Adaptive Gaussian Model for Satellite Image Debluring

IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 13, NO. 4, APRIL 2004 613

An Adaptive Gaussian Model forSatellite Image Deblurring

André Jalobeanu, Laure Blanc-Féraud, and Josiane Zerubia, Fellow, IEEE

Abstract—The deconvolution of blurred and noisy satelliteimages is an ill-posed inverse problem, which can be regularizedwithin a Bayesian context by using an a priori model of thereconstructed solution. Since real satellite data show spatiallyvariant characteristics, we propose here to use an inhomogeneousmodel. We use the maximum likelihood estimator (MLE) toestimate its parameters and we show that the MLE computedon the corrupted image is not suitable for image deconvolutionbecause it is not robust to noise. We then show that the estimationis correct only if it is made from the original image. Since thisimage is unknown, we need to compute an approximation ofsufficiently good quality to provide useful estimation results.

Such an approximation is provided by a wavelet-based deconvo-lution algorithm. Thus, a hybrid method is first used to estimate thespace-variant parameters from this image and then to compute theregularized solution. The obtained results on high resolution satel-lite images simultaneously exhibit sharp edges, correctly restoredtextures, and a high SNR in homogeneous areas, since the proposedtechnique adapts to the local characteristics of the data.

Index Terms—Deconvolution, estimation techniques, inhomoge-neous Gaussian models, maximum likelihood, satellite images.

I. INTRODUCTION

A. Problem Statement

THE GENERAL problem we deal with is the reconstructionof a satellite image from blurred and noisy data.

The degradation model is represented by the following equa-tion:

(1)

where is the observed data, is the original image, is anadditive noise and is assumed to be Gaussian, white and sta-tionary, of known variance , and is the convolution oper-ator. The point spread function (PSF) is assumed to be known(see Fig. 2).

We deal with a real satellite image deblurring problem, pro-posed by the French Space Agency (CNES). This problem ispart of a simulation for the SPOT 5 satellite. Both the originaland degraded images are provided by CNES. The noise standarddeviation and the PSF are also provided . In thiscase, is symmetric and separable w.r.t. lines and columns, but

Manuscript received November 28, 2000; revised March 17, 2003. This workhas been conducted within GdR ISIS (CNRS). The associate editor coordinatingthe review of this manuscript and approving it for publication was Prof. TimothyJ. Schulz.

The authors are with the Ariana-Joint Research Group CNRS/INRIA/UNSA,Sophia Antipolis, France (e-mail: [email protected]; [email protected];[email protected]).

Digital Object Identifier 10.1109/TIP.2003.819969

the formalism presented herein can be extended to a more gen-eral case for which these properties are not satisfied.

The deconvolution problem is ill-posed. Knowing the degra-dation model is not sufficient to obtain satisfactory results. It isnecessary to regularize the solution by introducing a priori con-straints [7]. The regularization constraint is a roughness penaltyon the solution. The regularized solution is then computed byminimizing the energy

(2)

where corresponds to the maximum a posteriori (MAP) esti-mate. The regularization constraint can be expressed as the priordistribution of the unknown image, within a Markov randomfield (MRF) framework [3], [7]

(3)

The data term corresponds to the likelihoodof , according to the statistics of the stationary Gaussian noise

(4)

The posterior distribution is given by

(5)

B. The Prior Model

The Markovian prior model of the unknown image is definedby the energy expressed in the Gibbsian probability of (3).This function could be quadratic (as suggested by Tikhonov in[20]) assuming that images are globally smooth, but it yieldsoversmooth solutions. A more efficient image model assumesthat only homogeneous regions are smooth, and that edges mustremain sharp. To get this edge-preserving regularization, it ispossible to use a nonquadratic -function, as introduced in [4]and [6]. This function is positive, symmetric, and increasing,and is applied to the nearest neighbor pixel differences. Prop-erties of -functions have been studied in order to preserve theedges, avoiding noise amplification [1]. This approach involvesa global parameter , which is automatically estimated [11].

