9 7

3.10 M ultiple-param eter regularized im age restoration, (a) D efocused Lena im age w ith r = 7 and contam inated by noise w ith SN R = 28 dB . (b) R estored im age by m ultiple-param eter restoration...97

3.11 M ultiple-param eter regularized im age restoration, (a) D efocused T ext im age w ith r = 5 and contam inated by noise w ith SN R = 33 dB . (b) R estored im age by m ultiple-param eter restoration... 102

4.1 T h e m a g n itu d e response | /T(cji, oj2) | o f (a) linear m otion blur w ith 8 units; (b) ex p on en tial blur; (c) defocusing w ith radius o f COC 7 units. I l l

4 .2 C om parison o f m odified W iener filtered and W iener filtered im ages. (a) T h e Lena im age, (b) N oise-contam inated linear m otion blurred Lena im age w ith J, = 14 and SN R = 30 dB . (c) R estored Lena im age by m odified W iener filter VV3. (d) R estored Lena im age by m odified W iener filter W 2. (e) Rest; <ed Lena im age by conventional W iener filter W \ ... 114;

4.3 C om parison o f m odified W iener filtered and W iener filtered im ages. (a) T h e im age Text, (b) N oise-contam inated defocused T ext im age w ith r — 3 and SN R = 15 dB . (c) R estored Lena im age by modified W iener filter W 3. (d) R estored Lena im age by m odified W iener filter

W 2. (e) R estored Lena im age by conventional W iener filter W\. . . . 115

5.1 Average filtering, (a) T h e im age Peppers, (b) N oise-contam inated P eppers im age w ith SN R = 10 dB . Restored im age w ith mask o f size (c) 3 x 3; (d) 5 x 5; (e) 7 x 7; (f) 9 x 9 ... 120

5.2 Frequence response of th e 1 x 3 average filter (th e sam pling frequency

f s is se t equal to 1 )... 123

5.3 G aussian filtering. R estored im age of the noise-contam inated im age P eppers (F ig. 5 .1 (b )) using a 7 x 7 Gaussian filter w ith <Jr equal to (a) 0.5 pixels; (b) 1 pixel; (c) 2 pixels and (d) 3.5 p ixels... 126

L I S T O F F I G U R E S xxi

5 .4 Suppression o f noise by R-filtering. (a) 1-D R-filtering. (b) 2-D R- filtering...127

5.5 T w o-step regularization sm oothing, (a) R estored im age in w hich an average filter is used in th e pre-sm oothing step, (b) R estored im age in which a G aussian filter is used in the pre-sm oothing ste p ... 128

5.6 Suppression of noise using nonlinear filters, (a) Lee filter, (b) Q uadratic filter, (c) C ubic filter... 134

5.7 Suppression of noise using nonlinear filters, (a) T h e original im age R ob ot, (b) N oise-contam inated R obot w ith SN R = 9 dB . (c) Lee filter, (d) Q uadratic filter, (e) C ubic filter...135

5.8 C om parison o f sm oothing filters (th is page and th e n ex t page), (a) T h e L ena im age, (b) N oise-contam inated Lena im age, SN R = 7 dB . (c) A verage filtering, (d) G aussian filtering, (e) W iener filtering, (f) M edian filtering, (g) Lee filtering, (h) 2-step average filtering, (i) 2- step G aussian filtering, (j) 2-step W iener filtering, (k) 1-D R -filtering. (1) 2-D R-filtering. (m ) Q uadratic filtering, (n) C ubic filtering. . . . 140

5.9 C om parison o f sm oothing filters (this page and the next page), (a) T he im age Airplane, (b) N oise-contam inated A irplane im age, S N R = 5 dB . (c) A verage filtering, (d) Gaussian filtering, (e) W iener filter­ ing. (f) M edian filtering, (g) Lee filtering, (h) 2-step average filter­ ing. (i) 2-step G aussian filtering, (j) 2-step W iener filtering, (k) 1-D R -filtering. (1) 2-D R -filtering. (m ) Q uadratic filtering, (n) C ubic filterin g ... 143

5.10 C om parison o f deblurring w ith and w ithout a pre-sm oothing step , (a.) T h e im age Airplane, (b) Linearly blurred Airplane im age w ith L — 7 and S N R = 15 dB . (c) R estored Airplane im age w ithout sm oothing.

