DEformable MOdels in Statistics for signal and image analysis. – DEMOS

In many fields of interests including biology, medical imaging or chemistry, observations are coming from n individuals curves or grey-level images. Such observations are typically high-dimensional data, and models involving such data have been recently extensively studied in statistics, and signal/image processing. In many situations the individual curves or images have a certain common structure that may lead to the assumption that the observations are random elements which vary around a common mean pattern (also called common shape).
Estimating such a mean pattern, or characterizing the mode of
variations and the law of the individual fluctuations around this
common shape is of fundamental interest to learn information from such data. Typically, this data exhibit a (classical) source of linear variation in amplitude/intensity, but also non-linear variations in time or space which can be modelled by the action of random deformation operators.

The aim of the DEMOS project is to propose new approaches adapted to the statistical analysis of signals and images which can be modelled as the random deformation of a stochastic process corrupted by an additive white noise. Estimating the deformations that may exist between similar signals or images is commonly referred to as the image warping or curve registration problem, and one of the main objectives of this project is to investigate the statistical aspects of such procedures which are widely used in signal and image processing. More generally, the goal of the DEMOS project is to provide new methodologies and new algorithms for the statistical analysis of the Mode of variations of objects (such as curves or images) lying in non-Euclidean spaces (such as Lie groups and Riemannian manifolds).

Studying such data relies on the use of modern statistical methods
such as the analysis of inverse problems with unknown operators, non-linear modes of variations for high-dimensional data and sparse PCA, statistical inference in group models for manifold valued variables and M-estimation techniques. Moreover, the DEMOS project is by its nature at the crossing of many research fields including statistics, signal and image processing and engineering. Therefore, potential applications of the methods that we plan to develop in the DEMOS project are numerous, including proteomic or genomic studies, pattern recognition, brain shape variability, and computational anatomy to name but a few. Therefore, a strong component of the project will be the development and the dissemination of publicly available computer codes to be used by other academics/institutions. In particular, one of our objectives is to develop MATLAB and R toolboxes to be used for a wide range of data sets and to disseminate our models and results.