Imaging and Computing Group

Imaging and Computing Seminar — Fall 2009

• Mo 09/21, at 3:30pm, in room 36-112

  Justin Romberg, EE, Georgia Tech.

  Topics in Compressed Sensing: Random Convolution and Dynamic Updating

  Abstract: In this talk, we will cover two topics related to compressed sensing (CS). The first has to do with making compressed sensing more "physical". Many of the essential results in CS rely on measurement devices which correlate signals against a series of random waveforms. While there are certain applications in which this can be accomplished, it often requires exotic new hardware. We will show that essentially the same theory can be developed for systems which convolve a signal with a random pulse and then sample the result, a framework which is directly applicable to a number of "active imaging" applications. We will also discuss how mathematical methods from CS can be applied to channel estimation in multiple-input multiple-output (MIMO) systems.

  In the second part of the talk, we will discuss recent progress on algorithms aimed at making compressive sampling "dynamic". We will show how the solutions to L1 optimization programs can be efficiently updated as 1) the signal we are measuring changes, and 2) new measurements are added, and stale ones are removed. The algorithms are based on homotopy methods, and are somewhat analogous to recursive least-squares in that they can be reduced to a series of low-rank updates.

• We 09/23, at 4:00pm, in room 2-146 (joint with the Numerical PDE seminar)

  Gabriel Peyre, Ceremade (applied math), U. Paris-Dauphine.

  Sparse Processing of Images

  Abstract: In this talk, I will review recent work on the sparse representations of natural images. I will focus on the application of these emerging models for the resolution of various imaging problems, which include compression, denoising and super-resolution of images, as well as compressive sensing and compressive wave computations.

  Natural images exhibit a wide range of geometric regularities, such as curvilinear edges and oscillating textures. Adaptive image representations select bases from a dictionary of orthogonal or redundant frames that are parameterized by the geometry of the image. If the geometry is well estimated, the image is sparsely represented by only a few atoms in this dictionary. The resolution of ill-posed inverse problems in image processing is then regularized using sparsity constraints in these adapted representations.

• Th 10/01, at 4:15pm, in room 36-112

  Vladimir Druskin, Schlumberger-Doll Research, Cambridge.

  Optimal finite-difference grids for Neumann-to-Dirichlet operators.

  Abstract: For many applications the solution of a partial differential equation is produced by local sources and is needed only at receiver locations, and not in the entire domain. We introduce and discuss new developments with a rigorous approach to targeted grid refinement which is based on model reduction in the spectral domain, and gives exponential supecrconvergence of the Neumann-to-Dirichlet map. The technique uses simple second order finite-difference approximations with optimized placement of the grid points. The fact that the NtoD map is well approximated makes the technique ideal for inverse problems, domain decomposition and absorbing boundary condition.

  Contributors: Sergey Asvadurov, Liliana Borcea, Murthy Guddatti, Fernando Guevara Vasquez, David Ingerman, Leonid Knizhnerman, Alexander Mamonov and Shari Moskow

• Th 10/22, at 4:15pm, in room 36-112

  Cynthia Rudin, Sloan school of management, MIT.

  Dynamics of AdaBoost

  Abstract: AdaBoost (Freund and Schapire 97) is one of the most successful and popular machine learning algorithms, though some of its basic convergence properties were not understood until recently. I will discuss recent results on AdaBoost's convergence, which were obtained by analyzing an iterated map that is derived from the algorithm. This map exhibits cyclic behavior that can be understood analytically. This approach allows us to solve a well-studied problem of machine learning, namely whether AdaBoost achieves a "maximum margin" solution.

  This talk is designed for a general mathematical audience, and no prior knowledge of machine learning is assumed.

• Th 10/29, at 4:15pm, in room 36-112

  Stephen Boyd, EE, Stanford.

