One of the primary objectives of visual computing has been the development of representations and algorithms that enable computer systems to acquire, process, and render shapes efficiently. While numerous representations and algorithms have been proposed, many involve non-differentiable components, making them incompatible with gradient-based optimization methods. The first part of this talk centers around developing differentiable representations and rendering techniques for shapes. We present our work on designing differentiable parameterized families of homeomorphisms and diffeomorphisms, possibly with additional symmetry constraints, which are used to deform a template shape having desired topological properties. This enables us to address shape search problems using gradient-based optimization methods. We showcase our results for various applications, including searching for curve embeddings whose perspective projections resemble a target image, untangling knots, parametrization of prototiles belonging to specific isohedral tiling classes, and density estimation on identification spaces. In the subsequent part of this talk, we focus on learning-based frameworks, which generate differentiable algorithms for shape-processing problems. Specifically, we will elaborate on our work on surface denoising, surface correspondence, temporal surface blending, and curve extraction problems. The talk concludes with a discussion on future directions and potential areas of exploration for advancing the field.
