Short-course Compressive Sensing of Videos
|
|
- Audra Sharp
- 5 years ago
- Views:
Transcription
1 Short-course Compressive Sensing of Videos Venue CVPR 2012, Providence, RI, USA June 16, 2012 Organizers: Richard G. Baraniuk Mohit Gupta Aswin C. Sankaranarayanan Ashok Veeraraghavan
2 Part 2: Compressive sensing Motivation, theory, recovery Linear inverse problems Sensing visual signals Compressive sensing Theory Hallmark Recovery algorithms Model-based compressive sensing Models specific to visual signals
3 Linear inverse problems
4 Linear inverse problems Many classic problems in computer can be posed as linear inverse problems Notation Signal of interest Observations x 2 R N y 2 R M measurement matrix Measurement model Problem definition: given y = x + e measurement noise y, recover x
5 Linear inverse problems y = x + e x 2 R N y 2 R M Problem definition: given y, recover x Scenario 1 We can invert the system of equations ˆx = M N 1 y Focus more on robustness to noise via signal priors
6 Linear inverse problems y = x + e x 2 R N y 2 R M Problem definition: given y, recover x Scenario 2 Measurement matrix has a (N-M) dimensional null-space Solution is no longer unique Many interesting vision problem fall under this scenario Key quantity of concern: Under-sampling ratio M/N
7 Image super-resolution Low resolution input/observation 128x128 pixels
8 Image super-resolution y = x + e 2x super-resolution y (1,1) =(x 1,1 + x 1,2 + x 2,1 + x 2,2 ) /4
9 Image super-resolution y = x + e 4x super-resolution y (1,1) =(x 1,1 + x 1, x 4,3 + x 4,4 ) /16
10 Image super-resolution y = x + e Super-resolution factor D Under-sampling factor M/N = 1/D 2 General rule: The smaller the under-sampling, the more the unknowns and hence, the harder the superresolution problem
11 Many other vision problems Affine rank minimizaiton, Matrix completion, deconvolution, Robust PCA Image synthesis Infilling, denoising, etc. Light transport Reflectance fields, BRDFs, Direct global separation, Light transport matrices Sensing
12 Sensing visual signals
13 High-dimensional visual signals Reflection Fog Volumetric scattering Human skin Sub-surface scattering Refraction Electron microscopy Tomography
14 Adelson and Bergen (91) The plenoptic function Collection of all variations of light in a scene time (1D) space (3D) spectrum (1D) angle (2D) Different slices reveal different scene properties
15 The plenoptic function time (1D) space (3D) angle (2D) spectrum (1D) High-speed cameras Hyper-spectral imaging Lytro light-field camera
16 Sensing the plenoptic function High-dimensional 1000 samples/dim == 10^21 dimensional signal Greater than all the storage in the world Traditional theories of sensing fail us!
17 Resolution trade-off Key enabling factor: Spatial resolution is cheap! Commercial cameras have 10s of megapixels One idea is the we trade-off spatial resolution for resolution in some other axis
18 Spatio-angular tradeoff [Ng, 2005]
19 Spatio-angular tradeoff [Levoy et al. 2006]
20 Spatio-temporal tradeoff Stagger pixel-shutter within each exposure [Bub et al., 2010]
21 Spatio-temporal tradeoff Rearrange to get high temporal resolution video at lower spatialresolution [Bub et al., 2010]
22 Resolution trade-off Very powerful and simple idea Drawbacks Does not extend to non-visible spectrum 1 Megapixel SWIR camera costs k Linear and global tradeoffs With today s technology, cannot obtain more than 10x for video without sacrificing spatial resolution completely
23 Compressive sensing
24 Sense by Sampling sample
25 Sense by Sampling sample too much data!
26 Sense then Compress sample compress JPEG JPEG2000 decompress
27 Sparsity pixels large wavelet coefficients (blue = 0)
28 Sparsity pixels large wavelet coefficients (blue = 0) wideband signal samples frequency large Gabor (TF) coefficients time
29 Concise Signal Structure Sparse signal: only K out of N coordinates nonzero sorted index
30 Concise Signal Structure Sparse signal: only K out of N coordinates nonzero model: union of K-dimensional subspaces aligned w/ coordinate axes sorted index
31 Concise Signal Structure Sparse signal: only K out of N coordinates nonzero model: union of K-dimensional subspaces Compressible signal: sorted coordinates decay rapidly with power-law power-law decay sorted index
32 Concise Signal Structure Sparse signal: only K out of N coordinates nonzero model: union of K-dimensional subspaces Compressible signal: sorted coordinates decay rapidly with power-law model: ball: power-law decay sorted index
33 What s Wrong with this Picture? Why go to all the work to acquire N samples only to discard all but K pieces of data? sample compress decompress
34 What s Wrong with this Picture? linear processing linear signal model (bandlimited subspace) nonlinear processing nonlinear signal model (union of subspaces) sample compress decompress
35 Compressive Sensing Directly acquire compressed data via dimensionality reduction Replace samples by more general measurements compressive sensing recover
36 Sampling Signal is -sparse in basis/dictionary WLOG assume sparse in space domain sparse signal nonzero entries
37 Sampling Signal is -sparse in basis/dictionary WLOG assume sparse in space domain Sampling measurements sparse signal nonzero entries
38 Compressive Sampling When data is sparse/compressible, can directly acquire a condensed representation with no/little information loss through linear dimensionality reduction measurements sparse signal nonzero entries
39 How Can It Work? Projection not full rank and so loses information in general Ex: Infinitely many (null space) s map to the same
40 How Can It Work? Projection not full rank and so loses information in general columns But we are only interested in sparse vectors
41 How Can It Work? Projection not full rank and so loses information in general columns But we are only interested in sparse vectors is effectively MxK
42 How Can It Work? Projection not full rank and so loses information in general columns But we are only interested in sparse vectors Design so that each of its MxK submatrices are full rank (ideally close to orthobasis) Restricted Isometry Property (RIP)
43 Restricted Isometry Property (RIP) Preserve the structure of sparse/compressible signals K-dim subspaces
44 Restricted Isometry Property (RIP) Stable embedding RIP of order 2K implies: for all K-sparse x 1 and x 2 K-dim subspaces
45 RIP = Stable Embedding An information preserving projection preserves the geometry of the set of sparse signals RIP ensures that
46 How Can It Work? Projection not full rank and so loses information in general columns Design so that each of its MxK submatrices are full rank (RIP) Unfortunately, a combinatorial, NP-complete design problem
47 Insight from the 70 s [Kashin, Gluskin] Draw at random iid Gaussian iid Bernoulli columns Then provided has the RIP with high probability
48 Randomized Sensing Measurements = random linear combinations of the entries of the signal y = x No information loss for sparse vectors whp measurements sparse signal nonzero entries
49 CS Signal Recovery Goal: Recover signal from measurements Problem: Random projection not full rank (ill-posed inverse problem) Solution: Exploit the sparse/compressible geometry of acquired signal
50 Random projection not full rank CS Signal Recovery Recovery problem: given find Null space Search in null space for the best according to some criterion ex: least squares (N-M)-dim hyperplane at random angle
51 Signal Recovery Recovery: given (ill-posed inverse problem) find (sparse) Optimization: Closed-form solution:
52 Signal Recovery Recovery: given (ill-posed inverse problem) find (sparse) Optimization: Closed-form solution: Wrong answer!
