Image low-rank, sparse and image deraining algorithm

1. Low-rank and sparse characteristics of image matrix

1.1 Rank of the image

Ok, since the rank can measure the correlation, the correlation of the matrix actually represents the structural information of the matrix. If the correlation between the rows of the matrix is ​​strong, it means that the matrix can actually be projected into a lower-dimensional linear subspace, that is, it can be fully expressed with a few vectors, and it is of low rank. So the summary point is: if the matrix expresses structural information, such as images, user-product recommendation tables, etc., then there is a certain correlation between the rows of the matrix, and the matrix is ​​generally of low rank.
If X is a numerical matrix with m rows and n columns, rank(x) is the rank of x, and if rank(X) is much smaller than m and n, then x is said to be a low-rank matrix. Each row or column of a low-rank matrix can be linearly represented by other rows or columns, which shows that it contains a lot of redundant information. Using this redundant information, the data can be restored, and features can also be extracted from the data.

To sum up: the measure of the rank of a matrix is ​​actually the correlation between the rows and columns of the matrix. A matrix is ​​full rank if its rows or columns are linearly independent. The number of rows or columns of nonzero elements determines the rank.

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