Non-negative tensor factorization with rank-revealing constraints
Estimation of the rank is a key issue of tensor factorization, especially when solving approximations problems. In this project, we design a family of optimization methods based on the beta-divergence, to solve non-negative tensor approximation problems with a variety of noise statistics.
Motivated by mixing models, we place a minimum-volume constraint on the sources. This penalized variant circumvents possible rank deficiencies in the case of, e.g., very similar materials with almost colinear spectral signatures.
The constraint also successfully estimates the number of sources in the image.
Super-resolution and unmixing in remote sensing
In remote sensing, the problem of super-resolution aims at reconstructing a high-resolution data cube from a hyperspectral and a multispectral image of the same scene. The hyperspectral image is spatially-degraded but has as high spectral resolution, while the multispectral image has high spatial resolution, at the cost of a low spectral resolution.
The super-resolution image of interest can then be used to extract materials and corresponding abundances, in a process termed unmixing.
This project provides tensor-based algorithms for the super-resolution and unmixing tasks. Proposed models are assorted with recovery guarantees. The performance of the methods are assessed on real datasets in a variety of cases, for instance when several degradation operators are unknown or when the acquisitions are subject to some inter-image variability.
Cramér-Rao bounds for coupled tensor models
In estimation theory, the Cramér-Rao bound is perhaps one of the most widely used tools for assessing the performance of an estimator. For coupled models, the literature refers to the so-called Constrained Cramér-Rao bound (CCRB), that is able to model the interactions between the various sources of information.
Due to the emergence of coupled tensor models in a variety of applications (e.g., remote sensing, medical imaging, spectrum cartography, chemometry), the
CCRB for such models is of great interest.
In this project, I provide closed-form of the CCRB for coupled tensor models used in remote sensing super-resolution and unmixing.
I assessed the performance of the estimators designed in the said project, showing that they reach the bounds.
I also designed a new bound subject to random equality constraints, and showed that is is usually tighter than the standard tool.
Multi-frame super-resolution in medical imaging
MRI is a very versatile imaging device that is able to provide 2D or 3D images of an organ of interest.
However, the acquisition of the latter suffer from a long acquisition time. Furthermore, patient motion can highly degrade the quality of the acquisition.
Finally, the machine limitation is often reached for very precise brain details.
This project, in collaboration with INSERM and Nancy University Hospital, aims at reconstruction a 3D image of a brain using various acquisitions.
The observations are downsampled and interpolated in arbitrary orientations.
The aim is to reconstruct a high-quality image in a reduced computation time, using tensor low-rank modelling.