Seminars and Colloquia by Series

Local, Non-local and Global Methods in Image Reconstruction

Series
Applied and Computational Mathematics Seminar
Time
Monday, March 28, 2011 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Yifei Lou – GaTech ECE (Minerva Research Group)
Image restoration has been an active research topic in imageprocessing and computer vision. There are vast of literature, mostof which rely on the regularization, or prior information of theunderlying image. In this work, we examine three types of methodsranging from local, nonlocal to global with various applications.A classical approach for local regularization term is achieved bymanipulating the derivatives. We adopt the idea in the localpatch-based sparse representation to present a deblurringalgorithm. The key observation is that the sparse coefficientsthat encode a given image with respect to an over-complete basisare the same that encode a blurred version of the image withrespect to a modified basis. Following an``analysis-by-synthesis'' approach, an explicit generative modelis used to compute a sparse representation of the blurred image,and its coefficients are used to combine elements of the originalbasis to yield a restored image.We follows the framework that generates the neighborhood filtersto an variational formulation for general image reconstructionproblems. Specifically, two extensions regarding to the weightcomputation are investigated. One is to exploit the recurrence ofstructures at different locations, orientations and scales in animage. While previous methods based on ``nonlocal filtering'' identify corresponding patches only up to translations, we consider more general similarity transformation.The second algorithm utilizes a preprocessed data as input for theweight computation. The requirements for preprocessing are (1) fastand (2) containing sharp edges. We get superior results in theapplications of image deconvolution and tomographic reconstruction.A Global approach is explored in a particular scenario, that is,taking a burst of photographs under low light conditions with ahand-held camera. Since each image of the burst is sharp but noisy,our goal is to efficiently denoise these multiple images. Theproposed algorithm is a complex chain involving accurateregistration, video equalization, noise estimation and the use ofstate-of-the-art denoising methods. Yet, we show that this complexchain may become risk free thanks to a key feature: the noise modelcan be estimated accurately from the image burst.

Inversion of the Born Series in Optical Tomography

Series
Applied and Computational Mathematics Seminar
Time
Monday, March 14, 2011 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
John Schotland – University of Michigan, Ann Arbor
The inverse problem of optical tomography consists of recovering thespatially-varying absorption of a highly-scattering medium from boundarymeasurements. In this talk we will discuss direct reconstruction methods forthis problem that are based on inversion of the Born series. In previouswork we have utilized such series expansions as tools to develop fast imagereconstruction algorithms. Here we characterize their convergence, stabilityand approximation error. Analogous results for the Calderon problem ofreconstructing the conductivity in electrical impedance tomography will alsobe presented.

Statistical Shape Analysis of Target Boundaries in 2D Sonar Imagery

Series
Applied and Computational Mathematics Seminar
Time
Monday, March 7, 2011 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Darshan Bryner – Naval Surface Warfare Center/FSU
There are several definitions of the word shape; of these, the most important to this research is “the external form or appearance of someone or something as produced by its outline.” Shape Analysis in this context focuses specifically on the mathematical study of explicit, parameterized curves in 2D obtained from the boundaries of targets of interest in Synthetic Aperture Sonar (SAS) imagery. We represent these curves with a special “square-root velocity function,” whereby the space of all such functions is a nonlinear Riemannian manifold under the standard L^2 metric. With this curve representation, we form the mathematical space called “shape space” where a shape is considered to be the orbit of an equivalence class under the group actions of scaling, translation, rotation, and re-parameterization. It is in this quotient space that we can quantify the distance between two shapes, cluster similar shapes into classes, and form means and covariances of shape classes for statistical inferences. In this particular research application, I use this shape analysis framework to form probability density functions on sonar target shape classes for use as a shape prior energy term in a Bayesian Active Contour model for boundary extraction in SAS images. Boundary detection algorithms generally perform poorly on sonar imagery due to their typically low signal to noise ratio, high speckle noise, and muddled or occluded target edges; thus, it is necessary that we use prior shape information in the evolution of an active contour to achieve convergence to a meaningful target boundary.

