Some understanding of the rank sum of matrix and the rank of image

Reference link ：cv The paper （Low-rank relevant ）_weixin_30790841 The blog of -CSDN Blog

Related papers

Chapter one ：RASL: Robust Alignment by Sparse and Low-rank Decomposition for Linearly Correlated Images , This is my contact with Low-rank The first article , Article utilization Low-rank The algorithm of image alignment （Alignment） At the same time, it can also effectively block , From the effect of the experiment , The alignment effect and de occlusion effect of the algorithm are still very good . However, this algorithm can only deal with batch images , Cannot process a single picture , This also limits its application scenarios , Let me briefly introduce the implementation process of this article .

    First of all, I believe everyone is familiar with the concept of rank in the matrix , Suppose we are given the same person's 10 A picture , If this 10 Picture expression , light , The deflection angle of the face is exactly the same , And it is not interfered by any noise, and there is no occlusion . So take this 10 The pictures are pulled into columns to form a matrix , Ideally , The rank of this matrix should be 1, We can think of it as 10 The picture is perfectly aligned . If this 10 Only a few of the pictures are not aligned , It's covered , The position is offset or affected by light , So take this 10 A matrix of pictures put together , The rank of the matrix is certainly not 1. If this 10 A picture of a human face , The differences between them are very big , Then put them together to form a matrix , The rank may be full of paper . From this perspective , We can argue that low-rank It is a mathematical representation of image alignment . Due to the actual situation , Aligned pictures cannot be exactly the same , So it's impossible for 1, But we can relax the conditions , When the matrix composed of samples , Rank comparison is small , It can be considered that the effect of sample alignment is relatively good . This is the main mathematical idea of the article , It seems very simple, isn't it , But it is not so easy to realize , It involves a lot of mathematical operations .

Reference link ： The rank of a matrix , Eigenvalues and eigenvectors Basic concept of matrix _TranSad The blog of -CSDN Blog _ Rank sum eigenvalue of matrix

The rank of a matrix , Eigenvalues and eigenvectors

1. The rank of a matrix
 

Put one m*n Take out the matrix of , You can get its row vector group （ Each line is a vector , common m Vector ） Or column vector group （ Each column is a vector , common n Vector ）.

  Let's take the row vector group to analyze ：

  If this m Vector , They are all linearly independent of each other , Then the rank of this matrix is m. In fact, that is ： There are several linear independent , Then the rank of this matrix is . What is? “ Linear independence ”, for instance , vector [1,2,1] Sum vector [2,4,2] You can tell it at a glance *2 The relationship between , Then these two “ Current relevant ”; If it's a vector [1,2,1] Sum vector [1,5,2], Then these two matrices are linearly independent .

  To judge a1,a2,a3 Whether it is linear or not , Can make k1a1+k2a2+k3a3=0, Judge whether by solving the equation a1,a2,a3 Linearly independent . if k1=k2=k3=0, Then linearly independent .

  There's a rule here ： Whether we take the row vector group or the column vector group of the matrix to calculate , The final rank is the same .

  I understand it ： The rank of a matrix is the number of uncorrelated vectors in the matrix .

2. Eigenvalues and eigenvectors of matrices
 

First of all, clear , Matrix times vector , It can stretch and rotate vectors .

  For example, I now have a vector （1,1）

  Now it is like this ：

If you multiply a matrix by this vector （1,1）, For example, multiply this matrix ：

  We can see that we will get the result （2,1）, This is the effect drawn on the picture ：

 

           It can be found that we only use a diagonal matrix here , Realize an extension of the original vector on the abscissa . Think more about , In fact, if the upper right corner of the matrix is not 0, Or the lower left corner is not 0（ Assume a positive value ）, Will cause the original vector to rotate to the right or to the left . It can be concluded that ： A matrix can act on a vector , Make it rotate or stretch .

