Développement de techniques de diagnostic non intrusif par tomographie optique

Abstract

Whether in the fields of industrial process or medical imaging, there has been an increasing development of optical diagnostic techniques over the past two decades. Enthusiasm for these methods mainly relies on the fact that they are non-intrusive, they use radiation sources safe to human and environment, and they are relatively inexpensive and easy to implement compared to other imaging techniques. One of these techniques is the Diffuse Optical Tomography (DOT). This three-dimensional imaging method consists in characterizing radiative properties of a participating medium from optical measurements in near-infrared provided by a set of sources and sensors located on the frontier of the probed medium. It is especially based on a forward model of light propagation in the participating medium, providing the predictions, and a minimization algorithm of a cost function which integrates the predictions and measurements, allowing the reconstruction of parameters of interest. In this work, the forward model is the diffuse approximation of the radiative transfer equation in the frequency domain while parameters of interest are the spatial distributions of absorption and reduced scattering coefficients. The objective of the work presented in this thesis is the development of a robust inverse method for the solution of the DOT problem in the frequency domain. To meet this objective, this work is structured in three main parts. First, a comparison between the damped Gauss-Newton and Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithms is proposed in the two-dimensional case. Two regularization methods are combined for each of the two algorithms, namely the mesh-based reduction of the control space dimension and the Tikhonov penalization for the damped Gauss-Newton algorithm, and the regularizations respectively based on the mesh and the use of Sobolev gradients, uniform or spatially dependent, when extracting the cost function gradient, for the BFGS method. Numerical results indicate that the BFGS algorithm outperforms the damped Gauss-Newton algorithm in term of the quality of the obtained reconstructions, CPU time and the ease of selecting the regularization parameter. Second, a study on the quasi-independence of the optimal Tikhonov penalization parameter with respect to the dimension of the control space in inverse problems of space-dependent function estimation is performed. This study follows an observation made in the first part of this work where the Tikhonov parameter, determined by the “L-curve” method, is found to be independent of the dimension of the control space in the under-determined case. This assumption is demonstrated theoretically and verified numerically on a linear inverse heat conduction problem then on the non-linear inverse problem of the DOT. The numerical verification is based on the determination of an optimal Tikhonov parameter, defined as the one which minimizes the misfit between the target and the reconstruction. The theoretical demonstration relies on the discrepancy principle in the linear case, while it mainly relies on the assumption that radiative functions to be reconstructed are normally distributed random variables in the non-linear case. In conclusion, this thesis demonstrates that the Tikhonov parameter can be determined using a control space parameterization associated with a coarse mesh in order to lower computational time. Third, a wavelet multi-scale method associated with the BFGS algorithm is developed. This method, which relies on a reformulation of the original inverse problem into a sequence of sub-inverse problems of different scales using wavelet transform, from the largest scale to the smallest one, enables to cope with the local convergence property of the optimizer and the presence of numerous local minima in the cost function. Numerical results show that the proposed method is more stable with respect to the initial guess of the radiative properties and provides more accurate final reconstructions compared to the ordinary BFGS algorithm while requiring similar computational times. The thesis is presented in the form of four articles. The first article is accepted in the International Journal of Thermal Sciences, the second is accepted in the journal Inverse Problems in Science and Engineering, the third is accepted in the Journal of Computational and Applied Mathematics and the fourth has been submitted in the Journal of Quantitative Spectroscopy & Radiative Transfer. Ten other articles have been published in conferences with reviewing committee. Pdf versions of these papers are available on the web site of the industrial research chair in technologies of energy and energy efficiency (t3e) (www.t3e.info).

Date	7 Aug 2015
Original language	French
Awarding Institution	École de technologie supérieure
Supervisor	Daniel Rousse (Supervisor), Benoit Rousseau (Co-supervisor) & Yann Favennec (Co-supervisor)