Generative Topology for Shape Synthesis

The Euler Characteristic Transform (ECT) is a powerful invariant for assessing geometrical and topological characteristics of a large variety of objects, including graphs and embedded simplicial complexes. Although the ECT is invertible in theory, no explicit algorithm for general data sets exists. In this paper, we address this lack and demonstrate that it is possible to learn the inversion, permitting us to develop a novel framework for shape generation tasks on point clouds. Our model exhibits high quality in reconstruction and generation tasks, affords efficient latent-space interpolation, and is orders of magnitude faster than existing methods. Understanding shapes requires understanding their geometrical and topological properties in tandem. Given the large variety of different representations of such data, ranging from point clouds over graphs to simplicial complexes, a general framework for handling such inputs is beneficial. The Euler Characteristic Transform (ECT) provides such a framework based on the idea of studying a shape from multiple directions--sampled from a sphere of appropriate dimensionality--and at multiple scales. In fact, the ECT is an injective map, serving as a unique characterisation of a shape (Ghrist et al., 2018; Turner et al., 2014). Somewhat surprisingly, this even holds when using a finite number of directions (Curry et al., 2022). Hence, while it is known that the ECT can be inverted, i.e. it is possible to reconstruct input data from an ECT, only algorithms for special cases such as planar graphs are currently known (Fasy et al., 2018).