In fact, if the parameter is estimated from small areas extractedfrom a large-size image, textured areas and homogeneous areasprovide different estimated values of [10]. Therefore, theseareas would be better reconstructed if processed separately.Real images cannot be efficiently represented by homogeneousmodels, since they are made of various textures, sharp edgesand constant areas. So the prior parameter should adapt

1057-7149/04$20.00 © 2004 IEEE

614 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 13, NO. 4, APRIL 2004

to the local structure of the image in order to provide abetter reconstruction (i.e., a less noisy result in homogeneousregions, and sharper details in other areas). However, an adaptiveregularizing model based on nonquadratic -functions seemsto be too computationally demanding, both for estimation andreconstruction. That is why we have chosen to use an adaptiveGaussian model instead, with a quadratic energy which allowsus to adjust the amount of regularization locally.

The energy is defined as follows:

(6)

where and are the adaptive parameters, with regard to(w.r.t.) columns and lines, and are first order deriva-tive operators (i.e., differences between neighbor pixels w.r.t.columns and rows). The variables are analogous to a contin-uous line process [7]: a low value of corresponds to an edgelocated between two pixels.

We have chosen to model the unknown image by an inhomo-geneous Gaussian MRF (IGMRF) [2], [19], since real imagescannot be efficiently described by homogeneous models, evenby using nonlinearity through -functions. The prior parametershave to adapt to the local structure of the image to enable the so-lution to be less noisy in constant areas and to exhibit sharperdetails in other regions.

C. Main Contributions

• Here, we address the problem of space-varying param-eter estimation for a Gaussian model by using a hybridapproach. We apply the maximum likelihood estimator(MLE) on complete data (i.e., the original image).The complete data are approximated by the result of awavelet-based deconvolution algorithm called complexwavelet packet thresholding (COWPATH). This way, weobtain a robust estimate of the prior model parameters.This method is hybrid since the estimation is not directlydone on the observed image, but on an intermediateimage.

• We propose an algorithm able to automatically restore ablurred and noisy image, since all the parameters are esti-mated by the MLE.

• The method described in this paper has been successfullyapplied to real satellite images and has demonstrated itssuperiority over nonadaptive techniques.

• The algorithm presented here is computationally efficientand is based on fast Fourier transforms.

D. Paper Organization

In Section II, we explain why we have chosen the completedata MLE instead of other estimators. Then, in Section III, wedetail how to estimate the parameters from the original image.Since this image is not known, we only have access to an approx-imation of it. In Section III-B, we study the robustness of theestimator to approximations, and then we evaluate the residualnoise. In Section IV, we show why it is impossible to use theMLE on observed data to estimate the parameters. Finally, inSection V, we detail the complete deconvolution algorithm and

give the computational cost. Section VI presents the results ofSPOT 5 satellite image deblurring and a comparison with ex-isting techniques.

II. CHOICE OF AN ESTIMATOR

The problem at hand is how to determine the optimal param-eters to obtain the best reconstructed image. The chosen esti-mator has to be adapted to the local statistical characteristics ofthe data while remaining robust to noise. In this section we givethe details of some classical estimators used for both homoge-neous and adaptive models. Then we explain why we prefer touse the complete data MLE. In the next section we discuss thedifficulty of implementing the MLE in the inhomogeneous case.

Essentially three kinds of statistical estimators have been usedfor parameter estimation problems. Let represent the parame-ters, then we have the following.

1) : MLE on , the observed data [21]. This es-timator has been successfully applied to the case ofhomogeneous models involving -functions, by using aMCMC method [11]. It needs sampling from both priorand posterior densities.

2) : MLE for the joint distribution of and[14]. Only an approximate optimization method can beused in practice, consisting of alternate maximizationsw.r.t. and . Optimizing w.r.t. is equivalent to theMAP criterion. Optimizing w.r.t. consists of computingthe MLE on the complete data , where is theimage restored with the current value of . It is a subop-timal method and the convergence is not guaranteed. Theadvantage of this method is that only prior sampling is re-quired, which substantially reduces the computation timein the case of deblurring.

3) : MLE computed on the original image. Theproblem is that the image is unknown. It can not beapproximated by its degraded version because isblurred and noisy, and so exhibits different properties.Parameters estimated on a blurred image have too high avalue and therefore lead to oversmooth solutions. Param-eters estimated on a noisy image are too low, and provideinsufficient regularization, leading to noisy solutions.This estimator is only significant in the case of completedata. Then it supplies good parameter estimates forimage segmentation [5]. As with the previous method, itonly requires sampling from the prior distribution.