(d) R estored A irplane w ith a G aussian pre-sm oothing filtering step. . 146

6.1 C om parison o f edge m aps obtained by Sobel operator and the pro­ posed 1-D first-order edge detector, (a) Original im age, (b) N oise- con tam in ated im age w ith pseudo w hite Gaussian noise at S N R = 10 dB level, (c) E dge m ap by Sobel operator, (d) Edge map by the proposed 1-D first-order edge detector (6 .1 0 )... 1.50

6.2 Edge m ap by th e proposed 2-D first-order FIR edge detector (6.14). (a) O riginal im age, (b) E dge m a p ... 153

L I S T O F F I G U R E S xxiii

6.3 Com parison o f zero-crossings obtained by LoG and th e proposed 2- D second-order ed ge detector designed by com bined SV D and B A m ethod, (a) O riginal im age, (b) N oise-contam inated im age w ith pseudo w h ite G aussian noise at SN R = 15 dB level, (c) Zero-crossings by LoG. (d) Zero-crossings by th e proposed 2-D second-order edge detector (6.14) designed by com bined SV D and B A m eth o d ...157

6 .4 Zero-crossings by a proposed 2-D second-order edge detector designed by th e B F G S m eth od , (a) O riginal im age, (b) N oise-contam inated im age w ith pseudo w h ite G aussian noise at SN R = 12 dB level, (c) Zero-crossings by the proposed 2-D second-order edge detector (6.17) designed by B F G S m eth o d ...159

7.1 (a) O riginal im age T ext, (b) Linearly blurred T ext w ith L = 8. (c) N oisy and linearly blurred T ext w ith L = 8 and SN R = 30 dB. (d) Graph of th e noise free lo g( V g(k, 129)). (e) Graph o f th e noise con tam in ated \ o g (V g(k, 1 2 9 ))... 166

7.2 (a) D efocused im age Lena (r = 5) and noise added at SN R = 35 dB. (b) Graph o f l o g ^ ^ A : , 6 5 ) ) ... 169

7.3 R estored im age by tw o-phase blind restoration m eth o d , (a) Linearly degraded Lena, (b) R estored Lena...171

7.4 R estored im age by tw o-phase blind restoration m ethod, (a.) E xpo­ nen tia lly degraded Lena, (b) Restored Lena... _1.73

L ist o f A bbreviations

XXV

[{ fi I i,>-r R egularization filter

1-D O ne-dim ensional

2-D T w o-dim ensional

* C onvolution operator (unless otherw ise sta ted )

A 1 T he transpose o f m atrix A

P SF Point spread function

D F T D iscrete Fourier transform

PO C S Projection onto convex sets

LoG Laplacian of Gaussian

AR, A utoregressive

A R M A A utoregressive m oving-average

FIR, F in ite im pu lse response

IIR Infinite im pu lse response

Lz Argum ent o f th e com p lex number z — |z |e J0, (i.e ., Lz — 0)

M SE M ean square error

N M SE N orm alized m ean square error

S N R Signal to noise ratio

IS N R Im provem ent signal to noise ratio

COC Circle of confusion

M M SE M inim um m ean square estim a te

LM M SE Linear m inim um m ean square estim a te

CLS C onstrained least-squares

B F G S Broyden-Fletcher-G oldfarb-Shannon

G C V G eneralized cross-validation

XXV11

A c k n o w le d g e m e n ts

T he author w ould like to thank his supervisor, Professor W .-S. Lu o f th e D epart­ ment, o f E lectrical and C om puter Engineering, for his encouragem ent, patien ce and advice during th e course o f this research, and for his help in th e preparation o f th is dissertation.

F in an cial assistance provided by Professor W .-S. L u’s research grants from N SER C and IRIS is also gratefully acknowledged.

1

C h a p te r 1

In tr o d u c tio n

1.1

D ig ita l Im a g e P r o c e ssin g

B y an im a g e w e m ean a tw o-dim ensional (2-D ) distribution of light intensity. Im a g e

p ro cessin g is th e treatm ent o f an im age to produce a second im age for a desired

purpose. T h is is often perform ed by using either graphic arts or com puter program s. A d ig ita l im a g e can be thought o f as a m atrix o f light in ten sity represented by a finite num ber o f b its. T h e elem en ts o f th e m atrix are called p ic tu r e ele m en ts or p ix els, each representing the in te n sity o f a sm all area o f th e corresponding coordinates in the digital im age. Im ages are usually digitized uniformly. It is a com m on practice th at th e sizes o f resulting d ig ita l im ages are taken to be at least 256 x 256 p ix els, thus a large am ount o f d a ta is involved in a digital im age. Fortunately, th e m a trix representation o f an im age p erm its th e use o f a digital com puter as w ell as ex istin g m atrix theory to develop various algorithm s useful in processing th e im age.

