Numerical methods for geometric partial differential equations: applications to freeform deformations in animation and nonrigid medical image registration

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Shapes may be described by using the familiar geometric notions of length, area, volume, angle and curvature. Changing shapes may be described by Geometric Partial Differential Equations: stretching or bending, smoothing rough edges, or mapping one onto another. The proposed research will build new tools for manipulating three dimensional shapes. To do this, we need to be able to numerically solve the equations which describe the transformations. This task is made difficult by the complexity of the equations and the large amounts of data. To date, simplified equations have been used which neglect some geometric detail. My research has found new representations of these equations which have overcome some of these difficulties. This work will allow for geometric detail to be preserved in shape manipulation. The proposal will allow for better geometric fidelity in three dimensional tools used in the following important applications: Freeform deformations in computer animation allow the artist to move an animated image by manipulating a simpler skeleton or cage. The geometric equations extrapolate the motion to the shape while preserving fine features. This saves considerable time and effort in animation. Nonrigid image registration tools are used to compare images of organs taken at successive times. For example, after a surgeon removes a tumour from a patient's brain, three dimensional images are taken of the brain immediately after surgery and at later times. In order to determine if the growth has stopped, the images are compared by mapping (warping) the later images onto the first one.