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IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 13, NO. 9, SEPTEMBER 2004 1213

Image Restoration Subject to aTotal Variation Constraint

Patrick L. Combettes, Senior Member, IEEE, and Jean-Christophe Pesquet, Senior Member, IEEE

Abstract—Total variation has proven to be a valuable concept inconnection with the recovery of images featuring piecewise smoothcomponents. So far, however, it has been used exclusively as an ob-jective to be minimized under constraints. In this paper, we pro-pose an alternative formulation in which total variation is used asa constraint in a general convex programming framework. Thisapproach places no limitation on the incorporation of additionalconstraints in the restoration process and the resulting optimiza-tion problem can be solved efficiently via block-iterative methods.Image denoising and deconvolution applications are demonstrated.

I. PROBLEM STATEMENT

THE CLASSICAL linear restoration problem is to find theoriginal form of an image in a real Hilbert space

from the observation of a degraded image

(1)

where is a bounded linear operator modeling theblurring process and models an additive noise com-ponent. Numerous approaches have been developed over thepast three decades to solve this problem (see, for instance, [2],[9], [16], [33], [35], [39], and the references therein). Roughlyspeaking, restoration problems are typically posed as optimiza-tion problems in which an appropriate objective function isminimized under certain constraints. Restricting ourselves toconvex problems, a general formulation is, therefore

Find such that (2)

where the objective is a convex functionand the constraint sets are closed convex subsets of

. These constraints arise from a priori knowledge about themodel (1) and the original image . For instance, the classicalformulation of [21] concerns problems with smooth images inwhich the energy of the noise is known. The goal is then tofind the smoothest image in terms of some high-pass filteringoperator which is consistent with (1) and thenoise information, hence and

Manuscript received June 18, 2003; revised December 3, 2003. The associateeditor coordinating the review of this manuscript and approving it for publica-tion was Dr. Giovanni Ramponi.

P. L. Combettes is with the Laboratoire Jacques-Louis Lions, UniversitéPierre et Marie Curie - Paris 6, 75005 Paris, France (e-mail: [email protected]).

J.-C. Pesquet is with the Institut Gaspard Monge and Centre National dela Recherche Scientifique (UMR–CNRS 8049), Université de Marne-la-Vallée,77454 Marne-la-Vallée, France (e-mail: [email protected]).

Digital Object Identifier 10.1109/TIP.2004.832922

in (2). In the Sobolev space ,where is an open subset of , the formulation of [21] reads

minimize subject to (3)

where denotes the Laplacian of . In ,smoothness can also be enforced with the objective

, where denotes the Eu-clidean norm in and the gradient of [20], [22], [34].

A. Total Variation Restoration

For images with sharp contours and block features, it was pro-posed in [32] (see also [26] and [31]) to formulate the restorationproblem as


If , the quantity

(5)

is the total variation of ; physically, is a measure of theamount of oscillations in . It should be noted that, under cer-tain assumptions [7], Lagrangian theory provides a conceptualequivalence between (4) and the unconstrained problem

minimize (6)

which is often implemented in practice (see also [30] for alterna-tive Lagrangian formulations). However, finding the exact La-grange parameter is a computationally intensive task. As aresult, the numerical value of in (6) is typically set in an adhoc manner, e.g., [23], [38], which can have a significant impacton the solution [1].

The formulation (4), and its variant (6), has proven particu-larly effective in denoising problems for piecewise smooth im-ages, and it has been the focus of a great deal of attention in recentyears in the image processing and applied mathematics commu-nities, e.g., [1], [5]–[8], [12], [14], [19], [23], [24], [28], [30], and[38]. At the same time, it has some inherent limitations.

• Staircase effect: The intensity levels of images producedby total variation minimization tend to cluster in patches.These artifacts have been observed by several authors, e.g.,[8], [39], and investigated for one-dimensional discretemodels in [25]. An explanation for the staircase effect isthat the algorithm produces a restored image whose total

1057-7149/04$20.00 © 2004 IEEE

1214 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 13, NO. 9, SEPTEMBER 2004

variation can be significantly below that of the originalimage .

• Knowledge of noise environment: The construction ofthe bound in (4) requires specific assumptions and in-formation about the noise [15], [36]. In some problems,such conditions may not be met.

In connection with the question of solving (4) exactly via nu-merical algorithms, it should also be noted that the nondiffer-entiable optimization algorithms currently in use are not easilyamenable to the incorporation of additional complex constraints[14].

