How to design a dictionary for the OMP algorithm which updates

Given an overcomplete dictionary matrix, where each of its columns represents an atom of a prototype signal. Given a signal y, it can be represented as a sparse linear combination of these atoms. The signal y can be expressed as y = Dx , or. By overcompleteness in the dictionary matrix, we mean that the number of atoms is much larger than the length of the signal y (whose length is obviously n), i.e., n<<k.

2. MP Algorithm (Matching Tracking Algorithm)

2.1 Algorithm Description

As one of the methods to sparsely decompose a signal, the signal is decomposed over the library of the complete dictionaries.

Assume that the represented signal is y and its length is n. Assume that H denotes the Hilbert space in which a set of vectors constitutes the dictionary matrix D, where each vector can be called an atom, whose length is the same as the length of the represented signal y, n, and which have been treated as normalized, i.e., |, i.e., unit vectors of length 1. MP Algorithm The basic idea of the MP algorithm is: from the dictionary matrix D (also known as the library of overcomplete atoms), select an atom (i.e., a certain column) that best matches the signal y, construct a sparse approximation, and solve for the signal residuals, and then continue to select the atom that best matches the signal residuals, and iterate repeatedly so that the signal y can be represented by these atoms by linearly summing them together with the final residual value. Obviously, if the residual value is within a negligible range, the signal y is a linear combination of these atoms. What if the atom that best matches the signal y is chosen? How to construct a sparse approximation and solve for the residuals? How to perform iterations? Let's detail the steps of signal decomposition using MP:[1] Calculate the inner product of the signal y with each column (atom) in the dictionary matrix, and select the atom with the largest absolute value, which is the one that best matches the signal y in this iterative operation. To put it in technical terms: the signal, from the dictionary matrix, selects the atom that most closely matches, satisfying, r0, the index of a column of the dictionary matrix. In this way, the signal y is decomposed into two parts: the vertical projection component at the most matched atom and the residual value, i.e. [2] The same decomposition as in step [1] is performed for the residual value R1f, then step K yields.