An efficient and robust reconstruction method for ...

An efficient and robust reconstruction method for optical tomography with the time-domain radiative transfer equation Yaobin Qiao Hong Qi* Qin Chen Liming Ruan* Heping Tan School of Energy Science and Engineering, Harbin Institute of Technology, Harbin 150001, China

*Corresponding author: Hong Qi School of Energy Science and Engineering, Harbin Institute of Technology 92, West Dazhi Street, Harbin, P. R. China, 150001 Tel: (86)-0451-86412638 Email: [email protected] (H. Qi)

Liming Ruan School of Energy Science and Engineering, Harbin Institute of Technology 92, West Dazhi Street, Harbin, P.R. China, 150001 Tel: (86)-0451-86412638 Email: [email protected] (L. M. Ruan)

Abstract An efficient and robust method based on the complex-variable-differentiation method (CVDM) is proposed to reconstruct the distribution of optical parameters in two-dimensional participating media. An upwind-difference discrete-ordinate formulation of the time-domain radiative transfer equation is well established and used as forward model. The regularization term using generalized Gaussian Markov random field model is added in the objective function to overcome the ill-posed nature of the radiative inverse problem. The multi-start conjugate gradient method was utilized to accelerate the convergence speed of the inverse procedure. To obtain an accurate result and avoid the cumbersome formula of adjoint differentiation model, the

CVDM was employed to calculate the gradient of objective function with respect to the optical parameters. All the simulation results show that the CVDM is efficient and robust for the reconstruction of optical parameters.

Keywords: Complex-variable-differentiation method, Optical tomography, Time-domain equation of radiative transfer, Multi-start iterative technique

1 Introduction With the advent of ultra-short pulsed lasers (with pulse width of 10-9~10-15 s), the propagation of pulsed laser in participating media is adding its appeal for many researchers and has been the subject of great interest in the radiation transfer community in the past decades [1-5]. Due to the transient effects caused by the ultra-short pulse laser can supply considerable time-resolved signals related to the internal structure of the participating media, the application of ultra-short pulse laser on optical tomography (OT) has flourished and attracted significant attentions in recent years. The OT, which reconstructs the spatial distribution of the optical properties in nonhomogeneous turbid media by analyzing light intensities measured on the boundary, is an effective probe technique and has wide potential applications in many scientific and engineering fields, such as nondestructive testing, infrared remote sensing, information processing, combustion diagnosing and biomedical imaging, to name a few [6-10]. Compared with other imaging technologies, OT is characterized as being noninvasive, portable, and producing real-time images of clinically relevant parameters. Measuring the absorption and scattering coefficients of vivo biological tissue with near-infrared light is of great interest in early tumor diagnosis, such as breast imaging and brain imaging [11]. Theoretically speaking, the OT can be classified into three different techniques according to the incident laser sources: the continuous laser (continuous technique) [12-14], the pulsed laser (time-domain technique) [15-21] and the modulated laser (frequency-domain technique) [22-29]. Compared with the continuous technique, the time-domain and frequency-domain techniques can provide more measurement information and high level of sensitivity, which are widely used in recent researches. Furthermore, the time-domain technique has more advantages because the radiation in arbitrary time can be evaluated easily and the Fourier transform can be avoided [30]. Many research groups have done considerable work in this field over the past decades

