Weighted manifolds arise naturally and frequently in Imaging. Such weights (or densities) may appear as uncertainties intrinsic to the acquiring of the image (for instance in Ultrasonography), in modeling textures and, in a variety of instances, as ad hoc tools employed at various stages of the implementation of a variety of tasks, such as smoothing, (elastic) registration, warping, segmentation, etc. Moreover, in the context of Medical Imaging, densities appear at even a more basic, intrinsic level: The density of many types of MRI images equals the very proton density.
We introduce a novel method of image sampling and reconstruction, based on viewing images as manifolds with density, and sampling them according to the generalized Ricci curvature introduced by Bakry, Emery and Ledoux. The quality of the reconstruction is guaranteed by a generalization of a classical result of Grove and Petersen. The new paradigm generalizes ideas and results that are by now common in Imaging and Graphics.
We apply the new algorithm to natural and range images, as well as cartoons, and show that the proposed method produces results similar to those obtained by employing more standard methods. A variation of this approach, due to Morgan and his students is also considered, as well as its application to the sampling and reconstruction of images and meshes.
Supported by Sonderforschungsbereich SFB/Transregio 109