53 Signal Recovery Recovery: given (ill-posed inverse problem) find (sparse) Optimization: Closed-form solution: Wrong answer!
54 Signal Recovery Recovery: given (ill-posed inverse problem) find (sparse) Optimization: find sparsest vector in translated nullspace
55 Signal Recovery Recovery: given (ill-posed inverse problem) find (sparse) Optimization: Correct! find sparsest vector in translated nullspace But NP-Complete alg
56 Signal Recovery Recovery: given (ill-posed inverse problem) find (sparse) Optimization: Convexify the optimization Candes Romberg Tao Donoho
57 Signal Recovery Recovery: given (ill-posed inverse problem) find (sparse) Optimization: Convexify the optimization Correct! Polynomial time alg (linear programming)
58 Compressive Sensing random measurements sparse signal nonzero entries Signal recovery via [Candes, Romberg, Tao; Donoho] optimization
59 Compressive Sensing random measurements sparse signal nonzero entries Signal recovery via iterative greedy algorithm (orthogonal) matching pursuit [Gilbert, Tropp] iterated thresholding [Nowak, Figueiredo; Kingsbury, Reeves; Daubechies, Defrise, De Mol; Blumensath, Davies; ] CoSaMP [Needell and Tropp]
60 Greedy recovery algorithm #1 Consider the following problem min ky xk 2 s.t kxk 0 apple K Suppose we wanted to minimize just the cost, then steepest gradient descent works as bx k = bx k 1 + T (y bx k 1 ) But, the new estimate is no longer K-sparse bx k =thresh bx k 1 + T (y bx k 1 ),K
61 Iterated Hard Thresholding update signal estimate prune signal es4mate (best K- term approx) update residual
62 Greedy recovery algorithm #2 Consider the following problem sparse signal 1 sparse Can we recover the support?
63 Greedy recovery algorithm #2 Consider the following problem sparse signal 1 sparse If then =[ 1, 2,..., N ] arg max h i,yi gives the support of x How to extend to K-sparse signals?
64 Greedy recovery algorithm #2 sparse signal K sparse residue: find atom: r = y bx k 1 k = arg max h i,ri Add atom to support: S = S [ {k} Signal estimate x (Least squares over support) k =( S ) y
65 Orthogonal matching pursuit goal: given y = columns of x, recover a sparse x are unit-norm initialize: bx 0 =0,r = y, ={},i=0 iteration: i = i +1 T r b = k = arg max{ b(1), b(2),..., b(n) } = S k (bx i ) =( ) y, (bx i ) c =0 Find atom with largest support Update signal estimate r = y bx i update residual
66 Specialized solvers CoSAMP [Needell and Tropp, 2009] SPG_l1 [Friedlander, van der Berg, 2008] FPC [Hale, Yin, and Zhang, 2007] AMP [Donoho, Montanari and Maleki, 2010] many many others, see dsp.rice.edu/cs and
67 CS Hallmarks Stable acquisition/recovery process is numerically stable Asymmetrical (most processing at decoder) conventional: smart encoder, dumb decoder CS: dumb encoder, smart decoder Democratic each measurement carries the same amount of information robust to measurement loss and quantization digital fountain property Random measurements encrypted Universal same random projections / hardware can be used for any sparse signal class (generic)
68 Universality Random measurements can be used for signals sparse in any basis
69 Universality Random measurements can be used for signals sparse in any basis
70 Universality Random measurements can be used for signals sparse in any basis sparse coefficient vector nonzero entries
71 Summary: CS Compressive sensing randomized dimensionality reduction exploits signal sparsity information integrates sensing, compression, processing Why it works: with high probability, random projections preserve information in signals with concise geometric structures sparse signals compressible signals
72 Summary: CS Encoding: = random linear combinations of the entries of measurements sparse signal nonzero entries Decoding: Recover from via optimization
73 Image/Video specific signal models and recovery algorithms
74 Transform basis Recall Universality: Random measurements can be used for signals sparse in any basis DCT/FFT/Wavelets Fast transforms; very useful in large scale problems
75 Dictionary learning For many signal classes (ex: videos, light-fields), there are no obvious sparsifying transform basis Can we learn a sparsifying transform instead? GOAL: Given training data x 1,x 2,...,x T xi 2 R N learn a dictionary D, such that s i are sparse. x i = Ds i D 2 R N Q s i 2 R Q
76 Dictionary learning GOAL: Given training data x 1,x 2,...,x T xi 2 R N learn a dictionary D, such that s i are sparse. x i = Ds i D 2 R N Q s i 2 R Q min D,S kx DSk F s.t 8i, ks i k 0 apple K
77 Dictionary learning min D,S kx DSk F s.t 8i, ks i k 0 apple K Non-convex constraint Bilinear in D and S
78 Dictionary learning min D,S kx DSk F + X k ks k k 1 Biconvex in D and S Given D, the optimization problem is convex in s k Given S, the optimization problem is a least squares problem K-SVD: Solve using alternate minimization techniques Start with D = wavelet or DCT bases Additional pruning steps to control size of the dictionary Aharon et al., TSP 2006
