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Wallace H. Coulter Department of Biomedical Engineering
Wallace H. Coulter Department of Biomedical Engineering
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ItemComputational Algebraic Geometry and Switching Surfaces in Optimal Control(Georgia Institute of Technology, 199912) Walther, Uli ; Georgiou, Tryphon T. ; Tannenbaum, Allen R.A number of problems in control can be reduced to finding suitable real solutions of algebraic equations. In particular, such a problem arises in the context of switching surfaces in optimal control. Recently, a powerful new methodology for doing symbolic manipulations with polynomial data has been developed and tested, namely the use of Groebner bases. In this paper, we apply the Groebner basis technique to find effective solutions to the classical problem of timeoptimal control.

ItemThe HamiltonJacobi skeleton(Georgia Institute of Technology, 199909) Siddiqi, Kaleem ; Bouix, Sylvain ; Tannenbaum, Allen R. ; Zucker, Steven W.The eikonal equation and variants of it are of significant interest for problems in computer vision and image processing. It is the basis for continuous versions of mathematical morphology, stereo, shapefromshading and for recent dynamic theories of shape. Its numerical simulation can be delicate, owing to the formation of singularities in the evolving front, and is typically based or, level set methods. However there are more classical approaches rooted in Hamiltonian physics, which have received little consideration in computer vision. In this paper we first introduce a new algorithm for simulating the eikonal equation, which offers a number of computational and conceptual advantages over the earlier methods when it comes to shock tracking. Next, we introduce a very efficient algorithm for shock detection, where the key idea is to measure the net outward flux of a vector field per unit volume, and to detect locations where a conservation of energy principle is violated. We illustrate the approach with several numerical examples including skeletons of complex 2D and 3D shapes.

ItemOn the Evolution of the Skeleton(Georgia Institute of Technology, 199909) August, Jonas ; Tannenbaum, Allen R. ; Zucker, Steven W.It is commonly held that skeleton variation due to noise is unmanageable. It is also believed that smoothing, invoked to combat noise, creates no new structures, as in the causality principle for smoothing images. We demonstrate that both views are incorrect. We characterize how smooth points of the skeleton evolve under a general boundary evolution, with the corollary that, when the boundary is smoothed by a geometric heat equation, the skeleton evolves according to a related geometric heat equation. The surprise is that, while certain aspects of the skeleton simplify, as one would expect, others can behave wildly, including the creation of new skeleton branches. Fortunately such sections can be flagged as ligature, or those portions of the skeleton related to shape concavities. Our analysis also includes junctions and an explicit model for boundary noise. Provided a smoothness condition is met, the skeleton can often reduce noise. However when the smoothness condition is violated, the skeleton can change violently, which, we speculate, corresponds to situations in which "parts" are created, e.g., when the handle appears on a rotating cup.

ItemOn the Laplace–Beltrami Operator and Brain Surface Flattening(Georgia Institute of Technology, 199908) Angenent, Sigurd ; Haker, Steven ; Tannenbaum, Allen R. ; Kikinis, RonIn this paper, using certain conformal mappings from uniformization theory, the authors give an explicit method for flattening the brain surface in a way which preserves angles. From a triangulated surface representation of the cortex, the authors indicate how the procedure may be implemented using finite elements. Further, they show how the geometry of the brain surface may be studied using this approach.

ItemCurve Evolution Models for RealTime Identification with Application to Plasma Etching(Georgia Institute of Technology, 199901) Berg, Jordan M. ; Yezzi, Anthony ; Tannenbaum, Allen R.It is desirable, in constructing an algorithm for realtime control or identification of free surfaces, to avoid representations of the surface requiring mesh refinement at corners or special logic for topological transitions. Level set methods provide a promising framework for such algorithms. In this paper we present: 1) a mathematical representation of free surface motion that is particularly wellsuited to realtime implementation; 2) a technique for estimating an isotropic and homogeneous normal velocity based on a simple measurement; and 3) an application to a semiconductor etching problem.

ItemAdaptive wavelet estimator for nonparametric density deconvolution(Georgia Institute of Technology, 1999) Pensky, Marianna ; Vidakovic, BraniThe problem of estimating a density g based on a sample X ₁ X ₂,…,X [subscript n] from p = q ∗ g is considered. Linear and nonlinear wavelet estimators based on Meyertype wavelets are constructed. The estimators are asymptotically optimal and adaptive if g belongs to the Sobolev space H[superscript α].Moreover, the estimators considered in this paper adjust automatically to the situation when g is supersmooth.

ItemOn the Affine Heat Equation for NonConvex Curves(Georgia Institute of Technology, 199807) Angenent, Sigurd ; Sapiro, Guillermo ; Tannenbaum, Allen R.In this paper, we extend to the nonconvex case the affine invariant geometric heat equation studied by Sapiro and Tannenbaum for convex plane curves. We prove that a smooth embedded plane curve will converge to a point when evolving according to this flow. This result extends the analogy between the affine heat equation and the wellknown Euclidean geometric heat equation.

ItemNonlinear Wavelet Shrinkage with Bayes Rules and Bayes Factors(Georgia Institute of Technology, 199803) Vidakovic, BraniWavelet shrinkage, the method proposed by the seminal work of Donoho and Johnstone is a disarmingly simple and efficient way of denoising data. Shrinking wavelet coefficients was proposed from several optimality criteria. In this article a wavelet shrinkage by coherent Bayesian inference in the wavelet domain is proposed. The methods are tested on standard DonohoJohnstone test functions.

ItemIntroduction to the Special Issue on Partial Differential Equations and GeometryDriven Diffusion in Image Processing and Analysis(Georgia Institute of Technology, 199803) Caselles, Vicent ; Morel, JeanMichel ; Sapiro, Guillermo ; Tannenbaum, Allen R.

ItemHyperbolic "Smoothing" of shapes(Georgia Institute of Technology, 199801) Siddiqi, Kaleem ; Tannenbaum, Allen R. ; Zucker, Steven W.We have been developing a theory of generic 2D shape based on a reactiondiffusion model from mathematical physics. The description of a shape is derived from the singularities of a curve evolution process driven by the reaction (hyperbolic) term. The diffusion (parabolic) term is related to smoothing and shape simplification. However, the unification of the two is problematic, because the slightest amount of diffusion dominates and prevents the formation of generic firstorder shocks. The technical issue is whether it is possible to smooth a shape, in any sense, without destroying the shocks. We now report a constructive solution to this problem, by embedding the smoothing term in a global metric against which a purely hyperbolic evolution is performed from the initial curve. This is a new flow for shape, that extends the advantages of the original one. Specific metrics are developed, which lead to a natural hierarchy of shape features, analogous to the simplification one might perceive when viewing an object from increasing distances. We illustrate our new flow with a variety of examples.