Robotics and Optimal Control. Verification and Reachable Sets. Zero-Sum Differential Games. This component will examine generalizations of the level set methods mentioned above, as well as other classes of algorithms including fast marching, ordered upwind, and sweeping and some of the viscosity solution theory underlying this ubiquitous equation. Applications will be explored depending on students' interests. In addition to attending lectures, students will be expected to: Read and discuss academic papers. Complete several homework assignments involving both paper and pencil and programming components.

Complete a course project which may be tailored to each student's own research or interests. Prerequisites: None officially. Students should have seen elementary differential equations, multivariable calculus and introductory numerical methods. Be ready to program. Homework assignments will most easily be solved with Matlab. Programming in Matlab is essentially the same as Java or C but with lots of useful visualization tools. Course projects may be programmed in the language of the student's choosing, should programming be required.

Familiarity with numerical methods for partial differential equations is not necessary. If you are in doubt, come to the first class or see me. There is no lecture Wednesday September 8 cancelled for CS orientation.

## CS D: Level Set Methods Class Home Page

There is no lecture Monday October 11 Thanksgiving Holiday. The official required textbook for the class, available at the bookstore. Excellent coverage of high order accuracy methods on structured grids, image processing and computational physics. The other level set book. Excellent coverage of fast marching methods, unstructured grids, and many other applications. The manual for the Matlab Toolbox used in the course.

Back cover copy This book is an introduction to level set methods and dynamic implicit surfaces.

These are powerful techniques for analyzing and computing moving fronts in a variety of different settings. While the book gives many examples of the usefulness of the methods for a diverse set of applications, it also gives complete numerical analysis and recipes, which will enable users to quickly apply the techniques to real problems. The book begins with the description of implicit surfaces and their basic properties, and then devises the level set geometry and calculus toolbox, including the construction of signed distance functions. Part II adds dynamics to this static calculus.

Topics include the level set equation itself, Hamilton-Jacobi equations, motion of a surface normal to itself, reinitialization to a signed distance function, extrapolation in the normal direction, the particle level set method, and the motion of codimension two and higher objects.

Part III is concerned with topics taken from the field of image processing and computer vision. These include the restoration of images degraded by noise and blur, image segmentation with active contours snakes , and reconstruction of surfaces from unorganized data points. Part IV is dedicated to computational physics. Next it discusses incompressible fluid dynamics, including a computer graphics simulation of smoke; free surface flows, including a computer graphics simulation of water; and fully two-phase incompressible flow.

Additional related topics include incompressible flames with applications to computer graphics and coupling a compressible and incompressible fluid. Finally, heat flow and Stefan problems are discussed. A student or researcher working in mathematics, computer graphics, science, or engineering interested in any dynamic moving front, which might change it's topology or develop singularities, will find this book interesting and useful.

## Level Set Methods

Review Text From the reviews: S. Osher and R. Fedkiw Level Set Methods and Dynamic Implicit Surfaces "This book is an introduction to level set methods and dynamic implicit surfaces. While it gives many examples of the utility of the methods to a diverse set of applications, it also gives complete numerical analysis and recipes, which will enable users to quickly apply the techniques to real problems. A student or researcher working in mathematics, computer graphics, science, or engineering interested in any dynamic moving front, which might change its topology or develop singularities, will find this book interesting and useful.

The authors, having contributed so profoundly, are in an excellent position to write an authoritative text. Without overly much cumbersome apparatus, the text introduces into an active area of scientific computing in a very informative way. This book is an excellent introduction to the field, allowing newcomers to quickly grasp the key ideas. It also covers, in sufficient detail, the level set approach to several problems arising in image processing and computational physics.

Together with many examples, the book also gives a complete description of numerical aspects which will enable users to quickly apply these techniques to real problems.

### Bibliographic Information

A student or researcher working in mathematics, computer graphics, science or engineering and interested in dynamically moving fronts which might change their topology or develop singularities, will find this book interesting and useful. This is a stimulating book for those interested in modeling dynamic interfaces with numerical methods that can handle topology changes and singularities.

Level set methods are such methods and have been successful in several applications. Review quote From the reviews: S. Muthsam, Monatshefte fur Mathematik, Vol. Rating details. Book ratings by Goodreads.

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