79 Dictionary learning min D,S kx DSk F + X k ks k k 1 Pros Ability to handle arbitrary domains Cons Learning dictionaries can be computationally intensive for high-dimensional problems; need for very large amount of data Recovery algorithms may suffer due to lack of fast transforms
80 Models on image gradients Piecewise constant images Sparse image gradients Natural image statistics Heavy tailed distributions
81 Total variation prior TV(I) = X u,v kri(u, v)k 2 = X u,v q I 2 x(u, v)+i 2 y(u, v) TV norm Sparse-gradient promoting norm Formulation of recovery problem min TV(x) s.t y = x
82 Total variation prior Optimization problem Convex Often, works better than transform basis methods Variants 3D (video) Anisotropic TV min TV(x) s.t y = x Code TVAL3 Many many others (see dsp.rice/cs)
83 Beyond sparsity Model-based CS
84 Beyond Sparse Models Sparse signal model captures simplistic primary structure wavelets: natural images Gabor atoms: chirps/tones pixels: background subtracted images
85 Beyond Sparse Models Sparse signal model captures simplistic primary structure Modern compression/processing algorithms capture richer secondary coefficient structure wavelets: natural images Gabor atoms: chirps/tones pixels: background subtracted images
86 Sparse Signals K-sparse signals comprise a particular set of K-dim subspaces
87 Structured-Sparse Signals A K-sparse signal model comprises a particular (reduced) set of K-dim subspaces [Blumensath and Davies] Fewer subspaces <> relaxed RIP <> stable recovery using fewer measurements M
88 Wavelet Sparse Typical of wavelet transforms of natural signals and images (piecewise smooth)
89 Tree-Sparse Model: K-sparse coefficients + significant coefficients lie on a rooted subtree Typical of wavelet transforms of natural signals and images (piecewise smooth)
90 Wavelet Sparse Model: K-sparse coefficients + significant coefficients lie on a rooted subtree RIP: stable embedding K-planes
91 Tree-Sparse Model: K-sparse coefficients + significant coefficients lie on a rooted subtree Tree-RIP: stable embedding [Blumensath and Davies] K-planes
92 Tree-Sparse Model: K-sparse coefficients + significant coefficients lie on a rooted subtree Tree-RIP: [Blumensath and Davies] stable embedding Recovery: inject tree-sparse approx into IHT/CoSaMP
93 Recall: Iterated Thresholding update signal estimate prune signal es4mate (best K- term approx) update residual
94 Iterated Model Thresholding update signal estimate prune signal es4mate (best K- term model approx) update residual
95 Tree-Sparse Signal Recovery signal length N=1024 random measurements M=80 target signal CoSaMP, (RMSE=1.12) Tree-sparse CoSaMP (RMSE=0.037) L1-minimization (RMSE=0.751)
96 Clustered Signals Probabilistic approach via graphical model Model clustering of significant pixels in space domain using Ising Markov Random Field Ising model approximation performed efficiently using graph cuts [Cevher, Duarte, Hegde, Baraniuk 08] target Ising-model recovery CoSaMP recovery LP (FPC) recovery
97 Part 2: Compressive sensing Motivation, theory, recovery Linear inverse problems Sensing visual signals Compressive sensing Theory Hallmark Recovery algorithms Model-based compressive sensing Models specific to visual signals
Compressive. Graphical Models. Volkan Cevher. Rice University ELEC 633 / STAT 631 Class
Compressive Sensing and Graphical Models Volkan Cevher volkan@rice edu volkan@rice.edu Rice University ELEC 633 / STAT 631 Class http://www.ece.rice.edu/~vc3/elec633/ Digital Revolution Pressure is on
More informationCompressive Sensing. A New Framework for Sparse Signal Acquisition and Processing. Richard Baraniuk. Rice University
Compressive Sensing A New Framework for Sparse Signal Acquisition and Processing Richard Baraniuk Rice University Better, Stronger, Faster Accelerating Data Deluge 1250 billion gigabytes generated in 2010
More informationRandomized Dimensionality Reduction
Randomized Dimensionality Reduction with Applications to Signal Processing and Communications Richard Baraniuk Rice University The Digital Universe Size: 281 billion gigabytes generated in 2007 digital
More informationThe Fundamentals of Compressive Sensing
The Fundamentals of Compressive Sensing Mark A. Davenport Georgia Institute of Technology School of Electrical and Computer Engineering Sensor explosion Data deluge Digital revolution If we sample a signal
More informationCombinatorial Selection and Least Absolute Shrinkage via The CLASH Operator
Combinatorial Selection and Least Absolute Shrinkage via The CLASH Operator Volkan Cevher Laboratory for Information and Inference Systems LIONS / EPFL http://lions.epfl.ch & Idiap Research Institute joint
More informationCompressive Sensing. Part IV: Beyond Sparsity. Mark A. Davenport. Stanford University Department of Statistics