s u b je c tiv e d ig ita l im a g e p ro cessin g and q u a n tita tiv e d ig ita l im a g e processing. Sub­

je c tiv e d ig ita l im age processing is designed to im prove human visual-perception of an im age. T h is involves a close exam ination o f the hum an visual system , especially in th e ad ap ta tio n and discrim ination aspects. T h e principal o b jectiv e of su b jective d igital im age processing is to process a given im age so that th e analyst can detect th e inform ation o f interest in th e resultant im age. It is m ainly a tr.ial-and-er.ror process. T h e m ain techniques em ployed in su b jective digital im age processing are m od ification techniques (e.g ., histogram equalization ), sm oothing techniques (e.g., m ed ian filtering and low pass filtering) and sharpening techniques (e.g., differentia­ tio n and highpass filterin g). T h ese techniques are often called im a g e en h an cem en t techn iqu es, and in th is regard im age enhancem ent is another nam e for su b jectiv e digital im age processing. Im age enhancem ent techniques have been w ell-developed [1, 2, 3, 4] and can be readily im plem ented on com puterized im age display system s.

Q u a n tita tiv e d igital im age processing techniques are based on m athem atical, m odels; th e resultant im ages are produced w ithout analyst intervention. An im ­ portant ex a m p le o f q u a n tita tiv e digital im age processing is im a g e resto ra tio n , m in ­ im iza tio n o f know n or unknown degradations and noise in an im age.

In th is th esis, we are m a in ly concerned w ith an alytic techniques and m a th em a t­ ical to o ls w hich are related to im age restoration, noise removal, edge d etection and identification.

3

n {*, y)

f i x , y ) - * 1J

Linear blur N oise

Figure 1.1: An im age form ation m odel.

1.2

P r o b le m s E n c o u n te r e d

1.2.1 Im age R estoration as an 111-Posed P rob lem

Im ages are produced to record or display useful inform ation about a phenom enon o f interest. Q u ite often, th e process o l im age form ation and recording is degraded. D u e to th e im perfection of im aging system s, recorded im ages are usually degraded versions o f th e original scenes. The basic goal of im age restoration is to reduce th e degradations in a recorded im ags so th at th e resulting im age w ill best approxim ate th e original scen e su b ject to som e criteria. T his requires th e know ledge o f th e im ag­ ing sy stem used and th e degradation m echanism involved. T h e im age degradation will be m odeled in this thesis by a linear blur and an additive noise process. T h e a dditive-noise assum p tion in th e im age form ation m odel is ju stified by th e nature of m ost im age sensors. M oreover, it is unanim ous in practice to m odel th e noise process as w h ite. As show n in Fig. 1.1, th e degradation m odel is given by th e expression

/

degraded sign a l), n ( x , y ) is th e ad d itive noise function, and h ( x , y \ a , f t ) is called th e tw o dim ensional (2-D ) im pu lse response or point spread function (P S F ) o f th e sy stem . A typical im age restoration problem is to find an e stim a te o f f ( x , y ) given th e degraded im age g ( x , y ), th e P S F /i(x , y\ a , ft), and in m any cases th e sta tistica l properties o f th e ad d itive noise process n ( x , y ) .

If th e im p u lse response in (1.1) is sh ift-in varian t (i.e ., s ta tio n a r y ), then

h ( x , y , a , ft) = h( x — a , y — ft). It follows that (1.1) becom es th e convolution in ­

tegral

/

-O O J — OO

w hich m ay b e regarded as a Fredholm integral equation of th e first kind [1, 5, ()]. T h ree observations concerning equation. (1.2) im m ediately follow. First of all, a solu tion f { x , y ) m ay not e x ist. Indeed, this w ill be th e case when g ( x , y ) is too rough and h is to o sm ooth or w hen h ( x , y ) and f ( x , y ) are orthogonal. Second, th e solu tio n (1 .2 ) m ay not be unique. T h is can be dem onstrated by m eans of th e R iem an n-L ebesq ue lem m a [7]. If h ( x , y ) is an integrable function, then it can be show n th a t [1]

f b f b

^lim Jhn J J h( x — a , y — ft) sin (//a ) sin (£ ft)d a d ft = 0 (1.3)

It follow s th a t a sinusoid o f high frequency can be added to th e o b ject f { x , y ) and th e resu ltin g su m is identical to th e degraded im age </(x, y). T h e last concern, which is alw ays a serious problem in m any practical applications, is th at a solution / ( x , y) does not d epend continuously upon th e recorded data g ( x , y ) . T h is m eans that sm all perturbations in g ( x , y ) m ight lead to large changes in f ( x , y ) . T his can also be seen in (1 .3 ). B ecause o f these reasons, th e problem o f solving equation (1 .2 ) is

5

called an ill-posed problem [8, pp. 9, 39, 49] — a com m on usage in m any diverse fields.

T hroughout th e th esis, it w ill be assum ed that th e im ages / ( x, y), g ( x , y ), and th e P S F h( x, y ) are o f finite ex ten t, and th at they all have support in th e first-quadrant. Sam pling th e equation (1.2) for a discrete approxim ation, we have

<?(M) = + (1-4)

* i

I

We assum e th a t th e size o f th e original im age m a trix [ f ( k , /)] is P ix Q, and th e size o f th e P S F m atrix / /p s f = [^(^, 01 J x ^ • Thus th e degraded im age m atrix [g(k, /)] and th e noise m atrix [n(fc, /)] are o f size ( P + J — 1) x ( Q + K — 1). Follow ing

H u n t’s approach [9], we express th e convolution form at o f th e discrete im age m o d el (1.4) in a vector-m atrix form by th e following steps.

Step 1: C hoose M > P T J — 1 and N > Q + K — 1. Form new ex ten d ed m atrices [/e], [/*c], [(Je\ and [ne] o f size M x N as follows:

0 < k < P - l , 0 < 1 < Q - 1 (1.5) otherw ise 0 < k < J - l , 0 < l < K - l (1.6) otherw ise 0 < k < P + J - l , 0 < l < Q + K - l (1.7) otherw ise 0 < k < P + J - l , 0 < l < Q + K - l (1.8) otherw ise

/«(*,/) =

Step 2: C onstruct colum n vectors fe , g ,;, n ,., of length M N by lexicographically ordering th e m atrices [ / e], [ge], and [rac], respectively. In this construction, the first row o f a m atrix b eco m es'th e first segm ent o f I,he corresponding vector, the second row th e second segm ent, etc. For exam p le,

M 0,0)

/e(0, 1)

/ e ( 0 , / V - 1)

/ e ( l , 0 )

h ( M - 1, N - 1)

T h e vectors

ge

ne

(1.9)

Step 3: C onstruct th e M N x M N m atrix H as

/ / = Ho Hm- I Hm-2 • • / / , n \ Ho I l x i- 1 ■ 112 h2 H i H o • • t h H m -1 H m — 2 H m- 3 • • • / / o (1.1.0)

7

where each / / , is form ed from th e *th row of th e ex ten d ed P S F m atrix [he], i.e .,

I I , =

he(i, 0) he( i , N - 1) he( i , N - 2) ••• he( i , l )

he(i, 1) he(i, 0) he( i , N - 1) ••• he(r,2 )

he(i, 2) ^e(i, 1) /»e(*,0) ••• he( i , 3 )

he( i , N - 1) he( i , N - 2 ) he( i , N - 3) ••• Ae(*,0)

Step 4: Finally, w rite (1.4) in th e vector-m atrix form

ge =

ne

T hroughout th is th esis, M and N w ill be chosen such th a t M J and N K ,

and for convenience, th ey are chosen to be equal. Thus, w ith uniform sam p lin g on an N x TV 2-D la ttic e , equation (1.2) can b e reduced to a discrete approxim ation o f a convolution form sim ilar to (1 .4 ), and then expressed in th e m atrix form

g = H f + n (1.13)

where / , g and n are lexicographically ordered vectors o f size N 2 x 1, and H is th e linear blurring (distortion) operator of size N 2 x N 2. N o te th a t m a trix H in (1.10) is block circulant w ith circulant blocks. For sim plicity, b lo ck circu lan t m a tr ix will be used instead o f b lo ck circulant m a tr ix w ith circulant blo ck s in th e rest o f this thesis. For a ty p ica l im age restoration task , N is a large integer ranging from 256 to 1,024. C onsequently, equation (1.13) represents a linear sy stem o f equation s o f

very large size, and careful treatm ent m ust be given in solving such a system for / . H unt [9] introdu ced m eth o d s for handling such large m atrices. In this thesis, the term “im age form ation m odel” always refers to (1.13) as we only deal with digital im ages.

T ikhonov [10, 11], T ikhonov and Arsenin [12] proposed an approxim ate solution to (1 .1 3 ) by m in im izin g th e functional

