Abstract
Overcomplete representation is attracting interest in image restoration due to its potential to generate sparse representations of signals. However, the problem of seeking sparse representation must be unstable in the presence of noise. Restricted Isometry Property (RIP), playing a crucial role in providing stable sparse representation, has been ignored in the existing sparse models as it is hard to integrate into the conventional sparse models as a regularizer. In this paper, we propose a stable sparse model with non-tight frame (SSM-NTF) via applying the corresponding frame condition to approximate RIP. Our SSM-NTF model takes into account the advantage of the traditional sparse model, and meanwhile contains RIP and closed-form expression of sparse coefficients which ensure stable recovery. Moreover, benefitting from the pair-wise of the non-tight frame (the original frame and its dual frame), our SSM-NTF model combines a synthesis sparse system and an analysis sparse system. By enforcing the frame bounds and applying a second-order truncated series to approximate the inverse frame operator, we formulate a dictionary pair (frame pair) learning model along with a two-phase iterative algorithm. Extensive experimental results on image restoration tasks such as denoising, super resolution and inpainting show that our proposed SSM-NTF achieves superior recovery performance in terms of both subjective and objective quality.
Highlights
Sparse representation of signals in dictionary domains has been widely studied and has provided promising performance in numerous signal processing tasks such as image denoising [1,2,3,4,5], super resolution [6,7,8], inpainting [9,10] and compression [11,12]
In this paper we focus on a stable sparse model and on the development of an algorithm that would learn a pair of non-tight frame based dictionaries from a set of signal examples
We propose the two-phase iterative algorithm for dictionary pair learning by dividing Problem (23) into two subproblems: The sparse coding phase which updates the sparse coefficients Y and thresholding values λ, and the dictionary pair update phase which computes Φ
Summary
Sparse representation of signals in dictionary domains has been widely studied and has provided promising performance in numerous signal processing tasks such as image denoising [1,2,3,4,5], super resolution [6,7,8], inpainting [9,10] and compression [11,12]. Common overcomplete systems differ from the traditional bases, such as DCT, DFT and Wavelet, because they offer a wider range of generating elements; potentially, this wider range allows more flexibility and effectiveness in signal sparse representation. It is a severely under-constrained illposed problem to find the underlying overcomplete representation due to the redundancy of the systems. When the underlying representation is sparse and the overcomplete systems have stable properties, the ill-posedness will disappear [13]
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