Compressive Sensing Matrix Generation by Random Methods

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1 Compressive Sensing Matrix Generation by Random Methods Charles Drake Poole December, 00 Abstract Compressed sensing is a method of recovering interesting images from sparse sampling. To get an ideal reconstruction we want to recover as many entries of x as possible with as few as K measurements.(emmanuel Candes) We want S + θ S,S < to hold for large values of S, ideally of the order of K. By THM. and. in Compressive sampling. (Int. Congress of Mathematics,, pp. -), they show that a trivial randomized matrix construction will obey the UUP for large values of S. Using this knowledge of compressed sensing matrices we explore the phase space of possible sensing matrices completely for a small problem, seeing which matrix is quantitatively best. We then try to extrapolate those findings onto larger sensing matrices where the phase space is too large to fully computationally explore and attempt to show this method leads to a better sensing matrix than a fully random matrix. With prior knowledge of the images general interesting features, we then compare results with different methods of randomly populating the sensing matrix in accordance with our earlier extrapolation to show the numerical quality, or performance of the given method. Objectives The objective is to find a sensing matrix, sometimes referred to as apertures, through random methods that accurately reproduces a given image and scaling for a finer resolution with more samples or larger sensors coverage. In a smaller phase space it is easier to exhaustively search for a quantitatively best sensing matrix for a given sensing problem. These low resolution sensing matrices should then be easily scaled for a higher resolution image of the same type by either spacing larger or increasing sensor density at the optimal locations. Since most images we re interested in aren t random noise, but have structure, a sparse sampling method can be manufactured to be good for certain problems.

2 Approaches Spray and Pray This method is can be explained in the previous two words. In more words, we just randomly populated a matrix and test it. We compare against previous matrices and if it s better, save it as the new best. Do this for a while. It s like a typical random approach, you can get lucky and have a great match at the beginning, or run forever. To prevent this we can limit the number of runs, and set a threshold for reconstruction. Some examples at the bottom of previous column. Guided Matrix Construction In this approach we build matrices manually, see what works, and then attempt to improve the functioning of the given matrix. Doing this is time consuming. Prohibitively so. Other approaches are easily automated. This one was dropped pretty quickly, some example matrices

3 0 0 0 Adaptive Approach Something that has interested me for x a long time is genetic algorithms. Breeding off of the previous generations good or best results, comparing, improving, randomly adapting. With this approach we start with a few random matrices, and then test them. Whichever is best we breed off a new set and retest Randomness Considerations Every project concerning randomness has to take into consideration what they want their randomness to be. Different distributions, algorithms, hardware will produce different results. Well, hardware should always be the same, but different methods of psuedo and quasi random generation produce different distributions. Some Examples,

4 Pseudo Pseudo-random suggests closeness to the real thing by apparent sameness. So, a pseudo-random is similar to a real random distribution, because it attempts to appear random, however since it s algorithmically produced it can no be considered truly random. Quasi Quasi-random numbers don t clump like random/pseudo-random. In order to have a more uniform distribution each subsequently generated value don t have serial independence over previously generated values. Therefore in a finite set of points voids and clumps are avoided. Typically this distribution takes some work, but several good algorithms and implementations have taken the work out of it, The main difference between the two is Quasi-random will have a uniform distribution, cause that s what we want. The pseudo-random will clump and cluster, and that can also be desirable.

5 Future Generating Randomness So far for quasi-random we ve played with a halton and a sobol method. Pseudorandom has been the Mersenne twister algorithm, because why use anything else? As the result of a previous project we have access to a hardware random number generator, the Entropy key, and of course any computer generates lots of random data. There s a lot of ways of obtaining random data and we d like to branch out and look at some of it. Algorithms In addition to the evolution of the matrices the method of reconstructing data we ve been using is really slow. Really really slow, it s hard to compile results when it takes - hours per reconstruction to produce a result. Parallelization There are several directions to take. Different methods of generating random data, different genetic algorithms, different adaptive algorithms. Of course all the analysis needs to be completed. We run multiple clusters, so Open MPI is a consideration.