Q ( f ) =

+ M \ C f \\2

For a given A, th e

f x

f x

T his

f x

9

th e reg u larization p a ra m eter. R egularization theory provides a form al basis for th e developm ent o f accep table solutions o f ill-posed problem s. P h illip s [13], Tikhonov [10, 11] and M iller [14] introduced the general concept o f a regularization operator in determ in ing regularized solutions for ill-posed problem s in th e early 1960’s. H unt [9] was one o f th e first to apply th e regularization theory to im age restoration. In [9], he also develop ed a com p utationally efficient technique for im age restoration based on the 2-D discrete Fourier transform (D F T ), a m eth o d which is still used today by m any researchers in this field. N o te that there is an in con sisten t usage o f the term s of regularization and constrained least-squares (CLS) m ethods. R egular­ ization m ethods provide estim a tes of th e im age problem (1.13) by im posing a set of constraints on th e problem such as th e functional (1.14). A g o o d e stim a te of (1.13) depends on an appropriate choice o f the corresponding regularization param eter A. T here are several m eth od s of finding A, Som e o f these are considered in C hapter 3. O ne m eth od o f finding A is known as th e CLS m eth od [9]. However, th e term CLS m ethod has b een used to describe a synonym o f regularization as a w hole, i.e., find­ ing th e solu tion and finding an o p tim a l A through th e Lagrange m ultiplier m eth od .

/

For exam p le, th e CLS m eth o d proposed by H unt is in fact a special application of th e id ea o f th e T ikhonov regularization. A nother exam p le is K atsaggelos’ paper [19] in which the T ikhonov m eth o d and th e CLS m ethod are equated.

E arly develop m ents in im age restoration are discussed in detail in references [l]-[4]. A fairly com p lete treatm ent o f regularization theory as applied to im age restoration can be found in [15, 16, 17]. During th e last decade itera tiv e restoration algorithm s includ ing basic itera tiv e algorithm s, m eth od o f projection on to convex sets (P O C S ), and regularized constrained iterative algorithm s have been stu died e x ten siv ely [18]-[23], Unser et. al. [24] presented a frequency-dom ain version o f th e

regularization theory and applied a class o f one-dim ensional (1-D ) recursive regu­ larization filters (R -filters) to so m e noise removal problems. R ecently this approach has b een ex ten d ed to a broader class of 1-D and 2-D R-filters that can b e used for im age restoration and other im age processing tasks [25, 26].

1.2.2

E dge D e tectio n

E dges in an im age correspond to in ten sity changes in th e physical properties of surfaces such as illum ination , reflectance and geom etry. E dge detection is m ainly a process th at d etects and localizes light-intensity changes. It is an im portant topic not on ly in im age processing, but also in pattern recognition, com puter vision, and robotics. A variety o f edge d etection schem es have been proposed in th e past two decades, and th ey work w ith various degrees of success for different im ages [27, 28, 29, 30, 31].

T h e edges in an im age can b e roughly m odeled by step functions, ramp functions or roof functions. N um erical differentiation is an obvious tool in d etectin g edges. Specifically, ed ges can b e detected by taking the gradient o f the im age and applying an appropriate threshold to th e resultant 2-D signal; or by taking th e Laplacian of th e im age and th en id en tifyin g its zero crossing points. C om m only used gradient operators in clu d e P rew itt, Sobel, R oberts and isotropic operators [4]. M ost o f these operators perform reasonably w ell on noise-free im ages whereas they do not work well for n oise-con tam in ated im ages. T his is because derivative operators enhance and em p h a size high frequency noise. O ne way to overcom e this difficulty is to sm o o th th e d a ta before differentiation is applied. An ex a m p le th at realizes this idea is th e L aplacian o f G aussian (LoG) operator proposed by Marr arid Hildreth

11

[27], which suppresses noise by first sm oothing the im age by convolving it w ith a G aussian function, and then applying a Laplacian to th e resultant im age. T h e edges o f th e im age at hand are finally identified by zero crossings o f th e output from LoG. It should be p ointed out that the above pre-sm oothing step ten d s to blur edges, and consequently th e LoG m ethod degrades th e lo ca liza tio n of edges. T hus, an ed ge d etector w hich can provide accurate locations o f edges for noisy im ages is desirable. Indeed, good localization o f the edge is one o f th e three criteria proposed by C anny [29] in obtain ing an op tim a l edge detector. C an ny’s other tw o criteria are good d e tec tio n ability and single response per edge. It is noted th a t C an ny’s d etector is o p tim a l only for a sm all fam ily of edge detectors (e.g ., LoG) in d etectin g stcp-function edges. N everth eless, C anny’s approach can be applied to different types o f edges, and th e approach leads to a new direction o f designing optim al edge d etectors. Follow ing the work [29] m any different op tim al edge detectors have b een designed w hich m odify and im prove C anny’s detector by adding m ore criteria, or by considering ram p-function and roof-function edges, rather th a n step edges. T h ese op tim al edge detectors include the op tim a l edge detector for ram p edges by Petrou and K ittler [32] and th e op tim a l infinite im pu lse response edge d etectio n filter by

