A variational method for nonrigid registration of multimodal image data is presented. A suitable deformation will be determined via the minimization of a morphological, i.e., contrast invariant, matching functional along with an appropriate regularization energy. The aim is to correlate the morphologies of a template and a reference image under the deformation. Mathematically, the morphology of images can be described by the entity of level sets of the image and hence by its Gauss map. A class of morphological matching functionals is presented which measure the defect of the template Gauss map in the deformed state with respect to the deformed Gauss map of the reference image. The problem is regularized by considering a nonlinear elastic regularization energy. Existence of homeomorphic, minimizing deformation is proved under assumptions on the class of admissible deformations. With respect to actual medical applications, suitable generalizations of the matching energies and the boundary conditions are presented. Concerning the robust implementation of the approach, the problem is embedded in a multiscale context. A discretization based on multilinear finite elements is discussed, and the first numerical results are presented.

A level set formulation of Willmore flow is derived using the gradient flow perspective. Starting from single embedded surfaces and the corresponding gradient flow, the metric is generalized to sets of level set surfaces using the identification of normal velocities and variations of the level set function in time via the level set equation. This approach in particular allows one to identify the natural dependent quantities of the derived variational formulation. Furthermore, spatial and temporal discretizations are discussed and some numerical simulations are presented.2000 Mathematics Subject Classification: 35K55, 53C44, 65M60, 74S05.

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A two-handed 3D styling system for fl'ee-form surfaces in a table-like Virtual Environment, the Responsive Workbench (I~WB)TM , is described. Intuitive curve and surface deformation tools based on variational modeling and interaction techniques adapted to 3D V R modeling applications are proposed. The user draws curves (cubic B-splines) directly in the Virtual Environment using a stylus as an inpue device, The era'yes are connected automatically, such that a curve network develops. A combination of automatic and usercontrolled topology extraction modules crea~es the connectivity information. The underlying surface model is based on B-spline surfaces, or, alternatively, uses multisided patches [20] bounded by closed loops of curve pieces. 1, STYLING IN VIRTUAL ENVIRONMENTSMany designers would like sketching free4brm shapes quickly in the conceptual design phase without using a complex CAD system [6]. Projection-based Virtual Environments like the Responsive Workbench (RYVB)TM t [19] offer the following advantages for sketching applications over traditinal desktop interfaces:3.The three-dimensionality of perception in combination with 3D intera.ction gives an immediate understanding of the shape.Regions can be identified, localized, and selected directly in space.The large, high-resolution projection plane enables the representation of objects on a scale which corresponds ~o the working region of both hands.

We present manifold next event estimation (MNEE), a specialised technique for Monte Carlo light transport simulation to render refractive caustics by connecting surfaces to light sources (next event estimation) across transmissive interfaces. We employ correlated sampling by means of a perturbation strategy to explore all half vectors in the case of rough transmission while remaining outside of the context of Markov chain Monte Carlo, improving temporal stability. MNEE builds on differential geometry and manifold walks. It is very lightweight in its memory requirements, as it does not use light caching methods such as photon maps or importance sampling records. The method integrates seamlessly with existing Monte Carlo estimators via multiple importance sampling.