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Signal Processing: Image Communication

Signal Processing: Image Communication 23 (2008) 433– 441

0923-59

doi:10.1

� Cor

E-m

journal homepage: www.elsevier.com/locate/image

Wavelet iterative regularization for image restoration with varyingscale parameter

Bin-bin Hao �, Min Li, Xiang-chu Feng

Department of Mathematics, Xidian University, (P.O.) Box 245-59 Xi’an, Shaanxi 710071, China

a r t i c l e i n f o

Article history:

Received 23 November 2007

Received in revised form

26 March 2008

Accepted 9 April 2008

Keywords:

Total variation

Iterative regularization method

Bregman distance

Inverse scale space

Shift invariant wavelet

Wavelet shrinkage

Image denoising

65/$ - see front matter & 2008 Elsevier B.V

016/j.image.2008.04.006

responding author. Tel.: +86 13571872584.

ail address: bbhao981@hotmail.com (B.-b. H

a b s t r a c t

We first generalize the wavelet-based iterative regularization method and the wavelet-

based inverse scale space to shift invariant wavelet-based cases for image restoration.

Then, a method to estimate the scale parameter is proposed from wavelet-based iterative

regularization; different parameters with different iterations are obtained. The wavelet-

based iterative regularization with the new parameter, which controls the extent of

denoising more precisely in the wavelet domain, leads to iterative global wavelet

shrinkage. We also obtain a time adaptive wavelet-based inverse scale space from the

iterative procedure with the proposed parameter. We provide a proof of the convergence

and obtain a stopping criterion for the iterative procedure with the new scale parameter

based on wavelet transform. The proposed iterative regularized method obtains quite

accurate results on a variety of images. Numerical experiments show that the proposed

methods can efficiently remove noise and well preserve the details of images.

& 2008 Elsevier B.V. All rights reserved.

1. Introduction

Variational regularization [1,9,21,22,24,25] method andwavelet shrinkage [16,18] are two main successfulapproaches in image restoration. During the last decade,the relations between them have become one of the mostactive areas of research in mathematical image processing[2,8,14,26,27].

In this paper we are concerned with the classicaldenoising problem of image degraded by additive whiteGaussian noise. We assume that the input image f ¼ g þ Zand f : O � R2

! R is composed of the original image g

and additive uncorrelated noise Z of variance s2.A variational approach to this problem is by solving thefollowing minimization problem:

u ¼ arg minu2BVðOÞ

JðuÞ þg2

Hðu; f Þn o

(1)

. All rights reserved.

ao).

the scale parameter g40 tunes the weight between theregularization term JðuÞ and fidelity term Hðu; f Þ. In the1990s, most of the research was focused on the regular-ization term. A main contribution is TV-based methodintroduced by Rudin–Osher–Fatemi [25]. That is

u ¼ arg minu

ZOjrujdx dyþ

g2k f � uk2

L2

� �(2)

This model performs very well for removing noise whilepreserving edges. However, it fails to separate welloscillatory component from high-frequencies compo-nents. To remedy this situation, Meyer [21] proposed thatone should focus on the role of the fidelity term. This hasinspired many new image denoising algorithms, e.g. [14].But it is pity that one finds some signals in the removedresidual part for these denoising algorithms. Thus, aniterative regularization method (IRM) [23], which replacesthe regularization term by a generalized Bregman distance[3], was proposed. Later, this IRM method was successfullygeneralized to a time-continuous inverse scale space (ISS)formulation [4,6], which starts with the initial imageuðx;0Þ ¼ 0 and approaches the noisy image f as time

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B.-b. Hao et al. / Signal Processing: Image Communication 23 (2008) 433–441434

increases. Also, the authors and colleagues further gen-eralized the IRM and the ISS method to wavelet-basedimage restoration [28].