Sarkar and Boyer [33]. *

1.2.3

Id en tification o f Im age M od el

T h e o b jectiv e o f im age restoration is to generate an e stim a te o f th e original scene that, is as good as possib le based on th e available (degraded) im age. T here are three m ajor ty p es o f degradations, nam ely, point degradations, spatial blurs, and noise. Point degradations are referred to as those th a t affect th e gray levels o f individu al

p ix els, w ith ou t introducing spatial blur, and they are in general nonlinear. A typical ex a m p le of p o in t degradation is film nonlinearity caused by film ’s sen sitiv ity to light in ten sity. T h e d en sity o f silver grains on developed film varies approxim ately logarith m ically w ith th e incident light intensity w ith saturation in both black and w h ite regions [34]. P o ten tia l sp atial blurs include relative linear m otion between the o b ject and th e im age, out-of-focus in an optical system , atm ospheric turbulence effects, geo m etric distortions, and sensor nonlinearities. N oise disturbances m ay be introdu ced by electron ic im aging sensors, transm ission devices, recording d evices, m easurem ent errors and q uan tization errors. T he o b jectiv e of im age restoration is to generate an e stim a te o f th e original scene as good as possible based on the available (degraded) im age. If th e im aging system is m odeled by taking all the above m en tio n ed degradations in to cons' nation, th e restoration problem would be too involved and there w ould be no general solution for the problem . In this thesis, it is assum ed th e im aging sy ste m is of th e form (1.13) where the spatial blur is shift-invariant, th e noise is ad d itive, and point degradation can be neglected. Under these assu m p tio n s, th e corresponding restoration problem becom es m athem atically m ore tractable.

A n oth er im p ortan t issu e w ith w hich we have to deal is that in practical, ap­ plication s blur characteristics are often unknown, hence th e blur operator H (or: equivalently th e blur P S F k ( k , l ) ) m ust b e identified from th e degraded im age g itself. In other words a practical im age restoration problem is to find an e stim a te of / g iv en th e degraded im age g only, w ithout priori know ledge o f th e blur degra­ dation and n oise degradation. In this case, the blur P S F and noise characteristics have t o be first estim a te d from th e im age to be restored. T h e associated im age restoration problem is therefore called a b lin d restoration problem . T here are three

13

approaches to a blind restoration problem . O ne approach is to lo ca te and exam in e sharp points and sharp lines in th e im age to be restored. B y definition, th e blur im pulse response is th e im age o f a point-source object. Therefore, a point source in th e degraded im age yield s a direct indication o f th e blur P S F . T his w ould b e th e case in an astronom ical im age, where the im age o f a faint star could b e used as an e stim a te o f th e blur P S F . T h e im age o f a sharp line can also be u tilized to determ in e th e blur P S F and details can be found in [35]. T h e second approach is m ainly to identify those blur P S F s whose spectra show a regular pattern o f zero crossings. Since these zeros can also be located using th e spectrum o f th e blurred im age, spectral or cepstral techniques [20] can be used to e stim a te th e P S F from the distance betw een spectral zeros o f tne blurred im age. T h ese techniques are e s­ p ecially effective for dealing w ith linear m otion blur and out-of-focus blur. D eta ils of this approach can be found in [2, 20, 36, 37]. In th e third approach, th e original im age (th e o b je ct) is first m odeled as a 2-D autoregressive (A R ) process and th en the identification problem is form ulated as a m axim um likelihood problem , w hich turns out to be equivalent to a 2-D autoregressive m oving-average (A R M A ) m odel identification problem . T h e earliest work in th is area was described by Tekalp et. al. [38, 39], in w hich m eth od s for th e identification of various blur P S F s are proposed. R ecently, follow ing th e work o f [38, 39] Lagendijk et. al. [20, 40] and B iem on d et. al. [41] proposed an ex p ecta tio n -m a x im iza tio n im age identification algorithm w hich can be applied to blurred im ages w ith relatively low signal to noise ratio (S N R ).

