Post-doctorant F/H

Le post-doctorant sera hébergé au laboratoire TéSA (https://www.tesa.prd.fr) et sera co-encadré par

Samy Labsir (https://sites.google.com/view/samylabsir/)

Julien Lesouple (http://perso.recherche.enac.fr/~julien.lesouple/fr/)

Jean-Yves Tourneret (https://perso.tesa.prd.fr/jyt/)

Context and problem

In this work, we propose to focus on well-known image processing estimation problem referred to as image registration. Given a set of (2D or 3D) moving images observed by an imaging sensor, the aim is to determine the geometric transformation allowing us to adjust these images to a reference (2D or 3D) image, in order to exploit multi-information of a scene. This problem of registration is classically encountered in a plethory of space applications typically when optical remote sensing systems detect Earth information.

We can cite the example of a SAR satellite observing a 3D earth image for oceanography tasks, under different angles [1]. Then each sensed image must be aligned to the true image. Also in aerospace defence applications, in the context of target tracking and facial recognition, video sequences containing the target of interest to track need to be adjusted to another sequence of reference [2].

Different state-of-the-art techniques exist to resolve this problem [3], classically based on the minimization of a cost function between the reference image and the transformed observed image. Then, the minimization is performed by an optimization algorithm. Nevertheless, these conventional methods have some limits. Indeed, it can be difficult to optimize the cost function because it is highly non-convex and can lead to several solutions [4]. Furthermore, in the context of 3D/2D registration, the model linking both reference image and observed image is ill-posed because the depth of the 3D image is not observable [5]. To overcome these analytical problems, deep learning and machine learning based techniques have recently appeared, [6] mainly based on SVM [7] and neural networks (NN) [8] substituting analytical models by parametric regression models. The latter are learned thanks to a dataset containing both pixel information of an arbitrary image to adjust and transformation relative to the reference image could which may also contain scale and depth parameters. Then, knowing a new image test, we can be able to predict the new transformation.

Starting from this assumption, two of the fundamental points to bear in mind:

1. the geometric transformation to predict have to be constrained by its geometrical properties. Indeed, it is basically an affine transformation containing both rotation and translation between the two images but also scale parameter. Then, learn this type of parameter needs to ensure the properties of this space are respected. In a mathematical point of view, this transformation is constrained to lie on the Lie group SE(2) or SE(3). When the scale parameter is also unknown, then the transformation belongs to the Lie group Sim(3) (space of similitude in dimension 3).
2. Parametric models used in machine learning need to learn hyperparameters which is classically a hard challenge typically for SVM. Furthermore, one of the main drawback is that this kind of approach, whatever SVM or NN, don't provide uncertainty on the estimation of the transformation, which is a crucial information in an operational context.

With this in mind, we propose in this work to deal with Lie groups approach. Mathematically speaking, ia Lie group is a space equipped with a Riemmanian structure but also with a structure of group [9]. Its advantage is to leverage mathematical tools (especially statistics and optimization) allowing us to overcome non-linearities induced by the parameters of the transformation (classically Euler angles). Machine learning methods on Lie groups exist in a supervised framework for tasks of estimation and filtering [10-11-12]. Nevertheless, in the context of supervised classification and prediction, works are almost non-existent and everything remains to be done.

In addition, an attention is paid to machine learning standing out from SVM and NN, more specifically to Gaussian process regression [13-14]. Indeed, these are powerful non-parametric prediction tools allowing us to learn correlation between measurements and model [15]. Another advantage is that, they are able to learn hyperparameter models and provide uncertainty on the prediction.

Hence, the challenge of this work is to combine the Lie group formalism with Gaussian process paradigm in order to design a new machine learning algorithm for image registration applications.

Objectives

The main objectives of this post-doctoral work are to develop new machine learning methods based on Gaussian processes on Lie groups. More precisely:

1) The main aim will be to familiarize with regression problems on Lie groups and geometric Gaussian process, i.e. define a non-parametric regression model on Lie groups.
2) The second aim will be to compute both likelihood and predictive distribution of the Gaussian processes in order to predict output and estimation of the kernel parameter. A challenge will be to perform a integral marginalization to compute them. To succeed that, two strategies could be considered: a) an analytical approximation on the integral to marginalize or b) an numerical approximation based on MCMC algorithms on Lie groups.
3) The proposed strategy will be numerically tested on simulated image data in a 3D/2D image registration context.