Package R1magic. April 9, 2013

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1 Package R1magic April 9, 2013 Type Package Title Compressive Sampling: Sparse signal recovery utilities Version 0.2 Date Author Maintainer Depends stats, utils Provides miminization routines and related utilities for compressive sampling. l-1, l-2 and TV (1-D) minimization, DFT basis matrix, sparse signal generator, mutual cumulative coherence and examples. License GPL (>= 3) LazyLoad yes NeedsCompilation no Repository CRAN Date/Publication :05:31 R topics documented: R1magic-package CompareL1_L2_TV DFTMatrix DFTMatrixPlain GaussianMatrix mutualcoherence objective1tv objectivel objectivel

2 2 R1magic-package oo solve1tv solvel solvel sparsesignal TV Index 12 R1magic-package Compressive Sampling: Sparse signal recovery utilities Details Provides miminization routines and related utilities for compressive sampling. l-1, l-2 and TV (1-D) minimization, DFT basis matrix, sparse signal generator, mutual cumulative coherence and examples. Package: R1magic Type: Package Version: 0.2 Date: License: GPL (>= 3) LazyLoad: yes Maintainer: <mehmet.suzen@physics.org> References Emmanuel Candes, Justin Romberg, and Terence Tao, Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information. (IEEE Trans. on Information Theory, 52(2) pp , February 2006) Emmanuel Candes and Justin Romberg, Quantitative robust uncertainty principles and optimally sparse decompositions. (Foundations of Comput. Math., 6(2), pp , April 2006) David Donoho, Compressed sensing. (IEEE Trans. on Information Theory, 52(4), pp , April 2006) Examples CompareL1_L2_TV1(100,10,0.1);

3 CompareL1_L2_TV1 3 CompareL1_L2_TV1 Compare L1, L2 and TV on a sparse signal. Compare L1, L2 and TV on a sparse signal. CompareL1_L2_TV1(N, M, per) N M per Size of the sparse signal to generate, integer. Number of measurements. Percentage of spikes. DFTMatrix0 Generate Discrete Fourier Transform Matrix using DFTMatrixPlain. Generate Discrete Fourier Transform Matrix (NxN). DFTMatrix0(N) N Integer value determines the dimension of the square matrix. It returns a NxN square matrix.

4 4 DFTMatrixPlain See Also DFTMatrixPlain Examples DFTMatrix0(2) DFTMatrixPlain Generate Plain Discrete Fourier Transform Matrix without the coefficient Generate plain Discrete Fourier Transform Matrix (NxN) without a coefficient. DFTMatrixPlain(N) N Integer value defines the dimension of the square plain DFT matrix. It returns a NxN square matrix. Examples DFTMatrixPlain(2)

5 GaussianMatrix 5 GaussianMatrix Generate Gaussian Random Matrix Generate Gaussian Random Matrix ( zero mean and standard deviation one.) GaussianMatrix(N, M) N M Integer value determines number of rows. Integer value determines number of columns. Returns MxN matrix. Examples GaussianMatrix(3,2) mutualcoherence Cumulative mutual coherence Generate vector of cumulative mutual coherence of a given matrix up to a given order. \ Mutual Cumulative Coherence of a Matrix A at order k is defined as M(A, k) = max p max p q,q Ω q < a p, a q > /( a p a q ) mutualcoherence(a, k) A k A matrix. Integer value determines number of columns or the order of mutual coherence function to.

6 6 objective1tv Returns k-vector References Compressed sensing in diffuse optical tomography \ M. Suzen, A.Giannoula and T. Durduran, \ Opt. Express 18, (2010) \ J. A. Tropp \ Greed is good: algorithmic results for sparse approximation, \IEEE Trans. Inf. Theory 50, (2004) Examples set.seed(42) B <- matrix(rnorm(100), 10, 10) # Gaussian Random Matrix mutualcoherence(b, 3) # mutual coherence up to order k objective1tv 1-D Total Variation Penalized Objective Function 1-D Total Variation Penalized Objective Function objective1tv(x, T, phi, y, lambda) x Initial value of the vector to be recovered. Sparse representation of the vector ( N x 1 matrix ) X=Tx, where X is the original vector T sparsity bases ( N x N matrix ) phi y lambda Returns a vector. Measurement matrix (M x N). Measurement vector (Mx1). Penalty coefficient.