[15-21, 30-35]. Among these research efforts, the diffusion approximation (DA), which is basically a special case of the first order spherical harmonics approximation, is always used to solve the forward problem of radiative transfer equation (RTE) because it captures the core characteristic of light migration in turbid media and is easy to be implemented. However, the DA model is very restrictive as some reasons [17, 30]. It fails to produce accurate estimations on the light propagation close to the sources and boundaries, and in cases in which the turbid medium contains low-scattering or non-scattering void-like regions, where the absorption and scattering coefficients are very low, such as the cerebrospinal fluid which surrounds the brain and fills the brain ventricles. In addition, the limitation of the DA model for the hyperbolic propagation of the collimated laser pulse at a finite velocity, is obvious. Therefore, more and more interests now turn to the light transport models based on solving the complete RTE gradually. The significance of solving complete RTE for the optical tomography has received increasing recognition and considerable attention in recent years. For example, Klose et. al. [13] employed an upwind-difference discrete-ordinates formulation of RTE as forward model for OT, and reconstructed the optical parameters of void-like media from the experimental data for the fist time. Boulanger et. al. adopted the time-domain RTE as forward model to reconstruct the absorption and scattering coefficients [18] and the refractive index [34] respectively, the results showed that the inside reconstruction was achieved better with more data were weighted in time. Guan et. al. [35] investigated the reconstruction of optical parameters based on time-domain RTE considering uniform refractive index and gradient refractive index respectively, and found that the reconstructed image in case of gradient refractive index agreed better with the original image than the uniform refractive index media. To date, although several inverse algorithms have been established to study the time-domain OT, accurate and efficient determination of the OT is still regarded as an unsolved problem and needs further research. It remains a challenging task to develop computationally efficient, accurate time-domain RTE-based reconstruction method. The most robust inverse methods for reconstruction have proved to be gradient-based [21]. For all the gradient-based inverse techniques, the core problem and difficulty of the reconstruction procedure is concerned with obtaining the gradient of the objective function (OF) with respect to optical parameters accurately and efficiently. Generally speaking, the most commonly used method for calculating the gradient in time-domain RTE-based OT problem is the adjoint differentiation (AD) in recent researches. For

instance, Klose et. al. [31] given a theoretical analysis and deducing of an AD scheme based on time-domain RTE to calculate the gradient for OT. Boulanger et. al. [18] employed the adjoint model and AD to calculate the gradient of OF and achieved the reconstruction of absorption and scattering coefficients in 1D, 2D, 3D media. Our group [32] proposed an AD scheme considering the derivatives of collimated intensity and diffusion intensity in the neighboring points to obtain accurate reconstructed results. Although successful applications of the AD on the OT have been obtained, some inherent drawbacks exist. The AD method requires a very complex and cumbersome formula which need deduced from the forward model, and has large consuming of computational time and compute memory. On the contrary, the finite difference (FD) approximations is a straightforward approach which can calculate the gradient by perturbing each component of the variable. However, it has an poor accuracy and suffer great influence of the disturbance size. To overcome this disadvantage, the complex-variable-differentiation method (CVDM) which aims to calculate the derivatives of real functions has received increasing attention and applied in the approach of inverse heat conduction problems [36, 37] more recently. The CVDM is a very promising approach since it is easy to use as the FD approximation and has no cancellation errors nor step-size-dependent problem [37]. To the best of our knowledge, few applications of the CVDM on OT problem have been reported. Thus, the motivation of this study is to develop an efficient and robust reconstruction algorithm based on the time-domain RTE, by means of the CVDM which can obtain the gradient of objective function easily and accurately. Firstly, the time-domain RTE with pulse laser source is described and an upwind-difference discrete-ordinate formulation is well established as forward model. Then the objective function with the regularization term of the generalized Gaussian Markov random field (GGMRF) model, is given to overcome the ill-posed problem. The multi-start conjugate gradient (CG) method, which can accelerate the convergence speed and enhance the accuracy of reconstructed results, is employed to solve the inverse problem. To obtain the gradient of objective function with respect to the optical parameters accurately and effectively, the CVDM is proposed and used in present research. Finally, the reconstruction results of optical parameters for the inhomogeneous media with varying absorption and scattering coefficients are presented to verify the proposed image-reconstruction algorithm.

2 Forward model 2.1 The time-domain radiative transfer equation The forward model aims to simulate the propagation of the photon in the media and calculate the measurement signals on the boundaries with the given collimated source and optical properties. The time-domain RTE is used as forward model in this work to obtain time-varying signals and avoid the restriction of the DA equation. The time-domain RTE in arbitrary direction Ω can ben written as [38]

 r  1 I  r, Ω, t   Ω I  r, Ω, t      r  I  r, Ω, t  s c t 4



4

I  r, Ω, t    Ω, Ω  dΩ

(1)

where r denotes the location and t denotes the time. c is the speed of light. I  r, Ω, t  is the radiative intensity.   r  and s  r  are the extinction coefficients and scattering coefficients, respectively.

  Ω, Ω  is the scattering phase function. Generally, the Henyey-Greenstein (H-G) phase function is used to define the scattering in biomedical tissues [39]. 1 g2

  Ω, Ω  

3 2

(2)

1  g  2 g  Ω Ω    2

where Ω and Ω represent the incident and scattering direction. The asymmetry factor g dominates the feature of the scattering function and it gets values between -1 and 1. g  0, g  0 , g  0 represent the isotropic, forward and backward scatterings, respectively. The intensity I with in the medium is composed of the collimated intensity Ic and the diffuse intensity Id as I  r, Ω, t   I c  r, Ω, t   I d  r, Ω, t 

(3)

The variation of the collimated intensity Ic in the medium obeys [38]: 1 I c  r, Ω, t   Ω I c  r, Ω, t      r  I c  r, Ω, t  c t

(4)

Considering the Eqs. (3) and (4), the Eq. (1) can be written as:

 r  1 I d  r, Ω, t   Ω I d  r, Ω, t      r  I d  r, Ω, t  s c t 4



4

I d  r, Ω, t    Ω, Ω  dΩ  Sc  r, Ω, t 

(5)

where Sc  r, Ω, t  is the source term resulting from the collimated intensity Ic. For a square pulse laser, the

collimated intensity in the direction Ω0 can be expressed as:

 I c  r, Ω, t   Iin exp   



s0

0



  r  dr   H  ct  s0   H  ct  ctp  s0    Ω  Ω0  

(6)

where tp is the pulse width, s0 is the distance of the collimated radiation in the direction Ω0 , and H is the Heaviside step function. Then Sc  r, Ω, t  resulting from the I c  r, Ω0 , t  can be written as:

Sc  r, Ω, t  

s  r   Iin exp  4 



s0

0



  r  dr   H  ct  s0   H  ct  ctp  s0    Ω0 , Ω  

(7)

The measurement signals used for the optical tomography is the exitance on the boundary [40]: P  r, t  



Ω n 0

I  r, Ω, t  Ω nd 

(8)

where n represents the normal vector of the boundary. 2.2 Solution method As the time-consuming of inverse problem is strongly depend on the efficiency of the solution method of forward model, the discrete ordinate method (DOM), which is computationally efficient [41], is employed to solve the Eq. (5). For the DOM, the position r and direction Ω are discretized into M  N grids and K solid angles. The radiative intensity I d  r, Ω, t  in the direction Ω k can be replaced by I k  r, t  . Converting the integral term into the sum of K discrete terms, Eq. (5) can be written as: I  r, t  I  r, t   r  1 I k  r, t   k k  k k     r  I k  r, t   s wk  I k   r, t    Ωk  , Ωk   Sc  r, t  c t x y 4 k 1 K



(9)

where  and  are the directional cosines of the light propagation direction Ω k , wk represents the weight associated with discrete direction Ω k . According to the different propagation directions, the following four upwind-difference formulas can be obtained:

1 k

 0, k  0 :

I k  r, t  x



I k ,i , j  I k ,i 1, j x

,

I k  r, t  y



I k ,i, j  I k ,i, j 1 y

(10a)

 2 k  0, k

 0:

 3 k  0, k  0 :  4  k

 0, k  0 :

I k  r, t  x I k  r, t  x I k  r, t  x







I k ,i 1, j  I k ,i, j x I k ,i 1, j  I k ,i, j x I k ,i, j  I k ,i 1, j x

,

,

,

I k  r, t  y I k  r, t  y I k  r, t  y







I k ,i, j  I k ,i, j 1 y I k ,i , j 1  I k ,i , j y I k ,i, j 1  I k ,i, j y

(10b)

(10c)

(10d)

The time discretization is performed using the backward difference: I k  r, t  t



I kn,i , j  I kn,i ,1j t

(n  2,..., nmax )

(11)

where I kn,i , j is the intensity of grid (i, j) for the direction Ω k at nth time step. By substituting the difference scheme of Eqs. (10) and (11) into Eq. (9), the following discretized equations can be obtained: (1) k  0 and k  0 :

I kn,i, j



S ci , j 

s 4

w  k

n 1 k , k  I k ,i , j

 I kn,i ,1j

 ct   k

x  I kn,i 1, j  k y  I kn,i , j 1

1  ct    k x  k y   a  s 

(12a)

(2) k  0 and k  0 :

I kn,i , j



S ci , j 

s 4

w 

n 1 k , k  I k ,i , j

s 4

w 

n 1 k , k  I k ,i , j

s 4

w 

n 1 k , k  I k ,i , j

k

 I kn,i ,1j

 ct   k

x  I kn,i 1, j  k y  I kn,i , j 1

1  ct   k x  k y   a  s 

(12b)

(3) k  0 and k  0 :

I kn,i, j



S ci , j 

k

 I kn,i ,1j

 ct   k

x  I kn,i 1, j  k y  I kn,i , j 1

1  ct   k x  k y   a  s 

(12c)

(4) k  0 and k  0 :

I kn,i, j



S ci , j 

k

 I kn,i ,1j

 ct   k

x  I kn,i 1, j  k y  I kn,i , j 1

1  ct   k x  k y   a  s 

For a point collimated source parallel to the x-axis on the boundary

 0, y0  , Sc can be discretized as:

(12d)

Sc,n k ,i , y0 

s, i, y

0

4

  Iin exp   m1, y0  m, y0 x 2  H  ct  xi   H ct  ctp  xi   k ,k  mi 









(13)

3 Inverse model The essence of OT is to estimate the optical parameters inside the medium by means of measuring the exitance of radiation on the boundary. The estimated optical parameters can be considered as approximation of the real parameters while the predicted exitance is consistent with the measured real exitance. So the OT technique can be converted to search the minimum value of objective function which is composed of the predicted exitance and measured exitance. 3.1 Objective function In the inverse model, the objective function is defined as a normalized-squared error between predicted and measured exitance as follows: 1 F (μ)  2

 s

d

n

2

 M sn,d  Psn,d  μ       (μ) n M s,d

(14)

where μ   a  r  , s  r  denotes the estimated optical parameters, Psn,d (μ) and M sn,d are the predicted and measured values of the dth detector at the nth time step with the sth incident light source, respectively.

 (μ) is the added penalty or regularization term. For the inverse OT problem is ill-posed, which means the unavoidable measurement noise and numerical computing errors often lead to unstable and inaccurate results, the generalized Gaussian Markov random field (GGMRF) model is adopted as regularization term [42] to overcome the ill-posed nature in present research:

 (μ)  

1 p

p

b

s r

μs  μr

p

(15)

{s , r}N

where p is the image sharpness parameter,  the scale parameter; N the set of all neighboring pixel pairs, r the neighboring position of s, and bs  r the weight coefficient. 本文中这些项的选取。 3.2 Multi-start CG Method The reconstruction of optical parameters is equivalent to minimizing the objective function F (μ) using the optimization technique. The Conjugate Gradient (CG) method, which is an effective optimization

technique, is applied to solve the minimization problem in this work. In the iteration procedure k, the updating directional vector of the optical parameters is determined by the gradient of the objective function which can be written as:

μk 1  μk   k dk

 

(16)

 

dk  F μk   k dk 1

with d0  F μ0

(17)

where d k is the descent direction vector and  k is the length of a step. This procedure is repeated until the

   

value of F μk 1  F μk

is smaller than a predefined value  .

Usually, the convergence speed of the CG method will slow down with the increasing of iterations k. Particularly when μ k is near the optimal solution, the convergence speed will become intolerable as the descent d k will be non-conjugated after the nth iteration [43]. The multi-start iteration technique, which stops the iteration at a fixed iteration number K and uses the current reconstructed results as an initial guess to renew the iteration, is employed to improve the convergence efficiency of CG. The flowchart of the multi-start CG method is shown in Fig. 1. The details of this algorithm are well documented in the literature [32], and hence are not repeated here.

