Shape from Single Scattering
for Translucent Objects
Chika Inoshita1 , Yasuhiro Mukaigawa1 , Yasuyuki Matsushita2 , Yasushi Yagi1
1

Osaka University

2

Microsoft Research Asia

Abstract. Translucent objects strongly scatter incident light. Scattering makes the problem of estimating shape of translucent objects difficult, because reflective or transmitted light cannot be reliably extracted
from the scattering. In this paper, we propose a new shape estimation
method by directly utilizing scattering measurements. Although volumetric scattering is a complex phenomenon, single scattering can be relatively easily modeled because it is a simple one-bounce collision of lightto a particle in a medium. Based on this observation, our method determines the shape of objects from the observed intensities of the single
scattering and its attenuation. We develop a solution method that simultaneously determines scattering parameters and the shape based onenergy minimization. We demonstrate the effectiveness of the proposed
approach by extensive experiments using synthetic and real data.

1

Introduction

In real-world scenes, there are many translucent objects around us, such as wax,
plastic products, and precious stones. Inside a translucent object, incident lightscatters and randomly travels. Although vision-based shape estimation techniques are maturing, it is still difficult to estimate the shape of translucent
objects. Passive approaches based on triangulation require a target objects tobe well-textured, and active methods tend to fail due to strong scattering of
the incident light. As pointed out by Godin et al . [1], the incident light shows
the brightest observation off the object surface for translucent materials due toscattering. Methods based on refractive or transmitted light cannot be directly
applied to translucent media where strong scattering exists. Figure 1 shows anexample of a failure mode for measuring translucent objects using a commercial 3D laser scanner, Konica-Minolta Vivid 9i. While it works well for humanskin, the shape of translucent objects is not accurately estimated due to strong
scattering. Shape estimation of translucent objects is still a wide open problem.
While scattering is well studied in computer graphics, it is less discussed
in computer vision because the complexity of light interactions renders inverse
problems intractable. Therefore, for vision applications, scattering has been regarded as a nuisance and eliminated by various approaches, e.g., polarization [2],coating with diffuse powder [3], and phase-shift measurements [4]. In contrast
to these approaches, we propose a new shape estimation method based on the

2

C. Inoshita, Y. Mukaigawa, Y. Matsushita, Y. Yagi(a) Translucent blocks
on the hand(b) Shape measurement result
by a laser range finder(c) Strong scatteringin translucent object

Fig. 1. Failure mode of shape measurement of translucent objects using a 3D laser
scanner.

observed scattering itself. In particular, we develop a method that uses the observation of single scattering, which occurs via a one-bounce collision of light toa particle, and its attenuation along the light path. We derive a solution methodfrom a scattering model that takes into account the refraction, an extinction
coefficient, and a phase function.
The primary contributions of our work are as follows. We propose a newshape estimation technique for translucent objects. It introduces the use of scattering as a beneficial signal for determining shape. The proposed method iseffective when the target object shows strong scattering where other shape estimation approaches cannot be applied. In fact, since there exist various objects inour daily life that exhibit strong scattering, our method has wide applicability.
This work is the first attempt to directly use the observed intensities of single
scattering for shape measurement. In addition, we develop an effective solution
method based on energy minimization for simultaneously estimating shape andscattering parameters.

2

Related work

Shape-from-intensity is a generic framework for shape measurement based on
observed intensities. Various optical phenomena, such as diffuse [5, 6] or specularreflection [7, 8], have been used as cues for shape estimation. Liao et al . [9]estimate object’s shape from light attenuation, which is inversely proportional
to the square of distance. Our method is related to Liao et al .’s method because
we also use the attenuation of the light although the setting is different.
Refractive lights have been used to measure the shape of transparent objects,
because most incident lights to the surface are transmitted inside. Miyazaki
and Ikeuchi [10] estimate the object surface by analyzing multiple interreflections in the object by a polarization ray-tracing method. Huynh et al . [11] use
multi-spectral polarization for obtaining surface normals. Kutulakos and Steger’s method [12] performs triangulation from the analyzed transmitted lightpath. Wetzstein et al . [13] use a light field probe to analyze the transmitted
light path. Trifonov et al .’s method [14] immerses target objects in fluid whoseShape from Single Scattering for Translucent Objects

