New PDE Based Models and Numerical Techniques in Level Set Surface Processing, Imaging Science and Materials Science

University Of California-Los Angeles, Los Angeles CA

Investigators

Abstract

Stanley J. Osher and Luminita A. Vese The investigators together with junior faculty, postdoctoral fellows, and students develop novel and efficient computational techniques for partial differential equations of fourth order arising in imaging science, including medical imaging, image processing, computer vision, and graphics, as well as material science. They begin with analysis done on their previous TV model by Yves Meyer. This gives a unique blend of nonlinear partial differential equation and functional analysis as well as new and very useful models for image decomposition. This leads the investigators to the analysis of more general fourth order equations used in imaging science, with geometric applications and interpretations, as well as new efficient computational and numerical methods to approximate these and other related partial differential equations. Problems considered include image decomposition into cartoon plus texture, Wulff evolution of shapes in crystal growth, image disocclusion by the Euler elastica model, image restoration via geometry of level sets, as well as reconstruction of surfaces from unorganized data points (such as LIDAR, LADAR, 3D cloud or geoscience data) in the form of patches belonging to the same unknown surface. This investigation dramatically advances the state-of-the-art in material science (e.g., microchip design), image analysis, computer vision, and graphics. These fields are of great strategic value in the US information technology industry, in nanoscience, in homeland security, and in medical imaging. The level set method and applications, developed by the investigators and collaborators, has impacted numerous areas of technology -- see, e.g., the Google website for close to 5,000 hits on "Level Set Methods" with a spectacular range of applications from Hollywood graphics to underwater explosions to control of passenger aircraft to developing new microchip designs and beyond. The investigators' new models reduce the burden of laborious human operations for the processing of large-scale data sets, enhance the efficiency of domain scientists as well as ordinary users, facilitate modeling and rendering tasks, and streamline the pipeline of information and materials processing.