稀疏表达以及相关改进

学习 http://blog.youkuaiyun.com/tiandijun/article/details/41578175  后随笔

1，sparse representation

解决问题：当数据量增大后，线性表达的基求解十分复杂，而且很多事多余的，稀疏表达可以解决这个问题。

稀疏直观理解就是在满足误差小和非零项尽可能多，非零项就是解决零范数问题，但是约束太强，是非凸问题，松弛约束之后就是1范数，这个是凸优化问题，然后就继续

松弛，有了p范数的问题。

如果从一个很大的dictionary里选择，很有可能会有一个与输入极为相似，这也就很有可能有sparse solution [0,...,1,...0]；

所以一个有过完备字典（over complete dictionary），稀疏解可以work

Limitations of sparse representation

改进

Nonlocally Centralized Sparse Representation

[1]W. Dong, L. Zhang and G. Shi, “Centralized Sparse Representation for ImageRestoration”, in ICCV 2011.

[2]W. Dong, L. Zhang, G. Shi and X.Li, “NonlocallyCentralized Sparse Representation for Image Restoration”,IEEE Trans. on ImageProcessing,vol. 22, no. 4, pp.1620-1630, April 2013.

NCSR: idea

NCSR: objective function

NCSR:solution

NCSR: The parameters and dictionaries

Gradient Histogram Preservation

[1] W. Zuo, L. Zhang, C. Song, and D. Zhang, “Texture Enhanced Image Denoisingvia

Gradient Histogram Preservation,” in CVPR 2013.

[2] W. Zuo, L. Zhang, C. Song, D. Zhang, and H.Gao, “GradientHistogram Estimation

and Preservation for Texture Enhanced Image Denoising,” in TIP 2014.

针对问题：

Like noise, textures are fine scale structures in images,and most of the denoising algorithms will remove the textures while removingnoise.

• Is it possibleto preserve the texture structures, to some extent, in denoising?

GHP：

Group sparsity

想法： 

通常认为一些东西的基（basis）和另一些东西不同，所以可以按照sample or feature分group.

An observation that features or data items within a group are expected to share the same sparsity pattern in their latent factor representation.

Different types of features, such as in CV: pixel values, gradient features, 3D pose features, etc. 同种feature组成一个group

Rank decomposition

rank 描述的是矩阵的相关性属性，其中的每一个atomy因为位置和排列呈现出二维的秩序，所以rank decomposition 是从2D角度剖析相似度的structure.