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Kiwoon Kwon, "The Second-Order Born Approximation in Diffuse Optical Tomography", Journal of Applied Mathematics, vol. 2012, Article ID 637209, 15 pages, 2012. https://doi.org/10.1155/2012/637209
The Second-Order Born Approximation in Diffuse Optical Tomography
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 where is a Lipschitz domain in , , is its boundary, is the unit outward normal vector on the boundary, is the photon density, is a source term, is a refraction parameter, and , , and are the absorption, reduced scattering, and diffusion coefficients, respectively. Assume that is a constant and , , are scalar functions satisfying for positive constants and . The unique determination of the optical coefficients is studied in electrical impedance tomography problem [2–5] and some elliptic problem , which is applicable to diffuse optical tomography problem also. Let us denote and to emphasize the dependence of on the optical coefficient .
Assuming we know some a priori information about the structural optical coefficients and the perturbation of the optical coefficients , the diffuse optical tomography problem is to find the perturbation of the optical coefficients from the difference between the perturbed and unperturbed photon density distribution on the boundary . The relation between and is given by the following Born expansion [7, 8]: where and is the Robin function for a source at , which is the solution of (1.1) for the optical coefficient when is the Dirac delta function. By definition of (1.4), the operator and are different in the following sense: Let the perturbation of the coefficients be when we neglect second-order terms and higher in the Born expansion (1.3). We can then formulate the linearized diffuse optical tomography problem to find from the following equation, which is the first-order Born approximation: This linearized diffuse optical tomography problem is simple to implement and widely used [9, 10].
In this paper, a new method, which is more accurate than the linearized method, will be suggested (1.6), which is based on the second-order Born approximation. And the method is faster than the full nonlinear method . Let the solution of the proposed method in this paper be , and let be sufficiently small. Then, the error for the linearized solution and the proposed solution is given by where and and are constants which are independent of . Hence, the error of the proposed solution in (1.7b) is of the order , which is higher than the order of the error of the linearized solution , .
The detailed statement with proof will be proved in Section 2. Numerical algorithm involving the detailed computation of the second-order term is given in Section 3. Numerical implementation of the proposed method and the linearized method is given in Section 4, and the conclusion of the paper is given in Section 5.
2. Error Analysis
Instead of solving linearized solution in (1.6), we suggest the second order solution satisfying or equivalently, In this section, we analyze the error for the linearized solution and the suggested solution .
Let ; then, the operator and , , are considered to be the operators from and , respectively, by the definition given in (1.4). For the detailed explanation about the definitions of higher-order Fréchet derivative in diffuse optical tomography and its relation to the Born expansion, see .
Proposition 2.1. Let be the solution of (1.1) for the given optical coefficients , , source , and modulating frequency . Then one gets the following relation between the operators between and , : for .
Proof. By the induction argument on and using (1.5), we get the following inequality: which is (2.3) for . Suppose that (2.3) holds for . Then we obtain Using (2.5) and the definition of the operator norm , we obtain (2.3) for . Therefore, by the induction argument, we have proved (2.3) for .
By , is bounded, and thus , , are also bounded by Proposition 2.1. Let us assume that there exists a bounded operator from to such that . is usually called the left inverse of . Let us denote for brevity.
Using Proposition 2.1 and the assumption on the left inverse, the main theorem of this paper is given as follows.
Theorem 2.2. Assume that there exists such that and is bounded, and let Then, where
Proof. By (1.3) and (1.6), we obtain Therefore we arrive at (2.7b) by the following inequality: From (2.7a) and (2.7b), we obtain the following upper bound of : Using (2.2) and (2.9), we obtain The second-order term on the righthand side of (2.12) is analyzed as follows: From (2.12), we obtain By using (2.3), (2.10), (2.11), (2.13), and the definition of , (2.7c) is achieved from (2.14) as follows:
3. Numerical Algorithm
Assume that we can measure the photon density distribution and on the entire boundary . That is to say, we have infinite detectors and one source. Then, the numerical algorithm is given as follows.
The detailed computation of the integral operators and , which is introduced in (1.5), is as follows:
Algorithm 1 is based on one source and infinite detectors. However, for practical reasons, we need to discretize Algorithm 1 to obtain the numerical algorithm for finite sources and finite detectors for finite frequencies. The following notations will be used for the discretization: (i) detector positions: for , (ii) source functions: (Dirac delta function) for , (iii) frequencies: for ,(iv) elements: for ,(v) nodes: for ,(vi) the measurement index: , (vii) the optical coefficient index: , where is 1 (the absorption coefficient) or 2 (the diffusion coefficient).