In the case of a global parameter , the first estimator givesthe best results. Both prior and posterior sampling are achievedin the frequency space by diagonalizing the covariance matrix,using a half-quadratic expansion of [11].

The two other estimators are easier to implement, as thereis no need for posterior sampling. It is possible to use a tableof precomputed values of prior expectations (and to interpolatethem to get an approximation for any parameter value) to speedup the algorithm. The problem raised by the second method isthat the estimation does not really take into account the dataimage, but only a restored version of it whose shape stronglydepends on the current values of the parameters. Thus, a badinitialization often leads to degenerated solutions.

JALOBEANU et al.: AN ADAPTIVE GAUSSIAN MODEL FOR SATELLITE IMAGE DEBLURRING 615

The last estimator is the fastest one, but requires animage close to the original image , which is unknown. There-fore, an approximation of has to be accurately determined(for instance by a nonparametric reconstruction algorithm) ifwe want the parameters obtained in this way to be significantfor regularization. That is why we choose to use the COW-PATH algorithm, proposed in [12], to compute the neededapproximation of . This algorithm has been chosen becausethe deconvolved result exhibits sharp textures and noise-freehomogeneous areas, and is sufficiently close to the originalimage to enable us to estimate the adaptive parameters from it.

In the following section, we detail how to estimate the inho-mogeneous parameters from complete data. This is a very effi-cient method, preferable to other ones because of its simplicityand accuracy.

III. ESTIMATION FROM THE ORIGINAL IMAGE

A. MLE on Complete Data

The MLE on complete data w.r.t. the image is

(7)

and the log-likelihood derivatives are

(8)

Therefore, the estimation problem consists of solving the system

(9)

The complete data formulation simplifies the estimationproblem, since the expectation term only depends on theparameters and the other term only depends on . To computeone step of a descent method to perform the estimation, it issufficient to compute the variance of each pixel difference w.r.t.the prior law (the mean is null). This can beachieved by sampling from this law with an optimized Gibbssampler [10] or by finding an approximate analytic expressionof this variance.

We propose to use the simplest approximation of thelocal variance, which provides good deconvolution results asshown in Fig. 6. We suppose that the variance of the gradient

is equal to the variance of the same gradient inthe homogeneous case, i.e., when all the parameters are equalto the corresponding . Since the covariance matrix of thehomogeneous prior distribution is diagonalized by a Fouriertransform, this variance can be calculated and it is equal to

[10]. This gives

(10)

Experimental studies have shown that this expression gives thesame estimation results as the inhomogeneous MLE, computedusing a Gibbs sampler and a Newton-Raphson gradient-descentalgorithm.

B. Robustness to Approximations

Since the true image is not provided, we compute the MLEdefined by (10) from an approximation we call . We have tocheck the sensitivity of the MLE in the complete data case tothe variations of the pixel differences of .

Let be the gradient error, i.e., the gradient of the residualnoise (difference between the approximated and the trueimages). It induces an error on the estimated parameters.By studying (10), we find that the relative errors are linked by

[10].The relative fluctuations can become very high for

small values of , because they are of the same order as .It corresponds to constant areas. If some noise is present on ,it induces an underestimation of in these areas, and finallyan insufficient regularization. This means that has to bevery smooth in these regions. Elsewhere is small, andtherefore the estimation is accurate.

To get a robust estimate, it is necessary to evaluate the mag-nitude of the error , which depends on the method chosento approximate . The algorithm COWPATH, used to compute

, induces a bounded error . Then we set all the gra-dients less than to zero to ensure a maximum regularization inconstant areas. We also set an upper bound forto avoid computational difficulties in the final deconvolution.Thus, we redefine the optimal weighting (10) to get a more ro-bust estimator in these regions

(11)

where is a hard thresholding function, i.e., it is equal to 0if and to elsewhere.

C. Evaluation of the Residual Noise

We have seen that the estimator seems to be quite sensitiveto the noise in constant areas. Therefore we have to evaluatethe residual noise, i.e., the noise remaining after the waveletshrinkage step, to determine the threshold of (11). A commonway to evaluate the residual noise of an algorithm is as follows:simulate some noise, apply the deblurring algorithm on it andthen study the result. This can be detailed as:

• simulate an image of a Gaussian white noise ;• apply the algorithm COWPATH [12] to deconvolve this

image, the result is denoted and corresponds to theresidual noise;

• compute the gradient image or , whosepixels are denoted ;

• compute the histogram of this gradient image.By studying the distribution of the gradients of the

residual noise, we verify that they have bounded values, i.e.,. Thus, the thresholding defined by (11) provides a

simple and efficient way to get a more robust MLE.But it is not the only method and many other ones could be

used. For example, we could compute the MLE in the incom-plete data case (the data are a noisy, but not blurred, version ofthe ground truth) to take into account the statistics of the residualnoise on the gradients, instead of filtering the gradients and thenapplying the MLE on the complete data.


IV. WHY NOT USE MLE ON THE OBSERVED DATA?

Usually, when only degraded observations are known, theparameters should be obtained by maximizing the likelihood onthe observed data as mentioned in Section II. In the followingsections, we show why this method is not appropriate in theparticular case of adaptive parameter estimation.

In Section IV-A we show that this estimation technique isintractable when applied to the observed data. Then, in Sec-tion IV-B, we discuss the significance of this estimator, ex-plaining why we do not use classical robustness criteria to assessits sensitivity to the noise. Finally, in Section IV-C, we showhow to compute the variance of this estimator when the originalimage is known. We choose a counter-example to illustrate theextreme sensitivity exhibited by this classical estimator. Sincethe estimation dramatically fails for the chosen particular case,it is likely that it may fail for other images, such as satelliteor aerial images for instance. This justifies our approach, con-sisting of using the MLE on complete data.

A. Complexity of the MLE on the Observed Data

To compute the MLE on the observed data , we have tomaximize the likelihood w.r.t. the parameters, which is propor-tional to , where and are the partition functionsrelated to the posterior and prior distributions [10], [11]. Thesedistributions are given by (3) and (5).

We have to solve the following equation w.r.t. each

(12)

where the expectations are taken w.r.t. the prior and posteriordensities. Sampling from these densities can only be achievedby using classical samplers such as Metropolis dynamics [15]or Gibbs sampler [7] because the model is inhomogeneous. Thelocal conditional distribution for each site can be computed,taking advantage of the Gaussian form of the probability dis-tribution. The posterior Gaussian MRF takes into account thedata, through the blur operator , inducing a high order neigh-borhood. Thus, sampling really becomes intractable in this case.

The expectations of (12) are second order moments ofGaussian variables. We have shown [10] that to obtain abounded expectation accuracy (necessary toensure the convergence of the descent algorithm used forestimation), we need to generate more than 20 000 samplesfor each step. For large size images, the estimation becomesmore than 100 000 times longer compared to the homogeneousequivalent model (because, for homogeneous models, thesquared pixel differences are averaged on the entire image).

B. Significance of the Local MLE

The MLE is often used in parameter estimation problemsbecause it is asymptotically optimal, since it is unbiased andefficient for large data records [13], [16]. Its probability den-sity function is Gaussian, centered on the true optimal parametervalue. Observations are noisy and therefore estimations madeover are also corrupted by noise. To check the robustness

of the MLE, its variance has to be estimated. The Cramer–Raobound (CRB) gives a lower bound of this variance

(13)

where is one of the parameters and denotes the expectationtaken w.r.t. the law . Again estimating this expectationis intractable.

In fact, we did not consider the CRB because this bound onlydepends on the parameters of the prior and posterior laws ( ,and ). It obviously does not take into account the real image .So it gives the maximum accuracy of the MLE considering that

is modeled by its prior distribution. However, because of itscomplexity, the satellite image is hardly perfectly describedby this distribution (i.e., it is impossible to generate a priorsample perfectly resembling a satellite image). So we prefer toconsider another robustness criterion, the MLE variance.

