Wavelet and PDE for Image Processing

In this series of papers, fundamental connections between the wavelet frame-based approach and variational methods were established. In particular, connections to the total variation model were established in [4,2], and to the Mumford-Shah model was established in [3]. Furthermore, we established a generic connection between iterative wavelet frame thresholding methods and general nonlinear evolution PDE models in [1]. We showed that wavelet frame transforms are discretizations of both variational and a generic type of nonlinear evolution PDE models that include the Perona-Malik equation and the shock filters.  Hence,  various PDE-based models can be understood as applying different thresholding\shimkage operators in the iterative thresholding algorithm on wavelet frame coefficients. This new understanding essentially merged the two seemingly unrelated areas: wavelet frame-based approach and PDE-based approach. It also gave birth to many innovative and more effective image restoration models and algorithms for various applications

    1. Bin Dong, Qingtang Jiang, Zuowei Shen, Image restoration: wavelet frame shrinkage, nonlinear evolution PDEs, and beyond, Multiscale Modeling and Simulation: A SIAM Interdisciplinary Journal, 15(1) (2017), 606-660. PDF
    2. Bin Dong, Zuowei Shen, Peichu, Xie; Image Restoration: A general wavelet frame based model and its asymptotic analysis, SIAM Journal on Mathematical Analysis, 49(1)(2017), 421-445. PDF
    3. Jianfeng Cai, Bin Dong, Zuowei Shen, Image restorations: a wavelet frame based model for piecewise smooth functions and beyond, Applied and Computational Harmonic Analysis, 41(1), (2016), 94-138. PDF
    4. Jianfeng Cai, Bin Dong, Stanley Osher, Zuowei Shen, Image restoration: total variation, wavelet frames, and beyond, Journal of the American Mathematical Society, 25 (4) (2012), 1033-1089. PDF