Tensor Factorization and Continuous Wavelet Transform for Model-free Single-Frame Blind Image Deconvolution Ivica Kopriva 1 and Qian Du 2 1 Ruđer Bošković Institute Bijenička cesta 54, 10000 Zagreb, Croatia 2 Department of Electrical and Computer Engineering Mississippi State University, MS 39762, USA E-mail: ikopriva@irb.hr, du@ece.msstate.edu Abstract Model-free single-frame blind image deconvolution (BID) method is proposed by converting BID into blind source separation (BSS), whereas sources represent the original image and its spatial derivatives. Continuous wavelet transform (CWT) is used to generate multi- channel image necessary for BSS. As opposed to an approach based on the Gabor filter bank, this brings additional options in adaptability to the problem at hand: through the choice of wavelet function and variation of the scale of the CWT. BSS is performed through orthogonality constrained factorization of the 3D multichannel image tensor by means of the higher- order-orthogonal-iteration algorithm. The proposed method virtually requires no information about blurring kernel: neither model nor size of the support. The method is demonstrated on experimental gray scale images degraded by de-focusing and atmospheric turbulence. A comparable or better performance is demonstrated relative to blind Richardson-Lucy method that, however, requires a priori information about parametric model of the blur. 1. Introduction Degradation of the spatial resolution of an image is caused by various sources (either individually or combined): atmospheric turbulence, defocusing, relative motion between image and object planes, aberrations, etc. Restoration of the original image from its blurred version is referred to as image restoration or image deconvolution [1, 2]. In non-blind deconvolution the blurring kernel is given [1], while in blind deconvolution it is unknown [2]. In majority of the cases it is, however, assumed that parametric model of the blurring kernel is known which leads to model- based blind image deconvolution (BID) [3-5]. This however presumes that source of the image degradation is known and that is not always the case in practice. Also, it is not always possible to have at disposal multiple frames as required by multi-frame BID [3,4]. Due to these reasons it has been exploited previously whether single-frame model-free BID problem is possible to solve? To this end, several algorithms for single-frame model-free BID have been demonstrated [6-10]. All these approaches have in common the following features: (1) BID is converted into BSS problem through the implicit use of the Taylor expansion of the shifted original image around origin in the image forming convolution equation [11]; (2) Gabor filter bank is used to generate multichannel version of the single-frame image. Hence, it is implicitly assumed that original image has certain degree of smoothness that limits performance of model-free BID in scenarios where degradation is strong and/or original image is composed of sharp edges [10]. Herein, we propose continuous wavelet transform (CWT) [12] as a substitute for Gabor filter bank with the following improvements: (1) since scale of the CWT is continuous, number of multichannel images can be varied by varying resolution and support of the scale; (2) selection of wavelet function gives additional degree of adaptability to the problem at hand. Since in considered problem sources represent original image and its spatial derivatives they are neither sparse nor statistically independent. Therefore, sparseness [6] and statistical independence [7] based approaches yield suboptimal result in multi-frame BID problem. Therefore, as in [9,10] we use tensor factorization (TF) with orthogonality constraints imposed on the factors and core tensor of the Tucker3 model of the multichannel image tensor. In comparison with the independent and sparse component analysis, TF approach yields constraints-relaxed solution of the model-free BID problem. The rest of the paper is organized as follows. Model-free BID problem with the TF based solution is formulated in section 2, and the CWT-based multichannel image generation is presented in section 3. The proposed method is demonstrated on experimental gray scale images degraded by defocusing and atmospheric turbulence in section 4, while conclusions are drawn in section 5. 2. Model-free Blind Image Deconvolution It is assumed that blurred gray scale image 1 2 0 I I R × + ∈ G , with I 1 and I 2 representing number of pixels, is degraded by space-invariant blur, also known as point spread function (PSF), and described by linear image forming convolution equation: