The last two terms are called regularization terms, and they aim to make the function smooth in extrinsic and intrinsic senses

The last two terms are called regularization terms, and they aim to make the function smooth in extrinsic and intrinsic senses

On Mesh Editing, Manifold Learning, and Diffusion Wavelets Raif M. Rustamov Drew University, Madison NJ 07940, USA rrustamov@drew.edu Abstract. We spell out a formal equivalence between the naive Laplacian editing and semi-supervised learning by bi-Laplacian Regularized Least Squares. This allows us to write the solution to Laplacian mesh editing in a closed form, based on which we introduce the Generalized Linear Editing (GLE). GLE has both naive Laplacian editing and gradient based editing as special cases. GLE allows using diffusion wavelets for mesh editing. We present preliminary experiments, and shortly discuss connections to segmentation. 1 Introduction A remarkable similarity exists between semi-supervised manifold learning and mesh editing: both seek to extrapolate data attached at some points to the whole manifold. Given a set of labeled samples, extrapolating labels throughout the entire sample space is the task of semi-supervised learning. The qualification “manifold” is added if the samples are assumed to belong to a manifold embedded into a high-dimensional space – attaching labels is equivalent to defining a function on this manifold. Editing a mesh involves determining the new locations of vertices given new locations of some of the vertices – the handles. The displacement vectors – the differences between new and old vertex positions – can be considered to define a function on the mesh. Thus, given the values of this function at the handles we are trying to extrapolate to the whole mesh – a task that would otherwise qualify as semi-supervised learning. If rotations at handles are also given, propagating them throughout the mesh is again an instance of semi-supervised learning. Does this similarity of the two fields extend beyond the objectives sought? Laplacian based approaches to mesh editing start by extracting the surface’s differential coordinates, and then reconstruct the surface by imposing the handle constraints and requiring that the differential coordinates are preserved as much as possible. The differential coordinates capture the local detail, so the more they are preserved, the more the shape is preserved. When viewed from this angle, Laplacian mesh editing seems to bear no resemblance to the methods of semi-supervised learning.