TreeSRNF: Square-Root Normal Fields for Generative Modelling of the Geometric and Structural Variability in Tree-like 3D Objects

We introduce a novel mathematical framework for analyz-ing and generating complex tree-shaped 3D objects, such as botanicaltrees and plants, which deform both in their 3D geometry and branchingstructure. Unlike previous works, which either consider only the skeletalstructure of tree-like objects or approximate their 3D geometry usingbranch thickness, the proposed framework accurately models both the3D geometry of the tree branches and the way they are interconnected.In this paper, we first generalize the Square Root Normal Fields (SRNF)representation, originally proposed for the statistical analysis of genus-0surfaces, to tree-shaped 3D objects. We then treat tree-shaped 3D ob-jects as points on a novel Riemannian tree-shape space equipped with anovel Riemannian metric that measures the amount of surface bendingand stretching, and structural changes one needs to apply to one 3D tree-shape to align it with another. This way, deformations become trajecto-ries in this novel tree-shape space. We analyze the theoretical propertiesof this novel tree-shape space and the corresponding metric and developalgorithms for computing point-wise and branch-wise correspondencesand geodesic paths between complex 3D trees. We finally show how touse these building blocks for (1) computing statistical summaries, i.e.means and modes of variation, of collections of tree-shaped 3D objects,and (2) synthesizing novel tree-shaped 3D objects by sampling fromprobability distributions fitted to a population of tree-shaped 3D objects.We demonstrate the performance and utility of the proposed frameworkon real and synthetic plants and botanical trees and show that it sig-nificantly outperforms the state-of-the-art. Additional results and sourcecode are available at https://tahmina979.github.io/TreeinSRNF/