Compressive Sensing Part IV: Beyond Sparsity Mark A. Davenport Stanford University Department of Statistics Beyond Sparsity Not all signal models fit neatly into the sparse setting The concept of dimension
More informationCompressive Sensing of High-Dimensional Visual Signals. Aswin C Sankaranarayanan Rice University
Compressive Sensing of High-Dimensional Visual Signals Aswin C Sankaranarayanan Rice University Interaction of light with objects Reflection Fog Volumetric scattering Human skin Sub-surface scattering
More informationCompressive Sensing. and Applications. Volkan Cevher Justin Romberg
Compressive Sensing and Applications Volkan Cevher volkan.cevher@epfl.ch Justin Romberg jrom@ece.gatech.edu Acknowledgements Rice DSP Group (Slides) Richard Baraniuk Mark Davenport, Marco Duarte, Chinmay
More informationCompressive Sensing: Opportunities and Perils for Computer Vision
Compressive Sensing: Opportunities and Perils for Computer Vision Rama Chellappa And Volkan Cevher (Rice) Joint work with Aswin C. Sankaranarayanan Dikpal Reddy Dr. Ashok Veeraraghavan (MERL) Prof. Rich
More informationELEG Compressive Sensing and Sparse Signal Representations
ELEG 867 - Compressive Sensing and Sparse Signal Representations Gonzalo R. Arce Depart. of Electrical and Computer Engineering University of Delaware Fall 211 Compressive Sensing G. Arce Fall, 211 1 /
More informationTutorial: Compressive Sensing of Video
Tutorial: Compressive Sensing of Video Venue: CVPR 2012, Providence, Rhode Island, USA June 16, 2012 Richard Baraniuk Mohit Gupta Aswin Sankaranarayanan Ashok Veeraraghavan Tutorial: Compressive Sensing
More informationMeasurements and Bits: Compressed Sensing meets Information Theory. Dror Baron ECE Department Rice University dsp.rice.edu/cs
Measurements and Bits: Compressed Sensing meets Information Theory Dror Baron ECE Department Rice University dsp.rice.edu/cs Sensing by Sampling Sample data at Nyquist rate Compress data using model (e.g.,
More informationCompressive Sensing: Theory and Practice
Compressive Sensing: Theory and Practice Mark Davenport Rice University ECE Department Sensor Explosion Digital Revolution If we sample a signal at twice its highest frequency, then we can recover it exactly.
More informationSparse Reconstruction / Compressive Sensing
Sparse Reconstruction / Compressive Sensing Namrata Vaswani Department of Electrical and Computer Engineering Iowa State University Namrata Vaswani Sparse Reconstruction / Compressive Sensing 1/ 20 The
More informationIntroduction to Topics in Machine Learning
Introduction to Topics in Machine Learning Namrata Vaswani Department of Electrical and Computer Engineering Iowa State University Namrata Vaswani 1/ 27 Compressed Sensing / Sparse Recovery: Given y :=
More informationCompressive Sensing: Opportunities and pitfalls for Computer Vision
Compressive Sensing: Opportunities and pitfalls for Computer Vision Rama Chellappa Joint work with Vishal Patel Jai Pillai Dikpal Reddy Aswin C. Sankaranarayanan Ashok Veeraraghavan Dr. Volkan Cevher (Rice)
More informationAgenda. Pressure is on Digital Sensors
Compressive Sensing Richard Baraniuk Rice University Acknowledgements For assistance preparing this presentation Rice DSP group Petros Boufounos, Volkan Cevher Mark Davenport, Marco Duarte, Chinmay Hegde,
More informationCompressive Sensing for High-Dimensional Data
Compressive Sensing for High-Dimensional Data Richard Baraniuk Rice University dsp.rice.edu/cs DIMACS Workshop on Recent Advances in Mathematics and Information Sciences for Analysis and Understanding
More informationCompressive Parameter Estimation with Earth Mover s Distance via K-means Clustering. Dian Mo and Marco F. Duarte
Compressive Parameter Estimation with Earth Mover s Distance via K-means Clustering Dian Mo and Marco F. Duarte Compressive Sensing (CS) Integrates linear acquisition with dimensionality reduction linear
More informationSignal Reconstruction from Sparse Representations: An Introdu. Sensing
Signal Reconstruction from Sparse Representations: An Introduction to Compressed Sensing December 18, 2009 Digital Data Acquisition Suppose we want to acquire some real world signal digitally. Applications
More informationOutline Introduction Problem Formulation Proposed Solution Applications Conclusion. Compressed Sensing. David L Donoho Presented by: Nitesh Shroff
Compressed Sensing David L Donoho Presented by: Nitesh Shroff University of Maryland Outline 1 Introduction Compressed Sensing 2 Problem Formulation Sparse Signal Problem Statement 3 Proposed Solution
More informationLecture 17 Sparse Convex Optimization
Lecture 17 Sparse Convex Optimization Compressed sensing A short introduction to Compressed Sensing An imaging perspective 10 Mega Pixels Scene Image compression Picture Why do we compress images? Introduction
More informationCompressed Sensing Algorithm for Real-Time Doppler Ultrasound Image Reconstruction
Mathematical Modelling and Applications 2017; 2(6): 75-80 http://www.sciencepublishinggroup.com/j/mma doi: 10.11648/j.mma.20170206.14 ISSN: 2575-1786 (Print); ISSN: 2575-1794 (Online) Compressed Sensing
More informationCollaborative Sparsity and Compressive MRI
Modeling and Computation Seminar February 14, 2013 Table of Contents 1 T2 Estimation 2 Undersampling in MRI 3 Compressed Sensing 4 Model-Based Approach 5 From L1 to L0 6 Spatially Adaptive Sparsity MRI
More informationTutorial on Image Compression
Tutorial on Image Compression Richard Baraniuk Rice University dsp.rice.edu Agenda Image compression problem Transform coding (lossy) Approximation linear, nonlinear DCT-based compression JPEG Wavelet-based
More informationModified Iterative Method for Recovery of Sparse Multiple Measurement Problems
Journal of Electrical Engineering 6 (2018) 124-128 doi: 10.17265/2328-2223/2018.02.009 D DAVID PUBLISHING Modified Iterative Method for Recovery of Sparse Multiple Measurement Problems Sina Mortazavi and
More informationADAPTIVE LOW RANK AND SPARSE DECOMPOSITION OF VIDEO USING COMPRESSIVE SENSING
ADAPTIVE LOW RANK AND SPARSE DECOMPOSITION OF VIDEO USING COMPRESSIVE SENSING Fei Yang 1 Hong Jiang 2 Zuowei Shen 3 Wei Deng 4 Dimitris Metaxas 1 1 Rutgers University 2 Bell Labs 3 National University
More informationDeconvolution with curvelet-domain sparsity Vishal Kumar, EOS-UBC and Felix J. Herrmann, EOS-UBC
Deconvolution with curvelet-domain sparsity Vishal Kumar, EOS-UBC and Felix J. Herrmann, EOS-UBC SUMMARY We use the recently introduced multiscale and multidirectional curvelet transform to exploit the
More informationCOMPRESSIVE VIDEO SAMPLING
COMPRESSIVE VIDEO SAMPLING Vladimir Stanković and Lina Stanković Dept of Electronic and Electrical Engineering University of Strathclyde, Glasgow, UK phone: +44-141-548-2679 email: {vladimir,lina}.stankovic@eee.strath.ac.uk
More informationNon-Differentiable Image Manifolds
The Multiscale Structure of Non-Differentiable Image Manifolds Michael Wakin Electrical l Engineering i Colorado School of Mines Joint work with Richard Baraniuk, Hyeokho Choi, David Donoho Models for
More informationSparse wavelet expansions for seismic tomography: Methods and algorithms
Sparse wavelet expansions for seismic tomography: Methods and algorithms Ignace Loris Université Libre de Bruxelles International symposium on geophysical imaging with localized waves 24 28 July 2011 (Joint
More informationDetecting Burnscar from Hyperspectral Imagery via Sparse Representation with Low-Rank Interference
Detecting Burnscar from Hyperspectral Imagery via Sparse Representation with Low-Rank Interference Minh Dao 1, Xiang Xiang 1, Bulent Ayhan 2, Chiman Kwan 2, Trac D. Tran 1 Johns Hopkins Univeristy, 3400
More informationParameter Estimation in Compressive Sensing. Marco F. Duarte
Parameter Estimation in Compressive Sensing Marco F. Duarte Systems of Lines: Application of Algebraic Combinatorics Worcester Polytechnic Institute, August 13 2015 Concise Signal Structure Sparse signal:
More informationLearning Splines for Sparse Tomographic Reconstruction. Elham Sakhaee and Alireza Entezari University of Florida
Learning Splines for Sparse Tomographic Reconstruction Elham Sakhaee and Alireza Entezari University of Florida esakhaee@cise.ufl.edu 2 Tomographic Reconstruction Recover the image given X-ray measurements
More informationMain Menu. Summary. sampled) f has a sparse representation transform domain S with. in certain. f S x, the relation becomes
Preliminary study on Dreamlet based compressive sensing data recovery Ru-Shan Wu*, Yu Geng 1 and Lingling Ye, Modeling and Imaging Lab, Earth & Planetary Sciences/IGPP, University of California, Santa
More informationDetection Performance of Radar Compressive Sensing in Noisy Environments
Detection Performance of Radar Compressive Sensing in Noisy Environments Asmita Korde a,damon Bradley b and Tinoosh Mohsenin a a Department of Computer Science and Electrical Engineering, University of
More informationThe Benefit of Tree Sparsity in Accelerated MRI
The Benefit of Tree Sparsity in Accelerated MRI Chen Chen and Junzhou Huang Department of Computer Science and Engineering, The University of Texas at Arlington, TX, USA 76019 Abstract. The wavelet coefficients
More informationRandomized sampling strategies
Randomized sampling strategies Felix J. Herrmann SLIM Seismic Laboratory for Imaging and Modeling the University of British Columbia SLIM Drivers & Acquisition costs impediments Full-waveform inversion
More informationRecovery of Piecewise Smooth Images from Few Fourier Samples
Recovery of Piecewise Smooth Images from Few Fourier Samples Greg Ongie*, Mathews Jacob Computational Biomedical Imaging Group (CBIG) University of Iowa SampTA 2015 Washington, D.C. 1. Introduction 2.
More informationStructurally Random Matrices
Fast Compressive Sampling Using Structurally Random Matrices Presented by: Thong Do (thongdo@jhu.edu) The Johns Hopkins University A joint work with Prof. Trac Tran, The Johns Hopkins University it Dr.
