Konstantin Mischaikow

Feature-based surface parameterization and texture mapping

Surface parameterization is necessary for many graphics tasks: texture-preserving simplification, remeshing, surface painting, and precomputation of solid textures. The stretch caused by a given parameterization determines the sampling rate on the surface. In this article, we present an automatic parameterization method for segmenting a surface into patches that are then flattened with little stretch. Many objects consist of regions of relatively simple shapes, each of which has a natural parameterization. Based on this observation, we describe a three-stage feature-based patch creation method for manifold surfaces. The first two stages, genus reduction and feature identification, are performed with the help of distance-based surface functions. In the last stage, we create one or two patches for each feature region based on a covariance matrix of the features surface points. To reduce stretch during patch unfolding, we notice that stretch is a 2 × 2 tensor, which in ideal situations is the identity. Therefore, we use the <i>Green-Lagrange tensor</i> to measure and to guide the optimization process. Furthermore, we allow the boundary vertices of a patch to be optimized by adding <i>scaffold triangles</i>. We demonstrate our feature-based patch creation and patch unfolding methods for several textured models. Finally, to evaluate the quality of a given parameterization, we describe an image-based error measure that takes into account stretch, seams, smoothness, packing efficiency, and surface visibility.

Chaos in the Lorenz equations: a computer-assisted proof

A new technique for obtaining rigorous results concerning the global dynamics of nonlinear systems is described. The technique combines abstract existence results based on the Conley index theory with computer- assisted computations. As an application of these methods it is proven that for an explicit parameter value the Lorenz equations exhibit chaotic dynamics.

Vector field design on surfaces

Vector field design on surfaces is necessary for many graphics applications: example-based texture synthesis, nonphotorealistic rendering, and fluid simulation. For these applications, singularities contained in the input vector field often cause visual artifacts. In this article, we present a vector field design system that allows the user to create a wide variety of vector fields with control over vector field topology, such as the number and location of singularities. Our system combines basis vector fields to make an initial vector field that meets user specifications.The initial vector field often contains unwanted singularities. Such singularities cannot always be eliminated due to the Poincaré-Hopf index theorem. To reduce the visual artifacts caused by these singularities, our system allows the user to move a singularity to a more favorable location or to cancel a pair of singularities. These operations offer topological guarantees for the vector field in that they only affect user-specified singularities. We develop efficient implementations of these operations based on Conley index theory. Our system also provides other editing operations so that the user may change the topological and geometric characteristics of the vector field.To create continuous vector fields on curved surfaces represented as meshes, we make use of the ideas of geodesic polar maps and parallel transport to interpolate vector values defined at the vertices of the mesh. We also use geodesic polar maps and parallel transport to create basis vector fields on surfaces that meet the user specifications. These techniques enable our vector field design system to work for both planar domains and curved surfaces.We demonstrate our vector field design system for several applications: example-based texture synthesis, painterly rendering of images, and pencil sketch illustrations of smooth surfaces.

Asymptotically autonomous semiflows: Chain recurrence and lyapunov functions

From the work of C. Conley, it is known that the omega limit set of a precompact orbit of an autonomous semiflow is a chain recurrent set. Here, we improve a result of L. Markus by showing that the omega limit set of a solution of an asymptotically autonomous semiflow is a chain recurrent set relative to the limiting autonomous semiflow. In the special case that there is a Lyapunov function for the limiting semiflow, sufficient conditions are given for an omega limit set of the asymptotically autonomous semiflow to be contained in a level set of the Lyapunov function.

Morse Theory for Filtrations and Efficient Computation of Persistent Homology

We introduce an efficient preprocessing algorithm to reduce the number of cells in a filtered cell complex while preserving its persistent homology groups. The technique is based on an extension of combinatorial Morse theory from complexes to filtrations.

Rigorous Numerics for Partial Differential Equations: The Kuramoto—Sivashinsky Equation

Abstract We present a new topological method for the study of the dynamics of dissipative PDEs. The method is based on the concept of the self-consistent a priori bounds, which permit the rigorous justification of the use of Galerkin projections. As a result, we obtain a low-dimensional system of ODEs subject to rigorously controlled small perturbation from the neglected modes. To these ODEs we apply the Conley index to obtain information about the dynamics of the PDE under consideration. We applied the method to the Kuramoto—Sivashinsky equation

Isolating neighborhoods and chaos

u_t = \left( {u^2 } \right)_x - u_{xx} - vu_{xxxx} , u(x,t) = u(x + 2\pi , t), u(x,t) = - u( - x,t).

Validated Continuation for Equilibria of PDEs

We obtained a computer-assisted proof of the existence of several fixed points for various values of ν > 0 .