Given compactly supported orthogonal wavelets basisC:¼fcðiÞji ¼ 1;2;3g with cðiÞj;k ¼ clð�Þ ¼ 2jcðiÞð2j

� �kÞ, l ¼ði; j; kÞ; j 2 Z; k 2 Z2; i ¼ 1;2;3 denoting f l ¼ h f ;cli andbf ¼ ff lgl, they proposed to replace BVðOÞ by B1

1ðL1ðoÞÞ,the wavelet-based iterative regularization method (W-IRM) is represented by the following formulation:

buðnÞ ¼ arg minbu Dbpðn�1Þ

J ðbu; buðn�1ÞÞ þg2kbf � buk2

L2

� �(3)

bpðnÞ ¼ bpðn�1Þþ gðbf � buðnÞÞ (4)

with nX1; buð0Þ ¼ 0; bpð0Þ ¼ 0 and DbpJ is the generalized

Bregman distance defined as DbpJ ðbu; boÞ ¼ JðbuÞ � JðboÞ�

hbu� bo; bpi. Denoting bvðnÞ ¼ bpðnÞ=g and after some simplifica-tion, (3) can be rewritten as

buðnÞ ¼ arg minbu JðbuÞ þ g2kbf þ bvðn�1Þ � buk2

L2

n o(5)

The authors also generalized the above W-IRM processto a time continuous ISS. For the singularity of qjulj at the

point ul ¼ 0;ffiffiffiffiffiffiffiffiffiffiffiffiffiu2l þ e

qis used to approximate julj, where e

is a small constant. Now, pel ¼ ul=ðu2

l þ eÞ anddpe

l=dul ¼ e=ðu2l þ eÞ3=2, let Dt ¼ g, nDt! t, a simple flow

called wavelet-based inverse scale space (W-ISS) flow [28]involving ul is obtained

dul

dt¼ðu2

l þ eÞ3=2

eð f l � ulÞ; ulð0Þ ¼ 0 (6)

In this paper, the W-IRM and the W-ISS are generalizedto the shift invariant wavelet-based cases, which reducethe artifacts introduced by the wavelet-based methods.Subsequently, we deduce a method to estimate the scaleparameter from shift invariant wavelet-based iterativeregularization. The shift invariant W-IRM with theproposed scale parameter that automatically controlsthe extent of denoising more precisely leads to iterativeglobal wavelet shrinkage. We provide a proof of theconvergence and an effective stopping criterion isobtained for the W-IRM with the new scale parameter,which is called the discrepancy principle in the theory ofiterative regularization of inverse problems. This para-meter can also be applied to W-ISS, obtaining a timeadaptive method for image restoration. Finally, somenumerical examples are presented and show that ourmethods reduce the optimal number of iterations.

The following is organized as follows. We summarizesome facts on wavelets and function spaces in Section 2.Section 3 is mainly concerned with the generalization toshift invariant W-IRM for image restoration beforeproposing a method to obtain a scale parameter andanalyzing the convergence results of the W-IRM with thenew scale parameter in Section 4. Section 5 showsnumerical results of image restoration. We give conclu-sions in Section 6.

2. Preliminary

In this section, we briefly recall some facts on waveletsand the equivalent Besov norms which are needed later on.

Besov norms can be expressed through waveletcoefficients [8,10,13,15,17]. A function f defined on theunit square O:¼½0;1Þ2, which can be extended periodicallyto all of R2. If the functions in C are smooth enough, thenone can often determine whether a function f is in theBesov space Ba

qðLpðOÞÞ by examining its wavelet coeffi-cients. In the case p ¼ q,

j f jBapðLpðOÞÞ �

Xi;j;k

2ðaþ1�2=pÞkjh f ;cðiÞj;kij

p

0@ 1A1=p

(7)

with i ¼ 1;2;3, 1ppp1; kX0. Here, ‘�’ represents theequivalence between the two norms.

Chambolle and Lucier in [10] remarked that whileusual definition of the Besov space norm is invariantunder translation of the function f, (7) is not. Onedrawback of traditional wavelet shrinkage is its depen-dence on translations. Coifman and Donoho presented in[11] translation invariant wavelet shrinkage based on thewavelet transform of different translations of the signal. Inpractical applications, Daubechies in [13] gave discretizedshift invariant wavelet transform with 2M rows of 2M

pixels and obtained the following Besov penalty term

j f jBapðLpðOÞÞ �

Xi;j;k

2ð jsþ2ð j�MÞÞjh f ;cðiÞ

j;k2j�M ijp

0@ 1A1=p

(8)

with s ¼ aþ 1� 2=p. To simplify the notation from nowon, we denote f l;ul and vl representing the shift invariantwavelet coefficients if not otherwise specified.