B. Proposed Approach

The goal of this paper is to propose an alternative frameworkfor incorporating total variation in the restoration process. Ourstarting point is the observation that, in the spirit of feasibilitymethods [9], [36], [40], one would ideally like to obtain a re-stored image in the convex set of im-ages whose total variation is consistent with that of the originalimage . Of course, in practice, is usually unknown, buta reasonable upper bound may be available for certain classesof images based on statistics of databases. Thus, under suitableassumptions, if is a binary image, then is the length ofthe boundary of its support [18, example 1.4]; if is a gray-levelimage, then is obtained by integrating the length of thelevel curves with respect to [41, theorem2.7.1] (see also [41, theorem 5.4.4]). Hence, constitutes ageometrical attribute that can be expected to exhibit limited vari-ance over certain classes of images, e.g., views of similar urbanareas in satellite imaging, tomographic reconstructions of similarcross sections in tomography, fingerprint images, text images, orface images. For instance, on a database of 564 images showingdifferent (from 19 to 48) views of 20 different human faces [37],we have computed the average value of the total variation foreach person as well as the proportion of views whose total vari-ation lies between 0.8 and 1.2 (as will be seen in SectionIV-A.3, this typically corresponds to the kind of deviation whichis acceptable in our framework). Fig. 1 shows that the propor-tion of images between these bounds is around 99%.

The information restricts the solutions to theconvex set

(7)

Given a pertinent objective , this set can be incorporated in(2) in conjunction with additional constraints arisingfrom a priori knowledge. This framework departs from the tra-ditional formulation (4) in that total variation is now used asa constraint rather than an objective. Under mild assumptionson , the resulting optimization problem (2) is much more at-tractive since, as a constraint, total variation can be efficientlyprocessed via subgradient projections in the context of block-it-erative outer approximation methods [11]. Furthermore, this ap-proach places no restriction on the number of constraints oron their analytical complexity.

It should be noted that the idea of using an upper bound onthe total variation of images also appears in [27] as a stoppingcriterion in a PDE-driven restoration scheme.

Fig. 1. For 20 different data sets, the percentage of images whose totalvariation falls within 20% of the average total variation of the set.

C. Organization of the Paper

In Section II, we provide the necessary background material.In Section III, the proposed problem formulation is formallystated and a block-iterative parallel algorithm is described tosolve it. Two numerical applications are considered in SectionIV, namely deconvolution of satellite images and denoising inthe presence of non-Gaussian noise. The proofs of technical re-sults are placed in the Appendix.

II. BACKGROUND

A. Notation

The underlying image space is a real Hilbert space withscalar product and norm . The distance from an image

to a nonempty set is . Givena continuous convex function and , the closedand convex set

(8)

is the lower level set of at height . A vector is asubgradient of at if

(9)

As is continuous, it always possesses at least one subgradientat each point . If is (Gâteaux) differentiable at , then itpossesses a unique subgradient at this point, namely its gradient

. The set of all subgradients of at is the subdifferentialof at and is denoted by . For background on convexanalysis, see [17] and [29].

B. Subgradient Projections

The reader will find a more detailed account of subgradientprojections in [4], [9], and [10]; we provide only the essentialfacts here.

Let be a nonempty closed and convex subset of and letbe a point in . Then, there exists a unique point suchthat is called the projection of onto

. Now, suppose that , where iscontinuous and convex (the identity shows that

COMBETTES AND PESQUET: IMAGE RESTORATION SUBJECT TO A TOTAL VARIATION CONSTRAINT 1215

such a representation always exists). Fix , a subgradient, and define

ifif

(10)

Then, and the projection of onto , i.e.

if

if(11)

is called a subgradient projection of onto . We note that com-puting requires only the knowledge of a subgradient ofat and is, therefore, much more economical than computingthe exact projection , as the latter amounts to solving a con-strained quadratic minimization problem. However, whenis easy to compute, one can set and obtain .

C. Image Spaces and Total Variation

Complements on functional spaces and total variation will befound in [18] and [41].

Let be an open subset of , let , and letbe a generic point in . As usual, denotes the

space of infinitely differentiable functions with compact supportin , and is the space of functions whose th power isabsolutely integrable on . Given and ,the function is the weak partial derivative of

with respect to if; the (weak) gradient of is

. The Sobolev space of index isis

a Hilbert space. Likewise,.

It will, henceforth, be assumed that is a nonemptybounded open set. An analog image is modeled as an element in

. Since is bounded, and the totalvariation of an image is finite and given by (5).