C. Variance of the MLE

First, let us define a general framework to compute the MLEvariance. We assume here we know . The observation is arealization of the random process , where .Then, the MLE is also a random variable

The variance gives the estimation error. We assume thatthe log-likelihood is locally quadratic w.r.t. near , andw.r.t. each component of near . The estimator and arelinked through the log-likelihood derivative . To expressthe variations of as a function of the variations of , we takethe Taylor expansion of around and . Then, to expressthe term . we take its Taylor expansion in (weassume that the mean value is close to , the parameterestimated from ). We can simplify the problem by remarkingthat (definition of the MLE). If wedefine and by

(14)

(15)

where these expressions are evaluated with and ,then the variance can be expressed as

(16)

If we denote , we have

(17)

We could use a stochastic method to estimate and but, asmentioned before, it is very time consuming. Therefore, we takethe particular case of a carefully designed image , which gives


mean estimated parameters that are equal to a constant value. Then, computations are simplified. The evaluation of (16)

finally shows that the fluctuation is high: it always remainsof the same order as (for comparison we havein the homogeneous case). How do we interpret this result? Itis not surprising that parameters adapt to the noise. Indeed, theestimation takes into account the observed data.

The parameter estimation problem (MLE on ) consists ofsolving the system (12). We have taken the same approximationas in Section III-A to compute the second order moments. Oneequation of the system (12) is , which corre-sponds to a particular parameter . , and are the vari-ances and the mean value of the pixel difference related to . Wehave plotted versus in Fig. 5, as functions of . Theintersection gives the MLE . Indeed, if there is some noise in

, the mean value is increased or decreased by . Thetwo shifted curves represent and . The limit of

for is finite. But for , we havebecause of the term . So is lower than for

, and it is a decreasing function.If is increased by the noise, the intersection of the curves

is shifted to the left, so the value is decreased. It means thatthe MLE provides less regularization in this case. So the dif-ference induced by the noisy observation cannot be sup-pressed by the regularizing function, and all the local variationsof preserve the noise instead of filtering it. This illustrates theundesirable properties of the MLE on observed data for inho-mogeneous models.

V. DECONVOLUTION ALGORITHM

A. Introduction

We choose to use the inhomogeneous Gaussian model de-fined by (6). The convexity enables us to compute the regular-ized solution using a deterministic optimization algorithm,by minimizing the following functional:

(18)

The proposed deconvolution algorithm can be summarized asfollows. First, we use an existing thresholding method in awavelet basis to estimate the original image from its observa-tion . The image obtained this way is not sufficiently sharp tosolve the deconvolution problem. However, we use this imageto determine the space-varying parameters of the model anduse them to get the final result, which is the major contributionof this paper.

The 3 main steps of the new algorithm are:

1) automatically threshold the image , deconvolvedwithout regularization, in a complex wavelet packet basis(algorithm COWPATH defined in [12]);

2) estimate the inhomogeneous parameters of the model [cf.(6)] on the result of the previous step, by a complete dataMLE [cf. (10)];

3) solve (18) by an accelerated descent algorithm to computethe final solution, .

B. Taking Into Account the Directions

A significant improvement of the method presented in theprevious sections consists of considering the orientation of thegradients. The model used for parameter estimation only takesinto account the horizontal and vertical directions. By also usingthe diagonal pixel differences, it is possible to introduce the di-agonal directions and without increasing the computationalcomplexity of the estimation step too much. This introducestwo new parameter fields and , related to pixel differencesalong these directions.

The nearest neighbor model is then replaced by aeight-neighbor model, in the restoration step only. Theestimation is done as seen previously, using a 4–neighbor MRFmodel.

In the restoration step, for each pixel, we use the four gradi-ents , , , and related to the vertical, horizontal, anddiagonal directions, respectively

(19)

to estimate the four associated parameters , , , andusing (10).

But they cannot be used directly in the new eight-neighbormodel by simply minimizing the corresponding functional withthe parameters estimated this way, because the estimator (10) isrelated to a four-neighbor MRF.

To combine all these parameters, we use a technique inspiredby the discretization of the divergence term pre-sented in [17]. This term occurs within a variational context inthe minimization of this functional

(20)

where , , and are continuous functions defined over a set. Indeed, the associated Euler–Lagrange equation is

(21)

The model presented here can be seen as a discretization ofthis functional. Taking eight instead of four neighbors finallyconsists of using a more accurate discretization.

The technique consists of using the direction of the gradientof the approximate original image at pixel

(22)

(To obtain more robust results, it is preferable to compute thegradients in this expression by smoothing along the orthogonaldirection.)


The value of determines how to distribute the parametersover each one of the four directions, in the termwhich is used in the deconvolution algorithm

(23)

where and.

For example, if , we only take into account the di-agonal parameters and . If there is a diagonally orientedtexture, this approach enables us to smooth only along the direc-tion of the texture, while the standard approach produces bothsmoothing along horizontal and vertical directions, which gen-erally can damage the texture.

