**Min Zhang 1, Yunhui Shi 1,\*, Na Qi <sup>1</sup> and Baocai Yin 1,2**


**\*** Correspondence: syhzm@bjut.edu.cn

Received: 18 January 2020; Accepted: 17 April 2020; Published: 22 April 2020

**Abstract:** The sparsity of images in a certain transform domain or dictionary has been exploited in many image processing applications. Both classic transforms and sparsifying transforms reconstruct images by a linear combination of a small basis of the transform. Both kinds of transform are non-redundant. However, natural images admit complicated textures and structures, which can hardly be sparsely represented by square transforms. To solve this issue, we propose a data-driven redundant transform based on Parseval frames (DRTPF) by applying the frame and its dual frame as the backward and forward transform operators, respectively. Benefitting from this pairwise use of frames, the proposed model combines a synthesis sparse system and an analysis sparse system. By enforcing the frame pair to be Parseval frames, the singular values and condition number of the learnt redundant frames, which are efficient values for measuring the quality of the learnt sparsifying transforms, are forced to achieve an optimal state. We formulate a transform pair (i.e., frame pair) learning model and a two-phase iterative algorithm, analyze the robustness of the proposed DRTPF and the convergence of the corresponding algorithm, and demonstrate the effectiveness of our proposed DRTPF by analyzing its robustness against noise and sparsification errors. Extensive experimental results on image denoising show that our proposed model achieves superior denoising performance, in terms of subjective and objective quality, compared to traditional sparse models.

**Keywords:** parseval frame; transform; sparse representation