          Is there a kind of vector , After I use a matrix to act , Its direction remains the same ？ Yes ！ In this way, eigenvalues and eigenvectors can be derived ：

          For a matrix A, We look for a non-zero vector x, bring A After action x（ namely Ax） And x parallel , And the size meets Ax=λx（λ It's a constant ）. Once such existence is found , be x It's the eigenvector ,λ It's characteristic value .

          Put it another way ： We use matrices A Deactivation （ multiply ） All the vectors in the world . Where the matrix A Most of the applied vectors will be rotated and stretched , Only the matrix A The eigenvector of can be alone （ Keep the original direction ）~

  Actually , The eigenvector represents “ In one direction ”, It can represent the key information of the matrix . So such two beings with the same characteristics interact , Naturally, the direction will not change . And they “ Direction ” The more consistent （ The more fellow believers ~）, The greater the effect ,λx The eigenvalue λ And the bigger , So eigenvalues λ The size of also represents the importance of this eigenvector （ Compared with other eigenvectors of this matrix ）. I remember learning and contacting PCA When it comes to dimensionality reduction , The larger the eigenvalue, the more important the eigenvector , Because such vectors can better represent image features , That's what it means .

          A matrix has multiple eigenvectors , We use feature space to represent all these . That is, the feature space contains all the feature vectors .

         Let's look at the definitions of eigenvalues and eigenvectors . Let's combine the understanding of matrix and spatial transformation , The effect of matrix on vector , It is equivalent to transforming the original space into a new space ; If we use the understanding that matrix is linear transformation （ Understand linear algebra visually （ One ）—— What is a linear transformation ？）, That is to say, the transformation of the original base .

          Understand from the perspective of space , Multiplying a vector by a coefficient is actually scaling the vector （ Length or positive and negative direction changes , But still in the same straight line ）, And matrix （ Matrix ） The function of is the transformation of space . If a matrix only scales a vector , Without changing the angle , Then this vector is called eigenvector , The scaling ratio is called eigenvalue .（ Reference link ：https://blog.csdn.net/qq_34099953/article/details/84291180）

Reference link ： The rank of image and the rank of matrix _ Blog of deep blind learning -CSDN Blog _ Rank of image matrix

2、 Rank of image rank sum matrix

The rank of a matrix = The largest linearly independent row （ Or column ） Number of vectors . For images , Rank can indicate the richness of information contained in the image 、
Redundancy 、 noise .

The smaller the rank ：

• The number of bases is small
• Large data redundancy
• Image information is not rich
• Less image noise

Low rank matrices

Concept

When the rank of the matrix is low （r << n, m）, It can be regarded as a low rank matrix . Low rank matrix means , There are more rows in this matrix （ Or column ） It's linear correlation , namely ： Information redundancy is large .

effect

• Using redundant information of low rank matrix , Missing data can be recovered , This problem is called “ Reconstruction of low rank matrix ” , namely ：“ Suppose the recovered matrix is low rank , Use the existing matrix elements , Recover the missing elements in the matrix ”, It can be applied to image restoration 、 Collaborative filtering and other fields .
• In deep learning , Too many parameters of convolution kernel , There is often large redundancy , namely ： Convolution kernel parameters are low rank . here , The convolution kernel can be decomposed with low rank , take k × k The convolution kernel of is decomposed into a k × 1 And a 1×k The core of , This can reduce the number of parameters 、 Speed up the calculation 、 Prevent over fitting .

Sparse coding

Why sparse coding ？
         First , There is a hypothesis （ It's also true , It's a priori knowledge ）—— The signal composition of nature is sparse . After sparse coding , Dimension reduction can be achieved , The role of improving computing efficiency .

Image and rank

problem ： How to define an image or an object with low rank attribute ？ Images with high rank attributes , If there is a missing phenomenon , How to recover , What methods exist to solve ？
   answer ： In image processing , The rank of an image can be simply understood as the richness of information contained in the image , Due to the similarity between local blocks 、 Repeatability , It often has the attribute of low rank ; The rank of the image is relatively high , It may be due to the influence of image noise , Through the limitation of low rank , It should be able to achieve a good denoising effect .