Observation matrix construction method based on low-coherence unit norm tight frame

Abstract

The invention relates to an observation matrix construction method based on a low-coherence unit norm tight frame, and belongs to the technical field of signal processing, and the method comprises thesteps: S1, initializing an initial observation matrix phi 0 into a random part Fourier matrix, and enabling the initial observation matrix phi 0 to serve as an initial alpha tight frame F; s2, calculating a Gram matrix corresponding to the framework F, and projecting the matrix to a structure constraint set of a tight framework by using a contraction function to generate a new Gram matrix; s3, updating the Gram matrix through a weighted iteration process; s4, reducing the rank of the new Gram matrix, calculating the square root of the new Gram matrix, and finding out a tight frame closest toa unit norm tight frame; and S5, solving the optimal target function to obtain an observation matrix. According to the method, the mutual interference coefficient between the observation matrix and the sparse basis is reduced, the dependence degree on signal sparsity is reduced, the problem that an ETF frame is difficult to construct is avoided, the initial observation matrix is initialized into apart of Fourier matrix, the calculation complexity is reduced, and the pressure of storage and processing equipment is reduced.

Description

technical field [0001] The invention belongs to the technical field of signal processing and relates to an observation matrix construction method based on a low-coherence unit norm tight frame. Background technique [0002] With the advancement of image processing technology, multimedia information dominated by image information has become the main form of information in people's lives. According to the Nyquist sampling theorem, if the original signal is to be completely restored, the sampling frequency of the original signal must be greater than or equal to twice the highest frequency of the original signal. The high-frequency sampling rate has brought enormous pressure to the hardware equipment, and some very high-frequency signals cannot be sampled. Such a signal processing method is to first sample and then compress, and this large amount of data compression will lead to a reduction in data accuracy, thus making the final reconstructed signal less effective. In additio...

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Problems solved by technology

However, the threshold selection of the algorithm depends entirely on experience, and the number of iterations is large. When the Gram matrix shrinks, the rank of the Gram matrix will change, and the algorithm is unstable.

In order to avoid the shortcomings of the threshold method, the Equiangular Tight Frame (ETF) method appeared. By gradually updating the Gram matrix, the off-diagonal elements of the Gram matrix are close to the ETF, thereby reducing the mutual coherence of the Gram matrix. , but the construction of ETF in this method is more difficult

Therefore, some scholars propose to ensure that the Gram matrix is ​​positive semi-definite by defining a new structural constraint space, and construct a new isometric compact frame, but the size of the frame is easily limited by the dimension of the observation matrix, and the number of algorithm iterations is large

In addition, there is a method to reduce coherence by inverting the matrix. However, the computational complexity of such methods is too high, and not all matrices are invertible.

There is also a direct construction of a tight frame using convex optimization constraints. Although the performance of the constructed frame is excellent, the construction process is relatively complicated, and the obtained observation matrix is ​​not suitable for noise-containing signals.

[0005] Although the above method of constructing the observation matrix achieves the purpose of reducing the coherence between the observation matrix and the sparse basis, the constructed observation matrix is ​​less robust, the construction process of the tight frame is difficult, or the computational complexity is too high

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