1.3

C o n tr ib u tio n s

T h is thesis is in ten d ed to be a self-contained description of som e filtering techniques for restoration, noise rem oval, edge d etectio n and blur identification o f degraded im ages. For im age restoration, we have developed a frequency-dom ain recursive regularization filter (R -filter) theory o f th e well-known regularization techniques to efficiently restore n oisy im ages w hose degradations are due, for exam p le, to cam era defocu sing, linear m o tio n , or exp on en tial blurring. T h e R -filters proposed here are derived by generalizing the 1-D recursive R-filters proposed by M. TJnser et. al. [24] in 1991 for noise rem oval purposes to various im age blur m echanism s and to e x te n d it further to th e 2-D case. U n ser’s spatial dom ain approach was transferred to frequency d om ain through th e use o f Parseval’ form ula. T h e filter can either be ap p lied in th e frequency dom ain or transferred back to th e spatial d om ain . In th e frequency dom ain , th e num ber o f operations carried out is sam e as th e D F T im p le­ m en ta tio n proposed by H unt [9]. In th e spatial dom ain, th e tim e required m ight be longer but on e has th e advantage o f adaptive filtering and region o f interest fil­ tering. For im age noise rem oval, several new nonlinear 2-D filters th at m ake use of lo ca l sta tistic s o f th e im age have been designed. M oreover, a tw o-step regular­ iza tio n approach is proposed for noise removal. For im age edge d etectio n , we have develop ed several design techniques for detectin g edges of noisy im ages by using linear, low-order, 2-D finite im p u lse response (F IR ) filters. T h e design techniques are based on sim ultan eou s consideration o f th e conventional ed ge d etectin g proce­ dures — th e pre-sm ooth ing step and th e differentiation step. It is also dem onstrated th a t b o th n oise rem oval and ed ge d etectio n tasks can also be perform ed within th e unified R -filter fram ework. F in ally a blur identification algorithm based upon th e

15

relative m inim um pattern o f spectra o f degraded im ages is proposed. T h e algorithm proposed is found to be successful when identifying linear m otion blurs, defocusing blurs, and blurs w ith circularly sym m etric P SF . M aking use of th e proposed blur identification algorithm , a tw o-phase blind restoration algorithm is presented. In the first phase o f th e algorithm , estim a ted power spectra o f subim ages o f th e orig­ inal im age are determ ined from th e degraded im age. In th e second phase, a blind restoration algorithm is em ployed to obtain a restored im age usinb estim a te d power spectra obtain ed in phase one.

1.4

O rg a n iza tio n o f th e T h e sis

T he principal o b jectiv es o f th is th esis are to present th e unified R-filter approach to im a g e restoration, noise rem oval and edge detection; nonlinear filtering for noise removal; low-order F IR filters for edge detection; and a tw o-phase blind restoration algorithm .

T h e rem ainder o f th is thesis is organized as follows. In C hapter 2 we provide a d etailed description o f th e 1-D R -filter theory. T h e 1-D R-filter theory on noise removal by U nser et. al. [24] is briefly reviewed. T h en it is exten d ed to in clu d e a n ontrivial 1-D deblurring m echanism . As a resu lt, th e generalized 1-D R -filters can be used to restore noise con tam in ated im ages th a t are further distorted by a linear m otion blur. It is show n th at 1-D R-filters can b e decom posed in to tw o co m p le­ m entary causal and anti-causal, recursive, stab le filters w ith identical coefficients. Furtherm ore, th e 1-D R -filter th eory is ex ten d ed to y ield a 2-D R-filter theory. B e­ ginning w ith th e im age restoration m odel (1.13), th e problem at hand is form ulated as a least-squares op tim iza tio n problem . T w o techniques for th e im p lem en tation

o f recursive 2-D R -filters are presented. T h e first is iterative in nature, w hile th e secon d is accom plish ed by approxim ating the square root o f m agnitude response of th e R -filter by a stab le causal filter. A pplications of th e proposed restoration algorithm s are illustrated using a num ber o f sam ple im ages.

C hapter 3 presents m eth o d s for determ ining op tim al regularization param eters a n d /o r o p tim a l stab ilizin g operators for im age restoration. T h e first part o f the chapter is d ev o ted to a stu d y on op tim al determ ination of th e regularization param ­ eter A by an L-curve approach. D efinition o f the L-curve is given and the Tikhonov regularization is discussed in som e d etail. Several m ethods for choosing suitable regularization param eters are review ed. It is then shown that th e regularization param eter corresponding to th e largest curvature o f th e L-curve gives a ‘nearly’ o p tim a l regularized solu tio n for a given im age restoration problem . D ealing with im ages blurred by linear m otion , an adaptive restoration m ethod is proposed where th e regularization param eter is determ ined by the proposed L-curve m ethod. T h e second part o f th e chapter presents several optim ization techniques for determ in ing th e regularization param eter and stabilizing operator. T h e generality o f th e pro­ posed techn iqu es im p lies th a t th ey can be applied to other 1-D and 2-D ill-posed problem s as w ell. Several sam p le im ages are used to illustrate th e techniques pro­ posed. Finally, th e conventional Tikhonov regularization m ethod where a sin gle regularization param eter is em p loyed is extended to a regularization schem e where m u ltip le regularization param eters can be incorporated. It is shown that w hen the proposed m u ltiple-param eter regularization m ethod is applied to an im age restora­ tio n problem , th e solu tion accuracy and sm oothness is better balanced. It is also show n th a t th e im p lem en ta tio n o f th e proposed algorithm requires 2-D D F T op­ erations only. Finally, th e R -fiiters for th e m ultiple-param eter im age restoration

1 7

problem are derived.

