A Simple Example of Compressed Sensing Theory

In compressed sensing theory, signal sampling plays a crucial role. Among various approaches, sparse sampling stands out as a common method. Specifically, sparse sampling involves collecting measurements at only a small subset of signal positions, thereby significantly reducing the required data volume for sampling. Following the sampling process, signal reconstruction becomes necessary. The Matching Pursuit (MP) algorithm serves as an effective reconstruction technique that can recover an approximate representation of the original signal from the known sampled data. The MP algorithm operates iteratively by selecting the dictionary atom that best matches the signal residual at each step, then updating the residual until convergence criteria are met. Therefore, sparse sampling and the MP algorithm represent two fundamental components of compressed sensing theory. Beyond these elements, compressed sensing theory also encompasses multiple stages including signal encoding, decoding, and transmission. Each stage requires careful design and optimization to ensure the overall system's performance and efficiency, often involving mathematical optimization techniques and computational implementations.