On the other hand, in the discrete setting that prevails in nu-merical applications, analog images are discretized on a finite

sampling grid. The total variation of the discrete imageis obtained by discretizing (5)

into (see [14])

(12)

Now, set . As is customary, a discrete imagewill be dealt with as a vector

in the usual Euclidean space through the columnstacking isometry [2]. In turn, upon intro-ducing suitable difference matrices in

in , and in ,the total variation of can be expressed as

(13)

Proposition 1: Let be either or . Then,is a continuous convex function.

III. NUMERICAL METHOD

A. Assumptions

As formulated in Section I-B, the restoration problem is tosolve (2) in a Hilbert space , where is aconvex function, is given by (7), and are closedconvex sets in . We saw in Section II-C that is eitheror , according as we consider the theoretical analog modelor the discrete model. As in [10] and [13], it is convenient tomodel the constraint sets as level sets, say

(14)

where are continuous convex functions from to .In view of Proposition 1, the set can also be put in this formatby setting . The problem is, therefore, to


We shall use the general convex programming framework de-veloped in [11], which is particularly well adapted to problemsof the type specified by (2) and (14). In order to avoid tech-nical complications, we shall focus on the numerical results ofimmediate computational interest in (the reader canrefer to [11] for the infinite dimensional results pertinent to theconvergence analysis in ). We now state our assumptionsformally.

Assumption 2:

1) is defined by (7), by (14), wherethe functions are finite and convex, and

.2) is convex and lower semicontin-

uous.3) There exists such that

is bounded and is differentiable and strictlyconvex on .

Proposition 3: Suppose that and thatis strictly convex, differentiable, and coercive, that is,

. Then items 2)–3) in Assump-tion 2 are satisfied.

B. Solution Algorithm

Recall from (11) that, if is a subgradient of at ,the subgradient projection of onto associated with is

if

if(16)


Given and convex weights , define

if

otherwise(17)

We are now ready to describe the algorithm.

Algorithm 4Fix . Let be the minimizer of over

and set .Take a nonempty index set .Set , where

a) for every is as in (16);b) lies in and ;c) is as in (17).

Set and.

Let be the minimizer of over .Set and go to .

Theorem 5: Suppose that Assumption 2 is satisfied and thatthere exists a strictly positive integer such that, for every

and every . Then, everysequence generated by Algorithm 4 converges to the unique so-lution to (2) or, equivalently, to (15).

The key step in the algorithm is Step , in which the up-date is obtained as the minimizer of over the intersec-tion of the two half spaces and , which contains . Spe-cific implementations of this basic minimization problem undertwo affine inequality constraints will be discussed in SectionsIV-A.2 and IV-B.2. Let us note that if one bypasses Step andreplaces Step by , one recovers an algorithm pro-posed in [10]. However, in this case, converges to anunspecified image in the feasibility set [10, theorem 3].

C. Practical Considerations

A few remarks are in order.

1) If the total variation set of (7) is selected at iteration ,the subgradient projection of is required. In viewof (16) and (13), we have

if

if(18)

Fig. 2. Original image. Pixel range [0, 255].

where the subgradient can be taken to be(see Section IV-A in [14])

(19)

with (20), as shown at the bottom of the page.2) The condition on imposes merely that every

index be used at least once within any consecutive it-erations. As a result, Algorithm 4 is very flexible in termsof parallel implementation. For instance, if parallelprocessors are available, one can choose to use all thesets simultaneously at iteration , i.e., ;if only one processor is available, one can sweep throughthe sets periodically as in the POCS algorithm [40], i.e.,

modulo . Intermediate block-iterativesweeping rules are described in [10].

3) If the projection of onto a selected set is easyto compute, one can take , which amountsto setting .

IV. SIMULATIONS RESULTS

A. Application to Satellite Imaging

1) Experiments: The original image shown in Fig. 2 isa SPOT-5 256 256 satellite image, hence , where

. The degraded image shown in Fig. 3 is obtained

ifotherwiseifotherwiseifotherwise

(20)


Fig. 3. Degraded image. Pixel range [4, 263].

Fig. 4. Standard total variation deconvolution by (4). Pixel range [0, 267].

by convolving with a 7 7 uniform blur and adding Gaussianwhite noise. The blurred-image to noise ratio is 30 dB. Thedegradation model is, therefore, (1), where the convolution ma-trix is assumed to be known.