If we use piecewise linear or bilinear finite element method, the finite element solution is represented by where is the piecewise linear or the bilinear function which is 1 on the th node and 0 on all the other nodes. Assume and are piecewise constant function, which is constant for each finite elements. Therefore, in diffuse optical tomography inverse problem, we have measurement information contents and variables to find.
We should discretize and to obtain a discretized version of Algorithm 1. Let the Jacobian and Hessian matrices, which is the discretization of integral operators and , be and . The relation between higher order derivatives for the diffusion operator and higher order terms of Born expansions including and is analyzed in .
Firstly, let us discretize , , and the Robin function as follows: Since we chose the source function as the Dirac delta function at the th source point, . However, we will discriminate these two functions in this paper, since they are different for general source function which is different from the Dirac delta function. We will use instead of for notational convenience.
Let the vector which corresponds to the discretization of in (3.3a) be defined as By the adjoint method , , where denotes complex conjugate. Likewise for (3.3a), let , , , and be the discretization of , , , and , respectively.
For a function and a measurable set , let us denote if the intersection of the support of and is not empty. The discretization of the linearized solution is attained by solving the following equation: where
The discretized solution is obtained by solving the following equation: where where , , , and are the discretization of corresponding terms in (3.1b) such that Even though the Hessian is not discretized, we obtain the following discretized numerical algorithm (Algorithm 2), expecting the Hessian is simply discretized and approximated in the next subsection:
3.2. Approximation of Hessian
In this subsection we approximate , by assuming and are constant in . The approximation is progressed in three ways.
First, we approximate the Robin function when by its leading term defined by where is the hypersurface area of the unit sphere in , and . Some important relations between and are found in .
Second, when , the Robin function and are approximated by constant values and in , respectively, where of the center of the element . That is to say, when , (3.9) is approximated as follows:
Third, when , we use the following lemma.
Lemma 3.1. Let the measurable set be contained in , , and ; then, the following inequality holds for : where is the volume of .
Proof. If a ball with a radius has the same volume as , we have for the space dimensions . Let the ball of radius with center be . Let , , and . Noting that , we obtain for all . Therefore, Equation (3.12b) is derived in the same manner.
4. Numerical Implementation
In the numerical implementation, the following parameters are used: (i),(ii),(iii),(iv),(v),(vi),(vii),(viii),(ix),(x),(xi) MHz,(xii),(xiii).
Since the diffusion coefficient is constant, the right-hand side is a column vector, Jacobian is a matrix, the Hessian is a third-order tensor, and is column vector in (3.5) and (3.7). is approximated by (3.11) and (3.16).
In the above setting, we reconstruct the obstacle which has different absorption coefficient (0.2 cm−1) compared to the background absorption coefficient (0.05 cm−1). Four cases of the obstacle are considered in Figures 1, 2, 3, and 4. The reconstruction of the absorption coefficient (cm−1) is implemented using two algorithms. One is the suggested Algorithm 2 based on the second-order Born approximation. The other is linearized method based on the first-order Born approximation, which is equivalent to the step I and II in Algorithm 2. We denoted these two methods in the figures: the 2nd order approximation and the 1st-order approximation, respectively. On the upper-left part of the figures, original and source/detector locations are plotted. The initial guess ( or ) for the absorption coefficient is plotted on the upper-right part of the figures. In the lowerleft and lowerright part of each figure, reconstructed absorption coefficients by the first approximation ( or ) and the second approximation ( or ) are plotted, respectively.
In all four cases, 10% noise is added. Truncated singular value decomposition(SVD) is used. is the number of largest singular values used in the truncated SVD method. We used the Tikhonov regularization parameter Jalpha as the value of the th largest singular values.
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 1 and 2 with Figures 3 and 4. And the resolution also depends on the size of obstacle, which is verified by comparing Figures 1 and 3 with Figures 2 and 4. Due to the diffusion property of near infrared light, the reconstructed image is much blurred especially in Figure 3. The sensitivity to the noise made some kind of irregular checkerboard pattern near the boundary (Figures 1, 3, and 4).
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).
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Copyright © 2012 Kiwoon Kwon. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.