Hyperparameter estimation in image restoration problems with partially known blurs

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1 Hyperparameter estimation in image restoration problems with partially known blurs Nikolas P. Galatsanos Illinois Institute o Technology Department o Electrical and Computer Engineering 330 South Dearborn Street Chicago, Illinois 6066 Vladimir Z. Mesarović Crystal Semiconductor Corporation 420 South Industrial Drive Austin, Texas Raael Molina, MEMBER SPIE Universidad de Granada Departamento de Ciencias de la Computación ei.a. E-807 Granada, Spain rms@decsai.ugr.es Abstract. This work is motivated by the observation that it is not possible to reliably estimate simultaneously all the necessary hyperparameters in an image restoration problem when the point-spread unction is assumed to be the sum o a known deterministic and an unknown random component. To solve this problem we propose to use gamma hyperpriors or the unknown hyperparameters. Two iterative algorithms that simultaneously restore the image and estimate the hyperparameters are derived, based on the application o evidence analysis within the hierarchical Bayesian ramework. Numerical experiments are presented that show the beneits o introducing hyperpriors or this problem Society o Photo-Optical Instrumentation Engineers. [DOI: 0.7/ ] Subject terms: image restoration; partially known blur; hyperparameter estimation; gamma hyperpriors; Bayesian estimation. Paper received July 3, 200; revised manuscript received Jan. 29, 2002; accepted or publication Feb. 4, Aggelos K. Katsaggelos Northwestern University Department o Electrical and Computer Engineering Evanston, Illinois Javier Mateos Universidad de Granada Departamento de Ciencias de la Computación ei.a. E-807 Granada, Spain Introduction Traditionally, image restoration algorithms have assumed exact knowledge o the blurring operator. In recent years, in particular in the ield o astronomical image restoration,,2 a signiicant eort has been devoted to solving the so-called blind deconvolution problem, in which it is assumed that little or nothing is known about the underlying blurring process see Res. 3,4 and reerences therein. In most practical applications, the point-spread unction PSF is neither unknown nor perectly known, 5 that is, usually some inormation about the PSF is available, but this inormation is never exact. The use o a PSF modeled by a known mean and an additive random error component has been addressed in the past see, or instance, Res However, in all these works the needed model parameters were assumed known. More recently, attempts were made to address the parameter estimation problem: in Res. 9,0 and Chapter III the expectation-maximization algorithm was used, and in Res. Chapter IV and 2 4 the estimation was addressed within the hierarchical Bayesian 5 ramework. However, in Res. 9,0, and Chapters III and IV, and 2 4 we observed that it was not possible to reliably estimate simultaneously the hyperparameters that capture the variances o the PSF error and the additive noise. In this paper we ameliorate the diiculties o estimating all the necessary hyperparameters by introducing gamma hyperpriors within the hierarchical Bayesian ramework. We derive two iterative algorithms that simultaneously estimate all the necessary hyperparameters and restore the image. The rest o this paper is organized as ollows: In Sec. 2 the image model, two models or the idelity to the data, and the hyperparameter model are discussed. In Sec. 3 the basic philosophy behind evidence analysis EA is briely presented and its application to the restoration problem rom partially known blur is discussed. Section IV presents two EA algorithms using the dierent proposed models. In Sec. 5 we present numerical experiments that compare the proposed approaches. Section 6 concludes the paper. 2 Components o the Hierarchical Model Let us now examine the components o the hierarchical model used or the restoration problem with partially known blur, that is, the image model, the observation model, and the model or the unknown hyperparameters. A commonly used model or the image prior in image restoration problems is the simultaneously autoregressive Opt. Eng. 4(8) (August 2002) /2002/$ Society o Photo-Optical Instrumentation Engineers 845

