Ph. D. Project
Low-rank matrix factorizations for polarimetric imaging. Application in bio-imaging.
2020/10/01 - 2023/09/30
Polarization information plays a crucial role in imaging. It unveils many important features of the observed scene, such as shape,
rugosity, orientation, physicochemical properties, etc. [1, 2] that are not readily accessible in conventional intensity imaging.
These features are essential in many application, notably for cancerous tissue characterization in bio-imaging [3]. Nonetheless, to
be fully exploited, polarization information requires the development of new signal processing tools taking into account its

This PhD project aims at developing new low-rank matrix approximation methods that exploit the polarization information
geometry. Expected contributions will focus on theoretical properties (e.g. unicity conditions) and methodological developments
(e.g. efficient algorithms). Within this PhD project, collaborations with biologists from CRAN will aim at demonstrating the
relevance of polarization for bio-medical imaging applications.

Many signal processing problems can be seen as the problem of factorizing a matrix into a product of two low-rank matrices:
blind source separation, data compression or missing data completion, among others. Such low-rank matrix factorization problems
typically exploit some given physical constraints, such as nonnegativity for nonnegative matrix factorization (NMF) [4].

To extend such fundamental tools to the polarimetric imaging case, we have recently introduced in [5] the notion of quaternion
nonnegative matrix factorization (QNMF). This new low-rank approximation method for polarimetric data exploits on the one
hand, an algebraic representation of polarization by using quaternions, and on the other hand, takes advantage of physical
constraints that generalize nonnegativity to the case of polarized signals.

First results obtained in [5] appear promising and pave the way to further developments centered around QNMF. On the theoretical
front, a first goal will be to obtain generic uniqueness conditions for the QNMF, building on recently developed convex-geometry
approaches for the NMF [6]. From a methodological viewpoint, a second objective will focus on the design of numerically
efficient algorithms for the factorization problem. The emphasis will be put on introducing statistical models and associated
criteria that take into account polarization information geometry.

Expected theoretical and methodological contributions will be validated through original applications in bio-imaging. These work
will be conducted in collaboration with the "Cibles moléculaires dans une démarche translationnelle" team at CRAN, with the aim to demonstrate the potential of polarization for bio-medical applications. Two emerging imaging modalities will be considered: polarimetric camera imaging and polarimetric spectro-microscopy.

[1] J. S. Tyo, D. L. Goldstein, D. B. Chenault, et al., "Review of passive imaging polarimetry for remote
sensing applications," Applied optics, vol. 45, no. 22, pp. 5453-5469, 2006.
[2] N. Ghosh and A. I. Vitkin, "Tissue polarimetry: Concepts, challenges, applications, and outlook," Journal
of biomedical optics, vol. 16, no. 11, p. 110 801, 2011.
[3] A. Pierangelo, A. Benali, M.-R. Antonelli, et al., "Ex-vivo characterization of human colon cancer by
mueller polarimetric imaging," Optics express, vol. 19, no. 2, pp. 1582-1593, 2011.
[4] D. D. Lee and H. S. Seung, "Learning the parts of objects by non-negative matrix factorization," Nature,
vol. 401, no. 6755, p. 788, 1999.
[5] J. Flamant, S. Miron, and D. Brie, Quaternion non-negative matrix factorization: Definition, uniqueness
and algorithm, 2019. eprint: arXiv:1903.10593.
[6] X. Fu, K. Huang, N. D. Sidiropoulos, et al., "Nonnegative matrix factorization for signal and data analytics:
Identifiability, algorithms, and applications," IEEE Signal Processing Magazine, vol. 36, no. 2, pp. 59-80,
low-rank approximation, polarimetric imaging, optimization, identifiability
Length : 3 years
Candidate profile : Master's degree or equivalent in signal processing / applied mathematics.
Biology, Signals and Systems in Cancer and Neuroscience