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A Python implementation of some geometric tools on Kendall 3D shape space for practical applications

Jorge Valero, Vicent Gimeno i Garcia, M. Victoría Ibáñez, Pau Martinavarro, Amelia Simó

TL;DR

This work addresses the challenge of analyzing geometric structures using Kendall's 3D Shape Space, and introduces tools designed to bridge the gap, translating complex mathematical abstractions into efficient, accessible software solutions for researchers.

Abstract

This work addresses the challenge of analyzing geometric structures using Kendall's 3D Shape Space. While Riemannian geometry provides a robust framework for shape analysis (independent of scale, position, and orientation) the transition from theoretical manifolds to practical computational workflows remains difficult. Although Geomstats is currently the leading Python library for manifold-based statistics, it lacks specific utilities required for advanced 3D shape analysis. This article introduces tools designed to bridge this gap, translating complex mathematical abstractions into efficient, accessible software solutions for researchers.

A Python implementation of some geometric tools on Kendall 3D shape space for practical applications

TL;DR

This work addresses the challenge of analyzing geometric structures using Kendall's 3D Shape Space, and introduces tools designed to bridge the gap, translating complex mathematical abstractions into efficient, accessible software solutions for researchers.

Abstract

This work addresses the challenge of analyzing geometric structures using Kendall's 3D Shape Space. While Riemannian geometry provides a robust framework for shape analysis (independent of scale, position, and orientation) the transition from theoretical manifolds to practical computational workflows remains difficult. Although Geomstats is currently the leading Python library for manifold-based statistics, it lacks specific utilities required for advanced 3D shape analysis. This article introduces tools designed to bridge this gap, translating complex mathematical abstractions into efficient, accessible software solutions for researchers.
Paper Structure (9 sections, 6 equations)

This paper contains 9 sections, 6 equations.

Table of Contents

  1. Introduction
  2. Mathematical Preliminaries: Kendall’s 3D Shape Space
  3. Orthonormal basis for simulation on the tangent space
  4. Sectional Curvature Calculation
  5. Deriving the Required Basis for the Tangent Space at $X$
  6. Change of Basis and Curvature Calculation
  7. Python Implementations
  8. For the orthonormal basis for simulation
  9. For the sectional curvature calculation