2 SAR model. 6 This model can be described by the ollowing conditional PDF: P const N/2 exp 2 Q 2, where R N represents the source image and is an unknown positive parameter that controls the smoothness o the image. For simplicity, but without loss o generality, we shall use a circulant Laplacian high-pass operator or Q throughout the rest o this paper. The space-invariant PSF is represented as the sum o a deterministic component and a stochastic component o zero mean, i.e., h h h, where h R N is the deterministic known component o the PSF and h R N is the random unknown error component modeled as zero-mean white noise with covariance matrix R h (/ ) I N N. For our problem, the image degradation can be described, in lexicographical orm, by the model 6 8 g H g, in which H H H, where g, g R N represent, respectively, the observed degraded image and the additive zero-mean white noise in the observed image, with covariance matrix R g (/ ) I N N. The matrix H is the known assumed, estimated, or measured component o the N N PSF matrix H; H is the unknown component o H, generated by h deined in Eq. 2. From Eqs. 2 4 it is clear that the orm o the conditional distribution o g is not simple. In act we are going to propose two dierent models or P(g,,, ). For the ixed- covariance model we assume that both the PSF noise h and the additive noise g are Gaussian. Then, since the vector is not a random quantity but rather a ixed one, it is straightorward to see rom Eq. 3 that P(g,,, ) is given by /2 P g,,, det R g exp 2 gàh t R g gàh. The conditional covariance R g in Eq. 5 is given by,7 R g FR h F t R g FFt I, where we have used the commutative property o the convolution and F denotes the circulant matrix generated by the image ; see Re.,4 or details For the averaged- covariance model we assume that the observations g are Gaussian, and instead o using FF t in the expression or the covariance we use its mean value rom the prior. Thus, or this model we get /2 P g,,, det R g where exp 2 g H t R g g H. R g N Qt Q I. Note that by using this approximation we have incorporated the uncertainty o the image prior model,, in the conditional distribution, which made the log P(g,,, ) unction quadratic with respect to. This yields a linear estimator or, as will be shown in the ollowing section. The Bayesian ormulation allows the introduction o inormation about parameters that have to be estimated by using prior distributions over them. 5 To do so we use, as hyperprior, the gamma distribution deined by P x x l x 2 /2 exp m x l x 2 x, where x,, denotes a hyperparameter, and the parameters l x and m x are explained below. The mean and the variance o a random variable x with PDF in Eq. 9 are given by l x E x 2m x l x 2 and l x Var x 2m 2 x l x According to Eq. 0, or l x large, the mean o x is approximately equal to /2m x, and its variance decreases when l x increases. Thus, /2m x speciies the mean o the gamma distributed random variable x, while l x can be used as a measure o the certainty in the knowledge about this mean. 3 Hierarchical Bayesian Analysis In this work, the joint distribution we use is deined by P g,,,, P g,,, P,, P P P. To estimate the unknown hyperparameters and the original image, we apply evidence analysis, since we have ound that it provides good results or restorationreconstruction problems. 8 According to the EA approach the simultaneous estimation o,,, and is perormed as ollows: 846 Optical Engineering, Vol. 4 No. 8, August 2002

3 Parameter estimation step: ˆ, ˆ, ˆ arg max P,, g,, arg max Restoration step:,, P g,,,, d. ˆ ˆ, ˆ, ˆ arg max P g, ˆ, ˆ, ˆ arg max P ˆ, ˆ, ˆ P g, ˆ, ˆ, ˆ, 2 3 The estimates ˆ, ˆ, and ˆ rom the parameter estimation step depend on the current estimate o the image. Likewise, the estimate ˆ rom the restoration step will depend on the current estimates o the parameters. Thereore, the above two-step procedure is repeated until convergence occurs. Using the two dierent choices or P(g,,, ) given in Eqs. 5 and 7, we will now proceed with the evidence analysis. 4 Proposed Algorithms 4. Evidence Analysis Based on the Fixed- Covariance Model Substituting Eqs. and 5 into, we obtain P g,,,, N/2 /2 det R g where exp 2 J,,, P P P, 4 J,,, Q 2 g H t R g g H Parameter estimation step To estimate ˆ, ˆ, ˆ, we irst have to integrate P(g,,,, ) in Eq. 4 over, that is, P,, g P P P N/2 /2 det R g exp 2 J,,, d. 6 To perorm the integration in Eq. 6, we expand J(,,, ) in Taylor series around a known (n), where (n) denotes the iteration index, i.e., J,,, (n) 0 8 i (n) is chosen to be the minimizer o J(,,, ) in Eq. 5, and that the Hessian matrix can be approximated by 2 J,,, (n) 2G (n) 2 Q t Q H tr g (n) H, 9 where we have not taken into account the derivatives o R g (n) with respect to.,4,7 Finally, substituting Eq. 7 into 6, and using the act that the actor det(r g ) /2 can be replaced by det(r g (n)) /2 because it depends weakly on compared to the exponential term under the integral,,7 Eq. 6 becomes P,, g P P P N/2 /2 det R g (n) det G (n) /2 exp 2 J (n),,,. 