Vortex evolution and stability of fish swimming

Series
Applied and Computational Mathematics Seminar
Time
Monday, February 28, 2011 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Fangxu Jing – USC Mechanical Engineering
Vortex dynamics and solid-fluid interaction are two of the most important and most studied topics in fluid dynamics for their relevance to a wide range of applications from geophysical flows to locomotion in moving fluids. In this talk, we investigate two problems in these two areas: Part I studies the viscous evolution of point vortex equilibria; Part II studies the effects of body elasticity on the passive stability of submerged bodies.In Part I, we describe the viscous evolution of point vortex configurations that, in the absence of viscosity, are in a state of fixed or relative equilibrium. In particular, we examine four cases, three of them correspond to relative equilibria in the inviscid point vortex model and one corresponds to a fixed equilibrium. Our goal is to elucidate some of the main transient dynamical features of the flow. Using a multi-Gaussian ``core growing" type of model, we show that all four configurations immediately begin to rotate unsteadily, while the shapes of vortex configurations remain unchanged. We then examine in detail the qualitative and quantitative evolution of the structures as they evolve, and for each case show the sequence of topological bifurcations that occur both in a fixed reference frame, and in an appropriately chosen rotating reference frame. Comparisons between the cases help to reveal different features of the viscous evolution for short and intermediate time ! scales of vortex structures. The dynamical evolution of passive particles in the viscously evolving flow associated with the initial fixed equilibrium is shown and interpreted in relation to the evolving streamline patterns. In Part II, we examine the effects of body geometry and elasticity on the passive stability of motion in a perfect fluid. Our main motivation is to understand the role of body elasticity on the stability of fish swimming. The fish is modeled as an articulated body made of multiple links (assumed to be identical ellipses in 2D or identical ellipsoids in 3D) interconnected by hinge joints. It can undergo shape changes by varying the relative angles between the links. Body elasticity is accounted for via the torsional springs at the joints. The unsteadiness of the flow is modeled using the added mass effect. Equations of motion for the body-fluid system are derived using Newtonian and Lagrangian approaches for both hydrodynamically decoupled and coupled models in 2D and 3D. We specifically examine the stability associated with a relative equilibrium of the equations, traditionally referred to as the ``coast motion" (proved to be unstable for a rigid elongated body model), and f! ound that body elasticity does stabilize the system. Stable regions are identified based on linear stability analysis in the parameter space spanned by aspect ratio (body geometry) and spring constants (muscle stiffness), and the findings based on the linear analysis are verified by direct numerical simulations of the nonlinear system.

Chemotaxis in active suspensions

Series
Applied and Computational Mathematics Seminar
Time
Monday, February 21, 2011 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Enkeleida Lushi – NYU Math Dept.
Micro-organisms are known to respond to certain dissolved chemicalsubstances in their environment by moving preferentially away or towardtheir source in a process called chemotaxis. We study such chemotacticresponses at the population level when the micro-swimmers arehydrodynamically coupled to each-other as well as the chemicalconcentration. We include a chemotactic bias based on the known bacteriarun-and-tumble phenomenon in a kinetic model of motile suspension dynamicsdeveloped recently to study hydrodynamic interactions. The chemicalsubstance can be produced or consumed by the swimmers themselves, as wellas be advected by the fluid flows created by their movement. The linearstability analysis of the system will be discussed, as well as the entropyanalysis. Nonlinear dynamics are investigated using numerical simulationin two dimensions of the full system of equations. We show examples ofaggregation in suspensions of pullers (front-actuated swimmers) anddiscuss how chemotaxis affects the mixing flows in suspensions of pushers(rear-actuated swimmers). Last, I will discuss recent work on numericalsimulations of discrete particle/swimmer suspensions that have achemotactic bias.

Blind Source Separation in Audio

Series
Applied and Computational Mathematics Seminar
Time
Friday, January 21, 2011 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Yang Wang – Michigan State University, Department of Mathematics
The blind source separation (BSS) problem, also better known as the "cocktail party problem", is a well-known and challenging problem in mathematics and engineering. In this talk we discuss a novel time-frequency technique for the BSS problem. We also discuss a related problem in which foreground audio signal is mixed with strong background noise, and present techniques for suppress the background noise.

