DETERMINISTIC TECHNIQUES FOR MULTICHANNEL BLIND IMAGE DECONVOLUTION W. Souidene 1,2 , K. Abed-Meraim 1 1 Telecom Paris, 46 rue Barrault 75013, Paris souidene, abed@tsi.enst.fr A. Beghdadi 2 2 L2TI, 99 avenue J.B Cl´ ement 93430, Villetaneuse beghdadi@l2ti.univ-paris13.fr ABSTRACT In this paper, we address four deterministic methods for blind multichannel identification in a blind image restoration frame- work. These methods are: SubSpace method (SS), Minimum Noise Subspace (MNS) and Symmetric MNS (SMNS) meth- ods, Cross Relation method (CR) and Least Squares Smooth- ing method (LSS). The latter is a new method that is intro- duced, here, for the first time, as an extension, from the 1-D to the 2-D case, of the least squares method by L. Tong et al. (1999). For each method, we detail its basic principle and provide a summary of its corresponding algorithm. In the noise free case, all the methods developed here offer a perfect channel identification. In the noisy case, these methods have a different behavior and their performance are compared in terms of channel identification by means of MSE. 1. INTRODUCTION The multichannel blind image restoration is an emerging re- search field that aims to restore an original image given sev- eral noisy blurred observations of this latter. This can be achieved three ways: we can directly restore the original im- age (equalization techniques); we can, first, identify the chan- nels and then restore the original image; finally, we can, jointly, identify the channels and restore the original image. Many techniques were proposed for each type of solution. Stochas- tic ones fail to perfectly restore the original image even in the noise free case due to finite sample size effect. In this article, we present a brief overview of the most famous deterministic methods for 2-D blind multichannel identification. We also introduce a new method, namely, the LSS one, that is an ex- tension of the method proposed in [7] from the 1-D to the 2-D case. The considered methods are part of the second family of techniques which consist in channel identification aiming at image restoration. Four deterministic techniques are pre- sented. These methods were first introduced for the 1D case [1] [6] [4] [7], and then some of them were extended to the image case [2] [3]. In this article, we propose the detailed al- gorithm of each technique and we compare their performance. 2. PROBLEM STATEMENT We deal with a SIMO system, this supposes that a single im- age passes through K independent channels and K different noisy blurred images are observed. In the sequel, the images are put into vector form. The notations used are: - f the original image of size m f × n f . - g 1 ,..., g K the K output images each of size m g × n g . - h 1 ,..., h K the K channel impulse responses each of size m h × n h with h i =[h i (1, 1),h i (1, 2),...,h i (m h ,n h )] T . - n 1 ,..., n K the additive noise in each channel. Note that the channel diversity is assumed large enough so that the K distinct channels do not share common zeros [3]. Given these notations we can write the system model 1 : g i (n 1 ,n 2 )= m h  l1=1 n h  l2=1 h i (n 1 ,n 2 )f (n 1 − l 1 +1,n 2 − l 2 + 1) for i =1 ...K. All the methods studied in this paper aim to identify the K channels, i.e. we directly search for h = [h T 1 ,..., h T K ] T . To do so, we process the observed images through windowed areas of size m w × n w . Let g i (n 1 ,n 2 ) be the data vector corresponding to a window of the i th image which last pixel is indexed by (n 1 ,n 2 ) and let f (n 1 ,n 2 ) be the corresponding filtered window of size (m w + m h − 1) × (n w + n h − 1): g i (n 1 ,n 2 )=[g i (n 1 ,n 2 ),...,g i (n 1 ,n 2 − n w + 1), g i (n 1 − 1,n 2 ),...,g i (n 1 − m w +1,n 2 − n w + 1)] T By stacking the K vectors g i into a single vector, we obtain: g(n 1 ,n 2 ) = [g T 1 (n 1 ,n 2 ),..., g T K (n 1 ,n 2 )] T = Hf (n 1 ,n 2 )+ n(n 1 ,n 2 ) with H =  H T 1 ,..., H T K  T , H i being the filter matrix as- sociated to h i . Our objective here is the blind identification of the channel parameter vector h given the blurred images g 1 ,..., g K . Indeed, channel estimation errors may result in significant degradation of the restored image quality. Figure 1 illustrates this effect in the mono-channel case. Let h be the original filter vector. We disturb it by adding ǫΔh where ǫ is a varying positive scalar and Δh a fixed random filter vector of unit norm. Hence, ˆ h = h + ǫΔh represents the blur func- tion used for image restoration by means of Lucy-Richardson method in [9]. The restored image quality is measured using the SSIM (Structural Similarity Index Measure) introduced in [8]. Figure 1 illustrates the degradation of the restored image quality due to channel identification errors. 1 For the sake of notational simplicity, we have adopted here a ’causal’ representation of considered filters 0-7803-9243-4/05/$20.00 ©2005 IEEE 439