C. Choosing the Best Initialization

Since the deconvolution method is iterative, the choice of theinitial image determines the speed of convergence. A bad ini-tialization often slows down the algorithm, even if the solutiondoes not depend on the followed path. Thus, we initialize usingan image close to the solution. This image is simply the approx-imate original image used for parameter estimation. Then, in thecase of the SPOT 5 simulation, 30–40 iterations are sufficient.

To understand why this initialization seems to be the bestone, let us recall that minimizing the functional correspondingto the adaptive regularization model is equivalent, in homoge-neous areas, to solving the heat equation. It is equivalent to a dif-fusion process, which smoothes these areas by averaging manypixels iterately. Starting with a noisy image is clearly not a goodinitialization, because many iterations are needed to achieve thediffusion process over constant areas (which can be very large)and thus to obtain smooth areas.

If we use the approximate original image obtained by awavelet thresholding algorithm, these areas are smooth, sothere is no need for additional smoothing. The only parts toprocess are the edges, which are not sufficiently sharp becauseof the insufficient spatial localization of the wavelets. Thus,the aim of the adaptive deconvolution step is to sharpen theedges, leaving other parts unchanged. Finally, we concludethat starting with the approximation image is much faster thanstarting with the observed image .

D. Optimization Method

We can use a conjugate gradient algorithm to optimize (18).This method can be improved by making some assumptions.

We have just seen that the aim of the adaptive deconvolutionalgorithm is to sharpen the edges. Edges correspond to highgradient values (i.e., low regularizing parameter values). It ispossible to accelerate the recovery of the edges by lowering

the corresponding regularizing parameters at the beginning ofthe algorithm and by restoring them progressively to theirestimated value. This ensures that we obtain the right solutionrelated to the estimated parameter values. This technique canspeed up the gradient descent step by a factor of two. This isimportant given that the limiting factor of the hybrid algorithm(wavelet thresholding followed by adaptive estimation anddeconvolution) is the adaptive estimation step. Indeed, this steprequires the computation of four discrete cosine transforms(DCT) per iteration and needs about 30 iterations.

E. Proposed Algorithm

We propose an hybrid algorithm able to automatically restorea blurred and noisy image. The first step consists of obtainingan approximation of the unknown image by thresholding theobservation deconvolved without regularization in a complexwavelet packet basis. The quality of this image is not sufficientas a solution of the deconvolution problem because it does notexhibit sharp edges. The second step is to estimate the adaptiveparameters from this approximate original image. We detail thispoint in the paper by justifying the choice of the estimator andstudying its robustness. Finally, the third step is the deconvolu-tion by the MAP computation minimizing (2), which is achievedby an accelerated descent method.

The proposed deconvolution method, called deconvolutionwith estimation of adaptive parameters (DEPA), consists of thefollowing steps (see Fig. 4).

1) Deconvolution of with algorithm COWPATH [12].2) Computation of the gradients.3) Estimation of the residual noise of the gradients:

a) simulation of a white Gaussian noise of variance ;b) deconvolution of the noise with COWPATH, with

the same parameters as for the image ;c) computation of the residual gradients;d) estimation of the variance of the residual gradi-

ents.

4) Thresholding of the gradients, using .5) Estimation of the IGMRF parameters using the MLE [cf.

(10)]; these two steps are combined in (11).6) Optional: computation of the diagonal parameters (see

Section V-B).7) Deconvolution by MAP estimation minimizing (18) (ini-

tialization using the result of COWPATH and minimiza-tion of (18) by a conjugate gradient algorithm).

F. Computational Cost

The cost of the proposed method is, where is the number of iterations in the adap-

tive deconvolution step, is the size of the kernel , andis the size of the image. In the case of the image provided byCNES, with and , we have , whichgives a total number of about 2780 op/pix. If we take into ac-count the directions, the total number is

(about 3080 op/pix for the chosen image).


Fig. 1. Original image X (128 � 64 area extracted from Nimes).

Fig. 2. Point spread function (PSF) h.

VI. SATELLITE IMAGE DEBLURRING

A. Evaluation of the Restoration Quality

The restoration quality can be assessed by visually comparingthe restored image with the original image (which is providedhere as a reference only, and is never used in the restoration pro-cedure). The same approach can be used to evaluate the effec-tiveness of the proposed algorithm w.r.t. existing deconvolutiontechniques.

