The 17th Meeting on Image Recognition and Understanding

Layered optical tomography of multiple scatteringmedia with combined constraint optimization

Bingzhi Yuan1,a) Toru Tamaki1,b) Bisser Raytchev1 Kazufumi Kaneda1

Yasuhiro Mukaigawa2

1. Introduction

In this paper, we describe a method to solve the optical

tomography by using an approximation to the path integral

to model the light in optical tomography. Since X-ray CT

relies on radioactive rays, optical tomography uses visible or

infrared ray has been developed over the last decades[1, 2]

for its safety and harmless. Our research aims to develop

an optical tomography method that use infrared or near in-

frared ray input and observed outgoing light at the opposite

side of the body, shown as in Figure 1(a), like as a source-

detector configuration that X-ray CT uses.

In our previous work[3], we use the concepts of path in-

tegral and light transport to model the light in optical to-

mography and solved the tomography problem.

In this paper, we extended our previous work[3]. By using

multiple configuration, as shown in Figure 1(c), we increase

the accuracy and by changing the solver we reduce the com-

putation cost successfully.

In our previous work[3], we took the following assump-

tions (see Fig. 1(b)) : (1) multiple scattering is dominant,

(2) forward scattering is also dominant relative to backward

scattering, (3) a material consists of many parallel layers

made of voxels, and (4) light is scattered from one layer to

another because forward scattering is assumed be dominant.

The material follow the assumptions above is the layered ma-

terial. With the layered material, we developed a constraint

optimization problem to solve the optical tomography.

The structure of rest part of the paper is listed as follow-

ing. In section 2, we review the method in our previous work

and show the changes. In section 3, we show the simulation

results and compare them with our previous work. Section

4 gives our conclusion.

2. Method

In our previous work[3], we proposed a model to describe

the transportation of light penetrate through the material

which follows the assumptions in last section. We’ll briefly

review it in next paragraph.

1 Graduate School of Engineering, Hiroshima University, Japan2 NAIST, Japana) yuan@eml.hiroshima-u.ac.jpb) tamaki@eml.hiroshima-u.ac.jp

Light source

Detector

(a)

light source at position

observation at position

layer 1

layer 2

layer m

(b)

…

…

…

…

…

… …

…

(c)

Fig. 1 Configurations of light sources and observations. (a)Source-detector configuration of CT. (b) A single config-uration with the layers model. A light source at positioni emits light to the first layer, then the light is scatteredto the next layer. At the last layer, output is observedat each position j. (c) Four configurations. The object isfixed while the light source and detector are rotated by 90degrees.

In our previous work[3], in a M by N layered material,

given an incident point i and the camera point j, the ob-

served light intensity Iij obtained by model in [3] is shown

as following

Iij =MN−2∑k=1

Iijk =MN−2∑k=1

I0 cijk aijk, (1)

Since the observed intensity is affected by all possible light

path that come in point i and observed in point j, the num-

ber of all possible light path is MN−2 and intensity for every

light path is Iijk. For a single light path ijk, its intensity

is affected by 3 parts, intensity of incident light which is I0,

intensity lost due to the scattering which is cijk and inten-

sity lost due to the attenuation which is aijk. In the layered

material, cijk is obtained by Gaussian model[4] and its mag-

nitude is control by σ2. σ2 describe how broad the light is

scattered. aijk describe the exponential attenuation when

light path ijk penetrate through the layered material and it

has the following formation

1

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MIRU2014 Extended Abstract

The 17th Meeting on Image Recognition and Understanding

aijk = e−dTijkσt . (2)

Here dijk describe the light path ijk and σt present the

extinction coefficient of the voxels in the layered material.

By keeping light source at the top of material, camera at

the bottom of material and then changing a pair of (i, j), the

positions of incident and outgoing points of light, we have

M2 observations Iij and equations to solve the following

least squares problem:

minσt

ftp, ftp =m∑i=1

m∑j=1

|Iij −mm−2∑k=1

I0cijk e−dT

ijkσt |2,

(3)

Once we have ftp, we can rotate the material 90 degrees for

3 times to obtain flr, fpt, frl. Then the final object function

f0 is

minσt

f0, f0 = ftp + flr + fpt + frl (4)

Due to the fact that the extinction coefficient must be pos-

itive and the numerical stability, we constrain Equ.4 with

0 � σt � u. � denotes generalized inequality that every

elements in a vector must satisfy the inequality

In our previous work[3], we use interior point method co-

operated with Newton’s method to solve the optimization

problem obtained from the optical tomography. Since New-

ton’s method employ second derivative of the cost function,

it is not the best choice for large scale material. There-

fore, in this paper, we replace the Newton’s method with

Quasi-Newton’s method and keep parameters same with [3].