Start Set initial guess μ 0 μ0  μ k

Calculate the predicted signals by forward model Measurement signals μ k 1  μ k   k d k

Calculate objective function F (μ k )

k  k 1

kK ?

No



   

F μ k 1  F μ k

Yes End

?

Fig. 1 Flowchart of the multi-start CG method

3.2 Calculation of the gradient using the CVDM Actually, the core problem and challenge of solving the inverse problem is to obtain the gradient of objective function with respect to the retrieval parameters. The finite difference (FD) method is a straightforward approach to compute the gradient by perturbing each component of the vector μ . dF F    h   F    h   d h

(18)

The value of variable h is small compare to the retrieval parameters. The merit of FD method is its ease of use. But its accuracy is poor and suffer greater influence from the value of h. To obtain an accurate gradient, the variable h always needs to choose different appropriate value for different magnitude of the retrieval parameters. Then the adjoint differentiation method is applied in the OT to calculate the gradient by researchers [31, 32]. According to the AD technique, the gradient of the objective function with respect to the optical parameters can be expressed as:

dF  dμ



dF Iin dIin μ

(19)

The sensitivity Iin μ of I in with respect to μ can be obtained by partially differentiating the forward model with respect to μ . dF dIin can be calculated using the derivative dF dIin1 by applying the chain rule:

dF dIin



n 1 dF I j

 dI j

n 1 j

Iin



F

dF

Iin

dIiN



F IiN

N  nmax

(20)

where F Iin can be obtained directly by partially differentiating the objective function F with respect to n 1 n n I in . The derivative Ii I j is calculated by differentiating the forward model with respect to I j .

As the calculation of the matrixes Iin μ and Iin1 I nj is related to the formula of forward model

directly, the complex and cumbersome formula of AD need be rededuced while the forward model changes. Moreover, the matrixes Iin μ and Iin1 I nj , which are very huge even with relatively rough coarse mesh, will lead the AD method to consume large amounts of computing times and employ huge computer memory [32]. To overcome the defects of AD and obtain accurate gradient, the complex-variable-differentiation method [44] is applied to compute the gradient of objective function F with respect to the optical parameters . With replacing the variable x by the complex variable x+ih ( h

x ), the real function f(x) can be expanded in a

Taylor series as f  x  ih   f  x   ihf   x  

 

h2 f   x   o h3 2

(21)

Since the value of h is very small, the first order derivative f   x  can be obtained as f  x 

Im  f  x  ih   h

(22)

where Im  f  x  ih  denotes the imaginary part of f  x  . From Eq. (22), it can be seen that the derivative only need to solve the complex function mathematically. This feature is very attractive for the function, which is so complex that obtaining its derivative is cumbersome or impossible. Unlike the FD method, the CVDM eliminates the truncation errors because it does not need difference two function. Furthermore, the derivatives become independent of the value of h [37]. To verify the



advantages of CVDM, a real function f  x   sin  x  1  e x



is chosen as a numerical example. Apparently,

the exact derivative of f  x  with respect to x equals: f  x 





cos  x  1  e x  sin  x  e x

1  e  x

2

(23)

The derivative f  1.0  0.46027676 can be obtained easily from Eq. (23) while x=1.0. Then the FD and CVDM are tested to calculate the derivative f   x  with different magnitude of h. As shown in Table 1, the value of f  1.0  can be obtained accurately by CVDM for the value of h varying between 10-30 and 10-3.