20 Taking 2 log on both sides o Eq. 20, and dierentiating with respect to,,, we obtain the ollowing iterations: (n ) 2m N Q (n) 2 tr G (n) Q t Q, (n ) 2m N g H (n) t R g (n) (n)2 F (n) F (n) t R g (n) g H tr (n) (n) (n) R g (n) tr G (n) H tr g (n) (n)2 F (n) F (n)t R g (n) H, (n ) N 2m g H (n) t 2 (n)2 R g (n) 2 22 g H tr (n) (n) (n) F (n) F (n) t R g (n) tr G (n) 2 H t (n)2 R g (n) H. 23 where the normalized conidence parameter x, x,,, is deined as J,,, J (n),,, (n) t J,,, (n) 2 (n) t 2 J,,, (n) (n). 7 Note that, in Eq. 7, N x N l x Restoration step According to Eq. 3, 24 Optical Engineering, Vol. 4 No. 8, August

4 ˆ ˆ, ˆ, ˆ arg maxp g, ˆ, ˆ, ˆ arg min H g t Rˆ g H g ˆ Q 2 log det Rˆ g, 25 where Rˆ g (/ ˆ ) FF t (/ ˆ ) I. The unctional in Eq. 25 is nonconvex and may have several local minima. In general, a closed-orm solution to Eq. 25 does not exist and numerical optimization algorithms must be used. A practical computation o Eq. 25 can be obtained by transorming it to the DFT domain.,4 (n ) 2m N (n) (n) tr R g (n) (n) Q t Q (n) 2 2 (n) 2 tr R g (n) tr G (n) 2 H th R g (n) g H (n) g H (n) t where x was deined in Eq. 24 and G (n) (n) Q t Q H tr g (n) H., Evidence Analysis Based on the Averaged- Covariance Model Using Eq. 7 as the likelihood equation, we derive another iterative parameter-estimation image-restoration algorithm or this problem. We ollow identical steps as in the previous section with R g N Qt Q I Parameter estimation step 26 To ind the estimates o the parameters, P(,, g) must be maximized. Taking the derivatives with respect to,, gives the ollowing iterations: (n ) 2m N Q (n) 2 tr R g (n) (n) (n) 2 Q t Q 2 tr R g (n) (n) (n) 2 Q t Q g H (n) t tr g H (n) (n) G Q t 2 Q H th R g (n) (n) (n) 2 Q t Q (n ) 2m 2 tr R g (n) (n) 2 (n) Q t Q g H (n), 27 g H (n) t N (n) (n) tr R g (n) tr G (n) 2 H th R g (n) (n) 2 (n) Q t Q, Image restoration step For the image estimation step, similar to Eq. 25, wecan write ˆ ˆ, ˆ, ˆ arg min H g t R ˆ g H g ˆ Q 2 ]. 3 Note that, since R ˆ g does not depend on, P(g, ˆ, ˆ, ˆ ) is quadratic with respect to. As a result, the image restoration step gives the linear estimate or ˆ ˆ, ˆ, ˆ H tr ˆ g H ˆ Q t Q H tr ˆ g g Comments on the Normalized Conidence Parameters There is a very interesting and intuitive interpretation o the use o the normalized conidence parameters x, x,, see Eq. 24 in the hyperparameter estimation procedures in Secs. 4.. and We are estimating the unknown hyperparameters by linearly weighting their maximum likelihood ML estimates with the prior knowledge we have on their means and variances. So, i we know, let us say rom previous experience, the noise or image variances with some degree o certainty, we can use this knowledge to guide the convergence o the iterative procedures. Note that, or a given hyperparameter x,,, x 0 means having no conidence on the mean o its hyperprior, so that the estimate o x is identical to the ML estimate in Re. 4. In contrast, when x, x,,, the value o x is ixed to the mean o its hyperprior. As will become evident in the ollowing section, an important observation is that neither o the two algorithms can reliably estimate the PSF and additive noise variances and simultaneously when no prior knowledge is introduced, that is, when 0.0. This was expected, since the sum o these noises appears in the data. However, introducing some even very little prior knowledge about one o the parameter aids both methods to accurately estimate both and while restoring the image. Notice that i we do not have any prior knowledge about the values o the hyperparameters, we can irst assume that there is no random component in the blur, that is, 848 Optical Engineering, Vol. 4 No. 8, August 2002