  Performance Bounds for Constrained Linear Stochastic Control

  Abstract: We develop computational bounds on performance for causal state feedback stochastic control with linear dynamics, arbitrary noise distribution, and arbitrary input constraint set. This can be very useful as a comparison to the performance of suboptimal control policies, which we can evaluate using Monte Carlo simulation. Our method involves solving a semidefinite program (a linear optimization problem with linear matrix inequality constraints), a convex optimization problem which can be efficiently solved. Numerical experiments show that the lower bound obtained by our method is often close to the performance achieved by several widely-used suboptimal control policies, which shows that both are nearly optimal. As a by-product, our performance bound yields approximate value functions that can be used as control Lyapunov functions for suboptimal control policies.

  Joint work with Yang Wang.

• Th 11/05, at 4:15pm, in room 36-112

  Tarek Habashy, Schlumberger-Doll Research, Cambridge.

  Overview of Inversion Approaches for Large-Scale Geophysical Measurements

  Abstract: In this presentation we review a number of inversion approaches for large-scale geophysical applications arising in oil exploration, such as cross-well measurements, surface-to-borehole, single-well, and surface EM. We will present a number of approaches that cover both model-based parametric methods and pixel-type imaging schemes. Much of the challenges facing these types of applications are related to the large computational cost of the forward modeling and the inherent non-uniqueness associated with the inversion of these measurements. We will cover some of the regularization approaches that can help in narrowing down the class of solutions that may have physical and practical usefulness. If time allows, we will also cover joint multi-physics inversion for the integration of electromagnetic with seismic or pressure measurements.

• Mo 11/09, at 4:30pm, in room 4-370 (joint with the Applied Math Colloquium)

  Alexander Barnett, Mathematics, Dartmouth.

  Robust and efficient computation of two-dimensional photonic crystal band structure using second-kind integral equations

  Abstract: Photonic crystals are dielectric structures with periodicity on the scale of the wavelength of light. They have a rapidly growing range of applications to signal processing, sensing, negative-index materials, and the exciting possibility of integrated optical computing. Calculating their `band structure' (propagating Bloch waves) is an elliptic PDE eigenvalue problem with (quasi-)periodic boundary conditions on the unit cell, i.e. eigenmodes on a torus. Since the material is piecewise homogeneous, boundary integral equations (BIE) are natural for high-accuracy solution.

  In such geometries BIEs are usually periodized by replacement of the free space Greens function kernel by its quasi-periodic cousin. We show why this approach fails near the (spurious) resonances of the empty torus. We introduce a new approach which cures this problem: imposing the boundary conditions on the unit-cell walls using layer potentials, and a finite number of neighboring images, resulting in a second-kind integral equation with smooth data. This couples to existing BIE tools (including high-order quadratures and Fast Multipole acceleration) in a natural way, allowing accuracies near machine precision. We also discuss inclusions which intersect the unit cell walls, and how we use a small number of evaluations to interpolate over the Brillouin zone to spectral accuracy. Joint work with Leslie Greengard (NYU).

• Th 11/12, at 4:15pm, in room 36-112

  Jerome Le Rousseau, Mathematical physics, Univ. of Orleans.

  Microlocal multi-product representation of solutions of hyperbolic PDEs with applications

• Th 11/19, at 4:15pm, in room 36-112

  Paul Hand, Courant Institute, NYU.

  Easing Cardiac Simulations by Homogenization

  Abstract: Computer simulations of cardiac tissue have the potential to provide a new environment for testing clinical treatments for heart disease and pathologies. Although the biophysics of ion flow at the cellular level is well understood, simulations that resolve individual cells are computationally intractable. Hence, there is a need for tissue level descriptions that capture the system's dynamics. The so-called bidomain equations provide this description, but they contain parameters which are difficult to measure. We will present a modification to how heart muscle is typically described. This modification, combined with the mathematical process of homogenization, will lead to cheaper computations of the bidomain parameters and a more mathematically natural framework for exploring the effects of different assumptions on the structure of heart tissue. Time permitting, we will also use homogenization to derive tissue-level equations for an alternative cardiac communication mechanism. Joint work with Charles Peskin and Boyce Griffith (NYU).

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