Bin Dong, University of Arizona / 北京国际数学中心

Title

Wavelet Frame Transforms and Differential Operators: Bridging Discrete and Continuum for Image Restoration

Abstract

Image restoration, including image denoising, deblurring, inpainting, computed tomography, etc., is one of the most important areas in imaging science. In image restoration, wavelet frame based models, such as the analysis based model, and differential operator based models, such as variational and PDE models, have been widely used and proven successful in many applications. These approaches were developed through different paths and generally provided understandings from different angles. Since both approaches are to model the same type of problems with success, it is natural to ask whether wavelet frame based approach is fundamentally connected with variational/PDE based approach when we trace all the way back to their roots.

My talk is based on a series of three papers ([1-3] below). In [1], we established connections between wavelet frame transforms and differential operators in variational framework. In [2], we established their connections for nonlinear evolution PDEs. Based on [1,2], we proposed a new piecewise smooth image restoration model based on wavelet frames in [3], and linked it with a brand new variational model, a special case of which resembles, but is superior to, the well-known Mumford-Shah model. The connections established in [1-3] provide us with new insights and inspiring interpretations of both wavelet frame and differential operator based approaches, which enable us to create new models and algorithms for image restoration that combine the merits of both approaches. The significance of our findings is beyond what it may appear. In fact, our analysis and discussions in [1-3] already indicate that wavelet frame based approach is a new and useful tool in numerical analysis to discretize and solve variational and PDE models in general, which enriches the existing theory and applications of numerical PDEs, variational techniques, wavelet frames, etc.

References

[1]. J. Cai, B. Dong, S. Osher and Z. Shen, Image restoration: total variation; wavelet frames; and beyond, Journal of AMS, 25(4), 1033-1089, 2012.

[2]. B. Dong, Q. Jiang and Z. Shen, Image restoration: wavelet frame shrinkage, nonlinear evolution PDEs, and beyond, preprint, December 2013.

[3]. Jian-Feng Cai, Bin Dong and Zuowei Shen, Image restorations: a wavelet frame based model for piecewise smooth functions and beyond, preprint, April 2014.


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