5 Algorithm EA EA2 Table Comparison between EA and EA2 algorithms. Restoration step Requires iterative optimization Linear closed-orm solution Parameter estimation step Complicated relation with Quadratic dependence on 0, and estimate and by any classical method, such as MLE. Once an estimate o is obtained which is in general a good one, since the noise model is usually accurate, while the prior image model is an approximation and the PSF variance is typically very small, it is used with a medium to high conidence parameter to guide the estimation o the partially known PSF with 0.0. Alternatively, we can run the algorithms with no prior knowledge o the parameters, that is, 0.0, using then a high conidence or the obtained value ( 0.7) and letting the algorithms to estimate the other two parameters with 0.0. Note that the two described methods to estimate all the parameters start by using good estimates o. 5 Numerical Experiments In all experiments presented in this section, the known part o the PSF was modeled by the Gaussian-shaped PSF deined as h i, j exp i2 j 2 2 or i, j 5, 4,..., 2 3,0,,...,4,5, Computational complexity (s/iteration) The irst o the two algorithms that simultaneously restore the image and estimate the associated hyperparameter uses the ixed- covariance model Eqs and 25 and is named EA. The second one, using the averaged- covariance model Eqs and 3, is named EA2. In Table some general comments on complexity and speed o the EA and EA2 algorithms are shown. Note also that since EA2 approximates FF t by Q t Q, EA will, in general, outperorm EA2 unless the real underlying image is a true realization rom the prior model see Experiment I below. A number o experiments were perormed with the proposed algorithms using a synthetic image obtained rom a prior distribution, the Lena image, and a real astronomical image to demonstrate their perormance. For both algorithms, the criteria x (n ) x (n) /x (n) 0 3, x,,, orn 250, whichever was met irst, were used to terminate the iterative process. The levels o the PSF and additive noise are measured using the signal-to-noise ratio SNR, i.e., SNR h h 2 /N and SNR g 2 /N, where h 2 and 2 are the energy o the known part o the PSF and o the original image, respectively. The perormance o the restoration algorithms was evaluated by measuring the improvement in SNR, denoted by ISNR and given by ISNR g 2 / ˆ 2, where, g, and ˆ are the original, observed, and estimated images, respectively. According to Eq. 2, the PSF deined in Eq. 33 was degraded by additive noise in order to obtain a PSF with SNR h 0, 20, and 30 db. Using these degraded PSFs, both synthetic and Lena images were blurred, and then Gaussian noise was added to obtain SNR g 0, 20, and 30 db, which produced a set o 8 test images. In all the experiments we assigned to the normalized conidence parameters x, x,,, the values 0.0,0.,...,.0, and we plotted the ISNR as a unction o two o the parameters x, x,,, while keeping the other constant. Experiment I. To compare the ISNR o the EA and EA2 algorithms assuming the means o the hyperpriors are the true values o the noise and image parameters, we used a synthetic image generated rom a SAR distribution with We have ound that the EA and EA2 algorithms give very similar results in terms o ISNR on this image, since ollows a SAR distribution and hence the two models are themselves very similar. Furthermore, we also ound that when gamma hyperpriors are used, both methods always gave accurate parameter estimates and the improvement in ISNR is always almost identical. We also noticed that when 0.0, the estimated value o was unreliable: it was always very close to zero ( ˆ was between two and three orders o magnitude lower than the true value, and urthermore, or most o the values o SNR g and SNR h, both algorithms stopped ater reaching the maximum number o iterations 250. The evolution o the ISNR or SNR h 0 db ( ) and SNR g 0 db ( ), or constant 0.0, is shown in Fig.. Similar plots are obtained or experiments with the other SNR h and SNR g. Experiment II. In this experiment both the Lena and the SAR image were used. Note that exact knowledge o the parameter is not possible or the Lena image. In order to include some prior knowledge about and, we used the ML estimates 8 or the known PSF problem, assuming that the blurring unction was the known part o the PSF. The ML estimates or the Lena image with SNR h 0 db and SNR g 0 db were 33. and , and or the SAR image SNR h 0 db and SNR g 0 db were and 270.0; their corresponding ISNR s were 3.2 and 3.92 db, respectively. We observed that including the previously obtained ML estimate as prior knowledge about slightly increases the quality o the resulting restoration. However, including the corresponding estimate o the value o decreases it. This is due to the act that this ML estimate is accurate or the image noise parameter, while it underestimates the variance value, as shown in Fig. 2. We also ound that, in most cases, EA perorms a bit better than EA2 algorithm or the Lena image. Here Optical Engineering, Vol. 4 No. 8, August