3. Shift invariant W-IRM and ISS

The denoising results often exhibit visual artifacts[10,13,12] especially near singularities, for example, Gibbsphenomena [12] in the neighborhood of the discontinu-ities, because of the lack of translation invariance of thewavelet basis. Chambolle and Lucier remarked that thesemi-norm on the right-hand side of (7) is waveletdependent [10]: if one changes the wavelets used in therepresentation, also the semi-norm is changed, in contrastto the intrinsic definition of the Besov space norm.Coifman and Donoho presented in [11] a translationinvariant wavelet shrinkage which is based on the wavelettransform of different translations of the signal, termed‘cycle spinning’. This method produces a reconstructionwith far weaker Gibbs phenomena. Noting B0

2ðL2ðOÞÞ ¼L2ðOÞ [8,20], from (8), we have

kð f þ vðn�1ÞÞ � uk2L2ðOÞ

�Xi;j;k

22ð j�MÞjhð f þ vðn�1ÞÞ � u;ci;j;k2j�M ij

2 (9)

and

jujB11;1ðOÞ�X

i;jXj0 ;k

22ð j�MÞjhu;ci;j;k2j�M ij (10)

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So, for each l ¼ ði; j; kÞ, the shift invariant wavelet-basediterative regularization inspired from [28] has the for-mulation of

uðnÞl ¼ arg minul

Pl

22ð j�MÞjulj

�þg2

Pl

22ð j�MÞj f l þ vðn�1Þ

l � ulj2

�vðnÞl ¼ f l þ vðn�1Þ

l � uðnÞl

8>>>>>><>>>>>>:(11)

Hence, the minimizer of (11) based on shift invariantwavelet has the formulation of

uðnÞl ¼

ð f l þ vðn�1Þl Þ

�1

gsignð f l þ vðn�1Þ

l Þ if j f l þ vðn�1Þl j4

1

g

0 if j f l þ vðn�1Þl jp

1

g

8>>>>>><>>>>>>:(12)

vðnÞl ¼ f l þ vðn�1Þl � uðnÞl (13)

In this discrete implementation, we will see that thesmall features vðn�1Þ

l of the image in some sense areincorporated into f l in the next iteration.

Fig. 1. Denoised results of ‘Shape’ image with s ¼ 20. From left to right,

SNR ¼ 25:6690; second row: W-ISS with SNR ¼ 24:8873, shift invariant W-ISS

The corresponding continuous ISS formulation i.e. shiftinvariant W-ISS should be obtained which has thefollowing form:

dul

dt¼ðu2

l þ eÞ3=2

eð f l � ulÞ; ulð0Þ ¼ 0 (14)

For the numerical examples, we use semi-explicit discretealgorithm to approximate (14)

uðnþ1Þl � uðnÞl ¼ Dt �

ððuðnÞl Þ2þ eÞ3=2

eð f l � uðnþ1Þ

l Þ (15)

Fig. 1 shows the denoising results with W-IRM, W-ISS,shift invariant W-IRM and shift invariant W-ISS for ‘Shape’image with different shapes and scales added Gauss whitenoise s ¼ 20. We use k f l � uðnÞl kL2

ps as the stoppingcriterion and choose g ¼ 0:001;Dt ¼ 0:001 and e ¼ 0:01manually. We can see some artifacts in the first columnfor the wavelet-based methods. For the richer class of theshift invariant wavelet coefficients in the second column,it shows fewer artifacts and has higher SNRs than the firstcolumn.

first row: W-IRM with SNR ¼ 24.9990, shift invariant W-IRM with

with SNR ¼ 25:5107.

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4. A new iterative parameter

In this section, we deduce the selection of a new scaleparameter from W-IRM. We provide a proof of theconvergence and obtain a stopping criterion for the W-IRM with the proposed scale parameter. The new para-meter-based method for image restoration automaticallycontrols the denoising extent in the wavelet domain andobtains quite accurate results.

