Random Convolution in Compressive Sampling Michael Fleyer

Standard Sampling Nyquist/Shannon sampling: Sufficient for perfect reconstruction, but not effective f

Compressive Sampling LPF ADC Compression/ Transform Coding Compressive Sampling No adaptation !!!

Compressive Sampling (cont.) Sensing: Assume: Then: Undersampling m<<n Is accurate reconstruction possible ? How to design the sampling scheme ?

CS example (Compressive Sensing Richard Baraniuk Rice University, Lecture Notes in IEEE Signal Processing Magazine Volume 24, July 2007) Compressive sensing measurement process with (random Gaussian) measurement matrix and discrete cosine transform (DCT) matrix. The coefficient vector S is sparse with K = 4.

Sparsity Orthogonal expansion DCT

Sparsity (cont.) Signal is S-sparse if it has at most S expansion coefficients The signal is compressible in the sense that the sorted magnitude of the decay quickly. In a standard approach coefficients are obtained adaptively !

Incoherence pair of orthobases of Sensing basis Representation basis

Incoherence (cont.) Example: Maximal incoherence 1) 2) Random matrices are largely incoherent with any fixed basis. With high probability

CS-required properties Sparsity in orthogonal basisIncoherence with

Sparse signal recovery Need to solve: Convex relaxation (other methods exist) P1: P0: NP-hard !!! is sparse

Reconstruction conditions E. Candès and J. Romberg, “Sparsity and incoherence in compressive sampling,” Inverse Prob., vol. 23, no. 3, pp. 969–985, 2007 Solution to P1 is obtained with probability exceeding if the number of measurements taken uniformly at random obeys: is S-sparse !

Linear Programming Define:

Example CS for Fourier sparse signal

Robust CS Non ideal sparsity Restricted Isometry Property (RIP) for all

RIP and CS Suppose. Then for all S-sparse signals Efficient discrimination of S-sparse signals in measurement space

General signal recovery E. Candès, J. Romberg, and T. Tao, “Stable signal recovery from incomplete and inaccurate measurements,” Comm. Pure Appl. Math., vol. 59, no. 8, pp. 1207–1223, Aug Assume. The solution of P1 obeys Where is the vector with all but the largest S components set to 0.

Recovery from noisy signals P1’: E. Candès, J. Romberg, and T. Tao, “Stable signal recovery from incomplete and inaccurate measurements,” Comm. Pure Appl. Math., vol. 59, no. 8, pp. 1207–1223, Aug Assume. The solution of P1’ obeys

Random sensing Form A by sampling… 1.n column vectors uniformly on the unit sphere 2.i.i.d entries from the normal distribution with zero mean and variance 1/m 3.a random projection P and normalize: 4.i.i.d entries from a Bernoulli distribution ( ) or other sub-gaussian distribution With overwhelming probability, all these matrices obey the RIP provided that

Random sensing (cont.) R extracts m coordinates uniformly at random RIP holds for Also holds for for generated by one of the above methods and for any with overwhelming probability. Universality

CS main results Previous results 1.Random waveforms. For random : 2.Random sampling from incoherent orthobasis. Select rows from orthogonal at random

CS by random convolution Compressive Sensing By Random Convolution Justin Romberg, submitted to SIAM Journal on Imaging Science Nonuniform sampler Random modulation preintegrator Universality Numerical structure Physically realizable H

CS by random convolution (cont.) For Notice that

CS by random convolution (cont.)

Subsampling Sampling at random locations (NUS) Randomly pre-modulated summation (RPMS)

Applications Radar Imaging Recover R with CS

Applications (cont.) Fourier Optics

Applications (cont.) Fourier Optics Fourier optics imaging experiment. (a) The “high-resolution" image we wish to acquire. (b) The high-resolution image pixellated by averaging over 4 4 blocks. (c) The image restored from the pixellated version in (b), plus a set of incoherent measurements

Coherence bounds Can be further reduced using H properties

Cumulative coherence Define: Where are rows of the matrix

Main Results Exact reconstructionRIP CS (NUS) * Random convolution NUS RPMS- Any orthogonal  ’ NUS- RPMS ** *M. Rudelson and R. Vershynin, On sparse reconstruction from Fourier and Gaussian measurements, Comm. on Pure and Applied Math., 61 (2008), pp **J. A. Tropp. Personal communication.

Example Fourier sparse signals sampled at random locations. Signals sparse at random orthobasis, sampled using random convolution.