We show the SNR values for all images. However, this is nota good quality criterion: it consists of averaging signal and noisevariances over the whole image. As human vision is more sensi-tive to the noise in homogeneous areas than it is along the edges[9], we could define a quality criterion which penalizes the noisemore in constant areas (where noise peaks clearly appear) [10].But such a criterion, defined as ad hoc, is not sufficiently usedin the image processing community, so we prefer not to showit in this paper. We recommend the reader to carefully examinethe different parts of the images (constant areas, sharp edges,small details and oriented textures) to observe how these partsare restored.

B. Results: SPOT 5 Simulation

Fig. 1 shows a 128 64 area extracted from the originalimage of Nîmes (SPOT 5 simulation at 2.5-m resolution, pro-vided by the CNES). Fig. 2 shows the PSF and Fig. 3 shows theobserved image . Fig. 6 shows the resulting imageobtained with the proposed algorithm, described in Section V.

C. Comparison With Different Methods

1) Quadratic regularization [20]: This produces resultsnearly equivalent to the parametric Wiener filter [8]. Itis also equivalent to isotropic diffusion [18]. The edgesare filtered as well as the noise, as seen on Fig. 9. It is

Fig. 3. Observed image Y (SNR = 15:4 dB).

Fig. 4. The proposed algorithm called “DEPA”.

therefore impossible to obtain sharp details and noisefreehomogeneous areas at the same time. Thus, the SNRremains low because of the insufficient noise removal inconstant areas.

2) Nonquadratic regularization [4], [6]: The resultingimage of the RHEA algorithm [11] exhibits sharp edges,compared to the quadratic regularization. However, somenoise remains in homogeneous regions and textures areattenuated. This result is illustrated by Fig. 10.

3) Complex wavelet packets [12]: COWPATH gives an ap-parently good result, since the textured and constant areasare well deconvolved. However, the edges remain blurred,as well as she smallest features. This is shown in Fig. 7.

4) Proposed method: Compared to the other ones, it is thefastest method. Even though the SNR is not the highest,the visual appearence is the most satisfactory. Indeed, thetextures and the oriented features are sharp and regular,while the homogeneous regions remain noise free, as seenon Fig. 6. Fig. 8 is provided to illustrate the improve-ment w.r.t. the reference image provided by COWPATH.As seen on this figure, the edges (especially the diagonalones) are sharper.


Fig. 5. Getting the MLE on the observed data.

Fig. 6. Deconvolved image using the proposed method (SNR = 17:1 dB).

Fig. 7. Deconvolved image using the COWPATH algorithm (SNR =17:4 dB): approximate original image.

Fig. 8. 40 � 40 areas extracted from Nimes, illustrating the improvementprovided by the proposed method (left) versus the existing COWPATHalgorithm (right).

VII. CONCLUSION

The main contribution of this paper is to propose an au-tomatic image deblurring algorithm based on an adaptiveGaussian model. This technique is robust and gives verygood deconvolution results on high resolution satellite data.Furthermore, the resulting images exhibit much sharper detailsthan images produced by concurrent deconvolution algorithms.

We have shown that, in the case of an inhomogeneous regu-larizing model, the MLE on observed data, which is an usual ap-

Fig. 9. Deconvolved image using quadratic regularization (SNR =15:9 dB).

Fig. 10. Deconvolved image using a nonquadratic regularization (SNR =17:2 dB).

proach for parameter estimation within a Bayesian framework,is not a robust estimator. We use the MLE on complete datainstead, assuming we know a good approximation of the orig-inal image. This approximation could be obtained by a wavelet-based thresholding algorithm. This hybrid approach, consistingof combining two different techniques to enhance the result ofeach one, is the essential novelty of the proposed method.

Furthermore, this new type of approach can be extended andthe results can be improved to handle more difficult cases (dif-ferent types of blur and higher noise variance).

The Gaussian adaptive model used for the unknown imageis the simplest model. The results could be improved by usingmore accurate modeling, for example by using a generalizedGaussian MRF instead of a Gaussian field.

To better take into account the textures which are charac-terized by longer distance interactions than edges, it should bepreferable to define a model with higher order interations. Thecurrent model is limited to the four nearest neighbors and theparameters are related to the first-order derivatives.

