1.
Scattering tomography
with path integral
Toru%Tamaki%(Hiroshima%University)%
%
%
Joint%work%with%
Bingzhi%Yuan,%Yasuhiro%Mukaigawa,%
and%many%others%

2.
Computed Tomography
Mul`plexing%Radiography%for%UltraVFast%Computed%Tomography%
J.%Zhang,%G.%Yang,%Y.%Lee,%S.%Chang,%J.%Lu,%O.%Zhou%
University%of%North%Carolina%at%Chapel%Hill%
h6ps://www.aapm.org/mee`ngs/07AM/VirtualPressRoom/LayLanguage/IIMul`plexing.asp%
h6p://www.sciencekids.co.nz/pictures/humanbody/braintomography.html

3.
Different Modalities
CT%:%h6p://www.hiroshimaVu.ac.jp/news/show/id/13475/dir_id/19%
PET%:%h6p://www.hiroshimaVu.ac.jp/hosp/hitokuchi/p_vdo8u6.html%%
MRI%:%h6p://www.hiroshimaVu.ac.jp/news/show/id/14427/dir_id/19%
CT%
(X%Ray)
MRI%
(Magne`c%field)
PET%
(Positron)
%
(infrared)

4.
Diffuse Optical Tomography (DOT)
,% −
−% 19 9 5 ,% ,% 3 ,%3% ,%
%h6p://www.mext.go.jp/b_menu/shingi/gijyutu/gijyutu3/toushin/07091111/008.htm
Input%
Light%pulse%
Detector%
Detector%
Time%
Time%
Lee,%Kijoon.%“Op`cal%Mammography:%Diffuse%Op`cal%Imaging%of%Breast%
Cancer.”%World%Journal%of%Clinical%Oncology%2.1%(2011):%64–72.%PMC.%Web.%8%
June%2015.%h6p://dx.doi.org/10.5306/wjco.v2.i1.64%

5.
Our research target
X"ray&CT Our&target DOT
No%Radia`on
Sharpness%
%
%
%
%
( )
Sca6ering No Yes So%much
Method Radon%trans. Ours PDE%by%FEM
h6p://www.sciencekids.co.nz/pictures/humanbody/braintomography.html
Lee,%Kijoon.%“Op`cal%Mammography:%Diffuse%Op`cal%Imaging%of%Breast%Cancer.”%World%Journal%of%Clinical%
Oncology%2.1%(2011):%64–72.%PMC.%Web.%8%June%2015.%h6p://dx.doi.org/10.5306/wjco.v2.i1.64%

6.
h6p://photosozaiVdatabase.com/modules/photo/photo.php?lid=664%
Sca6ering

7.
Volumetric%sca6ering%in%CG
1987
1993
1994
Tomoyuki%Nishita,%Yasuhiro%Miyawaki,%and%Eihachiro%Nakamae.%
1987.%A%shading%model%for%atmospheric%sca6ering%considering%
luminous%intensity%distribu`on%of%light%sources.%SIGGRAPH(Comput.(
Graph.%21,%4%(August%1987),%303V310.%DOI=10.1145/37402.37437%
h6p://doi.acm.org/10.1145/37402.37437%
Tomoyuki%Nishita,%Takao%Sirai,%Katsumi%Tadamura,%and%Eihachiro%Nakamae.%1993.%Display%of%the%earth%
taking%into%account%atmospheric%sca6ering.%In%Proceedings(of(the(20th(annual(conference(on(Computer(
graphics(and(interac>ve(techniques%(SIGGRAPH%'93).%ACM,%New%York,%NY,%USA,%175V182.%
DOI=10.1145/166117.166140%h6p://doi.acm.org/10.1145/166117.166140%
Tomoyuki%Nishita%and%Eihachiro%Nakamae.%1994.%Method%of%displaying%op`cal%effects%within%water%using%
accumula`on%buffer.%In%Proceedings(of(the(21st(annual(conference(on(Computer(graphics(and(interac>ve(
techniques%(SIGGRAPH%'94).%ACM,%New%York,%NY,%USA,%373V379.%DOI=10.1145/192161.192261%h6p://
doi.acm.org/10.1145/192161.192261%
Copyright%©%2011%by%the%Associa`on%for%Compu`ng%Machinery,%Inc.%(ACM).%

