PDE surface

Abstract: : Recent work by the authors and others has demonstrated the connections between the dynamic programming approach for two-person, zero-sum differential games and the new notion of viscosity solutions of Hamilton-Jacobi PDE, (Partial Differential Equations). The basic idea is that the dynamic programming optimality conditions imply that the values of a two-person, zero-sum differential game are viscosity solutions of appropriate PDE. This paper proves the above, when the values of the differential games are defined following Elliott-Kalton. This results in a great simplification in the statements and proofs, as the definitions are explicit and do not entail any kind of approximations. Moreover, as an application of the above results, the paper contains a representation formula for the solution of a fully nonlinear first-order PDE. This is then used to prove results about the level sets of solutions of Hamilton-Jacobi equations with homogeneous Hamiltonians. These results are also related to the theory of Huygen's principle and geometric optics.

Abstract: We present a novel geometric approach for solving the stereo problem for an arbitrary number of images (greater than or equal to 2). It is based upon the definition of a variational principle that must be satisfied by the surfaces of the objects in the scene and their images. The Euler-Lagrange equations which are deduced from the variational principle provide a set of PDE's which are used to deform an initial set of surfaces which then move towards the objects to be detected. The level set implementation of these PDE's potentially provides an efficient and robust way of achieving the surface evolution and to deal automatically with changes in the surface topology during the deformation, i.e. to deal with multiple objects. Results of a two dimensional implementation of our theory are presented on synthetic and real images.

Abstract: A Compendium of Partial Differential Equation Models presents numerical methods and associated computer codes in Matlab for the solution of a spectrum of models expressed as partial differential equations (PDEs), one of the mostly widely used forms of mathematics in science and engineering. The authors focus on the method of lines (MOL), a well-established numerical procedure for all major classes of PDEs in which the boundary value partial derivatives are approximated algebraically by finite differences. This reduces the PDEs to ordinary differential equations (ODEs) and thus makes the computer code easy to understand, implement, and modify. Also, the ODEs (via MOL) can be combined with any other ODEs that are part of the model (so that MOL naturally accommodates ODE/PDE models). This book uniquely includes a detailed line-by-line discussion of computer code as related to the associated equations of the PDE model.