O. Balima

Optical tomography reconstruction algorithm with the finite element method: An optimal approach with regularization tools

Optical tomography is mathematically treated as a non-linear inverse problem where the optical properties of the probed medium are recovered through the minimization of the errors between the experimental measurements and their predictions with a numerical model at the locations of the detectors. According to the ill-posed behavior of the inverse problem, some regularization tools must be performed and the Tikhonov penalization type is the most commonly used in optical tomography applications. This paper introduces an optimized approach for optical tomography reconstruction with the finite element method. An integral form of the cost function is used to take into account the surfaces of the detectors and make the reconstruction compatible with all finite element formulations, continuous and discontinuous. Through a gradient-based algorithm where the adjoint method is used to compute the gradient of the cost function, an alternative inner product is employed for preconditioning the reconstruction algorithm. Moreover, appropriate re-parameterization of the optical properties is performed. These regularization strategies are compared with the classical Tikhonov penalization one. It is shown that both the re-parameterization and the use of the Sobolev cost function gradient are efficient for solving such an ill-posed inverse problem.

NEW DEVELOPMENTS IN FREQUENCY DOMAIN OPTICAL TOMOGRAPHY. PART II. APPLICATION WITH A L-BFGS ASSOCIATED TO AN INEXACT LINE SEARCH

This second part deals with the application of the presented formulations for the reconstruction of optical properties in frequency domain optical tomography with the finite element method. We use the Limited memory BFGS algorithm with an inexact line search in order to avoid numerous evaluations of the objective function. Normalization of the objective function with measurements and independent scaling of its gradient are used to improve the quality of the reconstruction. The results show a better recovering of both the absorption and scattering coefficients.

Reduced modelling through identification on 2D incompressible laminar flows

This article presents a model reduction of laminar incompressible fluid flows in two-dimensional geometries cases. The modal identification method [Petit, Hachette, and Veyret, A modal identification method to reduce a high order model: application to heat conduction modelling, Int. J. Model. Simulation, 17, 242–250; Girault and Petit, Identification methods in non linear heat conduction, Part I: model reduction; Int. J. Heat Mass Transfer, 2005, 48, 105–118; Balima, Favennec, Girault, and Petit, The modal identification method in non linear heat transfer problems, Proceedings 13 International Heat Transfer Conference, Sydney, Australia, 2006] is used to derive a structure for the reduced models (RMs). These RMs are identified through a minimization algorithm where the cost function gradient is computed with the adjoint method [Balima, Favennec, Girault, and Petit, The modal identification method in non linear heat transfer problems, Proceedings 13 International Heat Transfer Conference, Sydney, Australia, 2006; Favennec, Girault, and Petit, The adjoint method coupled with the modal identification method for non linear model reduction, Inverse Probl. Sci. Eng., 2006, 14, 153–170; Balima, Réduction de modéle non linéaire par identification modale: application en thermique et comparaison avec la méthode POD-Galerkin, PhD thesis, Université de Poitiers, France, 2006]. Two laminar steady fluid flows in a pipe and in a square cavity (lid-driven cavity) are used to illustrate the study. The results show that the identified RMs are able to reproduce with accuracy the behaviour of the considered model.

A Modified Cost Function Based Optical Tomography Reconstruction Algorithm Optimized for Finite-Elements Methods

In optical tomography, the optical properties of the medium under investigation are obtained through the minimization of an objective function. Generally, this function is expressed as a discrete sum of the square of the errors between measurements and predictions at the detectors. This paper introduces a continuous form of the objective function by taking the integral of the errors. The novelty is that the surfaces of the detectors are taken into account in the reconstruction and a compatibility is obtained for all finite element formulations (continuous and discontinuous). Numerical tests are used to compare the reconstructions with both objective functions. It is seen that the integral approach leads to low values of objective functions those reconstructions may be affected by rounding errors. Scaling of the objective function and its gradient shows that both methods give comparable accuracy with an advantage to the continuous approach where the integral acts as a filter of noise.© 2011 ASME

THEORETICAL DEVELOPMENT FOR REFRACTIVE INDEX RECONSTRUCTION FROM A RADIATIVE TRANSFER EQUATION-BASED ALGORITHM

This study is devoted to the mathematics behind a reconstruction methodology based on the radiative transfer equation of a refractive index arbitrary distribution. The targeted algorith m should be of the least-squares and gradient type, r elying on the adjoint to the radiative transfer equ ation for varying refractive index, which is a novelty. P reliminary tests are demonstrated on generic phantoms.

