Closed surfaces with different shapes that are indistinguishable by the SRNF

summary:The Square Root Normal Field (SRNF), introduced by Jermyn et al. in [5], provides a way of representing immersed surfaces in $\mathbb{R}^3$, and equipping the set of these immersions with a “distance function" (to be precise, a pseudometric) that is easy to compute. Importantly, this distance function is invariant under reparametrizations (i.e., under self-diffeomorphisms of the domain surface) and under rigid motions of $\mathbb{R}^3$. Thus, it induces a distance function on the shape space of immersions, i.e., the space of immersions modulo reparametrizations and rigid motions of $\mathbb{R}^3$. In this paper, we give examples of the degeneracy of this distance function, i.e., examples of immersed surfaces (some closed and some open) that have the same SRNF, but are not the same up to reparametrization and rigid motions. We also prove that the SRNF does distinguish the shape of a standard sphere from the shape of any other immersed surface, and does distinguish between the shapes of any two embedded strictly convex surfaces

Elastic shape analysis of surfaces with second-order Sobolev metrics: a comprehensive numerical framework

This paper introduces a set of numerical methods for Riemannian shape analysis of 3D surfaces within the setting of invariant (elastic) second-order Sobolev metrics. More specifically, we address the computation of geodesics and geodesic distances between parametrized or unparametrized immersed surfaces represented as 3D meshes. Building on this, we develop tools for the statistical shape analysis of sets of surfaces, including methods for estimating Karcher means and performing tangent PCA on shape populations, and for computing parallel transport along paths of surfaces. Our proposed approach fundamentally relies on a relaxed variational formulation for the geodesic matching problem via the use of varifold fidelity terms, which enable us to enforce reparametrization independence when computing geodesics between unparametrized surfaces, while also yielding versatile algorithms that allow us to compare surfaces with varying sampling or mesh structures. Importantly, we demonstrate how our relaxed variational framework can be extended to tackle partially observed data. The different benefits of our numerical pipeline are illustrated over various examples, synthetic and real.Comment: 25 pages, 16 figures, 1 tabl

Elastic shape analysis of geometric objects with complex structures and partial correspondences

In this dissertation, we address the development of elastic shape analysis frameworks for the registration, comparison and statistical shape analysis of geometric objects with complex topological structures and partial correspondences. In particular, we introduce a variational framework and several numerical algorithms for the estimation of geodesics and distances induced by higher-order elastic Sobolev metrics on the space of parametrized and unparametrized curves and surfaces. We extend our framework to the setting of shape graphs (i.e., geometric objects with branching structures where each branch is a curve) and surfaces with complex topological structures and partial correspondences. To do so, we leverage the flexibility of varifold fidelity metrics in order to augment our geometric objects with a spatially-varying weight function, which in turn enables us to indirectly model topological changes and handle partial matching constraints via the estimation of vanishing weights within the registration process. In the setting of shape graphs, we prove the existence of solutions to the relaxed registration problem with weights, which is the main theoretical contribution of this thesis. In the setting of surfaces, we leverage our surface matching algorithms to develop a comprehensive collection of numerical routines for the statistical shape analysis of sets of 3D surfaces, which includes algorithms to compute Karcher means, perform dimensionality reduction via multidimensional scaling and tangent principal component analysis, and estimate parallel transport across surfaces (possibly with partial matching constraints). Moreover, we also address the development of numerical shape analysis pipelines for large-scale data-driven applications with geometric objects. Towards this end, we introduce a supervised deep learning framework to compute the square-root velocity (SRV) distance for curves. Our trained network provides fast and accurate estimates of the SRV distance between pairs of geometric curves, without the need to find optimal reparametrizations. As a proof of concept for the suitability of such approaches in practical contexts, we use it to perform optical character recognition (OCR), achieving comparable performance in terms of computational speed and accuracy to other existing OCR methods. Lastly, we address the difficulty of extracting high quality shape structures from imaging data in the field of astronomy. To do so, we present a state-of-the-art expectation-maximization approach for the challenging task of multi-frame astronomical image deconvolution and super-resolution. We leverage our approach to obtain a high-fidelity reconstruction of the night sky, from which high quality shape data can be extracted using appropriate segmentation and photometric techniques

Shape-Graph Matching Network (SGM-net): Registration for Statistical Shape Analysis

This paper focuses on the statistical analysis of shapes of data objects called shape graphs, a set of nodes connected by articulated curves with arbitrary shapes. A critical need here is a constrained registration of points (nodes to nodes, edges to edges) across objects. This, in turn, requires optimization over the permutation group, made challenging by differences in nodes (in terms of numbers, locations) and edges (in terms of shapes, placements, and sizes) across objects. This paper tackles this registration problem using a novel neural-network architecture and involves an unsupervised loss function developed using the elastic shape metric for curves. This architecture results in (1) state-of-the-art matching performance and (2) an order of magnitude reduction in the computational cost relative to baseline approaches. We demonstrate the effectiveness of the proposed approach using both simulated data and real-world 2D and 3D shape graphs. Code and data will be made publicly available after review to foster research