More informationAlgebraic Iterative Methods for Computed Tomography
Algebraic Iterative Methods for Computed Tomography Per Christian Hansen DTU Compute Department of Applied Mathematics and Computer Science Technical University of Denmark Per Christian Hansen Algebraic
More informationCompressed Sensing and Applications by using Dictionaries in Image Processing
Advances in Computational Sciences and Technology ISSN 0973-6107 Volume 10, Number 2 (2017) pp. 165-170 Research India Publications http://www.ripublication.com Compressed Sensing and Applications by using
More informationECE 8201: Low-dimensional Signal Models for High-dimensional Data Analysis
ECE 8201: Low-dimensional Signal Models for High-dimensional Data Analysis Yuejie Chi Departments of ECE and BMI The Ohio State University September 24, 2015 Time, location, and office hours Time: Tue/Thu
More informationThe convex geometry of inverse problems
The convex geometry of inverse problems Benjamin Recht Department of Computer Sciences University of Wisconsin-Madison Joint work with Venkat Chandrasekaran Pablo Parrilo Alan Willsky Linear Inverse Problems
More informationCompressive Sensing for Multimedia. Communications in Wireless Sensor Networks
Compressive Sensing for Multimedia 1 Communications in Wireless Sensor Networks Wael Barakat & Rabih Saliba MDDSP Project Final Report Prof. Brian L. Evans May 9, 2008 Abstract Compressive Sensing is an
More informationFace Recognition via Sparse Representation
Face Recognition via Sparse Representation John Wright, Allen Y. Yang, Arvind, S. Shankar Sastry and Yi Ma IEEE Trans. PAMI, March 2008 Research About Face Face Detection Face Alignment Face Recognition
More informationInverse Problems and Machine Learning
Inverse Problems and Machine Learning Julian Wörmann Research Group for Geometric Optimization and Machine Learning (GOL) 1 What are inverse problems? 2 Inverse Problems cause/ excitation 3 Inverse Problems
More informationP257 Transform-domain Sparsity Regularization in Reconstruction of Channelized Facies
P257 Transform-domain Sparsity Regularization in Reconstruction of Channelized Facies. azemi* (University of Alberta) & H.R. Siahkoohi (University of Tehran) SUMMARY Petrophysical reservoir properties,
More informationAdvanced phase retrieval: maximum likelihood technique with sparse regularization of phase and amplitude
Advanced phase retrieval: maximum likelihood technique with sparse regularization of phase and amplitude A. Migukin *, V. atkovnik and J. Astola Department of Signal Processing, Tampere University of Technology,
More informationIncoherent noise suppression with curvelet-domain sparsity Vishal Kumar, EOS-UBC and Felix J. Herrmann, EOS-UBC
Incoherent noise suppression with curvelet-domain sparsity Vishal Kumar, EOS-UBC and Felix J. Herrmann, EOS-UBC SUMMARY The separation of signal and noise is a key issue in seismic data processing. By
More informationA Study on Compressive Sensing and Reconstruction Approach
A Study on Compressive Sensing and Reconstruction Approach Utsav Bhatt, Kishor Bamniya Department of Electronics and Communication Engineering, KIRC, Kalol, India Abstract : This paper gives the conventional
More informationSPARSITY ADAPTIVE MATCHING PURSUIT ALGORITHM FOR PRACTICAL COMPRESSED SENSING
SPARSITY ADAPTIVE MATCHING PURSUIT ALGORITHM FOR PRACTICAL COMPRESSED SENSING Thong T. Do, Lu Gan, Nam Nguyen and Trac D. Tran Department of Electrical and Computer Engineering The Johns Hopkins University
More informationRobust image recovery via total-variation minimization
Robust image recovery via total-variation minimization Rachel Ward University of Texas at Austin (Joint work with Deanna Needell, Claremont McKenna College) February 16, 2012 2 Images are compressible
More informationCompressive Sensing based image processing in TrapView pest monitoring system
Compressive Sensing based image processing in TrapView pest monitoring system Milan Marić *, Irena Orović ** and Srdjan Stanković ** * S&T Crna Gora d.o.o, Podgorica, Montenegro ** University of Montenegro,
More informationTERM PAPER ON The Compressive Sensing Based on Biorthogonal Wavelet Basis
TERM PAPER ON The Compressive Sensing Based on Biorthogonal Wavelet Basis Submitted By: Amrita Mishra 11104163 Manoj C 11104059 Under the Guidance of Dr. Sumana Gupta Professor Department of Electrical
More informationGeneralized Principal Component Analysis CVPR 2007
Generalized Principal Component Analysis Tutorial @ CVPR 2007 Yi Ma ECE Department University of Illinois Urbana Champaign René Vidal Center for Imaging Science Institute for Computational Medicine Johns
More informationCompressive Topology Identification of Interconnected Dynamic Systems via Clustered Orthogonal Matching Pursuit
Compressive Topology Identification of Interconnected Dynamic Systems via Clustered Orthogonal Matching Pursuit Borhan M. Sanandaji, Tyrone L. Vincent, and Michael B. Wakin Abstract In this paper, we consider
More informationImage reconstruction based on back propagation learning in Compressed Sensing theory
Image reconstruction based on back propagation learning in Compressed Sensing theory Gaoang Wang Project for ECE 539 Fall 2013 Abstract Over the past few years, a new framework known as compressive sampling
More informationarxiv: v1 [stat.ml] 14 Jan 2017
LEARNING TO INVERT: SIGNAL RECOVERY VIA DEEP CONVOLUTIONAL NETWORKS Ali Mousavi and Richard G. Baraniuk arxiv:171.3891v1 [stat.ml] 14 Jan 17 ABSTRACT The promise of compressive sensing (CS) has been offset
More informationTomographic reconstruction: the challenge of dark information. S. Roux
Tomographic reconstruction: the challenge of dark information S. Roux Meeting on Tomography and Applications, Politecnico di Milano, 20-22 April, 2015 Tomography A mature technique, providing an outstanding
More informationAdaptive step forward-backward matching pursuit algorithm
ISSN 746-7659, England, UK Journal of Information and Computing Science Vol, No, 06, pp 53-60 Adaptive step forward-backward matching pursuit algorithm Songjiang Zhang,i Zhou,Chuanlin Zhang 3* School of
More informationCompressed Sensing for Rapid MR Imaging
Compressed Sensing for Rapid Imaging Michael Lustig1, Juan Santos1, David Donoho2 and John Pauly1 1 Electrical Engineering Department, Stanford University 2 Statistics Department, Stanford University rapid
More informationSparse Signals Reconstruction Via Adaptive Iterative Greedy Algorithm
Sparse Signals Reconstruction Via Adaptive Iterative Greedy Algorithm Ahmed Aziz CS Dept.,Fac. of computers and Informatics Univ. of Benha Benha, Egypt Walid Osamy CS Dept.,Fac. of computers and Informatics
More informationRECONSTRUCTION ALGORITHMS FOR COMPRESSIVE VIDEO SENSING USING BASIS PURSUIT
RECONSTRUCTION ALGORITHMS FOR COMPRESSIVE VIDEO SENSING USING BASIS PURSUIT Ida Wahidah 1, Andriyan Bayu Suksmono 1 1 School of Electrical Engineering and Informatics, Institut Teknologi Bandung Jl. Ganesa
More informationBlind Compressed Sensing Using Sparsifying Transforms
Blind Compressed Sensing Using Sparsifying Transforms Saiprasad Ravishankar and Yoram Bresler Department of Electrical and Computer Engineering and Coordinated Science Laboratory University of Illinois
More informationAlgebraic Iterative Methods for Computed Tomography
Algebraic Iterative Methods for Computed Tomography Per Christian Hansen DTU Compute Department of Applied Mathematics and Computer Science Technical University of Denmark Per Christian Hansen Algebraic
More informationLearning based face hallucination techniques: A survey
Vol. 3 (2014-15) pp. 37-45. : A survey Premitha Premnath K Department of Computer Science & Engineering Vidya Academy of Science & Technology Thrissur - 680501, Kerala, India (email: premithakpnath@gmail.com)
More informationReview and Implementation of DWT based Scalable Video Coding with Scalable Motion Coding.