However, the results of FD are satisfactory only for the value of h varying between 10-9 and 10-6, and become meaningless while the value of h become very little. So the CVDM is demonstrated to calculate the derivative accurately and be independent of the step-size. The CVDM is verified to be quite an efficient and accurate method to calculate the derivative of real function.



x Table1 First-order derivatives of f  x   sin  x  1  e



with different value of h

h

CVDM

FD

1.0×10-3

0.46027681

0.46018080

1.0×10-6

0.46027676

0.46027666

1.0×10-9

0.46027675

0.46027682

1.0×10-12

0.46027676

0.46035398

1.0×10-15

0.46027676

0.55511151

1.0×10-18

0.46027678

0.00000000

1.0×10-21

0.46027674

0.00000000

1.0×10-24

0.46027677

0.00000000

1.0×10-27

0.46027677

0.00000000

1.0×10-30

0.46027676

0.00000000

4 Results and discussions 4.1 Model for the simulation To demonstrate the performance of the proposed CVDM-based algorithm, the non-uniform spatial distribution of optical coefficients in inhomogeneous medium are reconstructed for test. As shown in Fig. 2, the two-dimensional inhomogeneous medium with size of 4  4 cm2 contains two square inclusions. The vertical incident point sources were set in the center of every edge, in turn, of the phantom model, and 16 detectors distributed evenly over the boundary. The measurement exitance in each detector is simulated by the forward model with 24 discrete ordinate quadratures, 21 21 grids and 150 time steps. The width of lase pulse is 0.1 ns and the detecting time equals

0.5 ns. To start the reconstruction, an uniform distribution of absorption and scattering coefficients, whose values were equal to the coefficients of the background, are selected as initial guesses. For biological tissues, the propagation of near infra-red light always present highly forward scattering. So the anisotropic factor of H-G phase function is assumed as g  0.8 .

Legend B Source

Detector A

Fig. 2 Schematic of the model with fibers

4.1 Performance of CVDM To test the performance of CVDM on the reconstruction of optical parameters, the gradient of objective function with respect to scattering coefficient is calculated using CVDM with different values of variable h. Assuming that the absorption coefficient was constant at a  0.05 cm1 . The scattering coefficients of background medium and two square inclusions are shown in the Table 2.

Table 2 scattering coefficient of the test medium

s cm1

Background medium

Inclusion A

Inclusion B

1.0

0.5

1.5

The convergence of objective function using CVDM with different value of h is shown in the Fig. 3. All of the convergence curves are in excellent agreement, which can prove that the calculation of gradient for OT problem using CVDM is independent of the variable h. The reconstructed image of scattering coefficient (see

Fig. 4) can obtain the distribution of two inclusions clearly. This is due primarily to the fact that the gradient of objective function can be obtained accurately using the CVDM. 0.055

h  104

h  108 Objective function F

h  1012

h  1016

0.05

h  1020

0.045

0.04

0

15

30 Iteration step k

45

60

Fig. 3 Convergence processes of the objective function with different variable h

4

4

s cm 1

s cm 1

1.5

3

1.3

1.1

2

y/cm

y/cm

3

1.3 1.2 1.1 2 1

0.9 0.9

1

1

0.7

0

0

1

2

3

4

0.5

0.8 0

0

x/cm (a)

1

2

x/cm

3

4

0.7

(b)

Fig. 4 Distribution of the scattering coefficient in the medium: (a) real distribution (b) reconstructed distribution

4.2 Simultaneous reconstruction of absorption and scattering coefficients Considering spatial inhomogeneous distribution of both the absorption and scattering coefficients, the

algorithm was tested to reconstruct the absorption and scattering coefficients simultaneously. Special emphasis was placed on two test cases: The scattering-dominated medium is setting as Test 1, and the medium in which the absorption coefficient is comparable to the scattering coefficient is setting as Test 2. The absorption and scattering coefficients of the test media are shown in the Table 3.

Table 3 Absorption and scattering coefficients of the test media

Test 1

Test 2

Background medium

Inclusion A

Inclusion B

a cm1

0.05

0.01

0.1

s cm1

1.0

0.5

1.5

a cm1

0.5

0.1

1.0

s cm1

1.0

0.5

1.5

4

a cm 1 0.1

4

s cm 1 1.5

0.09 3

3

0.08

1.3

0.06

2

y/cm

y/cm

0.07 1.1

2

0.05 0.9

0.04 1 0.03

1 0.7

0.02 0

0

1

2

x/cm (a)

3

4

0.01

0

0

1

2

x/cm (b)

