||As its core mission, computer graphics endeavors to deliver natural-looking and convincing graphic contents, such as images, videos and 3D models for various applications, including design, entertainment, education, simulation, etc. In many cases, “natural-looking” can be interpreted as low distortion with respect to some reference. Depending on the application, the distortion can be measured as the amount of feature stretching, non-feature noise, change of scale, self-overlapping, and so on. As distortions can largely affect human perception of the contents, we want to generate images and shapes with no distortion or controlled amount of distortions.
A basic requirement for shape modeling is the results should not have self-overlapping, in other words, the map should be bijective. Besides bijectivity, more general distortion measures, such as smoothness, conformality, area-preserving, rigidity, quasiconformal factor, etc., are used in different applications. Traditionally, the distortion measure of the results is defined as the sum of local distortions. And the distortion is minimized, so that overall, the resulting shape look natural. However, the distortion can be concentrated in some local regions and cause unnatural behavior of the results. While, in practice, the users usually want low distortions everywhere within the domain. This problem can be posed as a hard constrained optimization problem, where the distortions for any point in the domain is constrained to be smaller than some user specified bounds. This means that we need to solve an optimization problem with a great number of constraints, and most constraints are non-convex. Our mission is to design algorithms that can solve these optimization problem efficiently, in order to provide the users interactive tools for shape modeling. We would also like to investigate the theoretical aspects of these problems, such as the existence and uniqueness of the solution, given the user specified distortion constraints and other constraints.