Atlas-Aware Laplacian Smoothing

We consider an image, a result of sampling over a finite two dimensional regular grid a vector valued function of two variables. There is a wealth of work in both theory and application of image processing operations. Our approach is to map into the parameter space then perform image processing operations there. Laplacian smoothing is the building block for defining linear filters, which can be described in terms of polynomials of the Laplacian operator. We focus on extending basic Laplacian smoothing to continuous vector valued functions irregularly sampled on multi-chart parameterized surfaces. Our goal is to efficiently perform Laplacian smoothing with out introducing seam artifacts. We present an algorithm for Laplacian smoothing over a texture atlas which takes into account the discontinuities imposed by the charts. Our general approach is based on the finite volume method, and the graph Laplacian used in geometric signal processing. In image processing, the Laplacian of an image is obtained by convolving the image with the following 3×3 kernel: