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This Concept Map, created with IHMC CmapTools, has information related to: Compressive_Sensing, a signal of dimension Nx1 which is sparse or compressible, x is sparse or compressible, Basis Pursuit (BP) is given by <math xmlns="http://www.w3.org/1998/Math/MathML"> <mrow> <munderover> <mtext> min </mtext> <munderover> <mtext> x </mtext> <none/> <mtext> - </mtext> </munderover> <none/> </munderover> <mtext> </mtext> <mmultiscripts> <mrow> <mtext> || </mtext> <munderover> <mtext> x </mtext> <none/> <mtext> - </mtext> </munderover> <mtext> || </mtext> </mrow> <mtext> 1 </mtext> <none/> </mmultiscripts> <mtext> s.t. A </mtext> <munderover> <mtext> x </mtext> <none/> <mtext> - </mtext> </munderover> <mtext> =y </mtext> </mrow> </math>, approximately minimizes one-norm regularized least-squares problems using Spectral Projected Gradient, Basis Pursuit Denoise (BPDN) without noise it becomes Basis Pursuit (BP), the sparse or compressible stucture of the acquired signals by designing Non-adaptive sampling techniques, a MxN sensing matrix with M<<N, <math xmlns="http://www.w3.org/1998/Math/MathML"> <mrow> <mmultiscripts> <mtext> r </mtext> <none/> <mtext> t </mtext> </mmultiscripts> <mtext> = y - A </mtext> <mmultiscripts> <mtext> x </mtext> <none/> <mtext> t </mtext> </mmultiscripts> <mtext> + ( </mtext> <mmultiscripts> <mtext> I </mtext> <mtext> t </mtext> <none/> </mmultiscripts> <mtext> /N) </mtext> <mmultiscripts> <mtext> r </mtext> <none/> <mtext> t-1 </mtext> </mmultiscripts> </mrow> </math> where <math xmlns="http://www.w3.org/1998/Math/MathML"> <mrow> <mtext> η( . ; </mtext> <mmultiscripts> <mtext> θ </mtext> <mtext> t </mtext> <none/> </mmultiscripts> <mtext> ) </mtext> </mrow> </math>, <math xmlns="http://www.w3.org/1998/Math/MathML"> <mrow> <mtext> f= </mtext> <mmultiscripts> <mtext> Ψ </mtext> <none/> <mtext> H </mtext> </mmultiscripts> <mtext> x </mtext> </mrow> </math> where f, a signal of dimension Nx1 with M<<N, QP can be solved using Approximate Message Passing (AMP), Basis Pursuit (BP) can be solved using Iterative Soft Thresholding (IST), <math xmlns="http://www.w3.org/1998/Math/MathML"> <mrow> <mtext> (1- </mtext> <mmultiscripts> <mtext> δ </mtext> <mtext> k </mtext> <none/> </mmultiscripts> <mtext> ) </mtext> <mmultiscripts> <mtext> ||x|| </mtext> <mtext> 2 </mtext> <mtext> 2 </mtext> </mmultiscripts> <mtext> ≤ </mtext> <mmultiscripts> <mtext> ||Ax|| </mtext> <mtext> 2 </mtext> <mtext> 2 </mtext> </mmultiscripts> <mtext> ≤ (1+ </mtext> <mmultiscripts> <mtext> δ </mtext> <mtext> k </mtext> <none/> </mmultiscripts> <mtext> ) </mtext> <mmultiscripts> <mtext> ||x|| </mtext> <mtext> 2 </mtext> <mtext> 2 </mtext> </mmultiscripts> </mrow> </math> holds for all K-sparse vectors, Random matrices such as Gaussian matrices, QP can be solved using Iterative Soft Thresholding (IST), Iterative Soft Thresholding (IST) is given by <math xmlns="http://www.w3.org/1998/Math/MathML"> <mrow> <mmultiscripts> <mtext> x </mtext> <none/> <mtext> t+1 </mtext> </mmultiscripts> <mtext> = η( </mtext> <mmultiscripts> <mtext> A </mtext> <none/> <mtext> H </mtext> </mmultiscripts> <mmultiscripts> <mtext> r </mtext> <none/> <mtext> t </mtext> </mmultiscripts> <mtext> + </mtext> <mmultiscripts> <mtext> x </mtext> <none/> <mtext> t </mtext> </mmultiscripts> <mtext> ; </mtext> <mmultiscripts> <mtext> θ </mtext> <mtext> t </mtext> <none/> </mmultiscripts> <mtext> ) </mtext> </mrow> </math>, SPGL1 solves Basis Pursuit Denoise (BPDN), vectors to be strictly sparse, Fourier matrices with randomly selected rows for <math xmlns="http://www.w3.org/1998/Math/MathML"> <mrow> <mtext> M ≥ C . K </mtext> <mmultiscripts> <mtext> (log N) </mtext> <none/> <mtext> 4 </mtext> </mmultiscripts> </mrow> </math>, Φ is a MxN measurement matrix
