May 1, 2009 / Vol. 34, No. 9 / OPTICS LETTERS 1453

Maximum a posteriori blind image deconvolutionwith Huber–Markov random-field regularization

Zhimin Xu and Edmund Y. Lam*Imaging Systems Laboratory, Department of Electrical and Electronic Engineering, The University of Hong Kong,

Pokfulam Road, Hong Kong*Corresponding author: elam@eee.hku.hk

Received January 9, 2009; revised April 7, 2009; accepted April 7, 2009;posted April 9, 2009 (Doc. ID 106175); published April 30, 2009

We propose a maximum a posteriori blind deconvolution approach using a Huber–Markov random-fieldmodel. Compared with the conventional maximum-likelihood method, our algorithm not only suppressesnoise effectively but also significantly alleviates the artifacts produced by the deconvolution process. Theperformance of this method is demonstrated by computer simulations. © 2009 Optical Society of America

OCIS codes: 100.1455, 100.1830, 100.2000, 100.3020, 100.3190.

Digital images are obtained in areas ranging from ev-eryday photography to astronomy, remote sensing,medical imaging, and microscopy. In each case, thereis an underlying object or scene we wish to observe.However, the observation process is seldom perfect;there is always uncertainty in the measurements,manifested as blur, noise, and other degradations inthe recorded images. In the case of a linear shift-invariant imaging system, the relation between theobserved image i and the true image o in the samecoordinate frame is a convolution. That is,

i�x� = h�x� � o�x�, �1�

where h represents the point-spread function (PSF),x refers to a lexicographically ordered vector of thepixel coordinates in a 2D image, and the operator �denotes 2D convolution. In realistic scenarios, wemust also consider noise, which is often modeled by aspecific statistical distribution.

Our objective is to estimate o�x� and h�x� based ondegraded image i�x� and prior information about thetrue imaged scene. This estimate process is alsoknown as blind image deconvolution. Over the years,many blind deconvolution algorithms have been de-veloped, such as iterative [1–3] and noniterativemethods [4,5] (for a review see [6]). In this Letter, theBayesian image deconvolution model is used. Basedon Bayes’s rule, there are two possible estimates: themaximum likelihood (ML) estimate and the maxi-mum a posteriori (MAP) estimate. The ML methodinitially proposed by Richardson [7] and Lucy [8] isthe most well known for image deblurring. In [9],Fish et al. extended their idea to image reconstruc-tion in the “blind” situation. Nevertheless, owing tothe ill-posed nature of inverse problems, these algo-rithms often lack stability and uniqueness. The MAPmethod is closely related to the method of the ML es-timate, but the incorporation of a prior distributionover the original scene can be seen as a regulariza-tion of ML estimate, i.e.,

� = arg max�p�i���p���� �2�

�

0146-9592/09/091453-3/$15.00 ©

=arg max�

�log p�i��� + log p����, � = �o,h�, �3�

where the set � contains the unknown parameters forthe blind deconvolution problem and p denotes prob-ability.

Considering the cases where the data is contami-nated by Poisson noise, such as in astronomy and mi-croscopy, the intensity of each pixel x in the observedimage is a random variable that follows an indepen-dent Poisson distribution of a mean �h�x��o�x��. Thelikelihood can then be written as

p�i�o,h� = �x

�h�x� * o�x��i�x�

i�x�!exp�− �h�x� � o�x���.

�4�

Unpleasant artifacts (e.g., ringing) from the decon-volution process are often inevitable because of theloss of the high-frequency component in the recon-structed image. Besides, noise and discretization ofthe image and the PSF amplify errors in the decon-volution process. For instance, from the simulation, itcan be found that too many iterations in theRichardson–Lucy algorithm (RLA) will result in more“ringing” artifacts, especially on the location of inten-sity discontinuities. In this Letter, we use a non-Gaussian Markov random-field (MRF) model to sup-press the artifacts. MRF models are used in imageprocessing to define a distribution over neighbor-hoods for describing the local structure of an image,and that distribution is used to solve the image re-construction problem. Here we use a special form ofGibbs distribution to characterize the distribution ofthe MRF. This has been shown to successfully modelboth the smooth regions and discontinuities of im-ages in image decompression [10] and superresolu-tion image reconstruction from multiple low-resolution images [11]. In [12], a similar distributionis used for preserving edges in image restoration, butthe context is different from ours as we deal withPoisson noise in a “blind” situation. The form of the

distribution is explicitly given by

2009 Optical Society of America

1454 OPTICS LETTERS / Vol. 34, No. 9 / May 1, 2009

p�o� =1

Zexp− �

c�C��ac

To�� , �5�

where ��0, Z is a normalizing constant, ac is a lin-ear operator defined on a local group of points c calleda clique, and the function ��u�, which is also calledthe Huber function, is defined as

��u� = �u2,

�2 + 2���u� − ��, �u� � �

�u� � �. �6�

The Huber function is a good compromise betweenL1 and L2 norms. The Huber threshold � controls thetransition between these two norms. For small u, theHuber function becomes the usual L2 least-squarespenalty function, which exhibits the stability andrapid convergence characteristics; for large u, it re-duces to the noise insensitive L1 penalty function. Toinclude the measurement of discontinuities into thestatistical model, we use the linear operator ac tomeasure inconsistency of a local area.