8.
Tomoyuki%Nishita,%Yoshinori%Dobashi,%and%Eihachiro%Nakamae.%1996.%Display%of%clouds%taking%into%account%
mul`ple%anisotropic%sca6ering%and%sky%light.%In%Proceedings(of(the(23rd(annual(conference(on(Computer(
graphics(and(interac>ve(techniques%(SIGGRAPH%'96).%ACM,%New%York,%NY,%USA,%379V386.%
DOI=10.1145/237170.237277%h6p://doi.acm.org/10.1145/237170.237277%
Yonghao%Yue,%Kei%Iwasaki,%BingVYu%Chen,%Yoshinori%Dobashi,%and%Tomoyuki%Nishita.%2010.%Unbiased,%
adap`ve%stochas`c%sampling%for%rendering%inhomogeneous%par`cipa`ng%media.%ACM(Trans.(Graph.%29,%6,%
Ar`cle%177%(December%2010),%8%pages.%In%ACM(SIGGRAPH(Asia(2010(papers%(SIGGRAPH%ASIA%'10).%
DOI=10.1145/1882261.1866199%h6p://doi.acm.org/10.1145/1882261.1866199%
Volumetric%sca6ering%in%CG
1996
2010
Copyright%©%2011%by%the%Associa`on%for%Compu`ng%Machinery,%Inc.%(ACM).%

9.
Computer vision approach
to the inverse problem
Observed%
image
Forward%model%
Generated%by%Computer%Graphics%
Image
Computer%Graphics
GenerationAnalysis
Computer%Vision
Model%parameters%
2
minimizing%observa`ons%and%model%predic`on