3

refractive index is the same as the target and applies tomography for obtaining
shape. Hullin et al . [15] use a scan-line illumination for transparent objects immersed in fluorescent fluid. While these methods use refracted light, Morris andKutulakos [16] propose scatter-trace photography focusing on the transparent
object that reflects some of the incident light. While these methods are effective, it is not straightforward to apply them to translucent objects due to strong
scattering.
Shape measurement methods for objects in scattering media have been also
proposed. Narasimhan et al . [17] measure the shape of target objects in scattering media by light stripe scanning and photometric stereo. Kim et al . [18]remove scattering light by angular filtering and estimate the shape by tomography. Treibitz and Schechner [19] not only remove backscattering in an underwater scene, but also estimate the scene depth based on backscattering falloff.
Florescu et al . [20] apply the optical tomography using observed single scattering
based on a radiative transfer equation and reconstruct the attenuation parameter of the 3D volume. While these are related, these cannot be directly applied toestimate the shape of translucent objects. More recently, characteristics of scattering media have been actively studied. Mukaigawa et al . [21] propose a method
for separating single and multiple scatterings and analyze the light transport.
Narasimhan et al . [22] dilute scattering media to reduce multiple scattering for
estimating scattering parameters. The spatial distribution of scattering media,
such as smoke or milk drops, has also been studied [23, 24]. We develop our
method based on these studies on the characteristics of scattering.
Scattering has similar properties to interreflections. In fact, scattering can be
regarded as interreflections of light in a medium. In the study of shape estimation
with the existence of interreflections, there are three major approaches:
1. Elimination of the interreflections: Nayar et al . [25] show a method to remove
interreflections using high frequency illumination. After the elimination ofinterreflections, a classic technique, such as Lambertian photometric stereo,
can be well applied.
2. Modeling of interreflections: Nayar et al . [26] iteratively estimate the shape ofconcave surface by explicitly modeling the interreflection based on radiosity.
3. Use of the interreflection itself: Liu et al . [27] show that the light transportof the interreflections can be used as a cue for shape estimation.
Thus, similar approaches may be applicable for shape measurement of a translucent object. Our approach is close to the third approach where we wish to recover
shape from scattering itself.

3

Shape from Single Scattering

3.1

Background

When light travels through a translucent medium, the light scatters in themedium due to collisions with particles. Single scattering is a type of scattering,4

C. Inoshita, Y. Mukaigawa, Y. Matsushita, Y. Yagi

I
d2
d

θ
1

p(g, θ)(a) Scattering model(b) g > 0(c) g = 0(d) g < 0

Fig. 2. Illustration of the scattering model and examples of distribution profiles witha varying phase function.

in which light scatters at only once in the medium before reaching a viewer. Figure 2(a) shows a parametric scattering model. In scattering media, the incident
light exponentially attenuates with the length of light path by the LambertBeer law [28]. It also scatters in the media, and a Henyey-Greenstein phasefunction [29] is known as a good approximation for the phenomenon. With thisphase function, the observed intensity I of single scattering is described as [21,22]
I = sp(g, θ)e−σt (d1 +d2 ) ,1
1 − g2p(g, θ) =,4π (1 + g 2 − 2g cos θ) 32

(1)
(2)

where s is a scaling constant that includes the intensity of the incident lightand scattering coefficient, σt is an extinction coefficient, (d1 + d2 ) represents thelength of the light path in the medium, and p(g, θ) is the phase function. The
phase function represents the scattering distribution, and the distribution profile
is controlled by a parameter g (−1 ≤ g ≤ 1). Figures 2(b), (c), and (d) show
examples of the distribution profiles produced by varying g.
3.2

Formulation

We formulate the relationship between observed intensities of single scatteringand the shape of translucent target. Figure 3 shows our setting for the shapemeasurement. A translucent object is illuminated from one side and observed
from the top. We assume a homogeneous material as a target object and orthographic projection for both illumination and observation. In addition, we ignore
multiple scattering for now, but a method for handling it will be explained inlater sections.
Figure 4 illustrates a light path of single scattering in a medium. The incident
ray ix scatters at scattering point (x0 , y 0 ), and reaches to surface point (x, y).
n(x, y) represents the surface normal and ix = [1, 0, 0]T and iz = [0, 0, 1]T are
incident and exitant light vectors, respectively. r(x, y) is a unit scattering vector

Shape from Single Scattering for Translucent Objects

5izn(x, y)(x, y)

r(x, y)h(x, y)Camera(xi’, yi’)θpθrIncident
ray

diIllumination(x’, y’)

Scattering point

Incident
light

ix

z
y

x

Incident surface

Fig. 3. Setting for shape estimation. A translucent object is illuminated from the side and observed
from the top.

