Unveil Compressed Sensing

We discuss the applicability of compressed sensing theory. We take a genuine look at both experimental results and theoretical works. We answer the following questions: 1) What can compressed sensing really do? 2) More importantly, why? I. WHAT CAN COMPRESSED SENSING DO? Compressed sensing theory is described and studied as a panacea in many fields of science and engineering, evidenced by the website [2], which is built and managed by researchers of Rice University. We hereby take a genuine look at its performance from two most representative research results. A. Compressed Sensing Result of Rice University The research result of Rice University on compressed sensing is called “single-pixel camera” and is posted at website [3]. It is marketed by the company InView Technology ([4]). Fig. 1 demonstrates the performance of compressed sensing result by Rice University in comparison with Rapid technology which implements system compression method. All original image data and the Rice results are from their website [3]. We can see that, the Rice results lose color and shape. In striking contrast, Rapid results miraculously achieve visually lossless to original images. In addition, Rapid runs hundreds times faster than Rice product. The Rapid demo software and measurement data (partial samples) can all be downloaded from website [1]. B. Compressed Sensing Result of MIT The research result of MIT (Massachusetts Institute of Technology) on compressed sensing has been reported by MIT News at least three times, see website [5]. Fig. 2 demonstrates the performance of compressed sensing result by MIT in comparison with Rapid technology which implements system compression method. All original images and MIT results are from MIT News website [5] and thesis report [6] (page 90, Appendix B). Rapid demo software and measurement data (partial samples) can all be downloaded at the website [1]. Mug, original Compressed sensing System compression (measurements=40%) (measurements=25%) Soccer, original Compressed sensing System compression (measurements=40%) (measurements=25%) Dice, original Compressed sensing System compression (measurements=40%) (measurements=25%) Mandrill, original Compressed sensing System compression (measurements=40%) (measurements=25%) Fig. 1 Performance of compressed sensing by Rice University. II. WHY COMPRESSED SENSING PERFORM POOR? The poor performance of compressed sensing is caused by its lack of solid theoretical support. We hereby give detailed analysis over the mathematical framework of compressed sensing. We provide a counterexample which disproves the foundation theorem of compressed sensing theory. Furthermore, by a simple analysis, we can find that the main theoretical result which directs methodological practice of compressed sensing actually does not provide useful support for practical applications. A. Disprove the Fooundation Theorem The foundation theorem of compressed sensing theory, Theorem 1.1 in [7] states that “Suppose that the signal length N is a prime integer. Let Ω be a subset of {0, ... , N-1}, and let f be a vector supported on T such that . Then f can be reconstructed uniquely from Ω and ̂” It is further clarified that “Theorem 1.1 asserts that one can reconstruct f from 2|T| frequency samples (and that, in general, there is no hope to do so from fewer samples). In principle, we can recover f exactly by solving the combinatorial optimization problem ( ) ̂ ̂ , where is the number of nonzero terms ( ) .” This theorem is praised as “a very significant advance” in [10]. Since N is a prime number, we know that N = 2k+1 for an integer k. Let such that it has symmetric structure for Let n = |Ω|. We can show that a vector f supported on T such that is not necessarily the unique solution of problem ( ) and hence cannot be reconstructed from Ω and ̂ Let Ψ be the matrix that contains the n rows of Fourier matrix which are indexed by elements of Ω,