10.
Vision / CG approaches
University of British Columbia
oto of an acquisition rig for fluid phenomena, consisting of 5–16 strobe-synchronized con
ne of the cameras. Right: Reconstruction of an unsteady two-phase flow at different points in
e introduced in this paper manages to capture the fine-scale features in this challenging datas
me, 6 min/frame.
stic Tomography
n 3D Imaging of Mixing Fluids
Matthias B. Hullin Wolfgang Heidrich
y of British Columbia
mena, consisting of 5–16 strobe-synchronized consumer cameras. Middle:
of an unsteady two-phase flow at different points in time. Note how the novel
ture the fine-scale features in this challenging dataset. SAD regularizer, 100M
tur-
ight
stic
We
ial-
ude
gh-
ally,
can
hout
1 Introduction
The capture of dynamic 3D phenomena has been the subject of con-
siderable research in computer graphics, extending to both the scan-
ning of deformable shapes, such as human bodies (e.g. [de Aguiar
et al. 2007; de Aguiar et al. 2008]), faces (e.g. [Bickel et al. 2007;
Alexander et al. 2009]), and garments (e.g. [White et al. 2007;
Bradley et al. 2008]), as well as natural phenomena including liquid
surfaces [Ihrke and Magnor 2004; Wang et al. 2009], gases [Atch-
eson et al. 2008], and flames [Ihrke and Magnor 2004]. Access
to this kind of data is not only useful for direct re-rendering, but
also for deepening our understanding of a specific phenomenon.
The data can be used to derive heuristic or data-driven models, or
simply to gain a qualitative understanding of what a phenomenon
reconstructed view (60◦) 90◦
multiplication flame sheet blobs input
torch” dataset. The same two input images were used for all methods.
ews of the reconstructed flame are shown.
prove useful in more general contexts. This includes sparse-
view tomography problems in medical diagnostics [17] and
0◦ reconstructed view (30◦) reconstructed view (60◦) 90◦
input multiplication flame sheet blobs multiplication flame sheet blobs input
Figure 7. Views of three different reconstructions of the “torch” dataset. The same two input images were used for all methods.
Since all methods reconstruct the input views, only novel views of the reconstructed flame are shown.
reconstruction method (views)
RMS error
input all
prove useful in more general contexts. This includes sparse-
view tomography problems in medical diagnostics [17] and
reconstructed view (30◦) reconstructed view (60◦) 90◦
multiplication flame sheet blobs multiplication flame sheet blobs input
Views of three different reconstructions of the “torch” dataset. The same two input images were used for all methods.
methods reconstruct the input views, only novel views of the reconstructed flame are shown.
ruction method (views)
RMS error
input all
prove useful in more general contexts. This includes sparse-
view tomography problems in medical diagnostics [17] and
ew (30◦) reconstructed view (60◦) 90◦
eet blobs multiplication flame sheet blobs input
uctions of the “torch” dataset. The same two input images were used for all methods.
ws, only novel views of the reconstructed flame are shown.
RMS error
nput all
prove useful in more general contexts. This includes sparse-
view tomography problems in medical diagnostics [17] and
90°0° 30° 60°
input inputreconstructed
Gregson+%2012TOG%
Hasinoff%and%Kutulakos%2007PAMI%
No%sca6ering
Figure 2: Eight images captured during a single sweep of the laser plane.
off the mirror, the laser is spread into a vertical sheet of light by the
cylindrical lens. This sheet of light creates an illuminated plane of
smoke, which the camera views from a perpendicular direction.
During scanning, the galvanometer sweeps the laser sheet from
the front to the back of the smoke volume, while the camera cap-
tures images of the scattered light. The galvanometer then rapidly
jumps back to the front of the smoke volume and the process is re-
peated. In this work we capture 200 images during each volume
scan. At 5000 frames per second, this allows the volume to be
scanned at 25 Hz. We placed both the camera and galvanometer 3
meters from the working volume, and chose the camera lens, cylin-
drical lens, and galvanometer scan angle so that the camera field
of view and laser frustum were both 6 . This provided a working
volume of approximately 30x30x30 cm.
The galvanometer control software also triggers the camera, al-
lowing the frame capture to be synchronized with the position of
the laser slice. This assures that corresponding slices in successive
captured volumes are sampled at the same position.
The high frame rate and the small fraction of the total light scat-
tered to the camera require a relatively powerful laser. We use an
argon ion laser that produces 3 watts of light at a set of wavelengths
closely clustered around 500 nm.
3.1 Calibration
the laser scanner. Repeating this for another dot gives another such
line, and intersecting the two lines determines Lc.
Similarly, any two identified points on the laser line for a given
slice s, together with Lc, determines the plane Ps of illumination.
We observed that the intensity of illumination varied as a func-
tion of slice s and of elevation angle f within each laser slice. To
correct for this, we placed a uniform diffuse plane at an angle where
it was visible to both the laser and the camera and captured a se-
quence of frames. As the laser scans this reflectance standard, the
observed intensities are directly proportional to the intensity of laser
light for each laser plane and pixel position. Having calibrated the
laser planes Ps and the camera position, we can find the 3D point
v corresponding to each laser plane and pixel position, and apply
the appropriate intensity compensation when computing densities
along the ray from the laser position Lc through the point v. Al-
though this corrects for the relative intensity differences across the
laser frustum, we should note that we do not measure the absolute
intensity of the laser. The resulting density fields are therefore only
recovered up to an unknown scale factor.
We have assumed that image intensity values can be directly in-
terpreted as density values. This assumes that the effects of extinc-
tion and multiple scattering within the volume are negligible. Al-
though we found that this assumption held for the smoke volumes
we captured, this assumption will fail when the densities become
very high such as for thick smoke or dyes in water, or when the
Smoke%only
Hawkins+%2005TOG%An Inverse Problem Approach for Automatically Adjusting t
Rendering Clouds Using Photographs
Yoshinori Dobashi1,2
Wataru Iwasaki1
Ayumi Ono1
Tsuyoshi Yamamoto1
Yonghao
1
Hokkaido University 2
JST, CREST 3
The University
(a) (b)
(d) (e)
Figure 1: Examples of our method. The parameters for rendering clouds are estimated from the real phot
the top left corner of each image. The synthetic cumulonimbus clouds are rendered using the estimated par
Abstract
Clouds play an important role in creating realistic images of out-
door scenes. Many methods have therefore been proposed for dis-
playing realistic clouds. However, the realism of the resulting im-
ages depends on many parameters used to render them and it is
often difficult to adjust those parameters manually. This paper pro-
Links: DL PDF
1 Introduction
Clouds are important elements w
door scenes to enhance realism.
ten employed and the intensities
into account the scattering and ab
realistic clouds. However, one of
Dobashi+%2012TOG%
Narasimhan+%2006SIGGRAPH
Gkioulekasj+%2013SIGGRAPHAsia%
Appears in the SIGGRA
Inverse Volume Renderin
Ioannis Gkioulekas
Harvard University
Shuang Zhao
Cornell University
Kavi
Cornell
Figure 1: Acquiring scattering parameters. Left: Samples of two m
Uniform%parameter%es`ma`on

