On this page: Research Interests | Research Group | Motivation | Projects and Contributions | Additional Information & Resources
Signal and image processing using sparse, low-dimensional, and manifold-based signal models
Multiscale geometric analysis for image processing, compression, estimation, and computer vision
Low-rate signal sensing and acquisition; signal recovery from partial information; compressive sensing
Multiple signal processing, compression, acquisition, and recovery; light field imaging; sensor networks
Approximation theory and computational harmonic analysis; wavelets, curvelets, and wedgelets
Jae Young Park, Ph.D. 2013, University of Michigan (co-advised with Anna Gilbert)
Alejandro Weinstein, Ph.D. 2013, now Assistant Professor, Biomedical Engineering, Universidad de Valparaiso, Chile ( thesis)
Borhan Sanandaji, Ph.D. 2012 (co-advised with Tyrone Vincent), now Postdoctoral Scholar, UC Berkeley ( thesis)
Michael Coco, M.S. 2012 (co-advised with Lawrence Wiencke)
Effective techniques for processing and understanding data and information often rely on some sort of model that characterizes the expected behavior of the information. In many cases, the model conveys a notion of constrained structure or conciseness to the information; one may believe, for example, that a length-N data vector (or signal) has few degrees of freedom relative to its size N. Examples include bandlimited signals, images containing low-dimensional geometric features, or collections of signals observed from multiple viewpoints in a camera or sensor network.
The notion of conciseness is a very powerful assumption, and it suggests the potential for developing highly accurate and efficient data processing algorithms that capture and exploit the inherent structure of the model. In some cases, for example, the signals of interest can be expressed as sparse linear combinations of elements from some dictionary---the sparsity of the representation directly reflects the conciseness of the model and permits efficient algorithms for signal processing. Sparsity also forms the core of the emerging theory of
Compressive Sensing, which states that a sparse signal can be recovered from a small number of random linear measurements.
In other cases, however, sparse representations may not suffice to truly capture the underlying structure of a signal. Instead, the conciseness of the signal model may impose a low-dimensional geometric (often manifold-like) structure to the signal class as a subset of the high-dimensional ambient signal space. To date, the importance and utility of manifolds for signal processing has been acknowledged largely through a research effort into learning manifold structure from a collection of data points. While these methods have proved effective for certain tasks (such as classification and recognition), they also tend to be quite generic and fail to consider the geometric nuances of specific signal classes.
Projects and Contributions
The purpose of my research is to develop new methods and understanding for signal and data processing based on low-dimensional signal models, with a particular focus on the role of geometry. Below is a list of key topics (with links to selected publications):
Compressive Sensing and Dimensionality Reduction
E. J. Candès and M. B. Wakin, An introduction to compressive sampling, in IEEE Signal Processing Magazine, vol. 25, no. 2, pp. 2130, March 2008.
R. G. Baraniuk, V. Cevher, and M. B. Wakin, Low-dimensional models for dimensionality reduction and signal recovery: A geometric perspective, Proceedings of the IEEE, vol. 98, no. 6, pp. 959-971, June 2010.
M.B. Wakin, Concise Signal Models, Connexions modules endorsed by the IEEE Signal Processing Society.
Compressive Sensing of Analog Signals; Compressive Measurement Systems & Hardware
M. A. Davenport and M. B. Wakin, Compressive Sensing of Analog Signals Using Discrete Prolate Spheroidal Sequences, Applied and Computational Harmonic Analysis, vol. 33, no. 3, pp. 438-472, November 2012.
M. Wakin, S. Becker, E. Nakamura, M. Grant, E. Sovero, D. Ching, J. Yoo, J. Romberg, A. Emami-Neyestanak, and E. Candès, A Non-Uniform Sampler for Wideband Spectrally-Sparse Environments, IEEE Journal on Emerging and Selected Topics in Circuits and Systems, vol. 2, no. 3, pp. 516-529, September 2012. ( authors' copy)
J. Yoo, C. Turnes, E. Nakamura, C. Le, S. Becker, E. Sovero, M. Wakin, M. Grant, J. Romberg, A. Emami-Neyestanak, and E. Candès, A Compressed Sensing Parameter Extraction Platform for Radar Pulse Signal Acquisition, IEEE Journal on Emerging and Selected Topics in Circuits and Systems, vol. 2, no. 3, pp. 626-638, September 2012. ( authors' copy)
M. B. Wakin, J. N. Laska, M. F. Duarte, D. Baron, S. Sarvotham, D. Takhar, K. F. Kelly, and R. G. Baraniuk, An Architecture for Compressive Imaging, in IEEE 2006 International Conference on Image Processing -- ICIP 2006, Atlanta, GA, Oct. 2006.