4.1. Deduction of new parameter

In this subsection, a new parameter is obtained. Theability of noise removal depends on the parameter g forthe method introduced in Section 3. Large g correspondsto very little noise removal, and u is close to the noisyimage f. Small g yields a blurry, oversmoothed u [8,23].Fig. 2 shows the denoised images with different parametersusing the method proposed introduced in Section 3, fromleft to right, g ¼ 0:001;0:1;1, respectively. In the following,a new scale parameter estimation method is proposedfrom W-IRM. The Euler–Lagrange equation of (11) has theformulation of

signðulÞ � gð f l þ vðn�1Þl � ulÞ ¼ 0

where ula0. Denoting u�l and v�l the optimization solutionto (11), we have

signðu�lÞ ¼ signðgÞ � signð f l þ v�l � u�lÞ

¼ signð f l þ v�l � u�lÞ (16)

Multiply equation signðu�lÞ � gð f l þ v�l � u�lÞ ¼ 0 by ð f l þv�l � u�lÞ and sum up over all lXl

j f l þ v�l � u�lj ¼ g �Xl

ð f l þ v�l � u�lÞ2

Then, we have the following equation:

g ¼P

l j f l þ v�l � u�ljPl ð f l þ v�l � u�lÞ

2(17)

In numerical implementations, we use uðn�1Þl , vðnÞl to

approximate u�l; v�l , respectively, and accordingly gðnÞ

denotes g in (17). So, a new scale parameter that dependson wavelet coefficients is obtained. Applying the proposedscale parameter to W-IRM, with initial values uð0Þl ¼ 0 andvð0Þl ¼ 0, we obtain different scale parameters gðnÞ for

Fig. 2. Denoised results with different parameters. From left

different iterations. Eq. (11) should be written as

gðnÞ ¼P

lj f l þ vðn�1Þl � uðn�1Þ

l jPlð f l þ vðn�1Þ

l � uðn�1Þl Þ

2

uðnÞl ¼ arg minul

Pl

22ð j�MÞjulj þ

gðnÞ

2

Pl

22ð j�MÞj f l

�þ vðn�1Þ

l � ulj2

�vðnÞl ¼ f l þ vðn�1Þ

l � uðnÞl

8>>>>>>>>>>>><>>>>>>>>>>>>:(18)

which leads to

uðnÞl ¼

ð f l þ vðn�1Þl Þ

�1

gðnÞsignð f l þ vðn�1Þ

l Þ if j f l þ vðn�1Þl j4

1

gðnÞ

0 if j f l þ vðn�1Þl jp

1

gðnÞ

8>>>>>><>>>>>>:(19)

Or simply, uðnÞl ¼ S1=gðnÞ ð f l þ vðn�1Þl Þ where nX1, S denotes

soft wavelet shrinkage operator with threshold 1=gðnÞ, and

vðnÞl ¼ f l þ vðn�1Þl � uðnÞl (20)

For the update of (19) and (20), we have an equivalentresult

uðnÞl ¼

f l if j f lj41

ðn� 1ÞgðnÞ

nf l �1

gðnÞsignð f lÞ if

1

ngðnÞoj f ljp

1

ðn� 1ÞgðnÞ

0 if j f ljp1

ngðnÞ

8>>>>>>><>>>>>>>:(21)

and signðuðnÞl Þ ¼ signð f lÞ, if uðnÞl a0.

vðnÞl ¼

1

gðnÞsignð f lÞ if j f lj4

1

ngðnÞ

nf l if j f ljp1

ngðnÞ

8>>><>>>: (22)

The proof is similar to [28].Shift invariant W-IRM (18) leads to global soft wavelet

shrinkage (19) which is equivalent to firm shrinkage (21).Different iteration steps come into being different waveletthresholds.

The corresponding continuous ISS formulationwhich is not the main content of this paper is time

to right: scale parameter g ¼ 0:001;0:1;1, respectively.

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adaptive with DtðnÞ ¼ gðnÞ

dul

dt¼ðu2

l þ eÞ3=2

eð f l � ulÞ; ulð0Þ ¼ 0 (23)

For the numerical examples, we use semi-explicit discretealgorithm to approximate the above equation

uðnþ1Þl � uðnÞl ¼ DtðnÞ �

ððuðnÞl Þ2þ eÞ3=2

eð f l � uðnþ1Þ

l Þ (24)

4.2. Convergence analysis

IRM starts with the image uð0Þ ¼ 0 and approaches thenoisy image as time increases, with small scales incorpo-rate into large ones. Error estimation for the Bregmandistance and the convergence rates of variational regular-ization scheme have already been studied before [5,7,19].Subsequently, we give convergence analysis for thesolution defined by (19) and (20), and deduce a stoppingcriterion [23,4,6,28,19] for the iterative procedure.