Quasi-Newton’s method uses first derivative of the cost func-

tion to approximate to second derivative which can greatly

reduce the computation time when dealing with large scale

material.

3. Experiment

We evaluate the proposed method by numerical simula-

tion. Four kinds of materials of the size 10 × 10 shown

in Figure 2(a) are used. Each material has almost homoge-

neous extinction coefficients (in light gray) except few voxels

with much higher coefficients (in darker gray), which means

those voxels absorb light much more than others. Parame-

ters are set as follows: σ2 = 1.0 for scattering; u = 1.0 for

the upper bound.

Estimated results are shown in Figure 2(b) and (c): re-

sults in the second row (b) are obtained by previous work

[3], while results in the third row (c) are by our proposed

method. Our results (c) are much closer to the ground truth

(a) and better than the previous work (b). This is also con-

firmed by qualitative results in terms of root mean squares

error (RMSE) shown in Table 1. Table 2 shows computa-

tion time spent by the previous work and proposed method;

it is roughly reduced to 30% (by our unoptimized code in

MATLAB on a PC with Intel Xeon E5 2GHz).

4. Conclusion

In this paper, we have proposed an improved scattering to-

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0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.050

0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.050

0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.050

0.050 0.050 0.200 0.200 0.050 0.050 0.050 0.050 0.050 0.050

0.050 0.050 0.200 0.200 0.050 0.050 0.050 0.050 0.050 0.050

0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.050

0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.050

0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.050

0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.050

0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.050

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0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.050

0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.050

0.050 0.050 0.200 0.050 0.050 0.050 0.050 0.200 0.050 0.050

0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.050

0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.050

0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.050

0.050 0.050 0.200 0.050 0.050 0.050 0.050 0.200 0.050 0.050

0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.050

0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.050

0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.050

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0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.050

0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.050

0.050 0.050 0.200 0.200 0.200 0.200 0.200 0.200 0.050 0.050

0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.200 0.050 0.050

0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.200 0.050 0.050

0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.200 0.050 0.050

0.050 0.050 0.200 0.200 0.200 0.200 0.200 0.200 0.050 0.050

0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.050

0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.050

0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.050

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0.010 0.010 0.010 0.010 0.050 0.050 0.010 0.010 0.010 0.010

0.010 0.010 0.050 0.050 0.050 0.050 0.050 0.050 0.010 0.010

0.010 0.050 0.050 0.100 0.050 0.050 0.050 0.050 0.050 0.010

0.010 0.050 0.150 0.200 0.100 0.050 0.200 0.200 0.050 0.010

0.050 0.050 0.150 0.150 0.100 0.050 0.050 0.050 0.050 0.050

0.050 0.050 0.100 0.150 0.050 0.050 0.050 0.110 0.050 0.050

0.010 0.050 0.050 0.100 0.050 0.050 0.120 0.180 0.100 0.010

0.010 0.050 0.050 0.050 0.050 0.050 0.050 0.130 0.050 0.010

0.010 0.010 0.050 0.050 0.050 0.050 0.050 0.050 0.010 0.010

0.010 0.010 0.010 0.010 0.050 0.050 0.010 0.010 0.010 0.010

(a)

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0.046 0.051 0.052 0.052 0.052 0.047 0.050 0.049 0.050 0.052

0.049 0.054 0.052 0.052 0.052 0.052 0.049 0.052 0.053 0.049

0.053 0.050 0.062 0.060 0.044 0.047 0.048 0.052 0.055 0.053

0.058 0.059 0.114 0.129 0.083 0.068 0.057 0.050 0.046 0.044

0.057 0.058 0.132 0.147 0.082 0.068 0.058 0.050 0.046 0.045

0.051 0.047 0.080 0.079 0.041 0.045 0.049 0.052 0.056 0.054

0.047 0.048 0.062 0.061 0.047 0.049 0.051 0.053 0.055 0.050

0.050 0.050 0.051 0.051 0.051 0.052 0.051 0.052 0.052 0.054

0.052 0.055 0.045 0.046 0.056 0.055 0.051 0.052 0.053 0.053

0.046 0.051 0.045 0.046 0.053 0.049 0.052 0.052 0.053 0.053

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0.047 0.052 0.055 0.051 0.049 0.049 0.051 0.055 0.052 0.047