11.
INTRODUCTION TO
SCATTERING

12.
Scattering model
• Volumetric%sca6ering
Par`cipa`ng%medium%
(fog,%cloud,%smoke,%etc)%

13.
Types of Phase function
Forward%sca6ering
Backward%sca6ering
Phase%func`on
Isotropic%sca6ering%
input
Output%distribu`on

14.
Scattering model
Phase%func`on

15.
Rendering for CG
the%line%of%sight
screen
Numerical%integra`on%along

16.
Light in participating medium
A6enua`on
Absorp`on
OutVsca6ering
InVsca6ering
Mul`ple%sca6ering
Emission
Usually%ignored

17.
Absorption
(Microscopic)%BeerVLambert%Law
Absorp`on%coefficient% Infinitesimal%
path%length%
(Macroscopic)%BeerVLambert%Law
Integra`on%
from%%0%to%d
or
Exponen`al%a6enua`on
When%σa%is%
not%constant
op`cal%depth%
h6p://www.sciencekids.co.nz/pictures/humanbody/braintomography.html
CT:%es`ma`ng%
%XVRay%absorp`on

18.
Out-scattering
(Microscopic)%BeerVLambert%Law
Sca6ering%coefficient% Infinitesimal%
path%length%
(Macroscopic)%BeerVLambert%Law
Integra`on%
from%%0%to%d
or
Exponen`al%a6enua`on

19.
Attenuation = absorption + out-scattering
Absorp`on
OutVsca6ering
A6enua`on
Ex`nc`on%coefficient%
Exponen`al%a6enua`on

20.
Emission
Emission
Absorp`on%coefficient%
(Emission%of%absorbed%energy)%
Emi6ed%light%

21.
In-scattering
Phase%func`on%
(from%ω’%to%ω%at%x)%
(usually%common%for%all%x)
Incident%light%
(from%ω’%at%x%)
Integra`ng%all%incoming%lights%over%the%sphere
or

22.
Rendering equation
Light%transport%equa`on%
Volume%rendering%equa`on%(differen`al%form)
InVsca6ering%termA6enua`on%term

23.
DOT with Diffuse Approximation
InVsca6ering%termA6enua`on%termTimeVresolved%
observa`on
PDE%solved%by%FEM
No%forward%
sca6ering
Blurred%results
Problem
Diffuse%Approxima`on%
(first%order%spherical%
harmonics%approxima`on)
h6p://commons.wikimedia.org/wiki/File:Harmoniki.png
Main assumption

24.
Rendering equation
Light%transport%equa`on%
Volume%rendering%equa`on%(integral%form)
InVsca6ering%term
A6enua`on%
term
A6enua`on%of%
sca6ered%light

25.
Light Paths:
from light sources to cameras
Light%source
Camera%/%eye

26.
Light Paths:
from light sources to cameras
Light%source
Camera%/%eye

27.
Path (Ray) tracing:
from cameras to light source
Light%source
Camera%/%eye
Paths

28.
Path integral approach
introduced%by%Eric%Veach%(1997%Ph.D%thesis)
h6p://graphics.stanford.edu/papers/veach_thesis/
Pixel%value%of%pixel%j
Paths%of%length%kAll%possible%
paths%of%length%k
2
Path%measure
Product%of%measures%at%each%vertex%
All%possible%paths%
of%any%length% (path%space)
Monte%Carlo%
Integra`on

29.
Path integral for participating media
introduced%by%Mark%Pauly%(1999%Master%thesis)
h6p://graphics.stanford.edu/~mapauly/publica`ons.html
Pixel%value%of%pixel%j
measure
Area%measure%if%surface
Voxel%measure%if%volume
Metropolis%Light%Transport%for%Par`cipa`ng%Media%
Mark%Pauly,%Thomas%Kollig,%Alexander%Keller%
Eurographics%Workshop%on%Rendering,%2000%

30.
Details of paths in participating media
Anders%Wang%Kristensen,%Efficient%Unbiased%Rendering%using%Enlightened%Local%Path%Sampling,%Ph.D%thesis,%2010%URL%
A6enua`on
Geometric%term
BRDF%/%sca6ering%coeff,%phase%func`on
surface
inside
Cos%term
All%paths%of%length%k%
measure
Camera%response%
func`on
To%be%es`mated!