6 Fig. ISNR evolution with and or 0.0, using the real values o and or the SAR image with SNR h 0 db and SNR g 0 db, (a) or EA algorithm, (b) or EA2 algorithm. again, when 0.0, both algorithms result in an estimate o very close to zero ( ˆ was between one and three orders o magnitude less than the real value. Experiment III. In order to demonstrate that including accurate inormation about the value o the hyperparameters improves the results, we have tested both algorithms on the Lena image, assuming that the means o the hyperpriors or and are the true noise parameters. Since it is not possible to know the real value o or this image, we used 0.0, letting the algorithms estimate without prior inormation. Figure 3 shows the evolution o the ISNR or SNR h 0 db ( ) and SNR g 0 db ( ), or constant 0.0 or the EA and EA2 algorithms. Note that incorporating accurate inormation about the value o improves more the ISNR than including inormation about, especially or the EA algorithm. This is due to the act that the estimated value o is quite close to the real one even i 0.0, while i no knowledge about the value o is included, both methods give poor estimates ( ˆ was between one and three orders o magnitude lower than the real value when the algorithm stopped, and in most cases the imposed iteration limit was reached. Including knowledge about the real value o leads to more accurate estimations, since we are orcing Fig. 2 ISNR evolution with and or 0.0: or (a) EA and (b) EA2 algorithms, using the MLE estimated values o and or the Lena image with SNR h 0 db and SNR g 0 db; and or (c) EA and (d) EA2 algorithms, using the MLE estimated values o and or the SAR image with SNR h 0 db and SNR g 0 db. 850 Optical Engineering, Vol. 4 No. 8, August 2002

7 Fig. 3 ISNR evolution with and or 0.0, using the real values o and or the Lena image with SNR h 0 db and SNR g 0 db, (a) or EA algorithm, (b) or EA2 algorithm. Fig. 4 (a) Degraded Lena image with SNR h 0 db and SNR g 0 db; (b) restoration with EA algorithm, ISNR 3.26 db; (c) restoration with EA2 algorithm, ISNR 3.2 db. Optical Engineering, Vol. 4 No. 8, August

8 Fig. 5 (a) Observed Jupiter image; (b) restoration with EA algorithm; (c) restoration with EA2 algorithm. both algorithms to provide a estimate greater than zero. In most cases, also, EA perorms better than EA2 or this image. An example o the restoration o the degraded Lena image Fig. 4 a, SNR h 0 db, SNR g 0 db by the EA and EA2 algorithms with 0.0 and.0 is presented in Figs. 4 b and 4 c, respectively. Experiment IV. We also tested the methods on real images. Results are reported on a Jupiter image depicted in Fig. 5 a obtained at Calar Alto Observatory Spain, using a ground-based telescope, in August 992. For this kind o images there is no exact expression describing the shape o the PSF, although previous studies 9 have suggested the ollowing radially symmetric approximation or the PSF: h i, j i2 j 2 B, 34 R 2 where the parameters B and R were estimated rom the image to be B 3 and R However, the estimate o the PSF is not exact, since actors such as atmospheric turbulence introduce noise into it. Since the proposed methods do not provide reliable estimates simultaneously or both the PSF and additive noise variances, they were estimated in two steps. The algorithms were irst run with no prior knowledge about any o the hyperparameters, that is, 0.0 was used, in order to obtain an estimate o the noise variance. A high conidence was then given to the estimate o, viz., 0.8, and estimates o the other two parameters were obtained. The EA and EA2 algorithms were terminated ater 46 and 44 iterations, respectively, with the ollowing estimates: , , and 47.6 or the EA algorithm, and 2270., , and 47.5 or the EA2 algorithm. The resulting images are 852 Optical Engineering, Vol. 4 No. 8, August 2002

