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.

1. Esedoglu, S.; Tsai, Y.-H. Threshold dynamics for the piecewise constant Mumford-Shah functional. Journal of Computational Physics. 211:1 (2006), pp. 367-384.

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.

1. Chan, T. F.; Esedoglu, S. Aspects of total variation regularized L^1 function approximation. SIAM Journal on Applied Mathematics. 65:5 (2005), pp. 1817-1837.
2. Esedoglu, S.; Osher, S. Decomposition of images by the anisotropic Rudin-Osher-Fatemi model. Communications on Pure and Applied Mathematics. 57 (2004), pp. 1609-1626.

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.

1. Chan, T. F.; Esedoglu, S.; Nikolova, M. Algorithms for finding global minimizers of denoising and segmentation models. SIAM Journal on Applied Mathematics. 66 (2006), pp. 1632-1648.
2. Chan, T. F.; Esedoglu, S.; Nikolova, M. Finding the global minimum for binary image restoration. Proceedings of the ICIP 2005.
3. Bresson, X.; Esedoglu, S.; Vandergheynst, P.; Thiran, J. P.; Osher, S. Fast global minimization of the active contour/snake model. Journal of Mathematical Imaging and Vision. 28 (2007), pp. 151-167.
4. Kolev, K.; Klodt, M.; Brox, T.; Esedoglu, S.; Cremers, D. Continuous global optimization in multiview 3D reconstruction. Proceedings of EMMCVPR 2007.

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.

1. Chan, T. F.; Esedoglu, S.; Park, F. Image decomposition combining staircase reduction and texture extraction. Journal of Visual Communication and Image Representation. 18:6 (2007), pp. 464-486.

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.

1. Esedoglu, S.; Ruuth, S.; Tsai, R. Threshold dynamics for shape reconstruction and disocclusion. Proceedings of the ICIP 2005.
2. Esedoglu, S.; Ruuth, S.; Tsai, R. Threshold dynamics for high order geometric motions. Accepted for publication in Inrefaces and Free Boundaries.

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.

1. Esedoglu, S.; Greer, J. Upper bounds on the coarsening rate of discrete, ill-posed, nonlinear diffusion equations. Accepted for publication in Communications on Pure and Applied Mathematics.
2. Esedoglu, S.; Slepcev, D. Refined upper bounds on the coarsening rate of discrete, ill-posed diffusion equations. Submitted.
   
   Related older works on discrete, ill-posed diffusion equations:
3. Esedoglu, S. An analysis of the Perona-Malik scheme. Communications on Pure and Applied Mathematics. 54 (2001), pp. 1442-1487.
4. Esedoglu, S. Stability properties of the Perona-Malik scheme. SIAM Journal on Numerical Analysis. 44 (2006), pp. 1297-1313.

1. Chan, T. F.; Esedoglu, S.; Ni, K. Histogram based segmentation using Wasserstein distances. Proceedings of SSVM 2007.
2. Ni, K.;Bresson, X.; Chan, T. F.; Esedoglu, S. Local histogram based segmentation using Wasserstein distance. Preprint.

1. Elsey, M.; Esedoglu, S. Analogue of the total variation denoising model in the context of geometry processing. Submitted.

1. Esedoglu, S.; Shen, J. Digital image inpainting by the Mumford-Shah-Euler image model. European Journal of Applied Mathematics. 13 (2002), pp. 353-370.
2. Bertozzi, A.; Esedoglu, S.; Gillette, A. Inpainting by the Cahn-Hilliard equation. IEEE Transactions on Image Processing. 16:1 (2007), pp. 285-291.
3. Bertozzi, A.; Esedoglu, S.; Gillette, A. Analysis of a two-scale Cahn-Hilliard model for image inpainting. SIAM J. Multiscale Modeling and Simulation. 6:3 (2007), pp. 913-936.