C hapter 4 describes an im proved W iener filtering technique and its ap p lication to im age restoration. It is dem onstrated th at the conventional W iener filters can be im proved by takin g th e inform ation contained in th e spectra o f th e blurring operator, th e noise and th e im age into consideration. A m odified W iener filter is derived based on this idea togeth er w ith th e regularization concepts. It is dem onstrated th a t b etter quan tita tiv e resu lts can indeed b e achieved by using th e m odified W iener filter.

C hapter 5 is devoted to several filtering techniques for noise removal. In th e first part o f the chapter, average filtering and G aussian filtering are briefly review ed, and a tw o-step regularization approach is introduced. A p plication o f th e R -filtering techniques to noise rem oval is presented. T h e second part o f th e chapter provides a treatm ent o f th e noise removal problem using quadratic and cubic filters w hich u tilize local sta tistics o f th e im age.

In C hapter 6, num erical op tim iza tio n techniques are used to design edge d e tec ­ tors. H ere th e id ea is to integrate th e differentiation step w ith th e pre-sm ooth ing step to obtain a design specification for a band-lim ited differentiator. T h is leads to several im age d etectors th at are iow-order 1-D and 2-D F IR filters. T h ese filters are linear and shift-invariant, and can be im plem ented efficiently by discrete convo­ lution. E xperim ental results are presented to show th e perform ance o f th e design edge-d etector filters.

C hapter 7 proposes a tw o-phase blind restoration algorithm . We first present a practical identification m ethod by exam ining th e m inim um -pattern o f th e sp ectra of blurred im ages. T h e m eth o d is show n to b e successful in id en tifyin g linear m otio n blur, d efocu sin g blur, and those w ith circularly sym m etry P S F s. T h e identification procedure is th en incorporated in to th e proposed tw o-phase restoration m eth o d

w hich is described in th e second part of th e chapter.

C hapter 8 sum m arizes th e m ajor contributions o f th e thesis and discusses some- future research considerations in th e field.

19

C h a p te r 2

O n e -D im e n sio n a l and

T w o -D im e n sio n a l R -F ilte r s

2.1

O n e -D im e n sio n a l R -F ilte r s

2.1.1

In trod u ction

A problem is w ell-posed in th e sense defined by H adam ard [8] w hen its solution e x ists, is unique, and depends continuously on th e in itia l data. Ill-posed problem s fail to satisfy on e or m ore o f th ese criteria. It has been know n th a t im age restoration problem s are ill-p osed problem s [1, 10, 11, 12]. T h is can b e seen by considering an im aging sy stem m odeled as

g = H f + n (2.1)

T yp ically, im age restoration is a procedure th a t, for given degradation m echan ism

H and degraded im age g , one seeks a good approxim ation to th e original im age / .

In th e d iscrete version th e large size m atrix H as an a n alytical representation of th e degradation m echan ism is often ill-conditioned. As a result th e w idely accep tab le least-squares solu tion / = H^g — H^n, where H* = ( H TH ) ~ l H T is th e p se u d o ­

in v erse or th e M oore-P en rose g e n era lized in verse [42] o f H , is not o f use as it is too

sen sitiv e to th e sy stem noise n.

R egularization theory provides a m eth od to solve ill-posed problem s and to. co m p u te solu tions th a t satisfy a priori sm oothness constraints. In th is section, th e recent work o f U nser, A ldroubi, and Eden [24] on recursive regularization fil­ ters (R -filters) will b e exten d ed to a m ore general settin g to include nontrivial

1-D deblurring dynam ics in th e regularization filters. As a resu lt, th e generalized R -filters derived here can b e u tilized to restore noise contam inated im ages th at are further degraded by a 1-D m otio n blur. As a key in our develop m ent, a decom p osi­ tion theorem for th e 2-transform 1 of generalized R-filters is presented, w hich shows th at R -filters are sym m etric sta b le lowpass filters w ith an adjustable regularization param eter and can b e decom p osed into tw o com plem entary linear shift-invariant2,

1T he z-transform X ( z ) o f a sequence far(Ar)} is defined as

OO

* 0 0 = *(*)*“ *

k — — QQ

where z is a com plex variable. The z-transform o f the impulse response {/>(&)} o f a linear filter is often referred to as the system function or transfer function of the filter.

2T he class o f shift-invariant filters is characterized by the property that if { y( k) } is the response to {ar(fc)}, then { y ( k — /)} is the response to { x ( k — /)} where / is a positive or a negative integer. It can be shown that any linear shift-invariant filter is completely characterized by its impulse response { h( k ) } [43].