In recent years the heat equation on a weighted graph has been used to attack problems in image processing, including denoising, segmentation, and inpainting. In many cases the data used to build the graph is the set of patches or feature responses from a single image; and even then, patches or filter responses are usually only compared with their spatial neighborhoods. On the other hand, it has become practical to manipulate large collections of images, and using the statistics of large collections of images has become an important image processing tool. The PI will investigate how to use larger databases of images and image features in a PDE framework. For some of the proposed work, it will be necessary to invent new theory to lift notions of the smoothness of a surface embedded in Euclidean space to maps from the discrete grid into a weighted graph; and perhaps further to maps from a more general weighted graph into a weighted graph. Other proposed work will try to make use of (and improve) recent methods for sparse feature extraction, and solidify the theory underlying the NL-means method of Buades, Coll, and Morel.
A large class of popular image processing techniques making use of the theory of partial differential equations operate locally; that is, the behavior of each step of these algorithms at a given pixel in an image is determined solely by the neighboring pixel values. In recent years, it has become possible to manipulate large databases of images; and such databases of images have become available from many sources, including the world wide web. The PI will work towards extending the local techniques to make use of these large databases. The long term goal is image processing techniques which understand and utilize the content and context of images; such techniques would have many important applications, for example in medical imaging, hperspectral imaging, and computer vision in general.