1 2 3 4 50

102030405060708090

100110120130

the number of iteration

norm

2 of

f an

d u

Fig. 3. Form left to right: the first row: the results after 1, 2, 3 and 7 iterations w

row: the results after 1, 5, 10, 17 iterations with shift invariant W-IRM (12) and

f � uðnÞ with iterations using shift invariant W-IRM with iterative scale parame

Proposition 1. The updates gðnÞ;nX1 defined by (18) is

nondecreasing, and uðnÞl ! f l, as n!1.

Proof.

1

gðnÞ¼

Pl ð f l þ vðn�1Þ

l � uðn�1Þl Þ

2Pl j f l þ vðn�1Þ

l � uðn�1Þl j

¼

Pl1 1=gðn�1Þ2 þ

Pl2 ð f l þ vðn�1Þ

l Þ2P

l1 1=gðn�1Þ þP

l2 j f l þ vðn�1Þl j

where l1 denotes a subset of l that l1 ¼ fl : j f lj41=gðnÞg,and l2 denotes l2 ¼ fl : j f ljp1=gðnÞg. We note m ¼ #ðl1Þ,the number of l1. At the ðn� 1Þth iteration, 1=gðn�1Þ is aconstant, so we deduce that

1

gðnÞ¼

1

gðn�1Þ

mþ gðn�1Þ2P

l2ð f l þ vðn�1Þl Þ

2

mþ gðn�1ÞP

l2j f l þ vðn�1Þl j

¼: 1

gðn�1ÞCl

6 7 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2018

19

20

21

22

23

24

25

26

27

28

the number of iteration

norm

2 of

f an

d u

ith shift invariant W-IRM with iterative scale parameter (21); the second

(13), the third row: 30 iterations with shift invariant W-IRM, L2 norm of

ter, L2 norm of f � uðnÞ with iterations using shift invariant W-IRM.

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where

Cl ¼C1l

C2l

¼mþ gðn�1Þ2

Pl2 ð f l þ vðn�1Þ

l Þ2

mþ gðn�1ÞP

l2 j f l þ vðn�1Þl j

C1l � C2

l ¼ gðn�1Þ gðn�1ÞXl2

ð f l þ vðn�1Þl Þ

2

�Xl2

j f l þ vðn�1Þl j

!¼ gðn�1Þ

Xl2

ðgðn�1Þð f l þ vðn�1Þl Þ

2� j f l þ vðn�1Þ

l jÞ

p0

Because l2 ¼ fl : j f ljp1=gðn�1Þg, the last inequation ishold. So, 1=gðnÞ is nonincreasing, i.e. gðnÞ is nondecreasing.

Letting x be the limit of 1=gðnÞ, v�l the limit of vðnÞl and u�lthe limit of uðnÞl , from (20), we have u�l ¼ f l, that is

uðnÞl ! f l, as n!1. This ends the proof. &

Following, a monotonicity property of the residualbetween noisy image f and restored uðnÞ and of theBregman distances between clean image g and uðnÞ in

Fig. 4. Denoising results of ‘Shape’ image with s ¼ 20. From top to bottom, left

shift invariant W-IRM with iterative scale parameter (21) (SNR ¼ 26:8805; it

invariant W-ISS with iterative scale parameter (23) (SNR ¼ 27:5461; iter ¼ 4).

the wavelet frame is obtained with the similar deductionas [23]. Simultaneously, a similar stopping criterion in thesense of Bregman distance is obtained, which is well-known as the so-called discrepancy principle.

Proposition 2 (Monotonicity). bf denote the shift invariant

wavelet coefficients bf ¼ ff lgl the sequence HðbuðnÞ;bf Þ obtained

from the iterates of (19) and (20) is monotonically

nonincreasing that

HðbuðnÞ;bf ÞpHðbuðn�1Þ;bf Þ (25)

Moreover, if kbf � buðnÞkXkbf � bgk ¼ s, we have

DbpðnÞðbg; buðnÞÞpD

bpðn�1Þ

ðbg; buðn�1ÞÞ (26)

The proof is analogue to the case in [23].

The above estimate induces that the Bregman distanceDðbg; buðnÞÞ is decreasing at least as long as kbf � buðnÞk2

L2Xs2.

One could terminate the W-IRM at the iteration n suchthat kbf � buðnÞkL2

¼ s. It is easily deduced that for arbitrarysmall �, kbu� buðnÞkL2

p� which indicate that the results

to right, shift invariant W-IRM (12) and (13) (SNR ¼ 26:4390; iter ¼ 13),

er ¼ 2), shift invariant W-ISS (15) (SNR ¼ 26:2742; iter ¼ 16) and shift

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obtained from the stopping criterion approximate thenoise free image.

5. Numerical examples

In this section, we present some numerical examples ofimage denoising and decomposition with our proposedwavelet-based approach. We choose Haar basis and level 3for wavelet analysis in all the image restoration experi-ments.

Example 1. Fig. 3 displays the results with iterativeregularization based on shift invariant wavelet transformproposed in Section 3 and shift invariant W-IRM withiterative scale parameter proposed in Section 4 with‘cameraman’ image added Gauss white noise s ¼ 20. Wechoose g ¼ 0:001 manually for shift invariant W-IRM.Small scales are incorporated into the image, andeventually the ‘large part’ u converges to the originalimage. The figures show the convergence and the stabilityof the method. The last two plots show that k f � uðnÞkL2

decreases monotonically with the iteration, first dropping

Fig. 5. Denoising results of ‘Barbara (206:461, 81:336)’ image with s ¼ 20. Fro

(SNR ¼ 22:3633; iter ¼ 17), shift invariant W-IRM with iterative scale p

(SNR ¼ 22:4902; iter ¼ 19) and shift invariant W-ISS with iterative scale param

below s at the optimal iterate k ¼ 17 and 3, respectively.We should remark that shift invariant W-IRM withiterative scale parameter converges faster than shiftinvariant W-IRM.

Example 2. ‘Shape’ image is used adding Gauss whitenoise s ¼ 20. Fig. 4 shows the denoising results with shiftinvariant W-IRM, shift invariant W-IRM with iterativescale parameter, shift invariant W-ISS and shift invariantW-ISS with adaptive time step. We choose g ¼ 0:001, Dt ¼

0:001 for the general methods and e ¼ 0:01 manually. Weuse j f l � uðnÞl jL2

ps as the stopping criterion for all themethods. The optimal number of iteration we denote ‘iter’.We should remark that the adaptive scale parameter-based methods converge faster than the other twomethods correspondingly. We can see that small scales(Fig. 4) are more preserved with proposed iterative scaleparameter-based methods.

Example 3. Fig. 5 shows the denoising results of ‘Barbara’image with Gauss white noise s ¼ 20. The methods usedand the values of the parameter g and Dt for the general

m top to bottom, left to right, using shift invariant W-IRM (12) and (13)

arameter (21) (SNR ¼ 22:2448; iter ¼ 2), shift invariant W-ISS (15)

eter (23) (SNR ¼ 22:8451; iter ¼ 3).

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Fig. 6. Decomposite the texture image to the geometrical part u (the left column) and the textured part o ¼ g � u (right column) using shift invariant W-

IRM (the first row) and shift invariant W-IRM with iterative scale parameter (the second row).

B.-b. Hao et al. / Signal Processing: Image Communication 23 (2008) 433–441440

methods we choose are the same as those used inExample 2. For this example we choose e ¼ 1. j f l �uðnÞl jL2

ps is used as the stopping criterion for the methods.With proposed iterative scale parameter-based methods,textures are more preserved.

Example 4. In Fig. 6, it is demonstrated that shiftinvariant W-IRM can also perform well for image decom-position. Eqs. (19) and (20) evolve to separate the cleanimage into its geometrical part u and its texture parto ¼ g � u. We choose the scale parameter g ¼ 0:02 for thegeneral case. In Fig. 6, the first row shows the decom-position results using shift invariant W-IRM; the secondrow, shift invariant W-IRM with adaptive scale parameter.We can see that shift invariant W-IRM with iterative scaleparameter should decompose the image well with thegeometrical part containing fewer textured parts.

6. Conclusion

We have proposed shift invariant wavelet-basediterative regularization method and inverse scale spacemethod for image restoration which generalize the W-IRM

and W-ISS; further, a new iterative scale parameter withdifferent iteration steps is obtained. The application of theproposed scale parameter to shift invariant W-IRM leadsto global wavelet shrinkage with different thresholds fordifferent steps in the iteration. We provide a proof of theconvergence and obtain a stopping criterion for theproposed iterative procedure. We have seen with practicalexamples that the iterative scale parameter-based meth-ods could reduce the optimal number of the iterations,thus a fast and exact algorithm is obtained.

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