Multi-View Structure-from-Motion Enables Oriented Projective Shape Analysis in Three Dimensions
Abstract
Projective shape analysis provides a geometric framework for studying landmark configurations in digital images acquired by pinhole cameras. In the classical projective shape (PS) model, three-dimensional configurations (k-ads) are represented as points in (RP3)q, q = k − 5. A nonparametric test is developed in Patrangenaru et al. [12], for this framework, to determine whether an object matches a design blueprint, with each configuration reconstructed from a single uncalibrated stereo pair. Such two-view reconstructions are identified only up to a 3D projective transformation, which may reverse orientation, so that the sign-blind PS summary was the only available, before oriented projective shape (OPS) was recently considered. Multi-view Structure-from-Motion (SfM) technology removes this obstruction: its bundle adjustment is identified up to an orientation-preserving projective transformation. In this paper we revisit a well-cited three-cube object, from Patrangenaru et al. [12], with n = 8 SfM reconstructions built in Agisoft Metashape Professional 2.3.0, and validate our implementation by reproducing the published stereo construction from the original data. This allows us to perform what is to the best of our knowledge the first three-dimensional OPS analysis, compute its extrinsic total-variance index and perform statistical inference in this novel setting. Due to the high concentration of SfM data, the OPS index is asymptotically one-half the PS index, a structural consequence of concentration rather than a property of the object. Here our blueprint hypothesis is not rejected for any of the q = 14 non-frame landmarks, while the SfM reconstructions are about 26 times more concentrated than the stereo ones, a substantial gain in reconstruction precision. Sample-size, photograph-count, and frame-ordering analyses support the robustness of these conclusions.
Related
- Extrinsic MANOVA for Oriented Projective Shapes
- Quasi-projective dimensions over homological ring epimorphisms
- Local-Cohen-Macaulay DG-rings with unequal finitistic projective dimensions
- Higher smooth surgery structure sets of complex projective spaces, part II
- Structural Foundations of Nonlinear Systems with Unknown Inputs: The UID-Induced Normal Form and Minimal-Sensing Structure-from-Motion
BibTeX
@misc{alamoudi2026multiview,
title = {{Multi-View Structure-from-Motion Enables Oriented Projective Shape Analysis in Three Dimensions}},
author = {Musab Alamoudi and Robert L. Paige and Vic Patrangenaru},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.13263},
eprint = {2609.13263},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.13263},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - Multi-View Structure-from-Motion Enables Oriented Projective Shape Analysis in Three Dimensions
AU - Musab Alamoudi
AU - Robert L. Paige
AU - Vic Patrangenaru
PY - 2026
DA - 2026/09/07
PB - arXiv
DO - 10.48550/arxiv.2609.13263
UR - https://arxiv.org/abs/2609.13263
AB - Projective shape analysis provides a geometric framework for studying landmark configurations in digital images acquired by pinhole cameras. In the classical projective shape (PS) model, three-dimensional configurations (k-ads) are represented as points in (RP^3)^q, q = k − 5. A nonparametric test is developed in Patrangenaru et al. [12], for this framework, to determine whether an object matches a design blueprint, with each configuration reconstructed from a single uncalibrated stereo pair. Such two-view reconstructions are identified only up to a 3D projective transformation, which may reverse orientation, so that the sign-blind PS summary was the only available, before oriented projective shape (OPS) was recently considered. Multi-view Structure-from-Motion (SfM) technology removes this obstruction: its bundle adjustment is identified up to an orientation-preserving projective transformation. In this paper we revisit a well-cited three-cube object, from Patrangenaru et al. [12], with n = 8 SfM reconstructions built in Agisoft Metashape Professional 2.3.0, and validate our implementation by reproducing the published stereo construction from the original data. This allows us to perform what is to the best of our knowledge the first three-dimensional OPS analysis, compute its extrinsic total-variance index and perform statistical inference in this novel setting. Due to the high concentration of SfM data, the OPS index is asymptotically one-half the PS index, a structural consequence of concentration rather than a property of the object. Here our blueprint hypothesis is not rejected for any of the q = 14 non-frame landmarks, while the SfM reconstructions are about 26 times more concentrated than the stereo ones, a substantial gain in reconstruction precision. Sample-size, photograph-count, and frame-ordering analyses support the robustness of these conclusions.
ER -