Prof. Michael Kazhdan (Johns Hopkins University)
In these lectures we will take an in-depth look at the Poisson equation, with a focus on its use in the graphics community.
We will start by looking at the way in which a number of common gradient-domain image-processing techniques result in a Poisson-like equation (including stitching, contrast enhancement, and low dynamic-range compression). We will discuss common discretizations of the linear systems and will look, in detail, at implementations of a multigrid solver that supports out-of-core processing of images that are too large to fit into working memory.
From here we will move on to the 3D domain, where we will explore the Poisson equation within the context of surface reconstruction. In particular, this part will focus on implementations of a multigrid solver over an adaptive octree, where traditional assumptions about nesting functions spaces are not satisfied. We will show that even when we forgo regularity in favor of a more memory-friendly multiresolution hierarchy, we can still design a solver that is linear in the dimension of the system.
Finally, we will turn our attention solving the Poisson equation over 2D manifolds immersed in 3D. We will describe how to discretize the Laplace-Beltrami operator, taking into account the underlying metric structure, and we will present a new hierarchy of function spaces that trivially supports a linear-time multigrid solver. We will close by considering how the solver can be extended to support a time-varying metric, allowing us to explore (interactive) anisotropic and inhomogenous surface editing and evolution of surfaces under the action of simple geometric flows.