First, we assume that the above mentioned characteristics ofthe noise are known. The bound of (4) can, therefore, be de-rived from these informations [15], [36]. The standard total vari-ation problem (4) is solved using the adaptive level set methodof [14] and its solution is shown in Fig. 4. The total variation ofthe restored image is , which is much below the truevalue .

We now turn to an alternative scenario, in which no informa-tion is available about the noise but a bound is available on thetotal variation of the original image (e.g., as estimated froma pool of similar satellite images), which makes it possible toset up the problem in the proposed format (2). The objective ischosen to be , where .

Fig. 5. Quadratic deconvolution with bounded total variation. Pixel range[�55, 316].

Fig. 6. Quadratic deconvolution with bounded total variation and additionalconstraints. Pixel range [0, 259].

Note that since is not invertible, the residual energy functionis not strictly convex. We, therefore, introduce

the term to make strictly convex, in compliance withitem 3) in Assumption 2. Assuming no further information onleads to the minimization of over . The solution producedby Algorithm 4 in this case has total variation and isshown in Fig. 5. Next, we assume that additional information isavailable about , namely amplitude bounds and mean. Hence,we use the set of (7), as well as the sets and

in (2), where is the vector whoseentries are all equal to 1. The minimizer of over , ob-tained by Algorithm 4, has total variation and isshown in Fig. 6. It is apparent that the incorporation of additionalinformation has improved the restoration. Finally, to provide ev-idence of the value of total variation information, we display inFig. 7 the image obtained by minimizing over , i.e.,


Fig. 7. Quadratic deconvolution with no total variation constraint. Pixel range[0, 257].

Fig. 8. tv(x )=tv(�x) versus the iteration index n.

without the total variation constraint. This image has total vari-ation .

2) Implementation of Algorithm 4 and Convergence Pat-terns: Exact projections onto and are used since theyare easily computed, whereas a subgradient projection ontois employed. Now, let . Then, the initial pointis seen to be . Moreover, since the objectiveis quadratic, the expression of at Step of Algorithm 4can be obtained in closed form [13]. Thus, the th iteration canbe executed as described in the following pseudocode, wherethe subscript has been dropped (see Section IV-C in [13] fordetails).

1) For every , set , where, if otherwise.

2) Choose convex weights conforming to Step b).Set and .

3) If , exit iteration. Otherwise, set, and .

Fig. 9. kx � �xk=k�xk versus the iteration index n.

Fig. 10. Normalized error kx � x k=kx k as a function of �=�� .

4) Set .5) Set , and

.6) Set

if andif and

if and

To illustrate the convergence behavior of the method in thecase of the last experiment corresponding to Fig. 6, we displaythe evolution of the total variation and of the root mean-squareerror in Figs. 8 and 9. Both indicate that convergence occursroughly around the 50th iteration.

3) Sensitivity Assessment: In practice, the value of maynot be known precisely and it is, therefore, important to eval-uate its impact on the solution. Let us denote by the solutionproduced by the algorithm with the constraint setsfor a given bound in (7) and by the true bound.The normalized root mean square error as afunction of is displayed in Fig. 10. This curve shows thatthe method is robust in the sense that, as varies from 0.82to 1.21, the resulting error does not exceed 5%.


Fig. 11. Original image. Pixel range [0, 255].

Fig. 12. Noisy image. Pixel range [�261, 460].

B. Denoising Application

1) Experiments: Adding i.i.d. Laplacian noise to the orig-inal 128 128 gray-level image shown in Fig. 11 results inthe image shown in Fig. 12. The image-to-noise ratio is 1 dB.Here, , where .

We first assume knowledge of the noise statistics and noknowledge of the total variation bound. This corresponds to thestandard total variation problem (4), whose solution, obtainedby the method of [14], is shown in Fig. 13.

A second scenario consists of assuming no probabilisticknowledge of the noise and applying Algorithm 4 with the sameconstraints sets as in Section IV-A.1. A common objective indenoisingproblemsis the thpowerof the normof theresidual,namely

(21)

Fig. 13. Standard total variation denoising by (4). Pixel range [�113, 307].

Fig. 14. ` denoising with bounded total variation. Pixel range [�156, 277].

