2017.08-Osher_LOW DIMENSIONAL MANIFOLD MODEL for IMAGE PROCESSING
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摘要
- 本文提出了一个新的低维流形模型(low dimensional manifold model,简称LDMM)并将其用于图像处理问题中;
- LDMM基于这样的fact:很多自然图像的块流形有低维的结构;因此,可以用块流形的维数作为正则化来恢复图像。
- LDMM中的关键是在point cloud上解决laplace-beltrami方程,可通过点积分(point integral)方法解决;
- point interal方法正确的采用采样点的方法,结果比标准的graph laplacian好。
- 在图像去噪、修复、超分辨重建中的数值仿真表明LDMM是一个powerful method.
Introduction
- 图像恢复问题是不适定的,因此我们需要加正则化(也即先验);
- TV,可以很好的恢复图像的卡通部分,纹理部分却恢复的不好;
- nonlocal TV,可以很好的恢复纹理;
- 本文motivation:图像
f 的patch setP(f) 是欧式空间的一个点集,P(f)采样自一个低维的光滑流形M(f) ,成为patch manifold。本文想去恢复原始的图像,使得图像的patch manifoldM(f) 尽可能小,从而得到如下的优化模型mindim(M(f))+λ||y−Φf||22
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