9 shown respectively in Figs. 5 b and 5 c. It is clear that both algorithms provide good restorations, although the restoration provided by the EA algorithm seems to be better resolved. Alternatively, it is possible to estimate the additive noise variance using the ML approach as described in Re. 8, assuming that the PSF is known, as described by Eq. 34. This value is in turn used in the algorithms or the estimation o the remaining parameters. The experimental results provided very similar restorations in the two cases. 6 Conclusions In this paper we have extended the EA and EA2 algorithm and the EM algorithm rom our previous work in Res. 4 and 0, respectively, to include prior knowledge about the unknown parameters. The resulting parameter updates, in both EA and EA2 approaches, combine the available prior knowledge with the ML estimates in a simple and intuitive manner. Both algorithms showed the capability to accurately estimate all three parameters simultaneously while restoring the image, even with very low conidence in the prior knowledge. We have also shown that the image noise parameter obtained by the ML estimate or the exactly known PSF problem can be used to guide the estimates o the noise parameter or the partially known PSF problem. Acknowledgments This work has been supported by NSF grant MIP and by Comisión Nacional de Ciencia y Tecnología under contract TIC Reerences. R. Molina, J. Nuñez, F. Cortijo, and J. Mateos, Image restoration in astronomy: a Bayesian perspective, IEEE Signal Process. Mag. 8 2, J.-L. Starck, F. Murtagh, and A. Bijaoui, Image Processing and Data Analysis. The Multiscale Approach, Cambridge Univ. Press J. M. Conan, T. Fusco, L. Mugnier, E. Kersal, and V. Michau, Deconvolution o adaptive optics images with imprecise knowledge o the point spread onction: results on astonomical objects, in Astronomy with Adaptive Optics: Present Results and Future Programs, ESO/ESA L. Mugnier, J.-M. Conan, T. Fusco, and V. Michau, Joint maximum a posteriori estimation o object and PSF or turbulence degraded images, in Bayesian Inerence or Inverse Problems, Proc. SPIE 3459, R. Lagendijk and J. Biemond, Iterative Identiication and Restoration o Images, Kluwer Academic L. Guan and R. K. Ward, Deblurring random time-varying blur, J. Opt. Soc. Am. A 6, V. Z. Mesarović, N. P. Galatsanos, and A. K. Katsaggelos, Regularized constrained total least-squares image restoration, IEEE Trans. Image Process. 4 8, R. K. Ward and B. E. A. Saleh, Restoration o images distorted by systems o random impulse response, J. Opt. Soc. Am. A 2 8, V. Z. Mesarović, N. P. Galatsanos, and M. N. Wernick, Restoration rom partially-known blur using an expectation-maximization algorithm, in Proc. Thirtieth ASILOMAR Con., pp , Paciic Grove, CA, IEEE V. N. Mesarović, N. P. Galatsanos, and M. N. Wernick, Iterative LMMSE restoration o partially-known blurs, J. Opt. Soc. Am. A 7, V. Z. Mesarović, Image restoration problems under point-spread unction uncertainties, PhD Thesis, ECE Dept., Illinois Inst. o Technology, Chicago V. Z. Mesarović, N. P. Galatsanos, R. Molina, and A. K. Katsaggelos, Hierarchical Bayesian image restoration rom partially-known blurs, in Proc. Int. Con. on Acoustics Speech and Signal Processing, ICASSP-98, Vol. 5, pp , Seattle, IEEE N. P. Galatsanos, R. Molina, and V. Z. Mesarović, Two Bayesian algorithms or image restoration rom partially-known blurs, in Proc. Int. Con. on Image Processing, ICIP-98, Vol. 2, pp , Chicago, IEEE N. P. Galatsanos, V. Z. Mesarović, R. Molina, and A. K. Katsaggelos, Hierarchical Bayesian image restoration rom partially-known blurs, IEEE Trans. Image Process. 9 0, J. O. Berger, Statistical Decision Theory and Bayesian Analysis, Springer-Verlag, New York B. D. Ripley, Spatial Statistics, Wiley N. P. Galatsanos, V. Z. Mesarović, R. Molina, and A. K. Katsaggelos, Hyperparameter estimation using hyperpriors or hierarchical Bayesian image restoration rom partially-known blurs, in Bayesian Inerence or Inverse Problems, Proc. SPIE 3459, R. Molina, A. K. Katsaggelos, and J. Mateos, Bayesian and regularization methods or hyperparameters estimation in image restoration, IEEE Trans. Image Process. 8 2, R. Molina and B. D. Ripley, Using spatial models as priors in astronomical image analysis, J. Appl. Statist. 