TV optimization and graph-cuts

This paper describes some links between the minimization of the Total Variation and the minimization of some binary energies. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/63087/1/1042303_ftp.pd

ADI splitting schemes for a fourth-order nonlinear partial differential equation from image processing

We present directional operator splitting schemes for the numerical solution of a fourth-order, nonlinear partial differential evolution equation which arises in image processing. This equation constitutes the H−1-gradient flow of the total variation and represents a prototype of higher-order equations of similar type which are popular in imaging for denoising, deblurring and inpainting problems. The efficient numerical solution of this equation is very challenging due to the stiffness of most numerical schemes. We show that the combination of directional splitting schemes with implicit time-stepping provides a stable and computationally cheap numerical realisation of the equation

Finding the global minimum for binary image restoration

Restoring binary images is a problem which arises in various application fields. In our paper, this problem is considered in a variational framework: the sought-after solution minimizes an energy. Energies defined over the set of the binary images are inevitably nonconvex and there are no general methods to calculate the global minimum, while local minimziers are very often of limited interest. In this paper we define the restored image as the global minimizer of the total-variation (TV) energy functional constrained to the collection of all binary-valued images. We solve this constrained non-convex optimization problem by deriving another functional which is convex and whose (unconstrained) minimum is proven to be reached for the global minimizer of the binary constrained TV functional. Practical issues are discussed and a numerical example is provided. © 2005 IEEE

Global Minimizers of The Active Contour/Snake Model

The active contour/snake mode

Refined upper bounds on the coarsening rate of discrete, ill-posed diffusion equations

"We study coarsening phenomena observed in discrete, ill-posed diffusion equations that arise in a variety of applications, including computer vision, population dynamics and granular flow. Our results provide rigorous upper bounds on the coarsening rate in any dimension. Heuristic arguments and the numerical experiments we perform indicate that the bounds are in agreement with the actual rate of coarsening."http://deepblue.lib.umich.edu/bitstream/2027.42/64211/1/non8_12_002.pd

Threshold dynamics for anisotropic surface energies

We study extensions of Merriman, Bence, and Osher’s threshold dynamics scheme to weighted mean curvature flow, which arises as gradient descent for anisotropic (normal dependent) surface energies. In particular, we investigate, in both two and three dimensions, those anisotropies for which the convolution kernel in the scheme can be chosen to be positive and/or to possess a positive Fourier transform. We provide a complete, geometric characterization of such anisotropies. This has implications for the unconditional stability and, in the two-phase setting, the monotonicity, of the scheme. We also revisit previous constructions of convolution kernels from a variational perspective, and propose a new one. The variational perspective differentiates between the normal dependent mobility and surface tension factors (both of which contribute to the normal speed) that results from a given convolution kernel. This more granular understanding is particularly useful in the multiphase setting, where junctions are present.</p