31.
PROPOSED METHOD
TO THE INVERSE PROBLEM

32.
Discretization. How?
Ex`nc`on%coefficient%
at%voxel%b
Length%that%
the%path%xiV>xi+1%passes%voxel%b%
Z 1
0
t((1 s)xi + sxi+1)ds ⇡
X
b
t[b] dxi,xi+1
[b] = t · dxi,xi+1
Z 1
0
t((1 s)xi + sxi+1)ds ⇡
X
b
t[b] dxi,xi+1
[b] = t · dxi,xi+1
Z 1
0
t((1 s)xi + sxi+1)ds ⇡
X
b
t[b] dxi,xi+1
[b] = t · dxi,xi+1
%vectorize
Sparse%vector%of%coefficients
Sparse%vector%of%path%length
Inner%product%
of%sparse%vectors

33.
Discretization. How?
Ex`nc`on%coefficient%
at%voxel%b
Length%that%
the%path%xiV>xi+1%passes%voxel%b%
Z 1
0
t((1 s)xi + sxi+1)ds ⇡
X
b
t[b] dxi,xi+1
[b] = t · dxi,xi+1
I =
1X
k=1
X
⌦k
Le(x0, x1)
e t·dx0,x1
kx0 x1k2
"k 1Y
i=1
s(xi)fp(xi 1, xi, xi+1)
e t·dxi,xi+1
kxi xi+1k2
#
We(xk 1, xk)dµk
I =
1X
k=1
X
⌦k
Le(x0, x1)
e
R 1
0 t((1 s)x0+sx1)ds
kx0 x1k2
"k 1Y
i=1
s(xi)fp(xi 1, xi, xi+1)
e
R 1
0 t((1 s)xi+sxi+1)ds
kxi xi+1k2
#
We(xk 1, xk)dµk
Z 1
0
t((1 s)xi + sxi+1)ds ⇡
X
b
t[b] dxi,xi+1
[b] = t · dxi,xi+1
Z 1
0
t((1 s)xi + sxi+1)ds ⇡
X
b
t[b] dxi,xi+1
[b] = t · dxi,xi+1
before
a|er

34.
Addition of Vectorized paths
I =
1X
k=1
Le(x 1, x0)
X
⌦k
"k 1Y
i=1
s(xi)fp(xi 1, xi, xi+1)
# "
e t·(
Pk 1
i=0 dxi,xi+1
)
Qk 1
i=0 kxi xi+1k2/dVi
#
dA 1 dVk dAk+1
I =
1X
k=1
Le(x 1, x0)
X
⌦k
Hk e t·Dk
I =Le(x 1, x0)
X
⌦
Hk e t·Dk
I =
1X
k=1
X
⌦k
Le(x0, x1)
e t·dx0,x1
kx0 x1k2
"k 1Y
i=1
s(xi)fp(xi 1, xi, xi+1)
e t·dxi,xi+1
kxi xi+1k2
#
We(xk 1, xk)dµk
Assumed%to%be%known
Everything%else

35.
The forward model
x_kV1
x_k
x1
x0
Source%loca`on%s%
Is,t =Le
X
⌦
Hk(s,t) e t·Dk(s,t)
Observa`on%
loca`on%t%
Observa`on%at%s,t
x_kV1
x_k
x1
x0
Source%loca`on%s%
Observa`on%
loca`on%t%
x_kV1
x_k
x1
x0
Source%loca`on%s%
Observa`on%
loca`on%t%