6, R. Molina, B. D. Ripley, and F. J. Cortijo, On the Bayesian deconvolution o planets, in th IAPR Int. Con. on Pattern Recognition, pp Nikolas P. Galatsanos received the Diploma degree in electrical engineering in 982 rom the National Technical University o Athens, Athens, Greece, and the MSEE and PhD degrees in 984 and 989, respectively, rom the Department o Electrical and Computer Engineering, University o Wisconsin, Madison. He joined the Electrical and Computer Engineering Department, Illinois Institute o Technology, Chicago, in the all o 989, where he now holds the rank o associate proessor and serves as the director o the graduate program. His current research interests center around image and signal processing and computational vision problems or visual communications and medical imaging applications. He is coeditor o Recovery Techniques or Image and Video Compression and Transmission (Kluwer, Norwell, MA, 998). Dr. Galatsanos served as associate editor o the IEEE Transactions on Image Processing rom 992 to 994. He is an associate editor o the IEEE Signal Processing Magazine. Vladimir Z. Mesarović received the Diploma degree in electrical engineering rom the University o Belgrade in 992, and the MS and PhD degrees in electrical and computer engineering rom the Illinois Institute o Technology, Chicago, in 993 and 997, respectively. Since January 997, he has been with the Crystal Audio Division, Cirrus Logic, Austin, TX, where he is currently a project manager in the DSP Systems and Sotware Group. His research interests are in multimedia signal representations, coding and decoding, transmission, and their eicient DSP implementations. Raael Molina received a bachelor s degree in mathematics (statistics) in 979 and the PhD degree in optimal design in linear models in 983. He became proessor o computer science and artiicial intelligence at the University o Granada, Granada, Spain, in 2000, and is currently the dean o the Computer Engineering Faculty. His areas o research interest are image restoration (applications to astronomy and medicine), parameter estimation, image and video compression, and blind deconvolution. Dr. Molina is a member o SPIE, the Royal Statistical Society, and the Asociación Española de Reconocimento de Formas y Análisis de Imágenes (AERFAI). Optical Engineering, Vol. 4 No. 8, August

10 Aggelos K. Katsaggelos received the Diploma degree in electrical and mechanical engineering rom the Aristotelian University o Thessaloniki, Thessaloniki, Greece, in 979, and the MS and PhD degrees, both in electrical engineering, rom the Georgia Institute o Technology, Atlanta, Georgia, in 98 and 985, respectively. In 985 he joined the Department o Electrical Engineering and Computer Science at Northwestern University, Evanston, IL, where he is currently a proessor, holding the Ameritech Chair o Inormation Technology. He is also the director o the Motorola Center or Communications. During the academic year he was an assistant proessor at Polytechnic University, Department o Electrical Engineering and Computer Science, Brooklyn, NY. His current research interests include image and video recovery, video compression, motion estimation, boundary encoding, computational vision, and multimedia signal processing and communications. Dr. Katsaggelos is a Fellow o the IEEE, an Ameritech Fellow, and a member o the Associate Sta, Department o Medicine, at Evanston Hospital. He is an editorial board member o Academic Press, Marcel Dekker (Signal Processing Series), Applied Signal Processing, and the Computer Journal, and the editor-in-chie o the IEEE Signal Processing Magazine. He has served as an associate editor o several journals and is a member o numerous committees and boards. He is the editor o Digital Image Restoration (Springer-Verlag, Heidelberg, 99), coauthor o Rate-Distortion Based Video Compression (Kluwer Academic Publishers, 997), and coeditor o Recovery Techniques or Image and Video Compression and Transmission, (Kluwer Academic Publishers, 998). He has served as the chair and co-chair o several conerences. He is the coholder o eight international patents, and the recipient o the IEEE Third Millennium Medal (2000), the IEEE Signal Processing Society Meritorious Service Award (200), and an IEEE Signal Processing Society Best Paper Award (200). Javier Mateos received the Diploma and MS degrees in computer science rom the University o Granada in 990 and 99, respectively, and completed his PhD in computer science at the University o Granada in July 998. In 992, he became an assistant proessor in the Department o Computer Science and Artiicial Inteligence o the University o Granada, and he became associate proessor in 200. His research interests include image restoration and image and video recovery and compression. He is a member o the AERFAI (Asociación Española de Reconocimento de Formas y Análisis de Imágenes). 854 Optical Engineering, Vol. 4 No. 8, August 2002