Imaging via Compressive Sampling [Introduction to compressive sampling and recovery via convex programming]

There is an extensive body of literature on image compression, but the central concept is straightforward: we transform the image into an appropriate basis and then code only the important expansion coefficients. The crux is finding a good transform, a problem that has been studied extensively from both a theoretical [14] and practical [25] standpoint. The most notable product of this research is the wavelet transform [9], [16]; switching from sinusoid-based representations to wavelets marked a watershed in image compression and is the essential difference between the classical JPEG [18] and modern JPEG-2000 [22] standards. Image compression algorithms convert high-resolution images into a relatively small bit streams (while keeping the essential features intact), in effect turning a large digital data set into a substantially smaller one. But is there a way to avoid the large digital data set to begin with? Is there a way we can build the data compression directly into the acquisition? The answer is yes, and is what compressive sampling (CS) is all about

Quantitative Robust Uncertainty Principles and Optimally Sparse Decompositions

We develop a robust uncertainty principle for finite signals in C^N which states that for almost all subsets T,W of {0,...,N-1} such that |T|+|W| ~ (log N)^(-1/2) N, there is no sigal f supported on T whose discrete Fourier transform is supported on W. In fact, we can make the above uncertainty principle quantitative in the sense that if f is supported on T, then only a small percentage of the energy (less than half, say) of its Fourier transform is concentrated on W. As an application of this robust uncertainty principle (QRUP), we consider the problem of decomposing a signal into a sparse superposition of spikes and complex sinusoids. We show that if a generic signal f has a decomposition using spike and frequency locations in T and W respectively, and obeying |T| + |W| <= C (\log N)^{-1/2} N, then this is the unique sparsest possible decomposition (all other decompositions have more non-zero terms). In addition, if |T| + |W| <= C (\log N)^{-1} N, then this sparsest decomposition can be found by solving a convex optimization problem.Comment: 25 pages, 9 figure