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General Deformations of Point Configurations Viewed By a Pinhole Model Camera

Yirmeyahu Kaminski, Michael Werman

TL;DR

The paper investigates the recoverability of both deformation and 3D structure from monocular sequences of deforming point configurations under affine and general smooth deformations, for calibrated and uncalibrated cameras. It derives a fundamental essential matrix formulation for affine deformations and shows that two images are insufficient to recover deformation or original points, while three views generally leave a three-parameter ambiguity unless the two deformations are quasi identical. It further develops invariants of the shape that can be recovered from three views without calibration and analyzes the dimensionality of the solution sets, identifying regimes where unique or finite recovery is possible. For general smooth deformations, the authors model the motion as slow flows of a vector field and propose a successive affine approximation framework that enables reconstruction from calibrated data across multiple frames. Overall, the work provides rigorous conditions, dimension counts, and algorithmic pathways that clarify when and how deformation and structure can be recovered from monocular sequences of deformable configurations.

Abstract

This paper is a theoretical study of the following Non-Rigid Structure from Motion problem. What can be computed from a monocular view of a parametrically deforming set of points? We treat various variations of this problem for affine and polynomial deformations with calibrated and uncalibrated cameras. We show that in general at least three images with quasi-identical two deformations are needed in order to have a finite set of solutions of the points' structure and calculate some simple examples.

General Deformations of Point Configurations Viewed By a Pinhole Model Camera

TL;DR

The paper investigates the recoverability of both deformation and 3D structure from monocular sequences of deforming point configurations under affine and general smooth deformations, for calibrated and uncalibrated cameras. It derives a fundamental essential matrix formulation for affine deformations and shows that two images are insufficient to recover deformation or original points, while three views generally leave a three-parameter ambiguity unless the two deformations are quasi identical. It further develops invariants of the shape that can be recovered from three views without calibration and analyzes the dimensionality of the solution sets, identifying regimes where unique or finite recovery is possible. For general smooth deformations, the authors model the motion as slow flows of a vector field and propose a successive affine approximation framework that enables reconstruction from calibrated data across multiple frames. Overall, the work provides rigorous conditions, dimension counts, and algorithmic pathways that clarify when and how deformation and structure can be recovered from monocular sequences of deformable configurations.

Abstract

This paper is a theoretical study of the following Non-Rigid Structure from Motion problem. What can be computed from a monocular view of a parametrically deforming set of points? We treat various variations of this problem for affine and polynomial deformations with calibrated and uncalibrated cameras. We show that in general at least three images with quasi-identical two deformations are needed in order to have a finite set of solutions of the points' structure and calculate some simple examples.

Paper Structure

This paper contains 14 sections, 14 theorems, 33 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Affine Deformations
  3. Essential Matrix
  4. Deformation Recovery
  5. Beyond Essential Matrices
  6. Shape Recovery
  7. Critical Surface
  8. Invariant Shape
  9. Equations
  10. Dimension Analysis
  11. The case of two images
  12. Three images
  13. General Smooth Deformations
  14. Conclusion

Key Result

Lemma 1

When $t \neq 0$, the essential matrix of this pair of images is: $E \equiv [t]_{\times} A$, where $[t]_\times$ is the matrix of the cross product with $t$ in the standard basis of $\mathbb R^3$.

Figures (1)

  • Figure 1: The perspective image of the vertices of a cube, in black, deforming affinely to the blue points, the arrows are correspondences, epipolar lines.

Theorems & Definitions (29)

  • Lemma 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • proof
  • Theorem 4
  • proof
  • Definition 5
  • Lemma 6
  • ...and 19 more