36.
The inverse problem
t 0subject%to
Coefficient%must%be%posi`vemodelobserva`on
nstrained problem: Interior point
introduce a constrained optimization with inequality constraints of the form;
min
t
f0( t) subject to fi(x)  0, i = 1, · · · , m, (37)
t 2 <N⇥M
is a real vector and f0, · · · , fm : <N⇥M
! < are twice continuously differen-
dea is to approximate it as an unconstrained problem. Using Lagrange multipliers, we can
rite problem (37) as
min
t
f0( t) +
mX
i=1
I(fi( t)), (38)
Solved%by%the%logVbarrier%interior%point%method
Cost%func`on%with%constraints
The idea is to approximate it as an unconstrained problem. Using La
first rewrite problem (37) as
min
t
f0( t) +
mX
i=1
I(fi( t)),
where I : < ! < is an indicator function which keeps the solution insid
I(f) =
(
0, f  0
1, f > 0.
The problem (38) now has no inequality constraints, while it is not diffe
The barrier method26
is an interior point method which introduces a lo
to approximate the indicator function I as follows;
ˆI(f) = (1/t) log( f),
where t > 0 is a parameter to adjust the accuracy of approximation. The
to infinity rapidly as f goes close to 0 while it is close to 0 when f are fa
is differentiable, we have
min
t
f0( t) +
mX
i=1
(1/t) log( fi( t)),
or equivalently,
mX
Update%t%(larger)
Ini`al%value t 0
repeat
Solve
LogVbarrier%func`ons
by%QuasiVNewton

37.
Layer
2D3D
Target model
2D
Smallest%cost%
Rough%approxima`on%
Ideal%
Costly%in%space%and%`me
Second%op`on%
Less%expensive%
but%s`ll%large%cost

38.
Proposed 2D layered model
• Light%sca6ers:%
– Layer%by%layer%
– At%voxel%centers%
– Forward%only%
• Subject%to%a%simplified%
phase%func`on%
– With%uniform/known%
sca6ering%coefficients

39.
Enumerating paths
Source%loca`on%s%
Observa`on%
loca`on%t%
Source%loca`on%s%
Observa`on%
loca`on%t%
Exhaus`ve%(with%thresholding) (BiVdirec`onal)%Monte%CarloVlike

