We propose a new class of generative diffusion models, called functional diffusion. In contrast to previous work, functional diffusion works on samples that are represented by functions with a continuous domain. Functional diffusion can be seen as an extension of classical diffusion models to the infinite-dimensional domain. Functional diffusion is very versatile as images, videos, audio, 3D shapes, deformations, \etc, can be handled by the same framework with minimal changes. In addition, functional diffusion is especially suited for irregular data or data defined in non-standard domains. In our work, we derive the necessary foundations for functional diffusion and propose a first implementation based on the transformer architecture. We show generative results on complicated signed distance functions and deformation functions defined on 3D shape surfaces.

Signed Distance Functions (SDFs) are mathematical functions that describe the distance from a point in space to the nearest surface of an object. These functions are commonly used in computer graphics, computer-aided design (CAD), and computational geometry for representing and manipulating shapes and surfaces. The key feature of signed distance functions is that they not only provide the distance to the nearest surface but also indicate whether the point is inside or outside the object. The sign of the distance indicates the side of the surface: positive distances are typically associated with points outside the object, and negative distances are associated with points inside the object.

Given meshes sampled in a dynamic shape sequence, and limited (32) sparse correspondences between two meshes, we want to predict a deformation field. Specifically, the deformation field takes a point on the surface of the source frame as input and outputs a deformation vector which should map the point to the target frame.