sparse representation

Sparse Representation Definition
The mathematical essence of sparse representation is signal decomposition under sparse regularization constraints. With the continuous development of signal and image processing technology, how to use signal and image components (such as principal components, secondary components, independent components, sparse components, morphological components, etc.) to represent signals and images has become a lot of signal and image processing tasks. For example, research hotspots such as compression, reconstruction, noise suppression and feature extraction are of great significance. How to describe the energy of the signal and the information capacity of the image as much as possible through the minimum number of coefficients has become a problem in the study of signal and image representation methods. Different types of signals, the distribution of the coefficients of the image under different transformations will be different. The traditional signal representation theory is mostly based on the transformation of non-redundant orthogonal basis functions, such as Fourier transform, wavelet transform, etc. However, Fourier transform cannot express the time-frequency local properties of signals, and although wavelet transform has point singularity in processing However, the separable wavelets formed by one-dimensional wavelets have only limited directions, and cannot/optimally 0 represent high-dimensional functions with line or surface singularity. In order to better represent signals and images, In recent years, based on the orthogonal wavelet transform, many new transform methods have been proposed, such as Ridgelet, Curvelet, Bandelet, Contourlet [11], etc. Transformations. Based on these transformations, overcomplete redundant representations are commonly used. The basic idea is that the basis functions are replaced by overcomplete redundant function systems called dictionaries. The elements in the dictionary are called atoms. A signal is represented by a linear combination of atoms. The number of atoms is (much) larger than the dimension of the signal, thus creating redundancy. Because of this overcompleteness, there are many ways to represent a signal with the fewest coefficients The (sparsest) representation is the simplest and the one we consider the best. [1]
Sparse representation
model The general form of the existing sparse representation model is as follows:
X=argmin||y-Dx||k+λ||x||
Among them, y is the observation data, D is the dictionary, x is the sparse vector to be estimated, λ is the regular parameter, and k (1≤k<2) is the sparse measure. Among them, λ and k are unknown and need to be determined in advance (although k = 1 is usually taken, but the model is more flexible when k < 1). The theoretical study of the model mainly includes the approximation degree of the model solution and the l0 norm minimization solution, the uniqueness and stability of the sparse representation model solution, and so on. However, in some specific applications such as image enhancement and optimal allocation of measurement and control resources, the sparse metric is not the only and most important indicator. [2]
Model solution algorithm
The solution of the above model is divided into solution algorithms based on mathematical models, such as base pursuit, Focuss, Shrinkage, etc., and solution algorithms that do not consider mathematical models, such as matching pursuit algorithm family. However, there is one more problem to be solved in the existing algorithms, that is, the regular parameter λ and the parameter k representing the sparsity need to be pre-determined, and then solved. If the solution does not meet the requirements, readjust the values ​​of the two parameters until a satisfactory solution is obtained. This makes the model unable to achieve the degree of automation in application, which limits the application of sparse representation methods. [2] The
dictionary learning algorithm
was originally in the field of sparse representation research. Generally, it is assumed that the dictionary is known, and only the unknown sparse vector is solved. Some scholars have studied the selection and learning methods of dictionaries for situations where dictionaries are unknown. Existing dictionary learning methods can be divided into two types: training sample-based and parameterized dictionary-based. Among them, the latter is more difficult, and it is necessary to deeply analyze the characteristics and description methods of the studied signals. The process of dictionary learning generally adopts a two-step method, which is combined with the solution of the sparse representation model. [2]
References
[1] Li Ying, Zhang Yanning, Xu Xing. Morphological Component Analysis Based on Signal Sparse Representation: Progress and Prospects [D]. , 2009.
[2] Baidu Encyclopedia, wds1315, Sparse Representation, https:/ /baike.baidu.com/item/%E7%A8%80%E7%96%8F%E8%A1%A8%E7%A4%BA/16530498?fr=aladdin, 2017-05-19，2018-04-25.