[Special Time] A mathematical model for bunching and meandering instabilities during epitaxial growth of a thin film

Series
Applied and Computational Mathematics Seminar
Time
Thursday, January 13, 2011 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Michel Jabbour – University of Kentucky
Recent experiments indicate that one- and two-dimensionalinstabilities, bunching and meandering, respectively, coexist duringepitaxial growth of a thin film in the step-flow regime. This is in contrastto the predictions of existing Burton–Cabrera–Frank (BCF) models. Indeed, inthe BCF framework, meandering is predicated on an Ehrlich–Schwoebel (ES)barrier whereas bunching requires an inverse ES effect. Hence, the twoinstabilities appear to be a priori mutually exclusive. In this talk, analternative theory is presented that resolves this apparent paradox. Itsmain ingredient is a generalized Gibbs–Thomson relation for the stepchemical potential resulting in jump conditions along the steps that coupleadatom diffusions on adjacent terraces. Specialization to periodic steptrains reveals a competition between the stabilizing ES kinetics and adestabilizing energetic correction that can lead to step collisions. Theaforementioned instabilities can therefore be understood in terms of thetendency of the crystal to lower, away from equilibrium and in the presenceof dissipation, its total free energy. The presentation will be self-contained and no a priori knowledge of theunderlying physics is needed.

Shape Optimization of Chiral Propellers in 3-D Stokes Flow

Series
Applied and Computational Mathematics Seminar
Time
Monday, December 6, 2010 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Shawn Walker – LSU Mathematics Dept.
Locomotion at the micro-scale is important in biology and in industrialapplications such as targeted drug delivery and micro-fluidics. Wepresent results on the optimal shape of a rigid body locomoting in 3-DStokes flow. The actuation consists of applying a fixed moment andconstraining the body to only move along the moment axis; this models theeffect of an external magnetic torque on an object made of magneticallysusceptible material. The shape of the object is parametrized by a 3-Dcenterline with a given cross-sectional shape. No a priori assumption ismade on the centerline. We show there exists a minimizer to the infinitedimensional optimization problem in a suitable infinite class ofadmissible shapes. We develop a variational (constrained) descent methodwhich is well-posed for the continuous and discrete versions of theproblem. Sensitivities of the cost and constraints are computedvariationally via shape differential calculus. Computations areaccomplished by a boundary integral method to solve the Stokes equations,and a finite element method to obtain descent directions for theoptimization algorithm. We show examples of locomotor shapes with andwithout different fixed payload/cargo shapes.

Oscillatory component recovery and separation in images by Sobolev norms

Series
Applied and Computational Mathematics Seminar
Time
Monday, November 29, 2010 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Yunho Kim – University of California, Irvine
It has been suggested by Y. Meyer and numerically confirmed by many othersthat dual spaces are good for texture recovery. Among the dual spaces, ourwork focuses on Sobolev spaces of negative differentiability to recovertexture from noisy blurred images. Such Sobolev spaces are good to modeloscillatory component, on the other hand, the spaces themselves hardlydistinguishes texture component from noise component because noise is alsoconsidered to be a highly oscillatory component. In this talk, in additionto oscillatory component recovery, we will further investigate aone-parameter family of Sobolev norms to achieve such a separation task.

A quantitative rigidity result for the cubic to tetragonal phase transition in the geometrically linear theory with interfacial energy

Series
Applied and Computational Mathematics Seminar
Time
Monday, November 22, 2010 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Antonio Capella-Kort – Universidad Nacional Autónoma de México (UNAM)
We are interested in the cubic to tetragonal phase transition in a shape memory alloy. We consider geometrically linear elasticity. In this framework, Dolzmann and Mueller have shown the following rigidity result:The only stress-free configurations are (locally) twins (i.e. laminates of just two of the three Martensitic variants).However, configurations with arbitrarily small elastic energy are not necessarily close to these twins: The formation of microstructure allows to mix all three Martensitic variants at arbitrary volume fractions. We take an interfacial energy into account and establish a (local) lower bound on elastic + interfacial energy in terms of the Martensitic volume fractions. The model depends on a non-dimensional parameter that measures the strength of the interfacial energy. Our lower, ansatz-free bound has optimal scaling in this parameter. It is the scaling predicted by a reduced model introduced and analyzed by Kohn and Mueller with the purpose to describe the microstructure near an interface between Austenite and twinned Martensite. The optimal construction features branching of the Martensitic twins when approaching this interface.

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