The impulsive nature of the noise naturally suggests choosinga value of close to 1. We set , which secures thatAssumption 2 is satisfied by Proposition 3. The solution to (2),obtained by Algorithm 4 with the total variation constraint set

alone is shown in Fig. 14. The restoration displayed in Fig. 15is obtained by using the sets in (2). This experimentillustrates the benefits of added information. The important roleplayed by the total variation constraint set can be seen in Fig.16, which shows the image obtained by minimizing over

. The total variation of the images is in Fig. 11,in Fig. 13, in Fig. 14, in

Fig. 15, and in Fig. 16.2) Implementation of Algorithm 4 and Convergence Patterns:

Here, we have . The main difference with theexperiment of Section IV-A.1 is that an explicit solution atStep of Algorithm 4 is not available. We can, however,employ a Lagrangian formulation to reduce the problem to theminimization of a convex function of only two nonnegative


Fig. 15. ` denoising with bounded total variation and additional constraints.Pixel range [�1, 255].

Fig. 16. ` denoising with no total variation constraint. Pixel range [�4, 255].

real variables. Indeed, standard convex programming results[29, section 28] show that the solution and the associatedKarush–Kuhn–Tucker vector aresolutions to the saddle-point problem

(22)

Since the gradient of the objective is bijective,we can express the solution to the above minimization problemas a function of , namely

(23)

We then deduce as the solution to the following simple convexminimization problem in

Fig. 17. tv(x )=tv(�x) versus the iteration index n.

Fig. 18. kx � �xk=k�xk versus the iteration index n.

which can be solved by standard numerical methods. In turn, weobtain .

The asymptotic behavior of the algorithm in the experimentcorresponding to Fig. 15 is illustrated in Figs. 17 and 18, wherewe show the evolution of the total variation and of the root mean-square error, respectively. Convergence is achieved in roughly350 iterations.

APPENDIX

APPENDIX–PROOFS

Proof of Proposition 1: First, suppose that .Then, tv is defined by (5) and finite on (see Section II-C).Now, take and in and . Then, by linearity of

. Hence,, which proves

the convexity of tv. Next, since tv is finite and convex, it sufficesto show that it is sequentially lower semicontinuous to establishits continuity [17, Cor. I.2.5]. To this end, let be a se-

quence in such that and set . Then,


we must show that . Let us extract a subsequencesuch that . Then

(24)

Hence, . Therefore, it follows from [3, theo-rems 2.5.1 and 2.5.3] that we can extract a further subsequence

such that . It now follows fromFatou’s lemma [3, Lem. 1.6.8] that

(25)

which yields the desired inequality. Finally, when , tvis defined by (13) and, by [17, Cor. I.2.3], it is enough to show itsconvexity. This follows at once from the convexity of the normsand the linearity of the operators involved in (13).

Proof of Proposition 3: Since is convex and finite, it iscontinuous [17, Cor. I.2.3] and 2) holds. Furthermore, the coer-civity of implies that all its lower level sets are bounded, sothat 3) also holds.

Proof of Theorem 5: The result is an application of [11,theorem 6.4(i)]. First, it follows from [11, (6.10)] and Stepc) that at Step of Algorithm 4 can be written as

(26)

Moreover, by construction, lies in [11, Prop. 3.1(ii)].Hence, in view of [11, (5.4)] and item 3) in Assumption 2, Algo-rithm 4 is a special case of [11, Algorithm 6.4]. Next, it followsfrom Assumption 2, [11, Prop. 2.1(i)] and [11, Prop. 2.2(ii)],that assumptions [11, (A1)–(A3)] are satisfied with .We conclude by observing that, by [11, Prop. 4.7(ii)] and [29,theorem 24.7], all the conditions of [11, theorem 6.4(i)] are ful-filled.

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Patrick Louis Combettes (S’84–M’90–SM’96) is aProfessor with the Laboratoire Jacques-Louis Lions,Université Pierre et Marie Curie - Paris 6, Paris,France.

Dr. Combettes is the recipient of the IEEE SignalProcessing Society Paper Award.

Jean-Christophe Pesquet (S’89–M’91–SM’99) re-ceived the engineering degree from Supélec, Gif-sur-Yvette, France, in 1987, the Ph.D. degree from theUniversité Paris-Sud (XI), Paris, France, in 1990, andthe Habilitation à Diriger des Recherches from theUniversité Paris-Sud in 1999.

From 1991 to 1999, he was a Maître de Con-férences with the Université Paris-Sud and aResearch Scientist at the Laboratoire des Signauxet Systèmes, Centre National de la Recherche Sci-entifique (CNRS), Gif-sur-Yvette. He is currently a

Professor with Université de Marne-la-Vallée, France, and a Research Scientistat the Laboratoire d’Informatique of the university (UMR-CNRS 8049), Paris.

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