7 objectivel1 7 objectivel1 L-1 Penalized Objective Function L-1 Penalized Objective Function objectivel1(x, T, phi, y, lambda) x Initial value of the vector to be recovered. Sparse representation of the vector ( N x 1 matrix ) X=Tx, where X is the original vector T sparsity bases ( N x N matrix ) phi y lambda Returns a vector References Measurement matrix (M x N). Measurement vector (Mx1). Penalty coefficient. Emmanuel Candes, Justin Romberg, and Terence Tao, Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information. (IEEE Trans. on Information Theory, 52(2) pp , February 2006) objectivel2 L-2 Penalized Objective Function L-2 Penalized Objective Function objectivel2(x, T, phi, y, lambda)

8 8 oo x Initial value of the vector to be recovered. Sparse representation of the vector ( N x 1 matrix ) X=Tx, where X is the original vector T sparsity bases ( N x N matrix ) phi y lambda Measurement matrix (M x N). Measurement vector (Mx1). Penalty coefficient. Returns a vector. References Emmanuel Candes, Justin Romberg, and Terence Tao, Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information. (IEEE Trans. on Information Theory, 52(2) pp , February 2006) oo Frequency expression for DFT Frequency expression for DFT oo(p, omega) p omega Exponent Omega expression for DFT

9 solve1tv 9 solve1tv 1-D Total Variation Penalized Nonlinear Minimization 1-D Total Variation Penalized Nonlinear Minimization solve1tv(phi,y,t,x0,lambda=0.1) x0 Initial value of the vector to be recovered. Sparse representation of the vector ( N x 1 matrix ) X=Tx, where X is the original vector T sparsity bases ( N x N matrix ) phi Measurement matrix (M x N). y Measurement vector (Mx1). lambda Penalty coefficient. Defaults 0.1 Returns nlm object. solvel1 l1 Penalized Nonlinear Minimization l1 Penalized Nonlinear Minimization solvel1(phi,y,t,x0,lambda=0.1) x0 Initial value of the vector to be recovered. Sparse representation of the vector ( N x 1 matrix ) X=Tx, where X is the original vector T sparsity bases ( N x N matrix ) phi Measurement matrix (M x N). y Measurement vector (Mx1). lambda Penalty coefficient. Defaults 0.1

10 10 solvel2 Returns nlm object. solvel2 l2 Penalized Nonlinear Minimization l2 Penalized Nonlinear Minimization solvel2(phi,y,t,x0,lambda=0.1) x0 Initial value of the vector to be recovered. Sparse representation of the vector ( N x 1 matrix ) X=Tx, where X is the original vector T sparsity bases ( N x N matrix ) phi y Measurement matrix (M x N). Measurement vector (Mx1). lambda Penalty coefficient. Defaults 0.1 Returns nlm object.

11 sparsesignal 11 sparsesignal Sparse digital signal Generator. Sparse digital signal Generator with given thresholds. sparsesignal(n, s, b = 1, delta = 1e-07, nlev = 0.05, slev = 0.9) N Number of signal components, vector size. s Number of spikes, significatn components b Signal bandwidth, defaults 1. delta Length of discrete distances among components, defaults 1e-7. nlev Maximum value of insignificant component, relative to b, defaults to 0.05 slev Maximum value of significant component, relative to b, defaults to 0.9 TV1 1-D total variation of a vector. 1-D total variation of a vector. TV1(x) x A vector.

12 Index Topic package R1magic-package, 2 CompareL1_L2_TV1, 3 DFTMatrix0, 3 DFTMatrixPlain, 4 GaussianMatrix, 5 mutualcoherence, 5 objective1tv, 6 objectivel1, 7 objectivel2, 7 oo, 8 R1magic (R1magic-package), 2 R1magic-package, 2 solve1tv, 9 solvel1, 9 solvel2, 10 sparsesignal, 11 TV1, 11 12