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

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Stanley J. Osher and Luminita A. Vese The investigators together with junior faculty, postdoctoralfellows, and students develop novel and efficient computationaltechniques for partial differential equations of fourth orderarising in imaging science, including medical imaging, imageprocessing, computer vision, and graphics, as well as materialscience. They begin with analysis done on their previous TV modelby Yves Meyer. This gives a unique blend of nonlinear partialdifferential equation and functional analysis as well as new andvery useful models for image decomposition. This leads theinvestigators to the analysis of more general fourth orderequations used in imaging science, with geometric applicationsand interpretations, as well as new efficient computational andnumerical methods to approximate these and other related partialdifferential equations. Problems considered include imagedecomposition into cartoon plus texture, Wulff evolution ofshapes in crystal growth, image disocclusion by the Eulerelastica model, image restoration via geometry of level sets, aswell as reconstruction of surfaces from unorganized data points(such as LIDAR, LADAR, 3D cloud or geoscience data) in the formof patches belonging to the same unknown surface. This investigation dramatically advances thestate-of-the-art in material science (e.g., microchip design),image analysis, computer vision, and graphics. These fields areof great strategic value in the US information technologyindustry, in nanoscience, in homeland security, and in medicalimaging. The level set method and applications, developed by theinvestigators and collaborators, has impacted numerous areas oftechnology -- see, e.g., the Google website for close to 5,000hits on "Level Set Methods" with a spectacular range ofapplications from Hollywood graphics to underwater explosions tocontrol of passenger aircraft to developing new microchip designsand beyond. The investigators' new models reduce the burden oflaborious human operations for the processing of large-scale datasets, enhance the efficiency of domain scientists as well asordinary users, facilitate modeling and rendering tasks, andstreamline the pipeline of information and materials processing.