Comparison of Wavelet Transform and OMP Algorithm in Compressed Sensing

In this paper, we explore the comparison between wavelet transform and the Orthogonal Matching Pursuit (OMP) algorithm in compressed sensing, representing a relatively recent algorithmic advancement. Notably, wavelet transform serves as a mathematical tool that decomposes signals into a series of waveforms, which can be implemented through filter banks in code using functions like wavedec() and waverec() for decomposition and reconstruction. These waveforms find applications in signal processing, data compression, and noise reduction. Compressed sensing, an emerging signal processing technique, enables signal compression without information loss through sparse representation and random sampling, reducing storage and transmission costs. Here, we investigate how OMP algorithm implementation—typically involving iterative selection of dictionary atoms and residual updates—can be applied within wavelet transform and compressed sensing frameworks, along with comparisons to alternative algorithms like Basis Pursuit. By examining the working principles, computational complexity (typically O(k*m*n) for k-sparse signals), and application scenarios of these algorithms, we can better understand their advantages and limitations, laying groundwork for future research and practical implementations.