A reference (or approximate original image) is used to es-timate the adaptive parameters by complete data MLE. Thus,the estimation results strongly depend on the quality of this ap-proximation. We have shown that to get a robust estimate it isnecessary to threshold the residual noise which contaminates the


gradients of this approximate original image. It would be prefer-able to take into account the statistics of the residual noise of thisimage to better estimate the adaptive parameters.

Finally, as the proposed method essentially consists of twosteps—approximation by a wavelet-based method and adaptivedeconvolution—we wonder whether the output of the algorithmcould be injected back into the first step to obtain a better ap-proximation and then to provide better parameter estimates. Inthis case, it could be possible to build an iterative algorithm. Thestability of such a method is not certain and convergence studieshave to be carried out.

ACKNOWLEDGMENT

The authors would like to thank the French Space Agency(CNES) for providing the image of Nimes (SPOT 5 simulation),and J. Kalifa (Ecole Polytechnique) for the comparison with realwavelet packets.

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André Jalobeanu received the Master’s degree inastrophysics, image processing, and high-resolutionimaging from the in 1998 and the Ph.D. degree inimage deconvolution for remote sensing data in 2001,both from the University of Nice–Sophia Antipolis,France.

He has been a Visiting Scientist withUSRA/RIACS at NASA Ames, Moffett Field,CA, since 2002. His research focuses on solvinginverse problems using a Bayesian approach, rangingfrom 2-D image denoising and deconvolution to 3-D

surface reconstruction from multiple images. He currently works on modelingsurfaces of arbitrary topology, using wavelets on multiresolution meshesand adaptive models. He is also interested in complex wavelets, fractals,optimization techniques, parameter estimation, and stochastic methods.

Laure Blanc-Féraud received the Ph.D. degreein image restoration in 1989 and the “Habilitationá Diriger des Recherches” on inverse problems inimage processing in 2000, both from the Universityof Nice-Sophia Antipolis, France.

She is currently a researcher at CNRS, Sophia An-tipolis. Her research interests are inverse problemsin image processing by deterministic approach usingcalculus of variation and PDEs. She is also interestedin stochastic models for parameter estimation andtheir relationship with the deterministic approach.

She is currently working in the Ariana research group (i3S/INRIA) which isfocused on Earth observation.

Josiane Zerubia (S’81–M’81–SM’99–F’03)received the electrical engineering degree fromENSIEG, Grenoble, France, in 1981, and the Dr.Eng. degree in 1986, the Ph.D. degree in 1988, anda “Habilitation” in 1994, all from the University ofNice Sophia-Antipolis, France.

She has been a permanent Research Scientist atINRIA, Sophia-Antipolis, since 1989, and Directorof Research since July 1995. She was Head of theRemote Sensing Laboratory PASTIS at INRIA frommid-1995 to 1997. Since January 1998, she has been

in charge of a new research group working on remote sensing (ARIANA,INRIA-CNRS, University of Nice). She has been an Adjunct Professor atSup’Aero (ENSAE), Toulouse, France, since 1999. Previously, she was withthe Signal and Image Processing Institute, University of Southern California(USC), Los Angeles, as a post-doctorate. She also worked as a Researcherfor the University of Nice and CNRS from 1984 to 1988 and in the ResearchLaboratory of Hewlett-Packard, Grenoble, France, and Palo Alto, CA, from1982 to 1984. Her current research interest is image processing (imagerestoration, image segmentation or classification, line detection, perceptualgrouping, stereovision, super-resolution in 3-D) using probabilistic models orvariational methods. She also works on parameter estimation and optimizationtechniques.

Dr. Zerubia has been part of the IEEE IMDSP Technical Committee ofthe Signal Processing (SP) Society since 1997, a member-at-large of theBoard of Governors of IEEE SP Society since 2002, Associate Editor of theIEEE TRANSACTIONS ON IMAGE PROCESSING (T-IP) from 1998 to 2002, Areaeditor of T-IP since 2003, and Guest Editor for a Special Issue of the IEEETRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE in 2003.She has also been a member of the Editorial Board of the French Society forPhotogrammetry and Remote Sensing (SFPT) since 1998.

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Adaptive Gaussian Model for Satellite Image Debluring