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Shape Theory via the Atiyah--Molino Reconstruction and Deformations

Noémie C. Combe, Hanna N. Nencka

TL;DR

The paper introduces a geometric reconstruction framework that unifies Vaisman foliation theory and the Atiyah–Molino construction to address inverse problems. Central to the approach is the Hantjies tensor $H$, a curvature-like invariant that quantifies noise propagation and governs associativity in the reconstruction algebra; the transverse moment maps $\mu_i$ and their transversality condition $d\mu_1 \wedge d\mu_2 \neq 0$ provide global reconstruction criteria. When $H=0$, the Atiyah–Molino sequence splits holonomy-free and the space $M$ becomes diffeomorphic to $\mathbb{R}^2 \times \mathbb{R}^2$, enabling a unique, associative reconstruction, with toric symmetry yielding a further streamlined, unique solution. If $H\neq 0$, the non-associativity leads to a quasigroupoid of non-unique reconstructions parameterized by $[H] \in H^1(M, TM)$, describing deformation obstructions. The framework promises robust, noise-tolerant reconstructions with broad applications from medical imaging to quantum tomography and AI-driven manifold learning.

Abstract

Reconstruction problems lie at the very heart of both mathematics and science, posing the enigmatic challenge: \emph{How does one resurrect a hidden structure from the shards of incomplete, fragmented, or distorted data?} In this paper, we introduce a new approach that harnesses the profound insights of the Vaisman Atiyah--Molino framework. Our method renders the reconstruction problem computationally tractable while exhibiting exceptional robustness in the presence of noise. Central to our theory is the Hantjies tensor -- a curvature-like invariant that precisely quantifies noise propagation and enables error-bounded reconstructions. This synthesis of differential geometry, integral analysis, and algebraic topology not only resolves long-standing ambiguities in inverse problems but also paves the way for transformative applications across a broad spectrum of scientific disciplines.

Shape Theory via the Atiyah--Molino Reconstruction and Deformations

TL;DR

The paper introduces a geometric reconstruction framework that unifies Vaisman foliation theory and the Atiyah–Molino construction to address inverse problems. Central to the approach is the Hantjies tensor , a curvature-like invariant that quantifies noise propagation and governs associativity in the reconstruction algebra; the transverse moment maps and their transversality condition provide global reconstruction criteria. When , the Atiyah–Molino sequence splits holonomy-free and the space becomes diffeomorphic to , enabling a unique, associative reconstruction, with toric symmetry yielding a further streamlined, unique solution. If , the non-associativity leads to a quasigroupoid of non-unique reconstructions parameterized by , describing deformation obstructions. The framework promises robust, noise-tolerant reconstructions with broad applications from medical imaging to quantum tomography and AI-driven manifold learning.

Abstract

Reconstruction problems lie at the very heart of both mathematics and science, posing the enigmatic challenge: \emph{How does one resurrect a hidden structure from the shards of incomplete, fragmented, or distorted data?} In this paper, we introduce a new approach that harnesses the profound insights of the Vaisman Atiyah--Molino framework. Our method renders the reconstruction problem computationally tractable while exhibiting exceptional robustness in the presence of noise. Central to our theory is the Hantjies tensor -- a curvature-like invariant that precisely quantifies noise propagation and enables error-bounded reconstructions. This synthesis of differential geometry, integral analysis, and algebraic topology not only resolves long-standing ambiguities in inverse problems but also paves the way for transformative applications across a broad spectrum of scientific disciplines.

Paper Structure

This paper contains 14 sections, 7 theorems, 21 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Projections and Inverse Morphisms
  3. Centroids
  4. Moment Maps
  5. Proof of Proposition \ref{['P:2']}
  6. Geometric Structures and Foliations
  7. Structure of the Foliated Space
  8. Hantjies tensor fields
  9. The Setting
  10. Atiyah--Molino Space and Reconstruction Criterion
  11. Translating the construction to Atiyah--Molino framework
  12. Deformation Theory under Non-Vanishing Hantjies Tensor
  13. Toric Reconstruction in the Atiyah–Molino Setting
  14. Conclusion

Key Result

Proposition 1

(i) Let $O \subset \mathbb{CP}^3$ be a smooth, non-symmetric three dimensional object embedded in the complex projective space, and let $\Pi_1, \Pi_2: \mathbb{CP}^3 \dashrightarrow \mathbb{CP}^2$ be two independent rational projections onto distinct complex projective planes. Assume: Each projection $\Pi_i$ defines a holomorphic line bundle $L_i \to \mathbb{CP}^2$ with a connection $\nabla_i$ der

Figures (1)

  • Figure 1: Illustration of an imaging process: each projection image corresponds the line integrals of a molecule rotated by a three-dimensional rotation.

Theorems & Definitions (13)

  • Proposition 1
  • Proposition 2
  • proof
  • Theorem 1
  • Definition 1
  • Theorem 2
  • proof
  • Theorem 3
  • proof
  • Corollary 1
  • ...and 3 more