Slide of the talks: Toru Tamaki, Scattering tomography with path integral, Séminaire A3SI (Algorithmes, architectures, analyse et synthèse d’images), Laboratoire d'Informatique Gaspard-Monge (LIGM), ESIEE Paris, Université Paris-Est, 11-June-2015. Toru Tamaki, Scattering optical tomography with discretized path integral, Fachbereich Computerwissenschaften, Universität Salzburg, Austria, 03-December-2015. Toru Tamaki, Scattering optical tomography with discretized path integral, Departamento de Ciências da Informação e da Decisão em Saúde Faculdade de Medicina, Universidade do Porto, Porto, Portugal, 11-December-2015.

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

2. Computed Tomography
Multiplexing Radiography for Ultra-Fast Computed Tomography
J. