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Diffuse optical tomography problems rely on the solution of an optimization problem for which the dimension of the parameter space is usually large. Thus, gradient-type optimizers are likely to be used, such as the Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm, along with the adjoint-state method to compute the cost function gradient. Usually, the

Diffuse optical tomography (DOT) consists in reconstructing the spatial radiative property distributions of an optically thick participating (absorbing and scattering) medium,

Biomedical applications of the DOT include imaging of finger joints [

The evaluation of space-distributed physical properties from the knowledge of measurements is an inverse problem as opposed to forward problems where the state is computed from the knowledge of properties, boundary conditions, and so forth. Such inverse problems are usually difficult to solve due to their ill-posed nature [

The solution of the inverse problem is obtained through the minimization of a cost function that measures the misfit between the predictions (solution of the forward problem) and some experimental measurements. Though zero-order optimization methods may be used for relatively similar problems [

In order to cope with such instabilities, the ordinary Tikhonov approach consists in assuming that the properties are subject to a minimal norm in a given vectorial space which leads to enhancing the

Appropriate parameterization for the space-dependent properties may also affect the solution regularity. As a matter of fact, the amount of measurements data is likely to be insufficient to retrieve the large amount of parameters when considering, at the end, the discrete optimization problem on a fine grid. Thus, regularization by reduction of the dimension of the parameter space could be employed, which consists in searching the physical properties projected on a coarse finite element grid, as suggested for instance in [

This paper focuses on a different strategy to improve the reconstruction of space-distributed parameters when the data is corrupted by noise. The proposed strategy filters the effect of the noise, not on the data as it is done for instance when using the mollification method [

The underlying application of this study is optical tomography, that is, the reconstruction of radiative properties from light intensity measurements on some parts of the boundary. The paper is therefore organized as follows. Section

Let us define the domain of interest

The state

In order to use efficient optimization algorithms for large-scale inverse problems encountered in distributed parameter estimation, the cost function derivative or rather the cost function gradient has to be computed. To do so, the definition of the directional derivative of

Let a point

The application of the directional derivative (

The inner product

The differentiation of the state problem in the direction

Following Lions [

The adjoint-state problem is then identified through (

Next, if the adjoint problem (

As reported by Protas [

Consider

Consider

Most often, the

This way for writing down the cost function gradient is the ordinary one as first suggested by Lions [

In the present study, the choice of another inner product comes from the fact that we are faced with dealing with noisy measurements

In order to deal with the smoothing of the measurements noise that propagates in the adjoint system and then to the cost function gradient, the weighted Sobolev space is introduced.

Consider

This strategy has been applied in bidimensional optical tomography applications based on the radiative transfer equation [

The strategy that is developed here goes much further: it consists in filtering the cost function gradient where, and only where it is needed, the high frequency fluctuations exist, at the vicinity of the sensors.

In applications considered here, that is, in optical tomography, sensors are located on the boundaries, so that the effect of the noise appears on the boundaries before being diffused through the adjoint-state equation. Thus, the idea is to choose a filtering function whose value is high on the boundary and that continuously decreases within the medium. To do so, one uses the distance function definition.

Let

As an example, the filtering function can be written as

This makes the construction of the space-dependent weighted Sobolev space possible.

Consider

With the choice of the filtering function (

Using the

Taking into account that

Due to the inclusion

The inverse problem consists mathematically in minimizing a cost function which quantitatively measures the discrepancy between some optical measurements and related predictions over a set of sources, sensors, and angles of directional integration. In optical tomography, taking into account the reflection at the interface and assuming that the medium is convex, the measurable quantity is the exitance, or portion of it in a given solid angle

Schematic description of the experiment. Only one source is on at once, while others are off. For each source configuration number

The computation of the cost function gradient follows the generic methodology detailed in the previous section. The cost function derivative towards the direction

In order to derive the cost gradient for this specific application,

Consider

The

Consider

Consider

The generic equation for the cost gradient (

In these two equations, the direction

On one hand, (

Besides, the differentiation of the forward model enables rewriting previous equations too, for

