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1. Supervised Deep Learning of Elastic SRV Distances on the Shape Space of Curves

   https://doi.org/10.1109/CVPRW53098.2021.00499  

   Hartman, Emmanuel; Sukurdeep, Yashil; Charon, Nicolas; Klassen, Eric; Bauer, Martin (January 2021, Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR) Workshops)

   Motivated by applications from computer vision to bioinformatics, the field of shape analysis deals with problems where one wants to analyze geometric objects, such as curves, while ignoring actions that preserve their shape, such as translations, rotations, scalings, or reparametrizations. Mathematical tools have been developed to define notions of distances, averages, and optimal deformations for geometric objects. One such framework, which has proven to be successful in many applications, is based on the square root velocity (SRV) transform, which allows one to define a computable distance between spatial curves regardless of how they are parametrized. This paper introduces a supervised deep learning framework for the direct computation of SRV distances between curves, which usually requires an optimization over the group of reparametrizations that act on the curves. The benefits of our approach in terms of computational speed and accuracy are illustrated via several numerical experiments on both synthetic and real data. 

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2. Elastic Shape Analysis of Tree-Like 3D Objects Using Extended SRVF Representation

   https://doi.org/10.1109/TPAMI.2023.3334525  

   Wang, Guan; Laga, Hamid; Srivastava, Anuj (April 2024, IEEE Transactions on Pattern Analysis and Machine Intelligence)

   How can one analyze detailed 3D biological objects, such as neuronal and botanical trees, that exhibit complex geometrical and topological variation? In this paper, we develop a novel mathematical framework for representing, comparing, and computing geodesic deformations between the shapes of such tree-like 3D objects. A hierarchical organization of subtrees characterizes these objects - each subtree has a main branch with some side branches attached - and one needs to match these structures across objects for meaningful comparisons. We propose a novel representation that extends the Square-Root Velocity Function (SRVF), initially developed for Euclidean curves, to tree-shaped 3D objects. We then define a new metric that quantifies the bending, stretching, and branch sliding needed to deform one tree-shaped object into the other. Compared to the current metrics such as the Quotient Euclidean Distance (QED) and the Tree Edit Distance (TED), the proposed representation and metric capture the full elasticity of the branches (i.e., bending and stretching) as well as the topological variations (i.e., branch death/birth and sliding). It completely avoids the shrinkage that results from the edge collapse and node split operations of the QED and TED metrics. We demonstrate the utility of this framework in comparing, matching, and computing geodesics between biological objects such as neuronal and botanical trees. We also demonstrate its application to various shape analysis tasks such as (i) symmetry analysis and symmetrization of tree-shaped 3D objects, (ii) computing summary statistics (means and modes of variations) of populations of tree-shaped 3D objects, (iii) fitting parametric probability distributions to such populations, and (iv) finally synthesizing novel tree-shaped 3D objects through random sampling from estimated probability distributions. 

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3. On Length-Sensitive Fréchet Similarity

   https://doi.org/10.1007/978-3-031-38906-1_15  

   Buchin, Kevin; Fasy, Brittany Terese; Sereshgi, Erfan Hosseini; Wenk, Carola (July 2023, Lecture Notes in Computer Science)

   Morin, P; Suri, S (Ed.)

   Taking length into consideration while comparing 1D shapes is a challenging task. In particular, matching equal-length portions of such shapes regardless of their combinatorial features, and only based on proximity, is often required in biomedical and geospatial applications. In this work, we define the length-sensitive partial Fréchet similarity (LSFS) between curves (or graphs), which maximizes the length of matched portions that are close to each other and of equal length. We present an exact polynomial-time algorithm to compute LSFS between curves under and . For geometric graphs, we show that the decision problem is NP-hard even if one of the graphs consists of one edge. 

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4. Diffusing orbits along chains of cylinders

   https://doi.org/10.3934/dcds.2022121  

   Gidea, Marian; Marco, Jean-Pierre (January 2022, Discrete and Continuous Dynamical Systems)

   We develop a geometric mechanism to prove the existence of orbits that drift along a prescribed sequence of cylinders, under some general conditions on the dynamics. This mechanism can be used to prove the existence of Arnold diffusion for large families of perturbations of Tonelli Hamiltonians on A^3. Our approach can also be applied to more general Hamiltonians that are not necessarily convex. The main geometric objects in our framework are –dimensional invariant cylinders with boundary (not necessarily hyperbolic), which are assumed to admit center-stable and center-unstable manifolds. These enable us to define chains of cylinders, i.e., finite, ordered families of cylinders where each cylinder admits homoclinic connections, and any two consecutive cylinders in the chain admit heteroclinic connections. Our main result is on the existence of diffusing orbits which drift along such chains of cylinders, under precise conditions on the dynamics on the cylinders – i.e., the existence of Poincaré sections with the return maps satisfying a tilt condition – and on the geometric properties of the intersections of the center-stable and center-unstable manifolds of the cylinders – i.e., certain compatibility conditions between the tilt map and the homoclinic maps associated to its essential invariant circles. We give two proofs of our result, a very short and abstract one, and a more constructive one, aimed at possible applications to concrete systems. 

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5. Inferring 3D Shapes of Unknown Rigid Objects in Clutter through Inverse Physics Reasoning

   Song, Changkyu; Boularias, Abdeslam (April 2019, IEEE robotics automation letters)

   We present a probabilistic approach for building, on the fly, three dimensional (3D) models of unknown objects while being manipulated by a robot. We specifically consider manipulation tasks in piles of clutter that contain previously unseen objects. Most manipulation algorithms for performing such tasks require known geometric models of the objects in order to grasp or rearrange them robustly. One of the novel aspects of this work is the utilization of a physics engine for verifying hypothesized geometries in simulation. The evidence provided by physics simulations is used in a probabilistic framework that accounts for the fact that mechanical properties of the objects are uncertain. We present an efficient algorithm for inferring occluded parts of objects based on their observed motions and mutual interactions. Experiments using a robot show that this approach is efficient for constructing physically realistic 3D models, which can be useful for manipulation planning. Experiments also show that the proposed approach significantly outperforms alternative approaches in terms of shape accuracy. 

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