40.
Different directions
he pro-
material
a). The
ficients
coeffi-
t much
to lit a
etector
s from
unction
s set to
Figure
extinc-
gray) is
in the
se part
(voxels
blurred
mental
(a) (b)
(c) (d)
IV. NUMERICAL SIMULATIONS
A. Simple material
We describe numerical simulation results of the pro-
posed method for evaluation. A two-dimensional material
is divided into 10⇥10 voxels as shown in Figure 4(a). The
material has almost homogeneous extinction coefficients
of value 0.05 except few voxels with much higher coeffi-
cients of 0.2, which means those voxels absorb light much
more than other voxels. A light source is assumed to lit a
light at each of m positions to the top layer, and a detector
observes the outgoing light at each of m positions from
the bottom layer. The variance 2
of Gaussian function
(1) for scattering from one layer to the next layer is set to
2. The upper bound u in Eq. (8) is set to 1.
Estimated extinction coefficients are shown in Figure
4(b). The homogeneous part of the material with extinc-
tion coefficients of 0.05 (voxels shaded in light gray) is
reasonably estimated because estimated values fall in the
range between 0.04 to 0.06. In contrast, the dense part
of the material with extinction coefficients of 0.2 (voxels
shaded in dark gray) is not estimated well but looks blurred
vertically. This blur effect is due to the experimental
setting that the light paths go vertically from top to bottom
inside the material.
This effect can be verified by simply rotating the setting
by 90 degrees (now the configuration is a horizontal one)
and applying the proposed method. Figure 4(c) shows
the estimation result when the light source is on the left
(a) (b)
(c) (d)
Figure 4: Simulation results of a material of size 10 ⇥ 10.
(a) Ground truth of the distribution of extinction coeffi-
cients shown as values in each voxel as well as by voxel
colors. (b) Estimated extinction coefficients. A light source
and detector are above and blow the media. (c) Estimated
extinction coefficients. A light source and detector are left
IV. NUMERICAL SIMULATIONS
A. Simple material
We describe numerical simulation results of the pro-
posed method for evaluation. A two-dimensional material
is divided into 10⇥10 voxels as shown in Figure 4(a). The
material has almost homogeneous extinction coefficients
of value 0.05 except few voxels with much higher coeffi-
cients of 0.2, which means those voxels absorb light much
more than other voxels. A light source is assumed to lit a
light at each of m positions to the top layer, and a detector
observes the outgoing light at each of m positions from
the bottom layer. The variance 2
of Gaussian function
(1) for scattering from one layer to the next layer is set to
2. The upper bound u in Eq. (8) is set to 1.
Estimated extinction coefficients are shown in Figure
4(b). The homogeneous part of the material with extinc-
tion coefficients of 0.05 (voxels shaded in light gray) is
reasonably estimated because estimated values fall in the
range between 0.04 to 0.06. In contrast, the dense part
of the material with extinction coefficients of 0.2 (voxels
shaded in dark gray) is not estimated well but looks blurred
vertically. This blur effect is due to the experimental
setting that the light paths go vertically from top to bottom
inside the material.
This effect can be verified by simply rotating the setting
by 90 degrees (now the configuration is a horizontal one)
and applying the proposed method. Figure 4(c) shows
(a) (b)
(c) (d)
Figure 4: Simulation results of a material of size 10 ⇥ 10.
(a) Ground truth of the distribution of extinction coeffi-
cients shown as values in each voxel as well as by voxel
colors. (b) Estimated extinction coefficients. A light source
IV. NUMERICAL SIMULATIONS
A. Simple material
We describe numerical simulation results of the pro-
posed method for evaluation. A two-dimensional material
is divided into 10⇥10 voxels as shown in Figure 4(a). The
material has almost homogeneous extinction coefficients
of value 0.05 except few voxels with much higher coeffi-
cients of 0.2, which means those voxels absorb light much
more than other voxels. A light source is assumed to lit a
light at each of m positions to the top layer, and a detector
observes the outgoing light at each of m positions from
the bottom layer. The variance 2
of Gaussian function
(1) for scattering from one layer to the next layer is set to
2. The upper bound u in Eq. (8) is set to 1.
Estimated extinction coefficients are shown in Figure
4(b). The homogeneous part of the material with extinc-
tion coefficients of 0.05 (voxels shaded in light gray) is
reasonably estimated because estimated values fall in the
range between 0.04 to 0.06. In contrast, the dense part
of the material with extinction coefficients of 0.2 (voxels
shaded in dark gray) is not estimated well but looks blurred
vertically. This blur effect is due to the experimental
setting that the light paths go vertically from top to bottom
inside the material.
This effect can be verified by simply rotating the setting
by 90 degrees (now the configuration is a horizontal one)
and applying the proposed method. Figure 4(c) shows
the estimation result when the light source is on the left
side of the material, and the detector is on the right
side. The estimated extinction coefficients are blurred now
horizontally because light paths go from left to right.
A simple trick to integrate these two results is to take
the average of them, which is shown in Figure 4(d). The
(a) (b)
(c) (d)
Figure 4: Simulation results of a material of size 10 ⇥ 10.
(a) Ground truth of the distribution of extinction coeffi-
cients shown as values in each voxel as well as by voxel
colors. (b) Estimated extinction coefficients. A light source
and detector are above and blow the media. (c) Estimated
extinction coefficients. A light source and detector are left
and right sides of the media. (d) Average of (b) and (c).
Cost function values over iterations are shown in Figure
Averaging
Ver`cal%
es`mates%
Horizontal%
Es`mates%Toru%Tamaki,%Bingzhi%Yuan,%Bisser%Raytchev,%Kazufumi%Kaneda,%Yasuhiro%
Mukaigawa,%"Mul`pleVsca6ering%Op`cal%Tomography%with%Layered%Material,"%
The%9th%Interna`onal%Conference%on%SignalVImage%Technology%and%InternetV
Based%Systems%(SITIS%2013),%December%2V5,%2013,%Kyoto,%Japan.%(2013%12)%

41.
Joint optimization
he pro-
material
a). The
ficients
coeffi-
t much
to lit a
etector
s from
unction
s set to
Figure
extinc-
gray) is
in the
se part
(voxels
blurred
mental
(a) (b)
(c) (d)
Bingzhi%Yuan,%Toru%Tamaki,%Takahiro%Kushida,%Bisser%Raytchev,%Kazufumi%Kaneda,%Yasuhiro%Mukaigawa%and%
Hiroyuki%Kubo,%"Layered%op`cal%tomography%of%mul`ple%sca6ering%media%with%combined%constraint%
op`miza`on",%in%Proc.%of%FCV%2015%:%21st%KoreaVJapan%joint%Workshop%on%Fron`ers%of%Computer%Vision,%January%
28VV30,%2015,%Shinan%Beach%Hotel,%Mokpo,%KOREA.%

