Image segmentation and edge enhancement with stabilized inverse diffusion equations

Abstract

We introduce a family of first-order multidimensional ordinary differential equations (ODEs) with discontinuous right-hand sides and demonstrate their applicability in image processing. An equation belonging to this family is an inverse diffusion everywhere except at local extrema, where some stabilization is introduced. For this reason, we call these equations "stabilized inverse diffusion equations" (SIDEs). Existence and uniqueness of solutions, as well as stability, are proven for SIDEs. A SIDE in one spatial dimension may be interpreted as a limiting case of a semi-discretized Perona-Malik equation. In an experiment, SIDE's are shown to suppress noise while sharpening edges present in the input signal. Their application to image segmentation is also demonstrated.

Multiscale anisotropic diffusion for ringing artifact suppression in geophysical deconvolution data

“…However, the ringing artifacts produced by these deconvolution algorithms are rarely reported. Most of the research on ringing artifacts focuses on the enhancement of image compression techniques (Perona and Malik 1990;Weickert 1998;Pollak et al 2000;Lysaker et al 2003;Ding et al 2005). Popovici and Withers (2007) proposed using a bilateral filter to suppress ringing artifacts from compressed images.…”

“…A nonlinear anisotropic diffusion method was proposed by Perona and Malik (1990) and has been widely studied in a number of scientific fields over the past two decades (Black et al 1998;Pollak et al 2000;Chen and Hsu 2005;Ding et al 2005). The advantage of this method is that it can adaptively smooth noise while simultaneously preserving the actual model boundaries.…”

Nonlinear Modeling: Estimation and Prediction with Applications to GPS

“…at. [5] recently proposed an approach addressing robustness issues, and showed some remarkable results for a wide class of perturbation noises. The analysis remained as in all other cases, fundamentally deterministic, and also required knowledge of the stopping time for the evolution.…”

“…Partial support of this grant has also led to the publication of four journal papers [2,3,4,5] and three journal papers under review [6,7,8], as well as two invited book chapters [9,10] and several conference papers at ICASSP and ICIP [11,12,13,14] and invited national and international seminars, including an invitation for a one month visit in-the department of information sciences at University of Tokyo (Japan) (Nov. 1999), courtesy of the Japanese Society for Advancement of Science, and an invitation as principal guest speaker at a workshop on Mathematics in Imaging at University of Hong Kong (China) (Dec. 2000).…”

Semidiscrete and Discrete Well-Posedness of Shock Filtering

“…While this has already been demonstrated for nonlinear diffusion filtering [13,10], we plan to investigate a number of other PDEs in this manner, both in the one-and the higher-dimensional case. This should give important theoretical insights into the dynamics of these experimentally well-performing nonlinear processes.…”