Fig. 4. Light path in the target object. The
incident plane is almost planer, and incident light is assumed parallel to x-axis.
The illuminated ray reaches to surface point(x, y) changing the travel direction at the
scattering point (x0 , y 0 ).

pointing from the scattering point (x0 , y 0 ) to the surface point (x, y). The angle
between the incident vector ix and the scattering vector r(x, y) is denoted as θp .
θr represents the projected angle of θp on the plane spanned by r(x, y) and iz .
Our purpose is to estimate the height of the translucent object h(x, y) from theobserved intensities I(x, y) at the surface point (x, y) on the surface, where theheight of the incident ray is z = 0. Scattered incident ray ix is finally refracted
at the object surface. The angle of refraction obeys Snell’s law represented asn(x, y) × iz = ηn(x, y) × r(x, y),(3)where η is a refractive index, × represents a cross-product operator. The total
length of the light path becomes the sum of x0 , which corresponds to the distancefrom the incident point to scattering point and the distance from the scattering
point to the surface point, h(x, y)/sin θr . Since the intensity of single scattering
is modeled as Eq. (1), the observed intensity is represented by0h(x,y)I(x, y) = sFtin Ftout (x, y)p(g, θp )e−σt (x + sin θr ) ,(4)

where s is a scaling constant, Ftout (x, y) represents the Fresnel transmittance on
the surface point (x, y), and Ftin is the constant Fresnel transmittance on theincident point because the incident light is perpendicular to the incident plane.
Equation (4) shows that the observed intensity depends on both the geometricshape and scattering parameters including extinction parameter σt , refractive
index η, and phase function parameter g. Given these scattering parameters, theheight of the translucent object is determined up to an unknown offset due to s

6

C. Inoshita, Y. Mukaigawa, Y. Matsushita, Y. Yagibyh(x, y) =4sin θrlog s + log Ftin + log Ftout (x, y) + log p(g, θp ) − log I(x, y)σt−x0 sin θr .(5)Solution method

In the previous section, we described a basic theory of the shape from single
scattering. However, in reality, we cannot directly estimate the object heighth(x, y) by Eq. (5) due to the unknown parameters. In addition, the observed
intensities include the contributions not only from single scattering but also
from multiple scattering. In this section, we discuss a solution method for these
problems. In our method, we assume that the refractive index η is known because
it can be directly measured using a refractometer.
In our method, we employ an energy minimization approach for simultaneously determining both the shape and scattering parameters. When the unknown
parameters and height are correctly estimated, Eq. (4) should produce the intensity that becomes equivalent to the observed intensity I(x, y). Although we canestimate the unknown parameters by seeking parameters, which generate theobserved intensity, parameter estimation tends to be unstable due to a larger
number of unknowns than the captured intensity. To reliably derive a solution
to this problem, we use multiple n(2 ≤ n) images that are captured by changing
the height of the incident ray. Namely, we record multiple intensities Ii (x, y) withvarying heights of the incident rays z = di (i = 1, . . . , n) as shown in Fig. 4.
Now we have n intensity observations per scene point Ii (x, y) as
0Ii (x, y) = sFtin Ftout (x, y)p(g, θp )e−σt (xi +

h(x,y)−di)sin θr,i = 1, . . . , n.

(6)

We also take into account the signal-to-noise ratio of the observed intensities; thedarker observations suffer more from image noise while the brighter observations
are more reliable. We incorporate this as a weighting factor wi when determining the unknown parameters. Based on these, we define an energy function forcomputing heights h(x, y) and scattering parameters s, g, σt as
X X
2E(h(x, y), s, g, σt ) =wi(Ii (x, y) − Iigen (h(x, y), s, g, σt )) ,
(7)
i

x,y

where Iigen is the generated intensity based on Eq. (6), and wi is a weighting
factor for reducing the effect of noise. We define the weighting factor wi as
Px,y Ii (x, y)
wi = Pn P.
(8)
k=1
x,y Ik (x, y)
The energy function E evaluates the closeness between observed intensity andgenerated intensity based on Eq. (6). The minimizer of the energy function E

Shape from Single Scattering for Translucent Objects

7

gives us estimates of the height h(x, y) per-pixel and scattering parameters s, gand σt as{h(x, y), s, g, σt } =argmin E(h(x, y), s, g, σt ).

(9)h(x,y),s,g,σt

We describe the optimization method in the following section.

4.1

Implementation

This section describes the implementation details of the solution method. Our
method uses a non-linear optimization because of the non-convexity of Eq. (7)with respect to the unknown parameters. We now describe the computation
method of the initial guess of the height h(x, y) and the following optimization
strategy.

Estimation of initial shape To obtain an initial guess of the estimated parameters, we use the initial shape h0 (x, y) that can be computed by ignoringrefraction (η = 1). When η = 1, the scattering vector coincides with the output
vector iz , the 2D projection of the scattering point (x0 , y 0 ) becomes identical withthe surface point (x, y), and Fresnel transmittance Ftout (x, y) is constant because
the refraction is disregarded. Since the angle θp equals to π/2, phase functionp(g, θp ) becomes constant. Therefore, the intensity generated from initial heighth0 (x, y) is described as0

Ii0 (x, y) = Se−σt (h (x,y)+x−di ) , π
S = sFtin Ftout p g,
.
2

(10)

Here, unknown parameters are the height h0 (x, y), scaling constant S, and extinction coefficient σt . Using a pair of intensity observations Ii (x, y) and Ij (x, y)taken by different heights of incident rays di and dj , the extinction coefficient iscalculated asσt =log Ii (x, y) − log Ij (x, y)di − dj(di 6= dj ).

(11)

In practice, we take the average as the estimate of σt using all the pairs of di anddj . We employ an intensity at the incident point as the initial scaling constant
S as the scaling without attenuation. The initial guess of the height h0 (x, y) istherefore described ash0 (x, y) =1(log S − log Ii (x, y)) − x + di ,
σt

(12)

and is estimated using the parameters σt and S. We use this initial guess as an
input to the optimization: h(x, y) ← h0 (x, y).

8

C. Inoshita, Y. Mukaigawa, Y. Matsushita, Y. Yagi

Optimization Now we estimate the shape and parameters by minimizing Eq. (9)using h0 as the initial guess of the shape. The unknowns to be estimated are
per-pixel height h(x, y) and scattering parameters s, g, and σt . To efficiently
avoid local minima, we use a two-step approach for the optimization. Namely,
we first apply the Particle Swarm optimization [30] for limiting the search range
in a coarse manner, and then use the Nelder-Mead method [31] for finding the
optimal parameter set in a fine scale.
Extraction of single scattering As discussed above, actual observations consist of both single and multiple scatterings as shown in Fig. 1(c). To separate
the single scattering component from multiple scattering, we utilize a separation method [21] which uses projector as a light source as shown in Fig. 3. The
extracted single scattering component is used as input for our method. Readers
are referred to [21] for the details of the separation method.

5

Experiments

We assess the effectiveness of the proposed method using both synthetic andreal-world scenes. With the synthetic scenes, we assess the accuracy of the proposed method by comparing with the ground truth. For the real-world scenes,
we prepare two objects where the ground truth shape is available while thescattering parameters are unknown.
5.1

Synthetic scenes

For the synthetic scene experiment, we used one 1-D curved surface, one 1-D
discontinuous surface, and one 2-D discontinuous surface h(x) as the scenes;
Scenes A, B, and C. For each scene, we simulated intensity observations Ii (x)with varying heights of incident rays di = 0.2 × i[mm], i = 0, · · · , 9 using thescattering model of Eq. (6). We added Gaussian noise to the intensity observations at 5 levels (µ = 0 and σ = 0, 5, 10, 15, 20). The optical parameters are
consistently set to η = 1.2, g = 0.1, σt = 15, and s = 50000.
Figure 5(a) shows Scene A and (b) shows simulated intensity observation in
the case of Gaussian noise σ = 10. From this intensity, we estimate the object
shape. Figure 5(c) shows the estimated result of Scene A. The initial height isglobally skewed due to the inaccurate assumption of the refractive index η = 1
and local deformations due to observation noise. Especially, when Gaussian noiselevels are σ = 15 and 20, estimated shapes become noisier as the length oflight paths are greater because of the low S/N ratio. However, the optimized
results consistently agree well with the ground truth except for some fluctuations.
Figure 5(d) shows RMSE of the estimated height at each noise level. Although
the RMSE increases according to the noise amplitude, the overall errors are smalland show the accuracy of the method. Estimated scattering parameters and the
RMSE values are summarized in Table 1. With small noise, the scaling parameters and extinction coefficient σt are almost correct. The scattering parameter gShape from Single Scattering for Translucent Objects

2
0

0

2

4

6

8

x-axis length

10

12

2000

1000

[mm]

0

0

σ=0 (h0 )σ =5 (h0 )σ =10 (h0 )σ =15 (h0 )σ =20 (h0 )Height2

σ =0 (result)
σ =5 (result)
σ =10 (result)
σ =15 (result)
σ =20 (result)Ground truth

1
0
-1

4

6

8

x-axis length

10

12

[mm]

(b) Intensity observations[mm](a) Synthetic scene

2

[mm]

-2

I1
I2
I3
I4
I5
I6
I7
I8
I9
I10

I1
I2
I3
I4
I5
I6
I7
I8
I9
I10

3000

0.16

RSME of height

Height[mm]

Intensity of single scattering

4000

9

0.12
0.08
0.04

0

2

4

6

8

x-axis length(c) Estimated result
at each noise level

10

12

[mm]

0

σ =0 σ =5 σ =10 σ =15 σ =20

Variance of Gaussian noise(d) RMSE of heightat each noise level

Fig. 5. Estimated heights of Scene A

Table 1. Estimated parameters and RMSE of estimated height at each noise level.
RMSE of height
Extinction
ScalingParameter
coefficient σt [mm−1 ][mm]constant sgGround truth 5.0 × 104
0.1
1.5
σ=0
5.08 × 104
0.069
15.0
0.05 × 10−1σ=54.95 × 104
0.007
15.0
0.15 × 10−14
σ = 104.95 × 10
0.002
15.0
0.42 × 10−14
σ = 15
6.09 × 10−0.00216.0
1.64 × 10−1
σ = 206.33 × 104
0.003
16.1
1.90 × 10−1

has a larger deviation from its ground truth. Although g controls the scattering
distribution, it effects to both intensity scale and intensity attenuation depending
to object shape. Thus, estimating g becomes more difficult than estimating otherparameters.
Other experimental results using synthetic 1-D stepped shape (Scene B) and2-D pyramid like shape (Scene C). The ground truth and the estimated result in
the case of Gaussian noise σ = 10 are shown in Fig. 6(a) and Fig. 7(b). For theseasymmetric and discontinuous scenes as well, the estimation well converges nearto the ground truth.

Height

Initial height
Ground truth
Final result

5

0

-5 0

2

4

6

8

10

x-axis length

[mm]

12

Intensity of single scattering

C. Inoshita, Y. Mukaigawa, Y. Matsushita, Y. Yagi
[mm]

10

2500

I1
I2
I3
I4
I5
I6
I7
I8
I9
I10

2000
1500
1000
500
0

0

(a) Scene B and estimation result

2

4

6

8

x-axis length

10

12 [mm](b) Intensity observations

Intensity of single scattering

Fig. 6. Estimated heights of discontinuous Scene B

6500
5500

HeightHeight

4500

3

3

3500

2

2

2500

1

1

1500
3

0

x -a

xis

6 th
g

9
m]
[m

0
[mm]

0
[mm]

len

(a) Intensity observations(b) Scene C(c) Estimation result

Fig. 7. Estimated heights of discontinuous Scene C

CameraTelecentric
Lens
15cm

Projector
21cm

Fig. 8. Experimental setting. A projector is placed on the side of the object, and acamera is vertically placed, we used a telecentric lens for orthographic projection.

5.2

Real scenes

We also applied the proposed method to real-world scenes. Figure 8 shows theexperimental setting. A 3M MPro110 projector is placed on the side of the target
object, and a Point Grey Grasshopper camera which has a linear response sensor
is vertically placed to obtain a top view. To avoid the perspective effect of theTarget object

Intensity of single scattering

Shape from Single Scattering for Translucent Objects

Decomposed
single scattering

11

x35
10 3

30

Observed scattering

25

Multiple scattering

20
15
10
5

Single scattering

0
Observed scattering

5
0.5

0

Decomposed
multiple scattering

15
1.5

25 [mm]2.5

20
2x-axis length(b) Decomposition of
the scattering components[mm](a) Target object anddecomposition results

101Height
2.5

1.5
1.0
0.5
0.0
[mm]

Height2.0

Initial height
Ground truth
Final result

3
2
1
0
-1

0

5

10

15

25 [mm]20

x-axis length(c) 3D plot of the result(d) Estimation result

Fig. 9. Experimental result using a concave scene.

