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Convergence analysis for iterative data-driven tight frame construction scheme. (English) Zbl 1361.94006

Summary: Sparse modeling/approximation of images plays an important role in image restoration. Instead of using a fixed system to sparsely model any input image, a more promising approach is using a system that is adaptive to the input image. A non-convex variational model is proposed in [J.-F.Cai et al., Appl. Comput. Harmon. Anal. 37, No. 1, 89–105 (2014; Zbl 1336.94008)] for constructing a tight frame that is optimized for the input image, and an alternating scheme is used to solve the resulting non-convex optimization problem. Although it showed good empirical performance in image denoising, the proposed alternating iteration lacks convergence analysis. This paper aims at providing the convergence analysis of the method proposed in [loc. cit.]. We first established the sub-sequence convergence property of the iteration scheme proposed in [loc. cit.], i.e., there exists at least one convergent sub-sequence and any convergent sub-sequence converges to a stationary point of the minimization problem. Moreover, we showed that the original method can be modified to have sequence convergence, i.e., the modified algorithm generates a sequence that converges to a stationary point of the minimization problem.

MSC:

94A08 Image processing (compression, reconstruction, etc.) in information and communication theory
62-07 Data analysis (statistics) (MSC2010)

Citations:

Zbl 1336.94008
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References:

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