NEW DEVELOPMENTS IN FREQUENCY DOMAIN OPTICAL TOMOGRAPHY. PART I. FORWARD MODEL AND GRADIENT COMPUTATION

This paper deals with a gradient-based frequency domain optical tomography method where the collimated source direction is taken into account in the computation of both the forward and the adjoint models. The forward model is based on the least square finite element method associated to the discrete ordinates method where no empirical stabilization is needed. In this first part of the study, the forward model is highlighted with an easy handling of complex boundary condition through a penalization method. Gradient computation from an adjoint method is developed rigorously in a continuous manner through a Lagrangian formalism for the deduction of the adjoint equation and the gradient of the objective function. The proposed formulation can be easily generalized to stationary and time domain optical tomography by keeping the same expressions.

An Optical Tomography Reconstruction Algorithm With the Discontinuous Galerkin Method

In optical tomography, the optical properties are recovered through an iterative scheme, which consists in minimizing the errors between the measurements and the predications of a forward model. Traditionally, finite volumes or continuous finite elements formulations of light transport are used as a forward model for the predictions. we have introduced an integral form of the objective function that takes into account the surface of the detectors, and thus making compatible the inverse approach with all finite elements formulations (continuous and discontinuous). This present paper illustrates this novel approach by developing a Discontinuous Galerkin formulation as a forward model. Numerical tests are performed to gauge the accuracy of the method with a gradient-based algorithm. The results show that the reconstruction is accurate and can be affected by noise on the measurements as expected. A filtering of the gradient improves the quality and the accuracy of the reconstruction.Copyright

Optical Tomography With a Least Square Finite Element Formulation of the Collimated Irradiation in Frequency Domain

This paper presents a numerical study of optical tomography in frequency domain for the reconstruction of optical properties of scattering and absorbing media with collimated irradiation light sources. The forward model is a least square finite element formulation of the collimated irradiation problem where the intensity is separated into its collimated and scattered parts. This model does not use any empirical stabilization and moreover the collimated source direction is taken into account. The inversion uses a gradient type minimization method where the gradient is computed through an adjoint formulation. Scaling is used to avoid numerical round errors, as the output readings at detectors are very low. Numerical reconstructions of optical properties of absorbing and scattering media with simulated data (noised and noise-free) are achieved in a complex geometry with satisfactory results. The results show that complex geometries are well handled with the proposed method.Copyright

A Reconstruction Scheme Based on the Radiative Transfer Equation for Graded Index Semi-Transparent Media

Graded refractive index media appear in numerous industrial applications such as non-isothermal flows, optics material processing, biological imaging. Refractive index gradient has been an early help for combusting flow visualisation. The numerical treatment of radiative transport is difficult in such media due to the curvature of rays, especially when the media are not optically thick. Computer-aided remote probing (inversion) is done today with the help of the diffusion approximation adapted to varying refractive index media but is unsuitable for thin media. Therefore, it is important to develop an approach allowing the use of the radiative transport equation which is the most complete formalism for radiative transfer to date and to couple it to reconstruction schemes. The aim of this study is to demonstrate the reconstruction of an arbitrary refractive index distribution from a least-squares gradient-based iterative inversion algorithm taking advantage of the full transient Radiative Transfer Equation (tRTE). The finite-difference discrete-ordinates method for the tRTE and its adjoint has been implemented, accounting for spatial changes in the distribution of the refractive index in a semi-transparent medium. A least-squares gradient-based iterative algorithm has been designed and elementary tests have been carried to demonstrate reconstruction possibilities.Copyright