demo1: group-sparse thresholding, 1D data
This demo uses overlapping group-sparse thresholding (OGS) for 1D signal denoising. This demo illustrates the non-convex form of OGS, with two different non-convex penaltis: atan and MCP.
Po-Yu Chen and Ivan Selesnick, August 2013
Revised, May 2016
Reference: P. Y. Chen and I. W. Selesnick, "Group-Sparse Signal Denoising: Non-Convex Regularization, Convex Optimization," IEEE Transactions on Signal Processing, vol. 62, no. 13, pp. 3464-3478, July 1, 2014. doi: 10.1109/TSP.2014.2329274
Contents
Set up
clear close all rmse = @(err) sqrt(mean(abs(err).^2)); % Set random number generator for reproducibility rng('default') rng(1)
Make signal
% N : length of signal N = 100; % s : noise-free signal s = zeros(N,1); s(20+(-3:3)) = [2 3 4 5 4 3 2]; s(40+(-3:3)) = [3 -2 -4 5 2 4 -3]; s(60+(-3:3)) = [3 4 2 5 -4 -2 -3]; s(80+(-3:3)) = [3 -4 -2 5 4 2 -3]; % y : signal + noise y = s + (2/3) * randn(N, 1);
Plot signals
ylim1 = [-7 7]; figure(1) clf subplot(2, 1, 1) stem(s, 'marker', 'none') ylim(ylim1); title('Signal'); subplot(2, 1, 2) stem(y, 'marker', 'none') title(sprintf('Signal + noise (RMSE = %3.2f)', rmse(s - y))); ylim(ylim1); orient portrait print -dpdf figures/demo1_data
Denoising using OGS
Use the function ogs1. This is non-convex OGS with the default penalty 'atan'.
K = 5; % K : group size lam = 0.5; % lam : regularization parameter x = ogs1(y, K, lam); % OGS algorithm figure(2), clf subplot(2, 1, 1) stem(x, 'marker', 'none') ylim(ylim1); title(sprintf('OGS thresholding, atan penalty (default) (RMSE = %3.2f)', rmse(s - x)));
Denoising using OGS with 'mcp' penalty
Use the function ogs1 with optional arguments to specify penalty and rho parameter (rho between 0 and 1). We usually set rho between 0.8 and 1.0.
K = 5; % K : group size lam = 0.5; % lam : regularization parameter pen = 'mcp'; % mcp penalty rho = 0.9; % rho between 0 and 1 x = ogs1(y, K, lam, pen, rho); % OGS algorithm subplot(2, 1, 2) stem(x, 'marker', 'none') ylim(ylim1); title(sprintf('OGS thresholding, %s penalty (RMSE = %3.2f)', pen, rmse(s - x))); orient portrait print -dpdf figures/demo1_ogs