3

4

0.5

4

a cm 1 0.09

4

s cm 1 1.3

3

0.08

3

1.2

0.06

2

1.1

y/cm

y/cm

0.07 2

1

0.05 0.9

0.04

1

1 0.8

0.03 0

0

1

2

x/cm (c)

3

4

0.02

0

0

1

2

x/cm (d)

3

4

0.7

Fig. 5 Reconstruction results of Test 1. (a) and (b) show the real distributions of the absorption and

scattering coefficients respectively. (c) and (d) show the distributions of the reconstructed absorption and scattering coefficients a cm 1 1

4

4

s cm 1 1.5

0.9 3

3

0.8

1.3

0.6

2

y/cm

y/cm

0.7

1.1

2

0.5

0.9 0.4 1

1 0.3

0.7

0.2 0

0

1

2

x/cm (a)

3

4

0.1

4

a cm 1 0.9

3

0.8

0

0

1

2

x/cm (b)

3

4

4

s cm 1 1.3

3

1.2

2

1.1

y/cm

y/cm

0.7 0.6

0.5

2 1

0.5

0.9 0.4

1

1 0.8

0.3 0

0

1

2

x/cm (c)

3

4

0.2

0

0

1

2

x/cm (d)

3

4

0.7

Fig. 6 Reconstruction results of Test 2. (a) and (b) show the real distributions of absorption and scattering coefficients respectively. (c) and (d) show the distributions of reconstructed absorption and scattering coefficients

As shown in Figs. 6 and 7, the reconstructed results of both tests can obtain clear distribution images of the absorption and scattering coefficients. The shapes and locations of inclusions in both test media were well reconstructed using the algorithm based on the CVDM. The reconstructed results prove that the CVDM can calculate the gradient of objective function with respect to the absorption and scattering coefficients efficiently, which is suitable for solving the problem of OT. It is worth noting that the reconstructed results (see Fig. 6) of the Test 1 are worse than the results (see Fig.7) of Test 2. This can be attributed to that the little varying of

absorption coefficient has poor influence on the measurement signals when the absorption coefficient is very small. 4.3 Reconstructed results with measurement errors Measurement errors are inevitable in actual experimental measurements. To test the robustness of CVDM, random measurement errors were considered in the simulation. The measured values with random errors were obtained by adding normal distributed errors to the exact values: M sn,d  M sn,d  randn sn,d

(24)

where M sn,d and M sn,d are the measured value and the exact value of the dth detector at the nth moment when the sth light source is working, respectively. rand n is a standard normal distributed random number. The standard deviation of the measured value  sn,d , for a measurement error of  % at 99% confidence, is determined as:

 sn,d 

M sn,d   %

(25)

2.576

where 2.576 arises from the fact that 99% of a normally distributed population is contained within ±2.576 standard deviation of the mean. The real optical parameters of test medium is setting as the Test 2 in Table 3. The measurement data have been added 3%, 5% and 10% random errors to test the reconstruction algorithm based on the CVDM. To compare the overall accuracy of the reconstructed images, the normalized root-mean square error (NRMSE) is introduced and defined as [45]:    NRMSE=    

N

 ˆ

r

 r

r

N

 r

 r2



1/2

2

      

(26)

where ˆ r is the retrieval value of the optical parameter at the rth pixel,  r is the true value of the optical parameter at the rth pixel, and N is the total number of the pixels in the medium. As shown in Fig. 7, the reconstructed images can give an clear distribution of the absorption and scattering

coefficients even with the measurement errors. The values of NRMSE with different errors are given in Table 4. The values of NRMSE keep small in all tests, which indicates that the overall errors between the reconstructed images and the real images are little. So the conclusion can be reached that the reconstructed algorithm based on the CVDM has good robustness. The increasing of NRMES caused by the increasing of measurement errors show that the measurement errors has helped bluring the reconstruction image. 4

4

4

4

3

3

3

3

0.8

2

2

2

1

1

a cm 1

2

y/cm

y/cm

y/cm

0.7 0.6 0.5 0.4 1

1

0.3 0

0

1

2

3

4

0

x/cm (a)