Properties

The adjoint problem is eventually obtained identifying (

The optimization relies on the BFGS algorithm described by Liu and Nocedal [

The two-dimensional domain is a square of 2 cm length. Four collimated sources are located at the center of each side, while sensors are placed around sources towards twenty directions. The target properties to be reconstructed, ^{−1} and ^{−1} and two square inclusions, the former for which ^{−1} and ^{−1} and the latter for which ^{−1} and ^{−1}. Both the forward and the adjoint models are solved using

Numerical tests were performed for different levels of noise. First, a small noise of 20 dB is added to the synthetic data (this corresponds to 1% of noise). Then, a moderate noise of 15 dB is added to the data (approximately 3% of noise). Finally, the highest level of noise of 10 dB is considered (10% of noise).

Figure

2D RTE-based optical tomography. Evolution of the cost function with respect to iterations, for

Note that the BFGS optimizer is not based on matrix inversion, as it is the case for Gauss-Newton method and its derivative. It has been shown in other works that such a choice implicitly regularizes the inverse problem. Another regularization tool comes from the parameter mesh that is chosen sufficiently coarse to avoid the presence of too many local minima [

Note also that the choice of the stopping criterion plays a crucial role in the effectiveness of the algorithm: a too weak stopping criterion may lead to useless solution, while a too severe stopping criterion may lead to a divergent solution. In fact, the stopping criterion based on cost function stabilization is another dummy regularization that is to be used in practical applications. The determination of an appropriate value for this criterion is a challenging task in theory because it highly depends on the application. For the needs of this paper, the value equal to

Figures

2D RTE-based optical tomography. 20 dB noise. Evolution of errors

2D RTE-based optical tomography. 15 dB noise. Evolution of errors

2D RTE-based optical tomography. 10 dB noise. Evolution of errors

Figures

2D RTE-based optical tomography. 20 dB noise. Reconstructions obtained for

2D RTE-based optical tomography. 10 dB noise. Reconstructions obtained for

The radiative transfer equation (RTE) provides an equation of light propagation valid in most of participating (absorbing and scattering) media as long as the independent scattering regime is fulfilled [^{−3}]; ^{−2}]; ^{−1}]; ^{−1}];

Similarly to Section ^{−2} and ^{−2} for all

The cost function gradients with respect to

Therefore, if the inner product

The same algorithm as employed in Section ^{2}). The target properties to be reconstructed, ^{2} and ^{2}.

The stopping criteria are based on the cost function stabilization, with a critical value chosen equal to

Physical properties involved in the forward model (^{−1}, and ^{2} each (containing 16 discretization points), are used for this particular test. A reduction of the parameter space dimension is employed to improve the quality of the reconstructions obtained with the BFGS algorithm [

3D DA-based optical tomography. Synthetic data mesh: 132 651 nodes (a); mesh of the state and adjoint variable: 68 921 nodes (b); mesh associated with radiative properties

Figure

3D DA-based optical tomography. 10 dB noise. Evolution of errors

3D DA-based optical tomography. 10 dB noise. Reconstructions obtained in the plane passing through the center of the cube of which the normal vector

In this paper, inverse models based on the BFGS algorithm have been developed to solve optical tomography problems based on two different forward models: the bidimensional steady-state radiative transfer equation and the three-dimensional frequency diffuse approximation. A Sobolev filter function was defined and space-dependent Sobolev inner products were used when extracting the cost function gradients instead of the

Generally speaking, due to simplicity of implementation and effectiveness in filtering noise measurements locally that provides better reconstructions, space-dependent Sobolev gradients appear to be particularly attractive in the context of inverse problems. In the near future, the anisotropy factor will be integrated in the inverse problem based on the radiative transfer equation and the use of a logarithmic cost function will be considered as our more recent results suggest. Later, inversion numerical algorithms of the three-dimensional radiative transfer equation and diffuse approximation model will be integrated in a global numerical code. The latter will be based on the BFGS algorithm with the use of adimensionalisation of the radiative properties, reduction of the parameter space dimension, space-dependent Sobolev gradients, and a wavelet multiscale approach, as developed in another work.

Note that the proposed methods have been implemented within the FreeFem++ environment [

The authors declare that there is no conflict of interests regarding the publication of this paper and regarding the funding that they have received.

The authors thank the