J. A. Tropp, M. B. Wakin, M. F. Duarte, D. Baron, and R. G. Baraniuk, Random Filters for Compressive Sampling and Reconstruction, in IEEE 2006 International Conference on Acoustics, Speech, and Signal Processing -- ICASSP 2006, Toulouse, France, May 2006.
See also: Team website for Analog to Information project
See also: Compressive Imaging: A New Single Pixel Camera
Manifold-based Compressive Sensing and Manifold Learning
L. Carin, R.G. Baraniuk, V. Cevher, D. Dunson, M.I. Jordan, G. Sapiro, and M.B. Wakin, Learning Low-Dimensional Signal Models: A Bayesian Approach Based on Incomplete Measurements, IEEE Signal Processing Magazine, vol. 28, no. 2, pp. 39-51, March 2011.
M. B. Wakin, Manifold-Based Signal Recovery and Parameter Estimation from Compressive Measurements, Preprint, 2008.
R. G. Baraniuk and M. B. Wakin, Random Projections of Smooth Manifolds, Foundations of Computational Mathematics, vol. 9, no. 1, pp. 51-77, February 2009.
H. L. Yap, M. B. Wakin, and C.J. Rozell, Stable Manifold Embeddings with Structured Random Matrices, to appear in Journal of Selected Topics in Signal Processing.
C. Hegde, M. Wakin, and R. Baraniuk, Random Projections for Manifold Learning, in Neural Information Processing Systems -- NIPS, Vancouver, Canada, December 2007.
Multi-signal and Multi-image Compressive Sensing
A. C. Gilbert, J. Y. Park, and M. B. Wakin, Sketched SVD: Recovering Spectral Features from Compressive Measurements, Preprint, 2012.
J. Y. Park and M. B. Wakin, A Multiscale Algorithm for Reconstructing Videos from Streaming Compressive Measurements, to appear in Journal of Electronic Imaging. (See also: companion technical report.)
M. B. Wakin, A Study of the Temporal Bandwidth of Video and its Implications in Compressive Sensing, Colorado School of Mines Technical Report 2012-08-15, August 2012.
M. F. Duarte, M. B. Wakin, D. Baron, S. Sarvotham, and R. G. Baraniuk, Measurement Bounds for Sparse Signal Ensembles via Graphical Models, to appear in IEEE Transactions on Information Theory.
J. Y. Park and M. B. Wakin, A Geometric Approach to Multi-view Compressive Imaging, EURASIP Journal on Advances in Signal Processing, vol. 2012, article 37, 2012.
Algorithms for Sparse Signal Recovery
M. A. Davenport, D. Needell, and M. B. Wakin, Signal Space CoSaMP for Sparse Recovery with Redundant Dictionaries, Preprint, 2012.
A. J. Weinstein and M. B. Wakin, Recovering a Clipped Signal in Sparseland, to appear in Sampling Theory in Signal and Image Processing. (Old draft available here; final version to be posted soon.)
A.J. Weinstein and M.B. Wakin, Online Search Orthogonal Matching Pursuit, in IEEE Statistical Signal Processing Workshop -- SSP 2012, Ann Arbor, Michigan 2012.
M. A. Davenport and M. B. Wakin, Analysis of Orthogonal Matching Pursuit using the Restricted Isometry Property, IEEE Transactions on Information Theory, vol. 56, no. 9, pp. 4395--4401, September 2010.
E. J. Candès, M. B. Wakin, and S. P. Boyd, Enhancing Sparsity by Reweighted L1 Minimization, Journal of Fourier Analysis and Applications, vol. 14, no. 5, pp. 877--905, December 2008.
