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Ron DeVore University of South Carolina Richard Baraniuk Rice University

Ron DeVore University of South Carolina Richard Baraniuk Rice University

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## Ron DeVore University of South Carolina Richard Baraniuk Rice University

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**Compressive Sensing**Theory and Applications Ron DeVore University of South Carolina Richard Baraniuk Rice University**Agenda**• Introduction to Compressive Sensing (CS) [richb] • motivation • basic concepts • CS Theoretical Foundation [ron] • sparsity • encoding and decoding • restricted isometry principle (RIP) • recovery algorithms • CS Applications [richb]**Pressure is on Digital Sensors**• Success of digital data acquisition is placing increasing pressure on signal/image processing hardware and software to support higher resolution / denser sampling • ADCs, cameras, imaging systems, microarrays, … x large numbers of sensors • image data bases, camera arrays, distributed wireless sensor networks, … xincreasing numbers of modalities • acoustic, RF, visual, IR, UV, x-ray, gamma ray, …**Pressure is on Digital Sensors**• Success of digital data acquisition is placing increasing pressure on signal/image processing hardware and software to support higher resolution / denser sampling • ADCs, cameras, imaging systems, microarrays, … x large numbers of sensors • image data bases, camera arrays, distributed wireless sensor networks, … xincreasing numbers of modalities • acoustic, RF, visual, IR, UV = deluge of data • how to acquire, store, fuse, process efficiently?**Digital Data Acquisition**• Foundation: Shannon sampling theorem “if you sample densely enough(at the Nyquist rate), you can perfectly reconstruct the original data” time space**Sensing by Sampling**• Long-established paradigm for digital data acquisition • uniformly sampledata at Nyquist rate (2x Fourier bandwidth) sample**Sensing by Sampling**• Long-established paradigm for digital data acquisition • uniformly sampledata at Nyquist rate (2x Fourier bandwidth) too much data! sample**Sensing by Sampling**• Long-established paradigm for digital data acquisition • uniformly sampledata at Nyquist rate (2x Fourier bandwidth) • compress data (signal-dependent, nonlinear) sample compress transmit/store sparsewavelettransform receive decompress**Sparsity / Compressibility**largewaveletcoefficients largeGaborcoefficients pixels widebandsignalsamples frequency time**What’s Wrong with this Picture?**• Long-established paradigm for digital data acquisition • sampledata at Nyquist rate (2x bandwidth) • compressdata (signal-dependent, nonlinear) • brick wall to resolution/performance sample compress transmit/store sparse /compressiblewavelettransform receive decompress**Beyond Uniform Sampling**• Recall Shannon/Nyquist theorem • Shannon was a pessimist • 2x oversampling Nyquist rate is a worst-case bound for any bandlimited data • sparsity/compressibility irrelevant • Shannon sampling is a linear process while compression is a nonlinear process**Compressive Sensing (CS)**• Recall Shannon/Nyquist theorem • Shannon was a pessimist • 2x oversampling Nyquist rate is a worst-case bound for any bandlimited data • sparsity/compressibility irrelevant • Shannon sampling is a linear process while compression is a nonlinear process • Compressive sensing • new sampling theory that leverages compressibility • based on new uncertainty principles • randomness plays a key role**Compressive Sensing**• Directly acquire “compressed” data • Replace samples by more general “measurements” compressive sensing transmit/store receive reconstruct**Sampling**• Signal is -sparse in basis/dictionary • WLOG assume sparse in space domain sparsesignal nonzeroentries**Sampling**• Signal is -sparse in basis/dictionary • WLOG assume sparse in space domain • Samples sparsesignal measurements nonzeroentries**Compressive Data Acquisition**• When data is sparse/compressible, can directly acquire a condensed representation with no/little information lossthrough dimensionality reduction sparsesignal measurements nonzero entries**Compressive Data Acquisition**• When data is sparse/compressible, can directly acquire a condensed representation with no/little information loss • Random projection will work sparsesignal measurements nonzero entries [Candes-Romberg-Tao, Donoho, 2004]**Compressive Sensing**• Directly acquire “compressed” data • Replace samples by more general “measurements” random measurements transmit/store … receive reconstruct**Why Does It Work?