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Directional total generalized variation regularization

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Abstract

In inverse problems, prior information and a priori-based regularization techniques play important roles. In this paper, we focus on image restoration problems, especially on restoring images whose texture mainly follow one direction. In order to incorporate the directional information, we propose a new directional total generalized variation (DTGV) functional, which is based on total generalized variation (TGV) by Bredies et al. After studying the mathematical properties of DTGV, we utilize it as regularizer and propose the \(\hbox {L}^2\hbox {-}\mathrm {DTGV}\) variational model for solving image restoration problems. Due to the requirement of the directional information in DTGV, we give a direction estimation algorithm, and then apply a primal-dual algorithm to solve the minimization problem. Experimental results show the effectiveness of the proposed method for restoring the directional images. In comparison with isotropic regularizers like total variation and TGV, the improvement of texture preservation and noise removal is significant.

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Authors and Affiliations

  1. Department of Applied Mathematics and Computer Science, Technical University of Denmark, 2800, Kgs. Lyngby, Denmark

    Rasmus Dalgas Kongskov, Yiqiu Dong & Kim Knudsen

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  1. Rasmus Dalgas Kongskov
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  2. Yiqiu Dong
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  3. Kim Knudsen
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Correspondence to Yiqiu Dong.

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Communicated by Lothar Reichel.

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The work was supported by Advanced Grant 291405 from the European Research Council, Grant No. 4002-00123 from the Danish Council for Independent Research Natural Sciences, and Grant 11701388 from the National Natural Science Foundation of China.

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Kongskov, R.D., Dong, Y. & Knudsen, K. Directional total generalized variation regularization. Bit Numer Math 59, 903–928 (2019). https://doi.org/10.1007/s10543-019-00755-6

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