Theoretical Foundations and Analysis of Randomized Operators
R. Baraniuk, M. Davenport, R. DeVore, and M. Wakin, A Simple Proof of the Restricted Isometry Property for Random Matrices (aka "The Johnson-Lindenstrauss Lemma Meets Compressed Sensing"), Constructive Approximation, vol. 28, no. 3, pp. 253--263, December 2008.
J.Y. Park, H.L. Yap, C.J. Rozell, and M.B. Wakin, Concentration of Measure for Block Diagonal Matrices with Applications to Compressive Signal Processing, IEEE Transactions on Signal Processing, vol. 59, no. 12, pp. 5859-5875, December 2011.
A. Eftekhari, H. L. Yap, C. J. Rozell, and M. B. Wakin, The Restricted Isometry Property for Random Block Diagonal Matrices, Preprint, 2012.
A. Eftekhari, J. Romberg, and M. B. Wakin, Matched Filtering from Limited Frequency Samples, to appear in IEEE Transactions on Information Theory.
M. A. Davenport, P. T. Boufounos, M. B. Wakin, and R. G. Baraniuk, Signal processing with compressive measurements, IEEE Journal of Selected Topics in Signal Processing, vol. 4, no. 2, pp. 445-460, April 2010.
M. Davenport, M. Duarte, M. Wakin, J. Laska, D. Takhar, K. Kelly, and R. Baraniuk, The Smashed Filter for Compressive Classification and Target Recognition, in Computational Imaging V at IS&T/SPIE Electronic Imaging, San Jose, California, January 2007.
M. F. Duarte, M. A. Davenport, M. B. Wakin, and R. G. Baraniuk, Sparse Signal Detection from Incoherent Projections, in IEEE 2006 International Conference on Acoustics, Speech, and Signal Processing -- ICASSP 2006, Toulouse, France, May 2006.
C.W. Lim and M.B. Wakin, Automatic Modulation Recognition for Spectrum Sensing using Nonuniform Compressive Samples, in IEEE International Conference on Communications -- ICC 2012, Ottawa, Canada, June 2012.
C.W. Lim and M.B. Wakin, CHOCS: A Framework for Estimating Compressive, Higher-order Cyclostationary Statistics, in SPIE Defense, Security, and Sensing Symposium -- DSS 2012, Baltimore, Maryland, April 2012.
Compressive System Identification
B. M. Sanandaji, M. B. Wakin, and T. L. Vincent, Observability with Random Observations, Preprint, 2012.
B. M. Sanandaji, T. L. Vincent, and M. B. Wakin, Concentration of Measure Inequalities for Toeplitz Matrices with Applications, IEEE Transactions on Signal Processing, vol. 61, no. 1, pp. 109-117, January 2013. ( authors' copy) (See also companion technical report.)
B.M. Sanandaji, T.L. Vincent, and M.B. Wakin, Compressive Topology Identification of Interconnected Dynamic Systems via Clustered Orthogonal Matching Pursuit, IEEE 2011 Conference on Decision and Control and European Control Conference -- CDC-ECC, Orlando, Florida, December 2011.
B.M. Sanandaji, T.L. Vincent, M.B. Wakin, R. Toth, and K. Poolla, Compressive System Identification of LTI and LTV ARX Models, IEEE 2011 Conference on Decision and Control and European Control Conference -- CDC-ECC, Orlando, Florida, December 2011.
Manifolds and Multi-Image Processing
Non-differentiable Image Manifolds & Parameter Estimation
Single-Signal and Single-Image Processing and Compression
Multiscale Geometric Models for Approximation and Compression
Generalizations and Extensions to Higher Dimensions
Fundamentals of Representing and Encoding Information
M. B. Wakin, M. T. Orchard, R. G. Baraniuk, and V. Chandrasekaran, Phase and Magnitude Perceptual Sensitivities in Nonredundant Complex Wavelet Representations, in Asilomar Conference on Signals, Systems, and Computers, Pacific Grove, California, November 2003.
R.M. Castro, M.B. Wakin, and M.T. Orchard, On the Problem of Simultaneous Encoding of Magnitude and Location Information, in Asilomar Conference on Signals, Systems, and Computers, Pacific Grove, California, November 2002.
Additional Information & Resources
Email: mwakin (at) mines (dot) e d u