0.052 0.052 0.059 0.050 0.049 0.049 0.050 0.059 0.052 0.052

0.056 0.059 0.082 0.070 0.075 0.075 0.070 0.082 0.059 0.056

0.051 0.050 0.076 0.048 0.048 0.048 0.048 0.076 0.050 0.051

0.049 0.049 0.082 0.048 0.050 0.050 0.048 0.082 0.049 0.049

0.051 0.049 0.078 0.047 0.049 0.049 0.047 0.078 0.049 0.051

0.056 0.058 0.084 0.070 0.075 0.075 0.070 0.084 0.058 0.056

0.052 0.051 0.060 0.049 0.049 0.049 0.049 0.060 0.051 0.052

0.050 0.052 0.054 0.052 0.051 0.051 0.052 0.054 0.052 0.050

0.050 0.052 0.050 0.051 0.050 0.050 0.051 0.050 0.052 0.050

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0.047 0.054 0.054 0.056 0.056 0.055 0.056 0.055 0.050 0.042

0.055 0.055 0.062 0.066 0.066 0.067 0.065 0.065 0.053 0.050

0.058 0.067 0.105 0.138 0.176 0.168 0.127 0.116 0.064 0.057

0.048 0.052 0.079 0.075 0.074 0.075 0.078 0.132 0.064 0.059

0.045 0.052 0.082 0.076 0.075 0.076 0.080 0.166 0.065 0.055

0.049 0.052 0.080 0.075 0.074 0.075 0.078 0.141 0.063 0.060

0.059 0.066 0.107 0.137 0.175 0.165 0.125 0.122 0.064 0.057

0.057 0.055 0.065 0.064 0.063 0.063 0.063 0.068 0.052 0.051

0.055 0.058 0.060 0.064 0.064 0.064 0.063 0.056 0.055 0.051

0.053 0.056 0.051 0.053 0.053 0.052 0.053 0.049 0.052 0.048

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0.024 0.029 0.033 0.037 0.052 0.047 0.032 0.029 0.023 0.023

0.021 0.031 0.045 0.054 0.054 0.046 0.046 0.043 0.030 0.021

0.038 0.043 0.059 0.068 0.049 0.042 0.054 0.055 0.041 0.032

0.039 0.054 0.093 0.124 0.097 0.093 0.105 0.097 0.061 0.041

0.067 0.065 0.093 0.111 0.057 0.049 0.077 0.092 0.055 0.040

0.040 0.053 0.089 0.106 0.065 0.053 0.074 0.097 0.053 0.046

0.038 0.043 0.060 0.068 0.053 0.053 0.080 0.114 0.074 0.053

0.029 0.038 0.049 0.056 0.051 0.047 0.053 0.071 0.049 0.041

0.024 0.030 0.038 0.044 0.051 0.047 0.041 0.050 0.032 0.024

0.025 0.028 0.028 0.031 0.044 0.046 0.028 0.035 0.027 0.026

(b)

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0.049 0.050 0.050 0.051 0.053 0.048 0.051 0.047 0.048 0.053

0.049 0.050 0.051 0.051 0.051 0.050 0.048 0.048 0.052 0.050

0.044 0.057 0.058 0.051 0.044 0.046 0.046 0.046 0.053 0.055

0.055 0.046 0.187 0.200 0.057 0.060 0.056 0.046 0.047 0.047

0.056 0.047 0.194 0.190 0.052 0.053 0.052 0.076 0.041 0.039

0.053 0.047 0.058 0.053 0.041 0.044 0.050 0.045 0.055 0.054

0.045 0.048 0.060 0.055 0.045 0.047 0.049 0.048 0.054 0.049

0.052 0.052 0.041 0.053 0.052 0.051 0.049 0.048 0.051 0.050

0.049 0.052 0.049 0.047 0.054 0.052 0.048 0.049 0.052 0.049

0.048 0.051 0.052 0.050 0.052 0.049 0.051 0.047 0.047 0.054

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0.048 0.052 0.051 0.051 0.048 0.049 0.051 0.052 0.050 0.049

