Deformation of the Earth surface is an essential parameter allowing for seismic analyses. With the advent of very high
spatial resolution agile satellites (e.g. the constellation of Pléiades) and the availability of pre- and post-event
stereoscopic acquisitions, spaceborne observations have become the priviliged tool to map such deformations across very
large areas (cf. Vallage et al. ).
The state-of-the-art methods separate the computation of the deformation's planimetric component from the computation of the vertical component (i.e. altitude). As long as this separation simplifies the estimation problem, it is sub-optimal in terms of the output quality. The objective of this doctorate is to develop a more global estimation approach so as to augment the precision and reliability of the computed deformations as well as to render them more readable for experts outside Earth Sciences.
Following the state-of-the-art methods, a simple and ubiquitous approach to calculate the deformations along three dimensions is as follows :
• calculation of a pre- and post-event digital surface model (DSM) (cf. Rupnik et al. ) ; the differences of the two furnishes the altimetric components of the deformation ;
• calculation of an orthophotomap (a terrain-rectified image) for the pre- and post-event ;
• dense image matching between the pre- and post- orthophotomaps (cf. Rosu et al.  , Leprince et al. ) to retrieve the planimetric component of the deformations.
Thanks to the TOSCA programme funding, and the project « Calcul de topographie et mesure de déformation en science de la terre », this approach is today at disposition within the free open-source MicMac tool.
As long as this approach is operational, it is suboptimal due to the fact that it does not exploit all the available information and because the separation of the processing stages propagates the calculation errors across the processing chain. For instance,
• the calculation of two DSMs (pre- and post-event) must be done independently because given the surface
deformations one cannot carry out the dense image matching without taking relevant precautions; nonetheless, doing so, one neglects the fact that the two DSMs have a very similar geometry and their difference has essentially a low frequency nature (or quasi piece-wise constant).
• the errors present in the DSMs generate noise in the orthophotomaps; the noise then produces aliasing artefacts within the calculated displacement maps, and subsequently causes interpretation ambiguities.
3- Proposed approach
To simultaneously exploit all the information available, the following problem formulation is proposed:
• for each 3D point four parameters are estimated (ZPRE, ZPOST, ΔX, ΔY); globally, an unknown function from R2 → R4 is estimated;
• single-epoch image matching of pre-event (or post-event, respectively) images classically imposes a constraint on the ZPRE (respectively the ZPOST) coordinate;
• multi-epoch image matching of pre- and post-event imagery constrains a set of parameters, i.e. ZPRE, ZPOST, ΔX, ΔY;
• a specific regularisation term incorporates the a priori on the displacement field of different characteristics:
(1) the DSM ZPRE, ZPOST;
(2) the DSM evolution (ZPRE – ZPOST) and
(3) the planimetric displacement ΔX, ΔY.
Given the optimisation complexity, we aim to solve the problem with a variational approach ( Chambolle  ; Pock et
al. ), where the data term becomes function of a similarity measure such as the least squares matching (Gruen &
Baltsavias ). The convergence of the variational approaches (i.e. based on the gradient descent) necessitates a
good initial approximation of all unknown parameters. The traditional matching techniques (Hirschmueller ), e.g.
available as open-source code (Pierrot Deseilligny & Paparoditis ), can be employed to provide this initial solution.
The envisioned work tasks are:
• definition of the data terms adapted to matching using single- and multi-epoch images;
• definition of the regularisation terms adapted to diverse displacement fields;
• selection of the optimisation method (e.g. selection from the available variants of the conjugate gradient);
• implementation of the selected optimisation method;
• evaluation of the obtained results;
• iteration of the above.
In the first stage, to progressively increase the complexity, the variational approach would be implemented to solve the
classical single epoch image matching. Having completed the first iteration, one can also think of including into the
calculation some knowledge furnished by a human operator; for instance, an approximative position of the fault known
from a rough visual inspection of the automated result may serve to relax the regularisation in the second iteration (the
objective being to find a better estimate of the displacements on either side of the fault).
The developped methods will be implemented in the free, open-source MicMac which already today gives at disposal
modules to: geo-localise satellite images, generate single-epoch DSMs, and 2D displacement maps from a couple of
orthophotomaps. Consequently, the developped code will also be free and open-source.
It is expected that the candidate has:
• strong interest for applied research and applied mathematics;
• good general scientific knowledge;
• good programming skills, preferably in C/C++;
• ideally, some experience in image processing or photogrammetry/computer vision.
Master 2 or School of engineering