**• Random projection not full rank…… and so loses information in general**Why Does It Work?**• Random projection not full rank… … but preserves structureand informationin sparse/compressible signals models with high probability**Why Does It Work?**• Random projection not full rank… … but preserves structureand informationin sparse/compressible signals models with high probability K-dim hyperplanesaligned with coordinate axes K-sparsemodel**Why Does It Work?**• Random projection not full rank… … but preserves structureand informationin sparse/compressible signals models with high probability K-sparsemodel**CS Signal Recovery**• Random projection not full rank… … but is invertiblefor sparse/compressible signals models with high probability(solves ill-posed inverse problem) K-sparsemodel recovery**CS Signal Recovery**• Random projection not full rank • Recovery problem:givenfind • Null space • So search in null space for the “best” according to some criterion • ex: least squares (M-N)-dim hyperplaneat random angle**CS Signal Recovery**• Reconstruction/decoding: given(ill-posed inverse problem) find • L2 fast**CS Signal Recovery**• Reconstruction/decoding: given(ill-posed inverse problem) find • L2 fast, wrong**Why L2 Doesn’t Work**least squares,minimum L2 solutionis almost never sparse for signals sparse in the space/time domain null space of translated to(random angle)**CS Signal Recovery**• Reconstruction/decoding: given(ill-posed inverse problem) find • L2 fast, wrong • L0 number ofnonzeroentries**CS Signal Recovery**• Reconstruction/decoding: given(ill-posed inverse problem) find • L2 fast, wrong • L0correct, slowonly M=2K measurements required to perfectly reconstruct K-sparse signal stably[Bresler; Rice] number ofnonzeroentries**CS Signal Recovery**• Reconstruction/decoding: given(ill-posed inverse problem) find • L2 fast, wrong • L0 correct, slow • L1correct, efficient mild oversampling[Candes, Romberg, Tao; Donoho] linear program**Why L1 Works**minimum L1solution= sparsest solution (with high probability) if for signals sparse in the space/time domain**Universality**• Random measurements can be used for signals sparse in any basis**Universality**• Random measurements can be used for signals sparse in any basis**Universality**• Random measurements can be used for signals sparse in any basis sparsecoefficient vector nonzero entries**Compressive Sensing**• Directly acquire “compressed” data • Replace samples by more general “measurements” random measurements transmit/store … receive linear pgm**CS Hallmarks**• CS changes the rules of the data acquisition game • exploits a priori signal sparsity information • Universal • same random projections / hardware can be used forany compressible signal class (generic) • Democratic • each measurement carries the same amount of information • simple encoding • robust to measurement loss and quantization • Asymmetrical(most processing at decoder) • Random projections weakly encrypted**Recall: CS Hallmarks**• CS changes the rules of the data acquisition game • exploits a priori signal sparsity information • Universal • same random projections / hardware can be used forany compressible signal class (generic) • Democratic • each measurement carries the same amount of information • simple encoding • robust to measurement loss and quantization • Asymmetrical(most processing at decoder) • Random projections weakly encrypted**Gerhard Richter 4096 Farben / 4096 Colours**1974254 cm X 254 cmLaquer on CanvasCatalogue Raisonné: 359 Museum Collection:Staatliche Kunstsammlungen Dresden (on loan) Sales history: 11 May 2004Christie's New York Post-War and Contemporary Art (Evening Sale), Lot 34US$3,703,500**Rice Single-Pixel CS Camera**single photon detector imagereconstructionorprocessing DMD DMD random pattern on DMD array**Single Pixel Camera**Object LED (light source) Lens 2 Lens 1 Photodiode circuit DMD+ALP Board**Single Pixel Camera**Object LED (light source) Lens 2 Lens 1 Photodiode circuit DMD+ALP Board**Single Pixel Camera**Object LED (light source) Lens 2 Lens 1 Photodiode circuit DMD+ALP Board**Single Pixel Camera**Object LED (light source) Lens 2 Lens 1 Photodiode circuit DMD+ALP Board