42.
results of numerical simulation
Ground%truth
(a) (b) (c) (d) (e)
(a) (b) (c) (d) (e)
• 20x20%grid%of%20[mm]x20[mm]%
• Ex`nc`on%coefficients:%between%1.05%and%1.55%[mmV1]%
• Sca6ering%coefficients%(known):%1%[mmV1]%
• Phase%func`on%(known):%a%simplified%approxima`on
• Fixed%742%paths%for%each%s,t%
(exhaus`ve%with%th.)%
• Joint%op`miza`on%with%
interior%point%method%
Es`mated%results

43.
Shepp-Logan phantom
20%x%20

44.
Comparison with DOT
GT
Ours
DOT (GN)
DOT (PD)
(a) (b) (c) (d) (e)
ig 11 Numerical simulation results for a grid of size 24 ⇥ 24 with different inverse solver. Darker shades of gra
Ground%
Truth
Ours
DOT%
(GaussVNewton)
DOT%
(PrimalVDual%interior%point)
• 24x24%grid%of%24[mm]x24[mm]%
• 24x24x2%=%1152%triangle%meshes%for%FEM%
• 16%light%sources%and%16%detectors%around%the%square%
• With%EIDORS%(Electrical%Impedance%
Tomography%and%Diffuse%Op`cal%Tomography%
Reconstruc`on%So|ware)%
eidors3d.sourceforge.net%

45.
Accuracy and cost
(a) (b) (c) (d) (e)
RMSE
Ours 0.007662 0.01244 0.026602 0.021442 0.051152
2
= 0.4 (0.730%) (1.18%) (2.53%) (2.04%) (4.87%)
DOT (GN) 0.053037 0.060597 0.7605 0.059534 0.0855
(5.05%) (5.77%) (7.53%) (5.67%) (8.14%)
DOT (PD) 0.052466 0.0626 0.081081 0.066042 0.080798
(5.25%) (5.97%) (8.11%) (6.60%) (8.08%)
Computation
time [s]
Ours
257 217 382 306 5042
= 0.4
DOT (GN) 0.397 0.390 0.407 0.404 0.453
DOT (PD) 1.11 1.09 1.14 1.08 1.15
Table 2 RMSEs and computation time of the numerical simulations for five different media (a)–(e) with grid size of
4 ⇥ 24 with our proposed method and DOT with two solvers. Numbers in the brackets are relative errors of RMSE
o the background extinction coefficient values (i.e., 1.05).
GT
2 = 0.2
2 = 0.4
(a) (b) (c) (d) (e)
Fig 7 Numerical simulation results for a grid of size 20 ⇥ 20. Darker shades of gray represent larger values (more
light is absorbed at the voxel). The graybar shows the preojetcion from gray scale to value. The first row shows the
groundtruth of the five different media (a)–(e) used for the simulation. The second row shows the estimated results
when the 2
in the Gaussian phase function equals to 0.2. The last row shows the estimated results when the 2
= 0.4.
in the middle and bottom rows, and the root-mean-squared error (RMSE) shown in Table 5 is small
enough with 0.0075/1.05 = 0.7% of coefficient values. The other media, shown in columns (b)–
(e), have more complex distributions of the extinction coefficients. In the top part of Table 5, we
use the RMSEs to show the quality of the estimated results. The ratio in the brackets shows the
relative magnitude of RMSE compared with the majority extinction value or the background in the
Fig 7 Numerical simulation results for a grid of size 20⇥20. Darker shades of gray represent larger values (more light
is absorbed at the voxel). The bars on the side shows extinction coefficient values in greyscale. The first row shows
the ground truth of the five different media (a)–(e) used for the simulation. The second and third rows show estimated
results for 2 = 0.2 and 2 = 0.4, respectively, of the Gaussian phase function.
much higher coefficients of 1.2 (voxels shaded in dark gray), which means that those voxels ab-
sorb much more light than other voxels. The coefficients are estimated reasonably well as shown
in the middle and bottom rows, and the root-mean-squared error (RMSE) shown in Table 1 is small
enough with the relative error of 0.0075/1.05 = 0.7% to the background coefficient value. The
Our%implementa`on:%MATLAB

46.
Summary
• Op`cal%tomography%for%sca6ering%media%
• Observa`on%model:%path%integral%
• LeastVsquare%approach%with%constraints%
• Solver:%interior%point%method%
• Experiments:%numerical%simula`on%
• Future%work:%a%lotGT
GT
2 = 0.2