0.053 0.050 0.054 0.047 0.044 0.045 0.049 0.057 0.053 0.049

0.049 0.055 0.158 0.068 0.072 0.071 0.067 0.153 0.056 0.052

0.052 0.046 0.071 0.040 0.043 0.040 0.039 0.070 0.047 0.053

0.046 0.044 0.079 0.040 0.042 0.041 0.041 0.077 0.044 0.046

0.052 0.046 0.073 0.039 0.040 0.041 0.039 0.073 0.045 0.052

0.051 0.054 0.160 0.065 0.070 0.070 0.064 0.165 0.053 0.049

0.050 0.052 0.057 0.046 0.045 0.045 0.046 0.056 0.052 0.051

0.048 0.051 0.051 0.052 0.048 0.048 0.052 0.049 0.051 0.049

0.051 0.050 0.047 0.051 0.051 0.051 0.052 0.047 0.051 0.050

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0.047 0.049 0.051 0.052 0.049 0.049 0.057 0.050 0.051 0.045

0.054 0.048 0.051 0.047 0.046 0.051 0.053 0.050 0.048 0.052

0.055 0.058 0.181 0.196 0.208 0.198 0.185 0.227 0.047 0.045

0.050 0.049 0.058 0.051 0.047 0.052 0.054 0.172 0.055 0.062

0.039 0.049 0.062 0.054 0.050 0.057 0.058 0.180 0.052 0.050

0.052 0.046 0.058 0.049 0.048 0.053 0.056 0.172 0.056 0.060

0.052 0.054 0.193 0.202 0.205 0.185 0.168 0.261 0.041 0.039

0.054 0.045 0.051 0.048 0.047 0.051 0.054 0.054 0.046 0.050

0.047 0.049 0.049 0.052 0.048 0.057 0.060 0.043 0.048 0.047

0.051 0.053 0.048 0.049 0.051 0.049 0.053 0.042 0.054 0.051

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0.000 0.002 0.035 0.039 0.025 0.026 0.010 0.009 0.029 0.005

0.018 0.019 0.023 0.044 0.053 0.069 0.031 0.064 0.018 0.000

0.025 0.044 0.042 0.066 0.069 0.065 0.055 0.064 0.012 0.026

0.022 0.054 0.121 0.158 0.177 0.042 0.223 0.157 0.030 0.037

0.022 0.036 0.191 0.221 0.034 0.040 0.041 0.074 0.055 0.037

0.022 0.065 0.115 0.154 0.031 0.056 0.079 0.084 0.086 0.017

0.027 0.055 0.054 0.074 0.037 0.055 0.097 0.205 0.097 0.019

0.024 0.042 0.036 0.047 0.066 0.049 0.052 0.123 0.038 0.023

0.018 0.021 0.018 0.034 0.074 0.076 0.037 0.041 0.015 0.008

0.001 0.001 0.036 0.034 0.033 0.022 0.017 0.020 0.009 0.008

(c)

Fig. 2 Simulation results for σ2 = 1.0. (a) Ground truth of fourmaterials 1, 2, 3, and 4. (b) Results of [3]. (c) Results ofour method. Values in each voxel are estimated value ofσt, and darker gray represents larger value.

Table 1 RMSEs of results for four materials. The order is thesame with Figure 2(b) and (c).

method 1 2 3 4Fig. 2(b) [3] 0.016978 0.02731 0.043248 0.030447Fig. 2(c) ours 0.004987 0.01213 0.010154 0.022141

Table 2 Computation time (in seconds) of results for four mate-rials in Figure 2(b) and (c).

method 1 2 3 4Fig. 2(b) [3] 191.334 162.035 197.519 195.628Fig. 2(c) ours 55.602 55.602 70.531 76.773

mography with a layered model. We modified the optimiza-

tion problem we proposed in our previous work[3]. Then

we solved it by interior point method with a Quasi-Newton

method. In the experiments, results were much more im-

proved than the previous work as well as computational cost

is decreased.

References

[1] P. Gonzalez-Rodrguez, and A. D. Kim, Comparison of

light scattering models for diffuse optical tomography,

Optics Express, vol. 17, no. 1, pp. 8756–8774, 2009.

[2] S. R. Arridge, Optical tomography in medical imaging,

Inverse Problems, vol. 15, no. 2, pp. R41–93, 1999.

[3] T. Tamaki, B. Yuan, B. Raytchev, K. Kaneda, and

Y. Mukaigawa, Multiple-scattering optical tomogra-

phy with layered material, International Conference

on Signal-Image Technology Internet-Based Systems

(SITIS), pp. 93–99, Kyoto, Japan, Dec 2013.

[4] S. Premoze, M. Ashikhmin, R. Ramamoorthi, and S.

Nayar, Practical rendering of multiple scattering ef-

fects in participating media, Eurographics Symposium

on Rendering, Sweden, 2004

2

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MIRU2014 Extended Abstract