Research

Brief Research Description:

• Composite Dilation Wavelets: Composite dilation wavelets, introduced by Guo, Labate, Lim, Weiss and Wilson, are affine systems generated by translations from a full rank lattice and dilations from two countable sets of matrices. These affine systems form an orthonormal basis or a Parseval frame for the space of square integrable functions in dimension n. In my research, I have focused on the situation where both sets of dilation matrices are subgroups of the invertible nxn matrices. Currently, I am working on finding smoother compactly supported composite wavelets than the known Haar-type wavelets (joint work with Ilya Krishtal) and classifying the composite dilation Haar-type wavelets (joint work with Kyle Steffen). I am also interested in developing implementation algorithms for composite wavelets and smoothing MSF composite dilation wavelets.
• Compressed Sensing and High Dimensional Geometry: Donoho and Tanner established necessary and sufficient conditions on a measurement matrix such that the l_1 minimization produces the sparsest solution: if the measurment matrix, A, preserves k-neighborliness when projecting the standard simplex or the l_1 ball, this measurement matrix is one such that linear programming will provide the sparesest solution to the incidence y=Ax. Simultaneously, Candes, Romberg, Tao, Vershynin and others introduced the restricted isometry property that also gaurantees that solving the l_1 minimization problem will give you the minimum l_0 (sparesest) solution. Certain random matrix ensembles have been shown to satisfy both conditions. I have been working on comparability of these and other compressed sensing results (joint work with Coralia Cartis, Jared Tanner, and Andrew Thompson).

Publications:

• Crystallographic Haar-type Composite Dilation Wavelets (with Kyle R. Steffen), submitted, 2009 (preprint version)
• Matricial Filters and Crystallographic Composite Dilation Wavelets (with Ilya A. Krishtal), submitted, 2009 (preprint version)
• Phase Transitions for Greedy Sparse Approximation Algorithms (with Coralia Cartis, Jared Tanner, and Andrew Thompson) submitted 2009 (preprint version)
• On Support Sizes of Restricted Isometry Constants, (with Andrew Thompson) submitted 2009 (preprint version)
• Compressed Sensing: How Sharp is the Restricted Isometry Property?, (with Coralia Cartis and Jared Tanner) submitted 2009 (preprint version)
  • Extended technical report (formerly titled The restricted isometry property and l^q-regualrization: phase transitions for sparse approximation)
• Decay Properties of Restricted Isometry Constants, (with Coralia Cartis and Jared Tanner), IEEE Signal Processing Letters, 16(7), 572-575, 2009 (preprint version)
• Minimally Supported Frequency Composite Dilation Parseval Frame Wavelets, Journal of Geometric Analysis, 19(1), 19-35, 2009 (preprint version)
• Minimally Supported Frequency Composite Dilation Wavelets, Journal of Fourier Analysis and Applications, 15(6), 796-815, 2009 (preprint version)

• Phase Transitions for Restricted Isometry Properties, (with Coralia Cartis and Jared Tanner) Proceedings of Signal Processing with Adaptive Sparse Structured Representations, April 2009

Selected Conferences:

• MathFest and Project NExT Workshop, Portland, OR, August 2009
• 20 Years of Wavelets, Chicago, IL, May 2009
• SPARS'09, Workshop on Signal Processing with Adaptive Sparse Structured Representations, Saint-Malo, France, April, 2009
• AMS-MAA Joint Mathematics Meeting, Washington, D.C., January 2009
• Defense Threat Reduction Agency/National Science Foundation Algorithms Workshop, Baltimore, MD, November 2008
• MathFest and Project NExT Workshop, Madison, WI, July 2008
• Foundation of Computational Mathematics, Hong Kong, China, June 2008
• Classical and Modern Harmonic Analysis: from Theory to Numerical Computation, ICMS, Edinburgh, Scotland, May 2008
• AMS-MAA Joint Mathematics Meeting, Special Session on Wavelet Sets, San Diego, CA, January 2008
• Von Neumman Symposium: Sparse Approximation and High Dimensional Geometry, Snowbird, UT, July 2007
• STROBL07: Trends in Harmonic Analysis, Strobl, Austria, June 2007
• IPAM Short Course: Sparse Approximation and High Dimensional Geometry, Los Angeles, CA, May 2007
• AMS-MAA Joint Mathematics Meeting, New Orleans, LA, January 2007
• Washington University/University of Zagreb International Wavelet Workshop, St. Louis, MO, April 2006
• AMS Western Regional Meeting, Special Session on Wavelets, Eugene, OR, November 2005

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