0

1

2

3

4

0

x/cm (b)

0

0

0 1

1 2

x/cm (c)

2 3

3 4

4

0.2

4

1.2

3

3

3

3

1.1

1

1

0

y/cm

2

2

0

1

2

x/cm (d)

3

4

0

0

1

2

x/cm (e)

3

4

2

2

1

1

0

0

0

s cm 1

4

y/cm

4

y/cm

4

1

0.9

0

1

1

2

x/cm (f)

2

3

3

4

4

0.8

Fig. 7 Reconstructed results with measurement errors. (a) and (d) are the results with 3% error, (b) and (e) are the results with 5% error, (c) and (f) are the results with 10% error

Table 4 Influence of the measurement errors on the reconstructed results NRSME Error %

a

s

0

0.12640

0.08289

3

0.13362

0.09388

5

0.13680

0.09982

10

0.14182

0.10114

5 Conclusions An efficient and robust image reconstruction algorithm based on the CVDM for optical tomography was developed in this research. The time-domain radiative transfer equation was employed as the forward model to simulate the propagation of the photon in participating media, and was solved by the discrete-ordinates method to predict the exitance signals on the boundary of media. An objective function with a regularization scheme was defined by the predicted detector readings and simulated measurement detector readings. The generalized Gaussian Markov random field model was used as the regularization scheme to overcome the ill-posed nature of the inverse problem. The multi-start conjugate gradient method, which can accelerate the convergence speed and enhance the quality of the reconstructed image, was employed as optimization technique to minimize the objective function. The CVDM was employed to calculate the gradient of the objective function with respect to the absorption and scattering coefficients. The CVDM is step-size-independent, easy to use, more accurate than the FD method and can avoid the complex and cumbersome formula of the AD method. All the simulated results show the algorithm presented in this work is efficient and robust for reconstructing the distributions of the absorption and scattering coefficients in the inhomogeneous media, even with noisy data. In conclusion, the proposed methodology is proved to be a promising approach for solving the OT problems.

Acknowledgments The supports of this work by the National Natural Science Foundation of China (No. 51476043), the Major National Scientific Instruments and Equipment Development Special Foundation of China (No. 51327803) and the Foundation for Innovative Research Groups of the National Natural Science Foundation of China (No. 51421063) are gratefully acknowledged. A very special acknowledgement is made to the editors and referees who make important comments to improve this paper.

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Vitae of each author Yaobin Qiao obtained a BE in Zhengzhou University and a ME in Beihang University. He is currently pursuing a PhD of Engineering Thermophysics with Professor Hong Qi in Harbin Institute of Technology. His research focuses on the near-infrared optical tomography. Yaobin Qiao

Hong Qi is a Professor in the School of Energy Science and Engineering, Harbin Institute of Technology (HIT). He obtained a PhD in Engineering Thermophysics from HIT in 2008. Professor Qi has published more than 80 articles and obtained 22 patents. His research focuses on the near-infrared optical tomography, inverse radiation problem, Hong Qi

optimization algorithm, and radiative heat transfer.

Qin Chen obtained a BE in in Harbin Institute of Technology. She is currently pursuing a ME in Engineering Thermophysics with Professor Liming Ruan in Harbin Institute of Technology. Her research focuses on the infrared radiative transfer.

Qin Chen

Liming Ruan is a Professor in the School of Energy Science and Engineering, Harbin Institute of Technology (HIT). He obtained a PhD in Engineering Thermophysics from HIT in 1997. Professor Ruan has been a visiting scholar in Imperial College in 1999. His research mainly focus on the heat radiative transfer, radiative property of high Liming Ruan

temperature particle, target characteristics, radiative reverse problem, et al.

Heping Tan is a Professor in the School of Energy Science and Engineering, Harbin Institute of Technology (HIT). He obtained a PhD University of Poitiers in France in 1989. Professor Tan is the winner of National Outstanding Youth Fund. His research mainly focus on the numerical heat transfer, radiative transfer, target characteristics, Heping Tan

pneumatic radiation, et al.