With contributions by leading experts in geometric analysis, this volume is documenting the material presented in the John H. Barrett Memorial Lectures held at the University of Tennessee, Knoxville, on May 29 - June 1, 2018. The central topic of the 2018 lectures was mean curvature flow, and the material in this volume covers all recent developments in this vibrant area that combines partial differential equations with differential geometry.

Acta Numerica is an annual volume presenting survey papers in numerical analysis. Each year the editorial board selects significant topics and invites papers from authors who have made notable contributions to the development of that topic. The articles are intended to summarize the field at a level accessible to graduate students and researchers. Acta Numerica has proved to be a valuable tool not only for researchers and professionals wishing to develop their understanding of the subject and follow developments, but also as an advanced teaching aid at colleges and universities. Articles in previous volumes have been expanded into both monographs and textbooks, and many of the original articles themselves have been used as the prime resource for graduate courses.

PDE & Level Sets: Algorithmic Approaches to Static & Motion Imagery is specially dedicated to the segmentation of complex shapes from the field of imaging sciences using level sets and PDEs. It covers the fundamentals of level sets, different kinds of concepts of both geodesic curvature flows and planar flows, as well as the power of incorporation of regional-statistics in level set framework. In covering this material, this book presents segmentation of object-in-motion imagery based on level sets in eigen analysis framework, while also presenting classical problems of boundary completion in cognitive images, like the pop-up of subjective contours in the famous triangle of Kanizsa using surface evolution framework, or the mean curvature evolution of a graph with respect to the Riemannian metric induced by the image. All results are presented for modal completion of cognitive objects with missing boundaries.

This new edition of Professor Sethian's successful text provides an introduction to level set methods and fast marching methods, which are powerful numerical techniques for analyzing and computing interface motion in a host of settings. They rely on a fundamental shift in how one views moving boundaries; rethinking the natural geometric Lagrangian perspective and exchanging it for an Eulerian, initial value partial differential equation perspective. For this edition, the collection of applications provided in the text has been expanded, including examples from physics, chemistry, fluid mechanics, combustion, image processing, material science, fabrication of microelectronic components, computer vision, computer-aided design, and optimal control theory. This book will be a useful resource for mathematicians, applied scientists, practising engineers, computer graphic artists, and anyone interested in the evolution of boundaries and interfaces.

This book presents a self-contained introduction to the analytic foundation of a level set approach for various surface evolution equations including curvature flow equations. These equations are important in many applications, such as material sciences, image processing and differential geometry. The goal is to introduce a generalized notion of solutions allowing singularities, and to solve the initial-value problem globally-in-time in a generalized sense. Various equivalent definitions of solutions are studied. Several new results on equivalence are also presented. Moreover, structures of level set equations are studied in detail. Further, a rather complete introduction to the theory of viscosity solutions is contained, which is a key tool for the level set approach. Although most of the results in this book are more or less known, they are scattered in several references, sometimes without proofs. This book presents these results in a synthetic way with full proofs. The intended audience are graduate students and researchers in various disciplines who would like to know the applicability and detail of the theory as well as its flavour. No familiarity with differential geometry or the theory of viscosity solutions is required. Only prerequisites are calculus, linear algebra and some basic knowledge about semicontinuous functions.

Handbook of Biomedical Image Analysis: Segmentation Models (Volume I) is dedicated to the segmentation of complex shapes from the field of imaging sciences using different mathematical techniques. This volume is aimed at researchers and educators in imaging sciences, radiological imaging, clinical and diagnostic imaging, physicists covering different medical imaging modalities, as well as researchers in biomedical engineering, applied mathematics, algorithmic development, computer vision, signal processing, computer graphics and multimedia in general, both in academia and industry . Key Features: - Principles of intra-vascular ultrasound (IVUS) - Principles of positron emission tomography (PET) - Physical principles of magnetic resonance angiography (MRA). - Basic and advanced level set methods - Shape for shading method for medical image analysis - Wavelet transforms and other multi-scale analysis functions - Three dimensional deformable surfaces - Level Set application for CT lungs, brain MRI and MRA volume segmentation - Segmentation of incomplete tomographic medical data sets - Subjective level sets for missing boundaries for segmentation

At the summer school in Pisa in September 1996, Luigi Ambrosio and Norman Dancer each gave a course on the geometric problem of evolution of a surface by mean curvature, and degree theory with applications to PDEs respectively. This self-contained presentation accessible to PhD students bridged the gap between standard courses and advanced research on these topics. The resulting book is divided accordingly into 2 parts, and neatly illustrates the 2-way interaction of problems and methods. Each of the courses is augmented and complemented by additional short chapters by other authors describing current research problems and results.