| Some recent research projects. A complete list of papers/preprints is availabe at the Papers section of the website. |
Threshold Dynamics for Image Segmentation In joint work with Richard Tsai, we describe efficient algorithms, based on the threshold dynamics idea of Merriman, Bence, and Osher, for minimizing the piecewise constant Mumford-Shah functional of image segmentation. |
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Total Variation Model: L1 Fidelity and Anisotropic Versions In joint work with Tony F. Chan, we investigate the effects of replacing with the L1 norm the fidelity term in the Rudin, Osher, Fatemi (ROF) total variation based image denoising model. There are a number of interesting and desirable conseqences of this seemingly modest modification, such as contrast invariance. With Stan Osher, we consider anisotropic versions of the ROF model.
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Finding Global Minimizers of Segmentation Models As a byproduct of our work with Tony F. Chan on the total variation model with L1 fidelity, we realized that the piecewise constant Mumford-Shah model can be given a convex formulation. In the papers with Tony F. Chan and Mila Nikolova, we use this observation to give guaranteed algorithms for finding the global minimizer of segmentation models. In subsequent work, joint with Bresson et. al., we showed how to minimize the resulting convex energies using a dual formulation of Chan, Golub, Mulet and Chambolle that was initially used for total variation based energies.
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| High Order Image Denoising/Decomposition Models One of the well-known caveats of the total variation denoising model of Rudin, Osher, and Fatemi is the "staircasing" phenomena. This is an artifact that forms at regions of moderate gradient in the image, turning them into approximately piecewise constant regions separated by spurious edges. As proposed by Chambolle and Lions, one way to abate staircasing is via inclusion of higher order terms in the variational model. The resulting model can be solved via convex duality. |
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| Threshold Dynamics for High Order Geometric Motions Following on the work of Grzibovskis and Heintz, who proposed a threshold based algorithm for generating the Willmore flow, we devise algorithms for generating other high order geometric motions, such as motion by surface diffusion and Willmore flow with lower order terms, through alternating convolution with a Gaussian and simple thresholding. |
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Inverse Problems in Imaging
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Coarsening Rate of Ill-Posed Diffusion Equations Discrete, ill-posed diffusion equations arise in a variety of contexts: in models of granular flow where they describe the formation of shear bands in a granular material undergoing anti-plane shear, in image processing where they consitute one of the most well-known models -- called the Perona-Malik method -- for denoising images, and in population dynamics where they describe the chemotactic motion of certain types of bacteria. We study the coarsening phenomena observed in these models using a technique introduced by R. V. Kohn and F. Otto.
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| Histogram Based Image Segmentation | |
| Variational Models for Surface Fairing | |
Image Inpainting
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| Diffusion Generated Motion via Distance Functions | |
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Many common materials, such as most metals and ceramics, are polycrystalline: They are made up of tiny crystallites called grains that are distinguished from their neighbors by their differing crystallographic orientation. When these materials are heated (i.e. annealed) -- for instance during a manufacturing process -- grain growth occurs: The network of grains decreases its energy through a coarsening procedure, which involves the growth of some of the grains at the expense of others. Statistical measures of the grain network, such as the grain size distribution, have important implications for the macroscopic properties of the material, such as its conductivity and brittleness. As such, simulating how the grain network evolves is of great technological interest. We developed new, efficient, and accurate numerical methods for simulating grain growth and related dynamics. See this page for further details.
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Some of the research projects shown on this page were supported by the National Science Foundation through grants DMS-0410085 (later DMS-0605714), DMS-0713767, and DMS-0748333. In addition, some of the projects were supported by a contract from the Los Alamos National Laboratory, and some by an Alfred P. Sloan Foundation fellowship. |