0.8

Height0.7

8.00.60.8
0.6

6.00.50.4

0.4

4.00.3

0.2
0
250
200
200
150

150
100

100
50

50
0

(a) Target object

2.00.2
0.1

0.0 0
[mm]

0

(b) 3D plot of the merged result

Fig. 10. Experimental results using a asymmetric convex scene. (b) shows the 3D plotof the merged result.

imaging system, we use an Edmund optics telecentric lens for approximating
an orthographic projection. To perform a comprehensive analysis, we use twodifferent shapes of translucent objects; one is concave, and the other is convex.
We made concave object using same material as shown in Fig. 1 (b) and convex
object using a colored gelatin. The ground truth of these objects are known for
performing a quantitative evaluation. We set the refractive index η as 1.3. We

12

C. Inoshita, Y. Mukaigawa, Y. Matsushita, Y. Yagi

captured intensities of single scattering Ii (x, y) (di = 0.25 × i[mm], i = 0, · · · , 9)with shifting height of incident light.
Figure 9(a) shows the concave object. The object is illuminated by a projector from its side plane. Figures 9(a) and (b) show the decomposition of thescattering components. The red, green, and blue plots show the intensities ofsingle scattering, multiple scattering, and observed scattering, respectively. It
can be seen that single scattering is almost exponentially attenuated with theincreased distance from the incident point. Figure 9(c) shows the shape estimation result plotted as a 3D surface. We can see that our method is able toestimate the overall shape of the target except for the flat region. A possible
reason for the incorrect estimation is insufficient intensities of single scattering
for shape estimation because the light paths are longer for observing this region.
In addition, the shape nearby the incident plane did not match the ground truth
well due to the bright observation at the incident point and its glare. Estimatedscattering parameters are s = 12118, g = 0.042, and σt = 13.244.
We also show the result of the convex scene in Fig. 10. Figure 10(b) shows
the reconstruction result given by single scattering illuminated from front of thetarget object. Because of insufficient intensities of single scattering, estimatedheight has large error in back area. To reduce the error, we capture the intensities
of single scattering by illuminating from the other side of the target object, andmerge the two reconstruction results. Figure 10(c) shows the merged result.
Computation time In this experiment, it takes about 15 minutes to capture
images (6 images for extracting single scattering at each of 10 different depths),
and it takes about a few hours to compute the shape with an unoptimized Matlab
implementation. We compute estimates on Intel Core 2 Duo CPU (3.00 GHz)and 3GB RAM. The size of problem is 74 (71 points for h(x), s, g, and σt ) insynthetic data, 259 (256 points for h(x, y), s, g, and σt ) in symmetric real data,
and 873 (870 points for h(x, y), s, g, and σt ) in asymmetric real data.

6

Discussions

In this paper, we proposed a method for estimating the shape of translucent
objects based on the attenuation of single scattering. We showed synthetic andreal-world results that our method has a potential for accurate modeling oftranslucent objects, which has been difficult with other appearance-based methods. While the method works well for various translucent objects, the accuracy
suffers from low-intensity measurements and high signal-to-noise ratio when optically dense objects are measured, in which multiple scattering dominate theappearance. Another issue of the current approach is its high computational cost.
We are interested in looking into more in these aspects for making the approach
more practical.
Limitations There are a few limitations in the current method. These limitations will be removed in our future work.

Shape from Single Scattering for Translucent Objects

13

Object shape: Our current formulation requires that the incident plane isplaner and incident light in target object is parallel to the x-axis for extracting
single scattering using high frequency projection. When projected patterns interfere each other, single scattering cannot be extracted. This assumption needs
to be relaxed when it is applied to a more general shape.
Single scattering in an inhomogeneous material: Since our method uses
single scattering which exponentially attenuates with constant scattering parameters, we cannot estimate shape of an inhomogeneous material.

Acknowledgment
This work is partially supported by the Japan Society for Promotion of Science (JSPS) through “Funding Program for Next Generation World-Leading
Researchers (NEXT Program),” initiated by the Council for Science and Technology Policy (CSTP).

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