Project Title: Review and Implementation of DWT based Scalable Video Coding with Scalable Motion Coding. Midterm Report CS 584 Multimedia Communications Submitted by: Syed Jawwad Bukhari 2004-03-0028 About
More informationThe convex geometry of inverse problems
The convex geometry of inverse problems Benjamin Recht Department of Computer Sciences University of Wisconsin-Madison Joint work with Venkat Chandrasekaran Pablo Parrilo Alan Willsky Linear Inverse Problems
More informationSparse Models in Image Understanding And Computer Vision
Sparse Models in Image Understanding And Computer Vision Jayaraman J. Thiagarajan Arizona State University Collaborators Prof. Andreas Spanias Karthikeyan Natesan Ramamurthy Sparsity Sparsity of a vector
More informationIMAGE DENOISING TO ESTIMATE THE GRADIENT HISTOGRAM PRESERVATION USING VARIOUS ALGORITHMS
IMAGE DENOISING TO ESTIMATE THE GRADIENT HISTOGRAM PRESERVATION USING VARIOUS ALGORITHMS P.Mahalakshmi 1, J.Muthulakshmi 2, S.Kannadhasan 3 1,2 U.G Student, 3 Assistant Professor, Department of Electronics
More informationTotal Variation Denoising with Overlapping Group Sparsity
1 Total Variation Denoising with Overlapping Group Sparsity Ivan W. Selesnick and Po-Yu Chen Polytechnic Institute of New York University Brooklyn, New York selesi@poly.edu 2 Abstract This paper describes
More informationA Novel Image Super-resolution Reconstruction Algorithm based on Modified Sparse Representation
, pp.162-167 http://dx.doi.org/10.14257/astl.2016.138.33 A Novel Image Super-resolution Reconstruction Algorithm based on Modified Sparse Representation Liqiang Hu, Chaofeng He Shijiazhuang Tiedao University,
More informationSPARSE signal representations provide a general signal
IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 58, NO. 2, FEBRUARY 2012 1135 Rank Awareness in Joint Sparse Recovery Mike E. Davies, Member, IEEE, and Yonina C. Eldar, Senior Member, IEEE Abstract This
More informationCS 664 Segmentation. Daniel Huttenlocher
CS 664 Segmentation Daniel Huttenlocher Grouping Perceptual Organization Structural relationships between tokens Parallelism, symmetry, alignment Similarity of token properties Often strong psychophysical
More informationImage Reconstruction from Multiple Sparse Representations
Image Reconstruction from Multiple Sparse Representations Robert Crandall Advisor: Professor Ali Bilgin University of Arizona Program in Applied Mathematics 617 N. Santa Rita, Tucson, AZ 85719 Abstract
More informationCompressed Sensing and L 1 -Related Minimization
Compressed Sensing and L 1 -Related Minimization Yin Wotao Computational and Applied Mathematics Rice University Jan 4, 2008 Chinese Academy of Sciences Inst. Comp. Math The Problems of Interest Unconstrained
More informationStructure from Motion
Structure from Motion Outline Bundle Adjustment Ambguities in Reconstruction Affine Factorization Extensions Structure from motion Recover both 3D scene geoemetry and camera positions SLAM: Simultaneous
More informationOutlier Pursuit: Robust PCA and Collaborative Filtering
Outlier Pursuit: Robust PCA and Collaborative Filtering Huan Xu Dept. of Mechanical Engineering & Dept. of Mathematics National University of Singapore Joint w/ Constantine Caramanis, Yudong Chen, Sujay
More informationOpen Access Reconstruction Technique Based on the Theory of Compressed Sensing Satellite Images
Send Orders for Reprints to reprints@benthamscience.ae 74 The Open Electrical & Electronic Engineering Journal, 2015, 9, 74-81 Open Access Reconstruction Technique Based on the Theory of Compressed Sensing
More informationSignal Processing with Side Information
Signal Processing with Side Information A Geometric Approach via Sparsity João F. C. Mota Heriot-Watt University, Edinburgh, UK Side Information Signal processing tasks Denoising Reconstruction Demixing
More informationA fast algorithm for sparse reconstruction based on shrinkage, subspace optimization and continuation [Wen,Yin,Goldfarb,Zhang 2009]
A fast algorithm for sparse reconstruction based on shrinkage, subspace optimization and continuation [Wen,Yin,Goldfarb,Zhang 2009] Yongjia Song University of Wisconsin-Madison April 22, 2010 Yongjia Song
More informationMotivation. My General Philosophy. Assumptions. Advanced Computer Graphics (Spring 2013) Precomputation-Based Relighting
Advanced Computer Graphics (Spring 2013) CS 283, Lecture 17: Precomputation-Based Real-Time Rendering Ravi Ramamoorthi http://inst.eecs.berkeley.edu/~cs283/sp13 Motivation Previously: seen IBR. Use measured
More informationArray geometries, signal type, and sampling conditions for the application of compressed sensing in MIMO radar
Array geometries, signal type, and sampling conditions for the application of compressed sensing in MIMO radar Juan Lopez a and Zhijun Qiao a a Department of Mathematics, The University of Texas - Pan
