Mathematical Modeling in Image Restoration, Image Analysis and Beyond

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:Bin Dong(Peking University)
:2019-03-07 16:30
:数理大楼117

AbstractImage 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 approach PDE based approach (including variational models PDE models) are two of the most successful approaches are widely adopted in both academia industry. The development of the two approaches followed rather different paths: wavelet frame based approach uses the tools from applied harmonic analysis utilize sparsity to model images, while PDE based approach uses functions spaces relies on geometry. Therefore, the two approaches are often considered as different approaches.

 

The first half of this talk is based on a series of papers, where we established rigorous generic connections between wavelet frame PDE based approach. This includes connections of wavelet frame based approach to total variation model, the Mumford-Shah model, the total generalized variational model. Furthermore, connections of wavelet frame shrinkage to a rather general form of nonlinear evolution PDEs is also established, where the Perona-Malik equation, Osher-Rudin’s shock filters Navier-Stokes image inpainting equation are special cases. Other than the establishment of the links between the two approaches, brnew models for both approaches are also discovered, which combine merits from both approaches, thus outperform existing models in various applications in image restoration.

 

Our theoretical studies also enable us to connect mathematical modeling computations with deep learning. The connections not only can provide guidance to deep network design, which is a central task in deep learning, but also enable us to tackle challenging problems in applied computation mathematics. In the second half of my talk, I will present our recent work on bridging numerical differential equations with deep neural network design for various tasks of inverse problems, image processing analysis.

 

Speaker Introduction董彬,北京大学,北京国际数学研究中心长聘副教授、主任助理,北京大数据研究院深度学习实验室研究员、生物医学影像分析实验室副主任。2009年在美国加州大学洛杉矶分校数学系获得博士学位。博士毕业后曾在美国加州大学圣迭戈分校数学系任访问助理教授、20112014年在美国亚利桑那大学数学系任助理教授,2014年底入职北京大学。主要研究领域为应用调和分析、优化方法、机器学习、深度学习及其在图像和数据科学中的应用。在理论上,将图像领域独立发展近30年的两个数学分支(PDE/变分方法和小波方法)建立深刻的联系,改变了领域内对这两类方法的认识,拓宽了这两类方法的应用范畴。应用上,以数学理论为指导思想,为来源于医学影像、计算机视觉、深度学习等领域中的重要问题提供行之有效的解决方案。董彬在包括《Journal of the American Mathematical Society》、《Applied Computational Harmonic Analysis》、《SIAM系列期刊》、《Inverse Problems》、《Mathematics of Computation》、《Journal of the Royal Statistical Society Series B》、《MICCAI》、《ICML》、《ICLR》在内的国际重要学术期刊和会议上发表论文50余篇,拥有2项美国专利,现任期刊《Inverse Problems Imaging》编委。2014年获得香港求是基金会的求是杰出青年学者奖,2015年入选中组部第十一批千人计划青年人才。