Combining with the prior distribution presentedabove, from Eq. (3), we can get the negative log-likelihood

J��� = x���h�x� � o�x�� − i�x�� + i�x�log

i�x�

h�x� � o�x�

+ � c�C

��acTo�x�� . �7�

Maximizing �log p�i ���+log p���� is equivalent tominimizing J���. According to the expectation-maximization (EM) algorithm, the numerical itera-tive algorithm is then obtained,

ht�k+1��x� = h�k��x��o�k��− x� � i�x�

h�k��x� � o�k��x�� ,

h�k+1��x� =ht

�k+1��x�

x

ht�k+1��x�� , �8�

o�k+1��x� = o�k��x��h�k+1��− x� � i�x�

h�k+1��x� � o�k��x��

�1

�1 + � c�C

�ac���acTo�k��x��� , �9�

where the superscript indexes the number of the it-eration and ���u� refers to the first derivative of thefunction ��u�,

���u� = �2u, �u� � �

2�u

�u�, �u� � �� . �10�

For the selection of the relaxation factor � and theHuber threshold �, we use the method proposed in

[13] as a reference.

To demonstrate the effectiveness of our method, weexecute simulations with two kinds of images underthe Poisson noise process. First, we take the situationin astronomical image processing into account. It iswell known that the noise associated with the collec-tion of an astronomical image by a CCD camera ismostly Poissonian. A gray level image “Saturn” ofsize 256�256 is used. This image is degraded by an8�8 Gaussian blur kernel. Then simulated photonnoise is added to the degraded image by generating arandom number obeying a Poisson distribution withthe mean of the pixel value of this noiseless degradedimage. The numbers generated for all the pixels thenform the observed image. We compare our approachwith the conventional RLA. Figures 1(a)–1(d) showthe original image, the blurred and noisy image, therestored image with standard RLA, and the resultwith our algorithm, respectively. In the simulationwe choose �=0.06, �=0.008, and 200 iterations. As forthe linear operator ac, a 3�3 window is used for con-sistency measure,

ac = �−1

2− 1 −

1

2

− 1 6 − 1

−1

2− 1 −

1

2� . �11�

Comparing Fig. 1(c) with Fig. 1(d), it can be foundthat the artifacts on the surface of the planet, whichare caused by the noise, are apparently reduced. Tomeasure the improvement in the restored imagequality, the normalized mean-square error (NMSE)can be used and is defined as

NMSE =x

�o�x� − o�x��2

x

�o�x��2. �12�

Here, o�x� stands for the restored image. The NMSEof two different estimates versus the iteration num-ber are shown in Fig. 2, where the convergence is alsohighlighted.

We further illustrate the capabilities of our ap-proach for a scenario in microscopy. Owing to photon-limited detection, microscopical images (e.g., confocalmicroscopical images) are always corrupted by Pois-son noise. We use a synthetic phantom image com-

Fig. 1. Restorations of a simulated astronomy image: (a)original image, (b) blurred and noisy image, (c) image re-stored with RLA, and (d) image restored with ourapproach.

posed of five geometrical patterns for simulation.

May 1, 2009 / Vol. 34, No. 9 / OPTICS LETTERS 1455

This kind of image has been adopted to evaluate theperformances of image-restoration algorithms interms of qualitative and also quantitative measures[14]. The experimental settings are the same as for“Saturn.” As for the parameter, we use �=0.8 and �=0.006. The deconvolution results of standard RLAand our approach are shown in Figs. 3(c) and 3(d). Inthe RLA deconvolution, many artifacts, such asspeckles and ringings, appear inside the shapes andnear their edges. However, these are remarkably at-tenuated in the result with our method.

In conclusion, a MAP-based methodology for blinddeconvolution is proposed. It uses a non-GaussianMRF model based on the Huber function for stableimage reconstruction. For the sake of stability anduniqueness, many algorithms have been proposed toregularize RLA, such as L2 norm and L1 norm. Yet

Fig. 2. (Color online) NMSE of RLA and our approach.

Fig. 3. Restorations of a synthetic phantom image: (a)original image, (b) blurred and noisy image, (c) image re-stored with RLA, and (d) image restored with ourapproach.

these often oversmooth the edges in an image or arenot readily computed. Because of the Huber func-tion’s good combination of L1 and L2 norms, thisproperty helps to maintain robustness in the pres-ence of noise and discontinuities of an image like anL1 measure, and also stability and rapid convergencecharacteristic of an L2 measure. Meanwhile, the in-corporation of the interaction between pixels adap-tively adjusts the regularization during the deconvo-lution process. We have presented results on twodifferent kinds of data. Comparing the results withtraditional deconvolution techniques (e.g., RLA), ouralgorithm remarkably alleviates the noticeable arti-facts in the reconstructed images.

This work was supported in part by the ResearchGrants Council of the Hong Kong Special Adminis-trative Region, China under projects HKU 713906and 713408.

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