More informationCoded Acquisition of High Speed Videos with. Multiple Cameras
Coded Acquisition of High Speed Videos with Multiple Cameras CODED ACQUISITION OF HIGH SPEED VIDEOS WITH MULTIPLE CAMERAS BY REZA POURNAGHI, M.A.Sc. (Electrical Engineering) McMaster University, Hamilton,
More informationELEC Dr Reji Mathew Electrical Engineering UNSW
ELEC 4622 Dr Reji Mathew Electrical Engineering UNSW Review of Motion Modelling and Estimation Introduction to Motion Modelling & Estimation Forward Motion Backward Motion Block Motion Estimation Motion
More informationRecognition, SVD, and PCA
Recognition, SVD, and PCA Recognition Suppose you want to find a face in an image One possibility: look for something that looks sort of like a face (oval, dark band near top, dark band near bottom) Another
More informationStereo Vision. MAN-522 Computer Vision
Stereo Vision MAN-522 Computer Vision What is the goal of stereo vision? The recovery of the 3D structure of a scene using two or more images of the 3D scene, each acquired from a different viewpoint in
More informationDepartment of Electronics and Communication KMP College of Engineering, Perumbavoor, Kerala, India 1 2
Vol.3, Issue 3, 2015, Page.1115-1021 Effect of Anti-Forensics and Dic.TV Method for Reducing Artifact in JPEG Decompression 1 Deepthy Mohan, 2 Sreejith.H 1 PG Scholar, 2 Assistant Professor Department
More informationIterative CT Reconstruction Using Curvelet-Based Regularization
Iterative CT Reconstruction Using Curvelet-Based Regularization Haibo Wu 1,2, Andreas Maier 1, Joachim Hornegger 1,2 1 Pattern Recognition Lab (LME), Department of Computer Science, 2 Graduate School in
More informationDistributed Compressed Estimation Based on Compressive Sensing for Wireless Sensor Networks
Distributed Compressed Estimation Based on Compressive Sensing for Wireless Sensor Networks Joint work with Songcen Xu and Vincent Poor Rodrigo C. de Lamare CETUC, PUC-Rio, Brazil Communications Research
More informationEXAM SOLUTIONS. Image Processing and Computer Vision Course 2D1421 Monday, 13 th of March 2006,
School of Computer Science and Communication, KTH Danica Kragic EXAM SOLUTIONS Image Processing and Computer Vision Course 2D1421 Monday, 13 th of March 2006, 14.00 19.00 Grade table 0-25 U 26-35 3 36-45
More informationAn Approach for Reduction of Rain Streaks from a Single Image
An Approach for Reduction of Rain Streaks from a Single Image Vijayakumar Majjagi 1, Netravati U M 2 1 4 th Semester, M. Tech, Digital Electronics, Department of Electronics and Communication G M Institute
More informationCPSC 340: Machine Learning and Data Mining. Principal Component Analysis Fall 2016
CPSC 340: Machine Learning and Data Mining Principal Component Analysis Fall 2016 A2/Midterm: Admin Grades/solutions will be posted after class. Assignment 4: Posted, due November 14. Extra office hours:
More informationApplications: Sampling/Reconstruction. Sampling and Reconstruction of Visual Appearance. Motivation. Monte Carlo Path Tracing.
Sampling and Reconstruction of Visual Appearance CSE 274 [Fall 2018], Lecture 5 Ravi Ramamoorthi http://www.cs.ucsd.edu/~ravir Applications: Sampling/Reconstruction Monte Carlo Rendering (biggest application)
More informationSpatially-Localized Compressed Sensing and Routing in Multi-Hop Sensor Networks 1
Spatially-Localized Compressed Sensing and Routing in Multi-Hop Sensor Networks 1 Sungwon Lee, Sundeep Pattem, Maheswaran Sathiamoorthy, Bhaskar Krishnamachari and Antonio Ortega University of Southern
More informationSPARSE SIGNAL RECONSTRUCTION FROM NOISY COMPRESSIVE MEASUREMENTS USING CROSS VALIDATION. Petros Boufounos, Marco F. Duarte, Richard G.
SPARSE SIGNAL RECONSTRUCTION FROM NOISY COMPRESSIVE MEASUREMENTS USING CROSS VALIDATION Petros Boufounos, Marco F. Duarte, Richard G. Baraniuk Rice University, Electrical and Computer Engineering, Houston,
More informationWhen Sparsity Meets Low-Rankness: Transform Learning With Non-Local Low-Rank Constraint For Image Restoration
When Sparsity Meets Low-Rankness: Transform Learning With Non-Local Low-Rank Constraint For Image Restoration Bihan Wen, Yanjun Li and Yoram Bresler Department of Electrical and Computer Engineering Coordinated
More informationImage Reconstruction based on Block-based Compressive Sensing
Image Reconstruction based on Block-based Compressive Sensing Hanxu YOU, Jie ZHU Department of Electronic Engineering Shanghai Jiao Tong University (SJTU) Shanghai, China gongzihan@sjtu.edu.cn, zhujie@sjtu.edu.cn
More informationHyperspectral Data Classification via Sparse Representation in Homotopy
Hyperspectral Data Classification via Sparse Representation in Homotopy Qazi Sami ul Haq,Lixin Shi,Linmi Tao,Shiqiang Yang Key Laboratory of Pervasive Computing, Ministry of Education Department of Computer
More information