Diffuse optical tomography is used to find the optical parameters of a turbid medium with infrared red light. The problem is mathematically formulated as a nonlinear problem to find the solution for the diffusion operator mapping the optical coefficients to the photon density distribution on the boundary of the region of interest, which is also represented by the Born expansion with respect to the unperturbed photon densities and perturbed optical coefficients. We suggest a new method of finding the solution by using the second-order Born approximation of the operator. The error analysis for the suggested method based on the second-order Born approximation is presented and compared with the conventional linearized method based on the first-order Born approximation. The suggested method has better convergence order than the linearized method, and this is verified in the numerical implementation.

Diffuse optical tomography involves the reconstruction of the spatially varying optical properties of a turbid medium. It is usually formulated as inverse problem with respect to the forward problem describing photon propagation in the tissue for given optical coefficients [

The forward model is described by the photon diffusion equation with the Robin boundary condition. In the frequency domain, it is given by

Assuming we know some a priori information

In this paper, a new method, which is more accurate than the linearized method, will be suggested (

The detailed statement with proof will be proved in Section

Instead of solving linearized solution

Let

Let

By the induction argument on

By [

Using Proposition

Assume that there exists

By (

Assume that we can measure the photon density distribution

The detailed computation of the integral operators

Algorithm

the measurement index:

the optical coefficient index:

(I) Compute the solution

(II) Find

(III) Find

(IV) Compute

If we use piecewise linear or bilinear finite element method, the finite element solution is represented by

We should discretize

Firstly, let us discretize

Let the vector

For a function

The discretized solution

(I) Compute the solution

(II) Find

(III) Find

(IV) Compute

In this subsection we approximate

First, we approximate the Robin function

Second, when

Third, when

Let the measurable set

If a ball with a radius

Therefore, when

In the numerical implementation, the following parameters are used:

Since the diffusion coefficient

In the above setting, we reconstruct the obstacle ^{−1}) compared to the background absorption coefficient (0.05 cm^{−1}). Four cases of the obstacle ^{−1}) is implemented using two algorithms. One is the suggested Algorithm

In all four cases, 10% noise is added. Truncated singular value decomposition(SVD) is used.

As is shown in the figure, the discrimination between background and the obstacle is clearer in the second-order approximation than the first-order approximation. The reconstructed image resolution depends on the distance from the boundary of the tissue, which is verified by comparing Figures

We derived a new numerical method based on the second-order Born approximation. The method is a method of order 3, which is more accurate than the well-known linearized method based on the first-order Born approximation. The error analysis for the method is proved, and the computation of the second-order term is explained using some approximation and integral inequalities. The comparison between the suggested and the linearized method is implemented for four different kinds of absorption coefficients. In the implementation, the suggested method shows more discrimination between the optical obstacle and the background than the linearized method. If more accurate numerical quadrature with more efficient approximation of the Robin function is used, the efficiency of the present method will be elaborated. The simultaneous reconstruction of the absorption and the reduced scattering coefficients based on the proper approximation on the second derivatives of the Robin function would be an interesting topic.

This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2010-0004047).