The Generalized Beltrami Problem Concerning Geodesic Representation

The Problem of Partial Geodesic Representation

The original problem of geodesic representation, solved by Dini (1869) with reality restrictions and by Lie (1883) in complete generality, may be stated as follows : Find all pairs of surfaces S and Sx whose points may be put into correspondence in such a way that every geodesic on the one surface is pictured by a geodesic on the other. Apart from the trivial case where Sx is isometric with (ap...

Image retargeting via Beltrami representation

Image retargeting aims to resize an image to one with a prescribed aspect ratio. Simple scaling inevitably introduces unnatural geometric distortions on the important content of the image. In this paper, we propose a simple and yet effective method to resize an image, which preserves the geometry of the important content, using the Beltrami representation. Our algorithm allows users to interact...

The Discrete Geodesic Problem

We present an algorithm for determining the shortest path between a source and a destination on an arbitrary (possibly nonconvex) polyhedral surface. The path is constrained to lie on the surface, and distances are measured according to the Euclidean metric. Our algorithm runs in time O(n log n) and requires O(n2) space, where n is the number of edges of the surface. After we run our algorithm,...

Multiscale Representation of Deformation via Beltrami Coefficients

Geodesic Obstacle Representation of Graphs

An obstacle representation of a graph is a mapping of the vertices onto points in the plane and a set of connected regions of the plane (called obstacles) such that the straight-line segment connecting the points corresponding to two vertices does not intersect any obstacles if and only if the vertices are adjacent in the graph. The obstacle representation and its plane variant (in which the re...