Organic Priors in Non-Rigid Structure from Motion


ETH Zurich1
KU Lueven2

European Conference on Computer Vision (ECCV), 2022.
Tel-Aviv, Israel.
Oral Presentation

Paper
Supplementary
Video



Abstract

This paper advocates the use of organic priors in classical non-rigid structure from motion (NRSfM). By organic priors, we mean invaluable intermediate prior information intrinsic to the NRSfM matrix factorization theory. It is shown that such priors reside in the factorized matrices, and quite surprisingly, existing methods generally disregard them. The paper's main contribution is to put forward a simple, methodical, and practical method that can effectively exploit such organic priors to solve NRSfM. The proposed method does not make assumptions other than the popular one on the low-rank shape and offers a reliable solution to NRSfM under orthographic projection. Our work reveals that the accessibility of organic priors is independent of the camera motion and shape deformation type. Besides that, the paper provides insights into the NRSfM factorization---both in terms of shape and motion---and is the first approach to show the benefit of single rotation averaging for NRSfM. Furthermore, we outline how to effectively recover motion and non-rigid 3D shape using the proposed organic prior based approach and demonstrate results that outperform prior-free NRSfM performance by a significant margin. Finally, we present the benefits of our method via extensive experiments and evaluations on several benchmark datasets.





Non-Rigid Structure from Motion Factorization

The problem of recovering the 3D shape of a non-rigidly deforming object from its image feature correspondences across multiple frames is widely known as Non-Rigid Structure from Motion (NRSfM). One of the popular ways to solve NRSfM is the matrix factorization approach. For this, a measurement matrix W is provided as input, which basically is a matrix of image feature correspondences augmented along the columns of W for each keypoint. Another way to think about it is that each column of W is a two-dimensional trajectory of a point across all the frames ---also referred to as trajectory space representation (see Figure b). The goal of NRSfM factorization approach under orthographic camera projection matrix assumption is to decompose the matrix W into product of two matrices R and S. Where, R, S must be a rotation matrix and shape matrix, respectively. Another general assumption is that the measurement matrix is already mean-centralized, so that the camera matrix reduces to pure rotation matrix. The problem is challenging due to the inherent unconstrained nature of the problem itself, as many 3D configurations can have similar image projections. To date, no algorithm can solve NRSfM for all kinds of conceivable motion. Consequently, additional constraints, priors, and assumptions are often employed to solve NRSfM using the matrix factorization approach.

Figure. Classical NRSfM factorization setup.




Organic Priors

In this paper, we coined the term "organic priors" for NRSfM. By organic priors, we mean the priors that reside in the factorized matrices, which are the results of natural mathematical steps in NRSfM matrix factorization. We show that such priors carry over within the intermediate factorized matrices, and its existence is independent of camera motion and shape deformation type. We used such a word to describe it because they come naturally by properly conceiving the algebraic and geometric construction of classical NRSfM factorization approach. Surprisingly, most existing methods, if not all, ignore them. Our work shows how to procure organic priors and utilize them effectively.

Figure. Organic Rotation Priors





Reconstruction Results on MoCap Dataset.



Pickup


Stretch


Yoga

Face


Reconstruction Results on NRSfM Challenge Dataset




Paper

Sheet Stretch

Balloon



Authors




Citation


	@article{kumar2022organic,
		title={Organic Priors in Non-Rigid Structure from Motion},
		author={Kumar, Suryansh and Van Gool, Luc},
		booktitle={European Conference on Computer Vision (ECCV)},
		year={2022}
	}	     
					


Acknowledgements




The authors thank Google for their generous gift ("ETH Zurich Foundation", 2020-HS-411).



Copyright © 2022 Suryansh Kumar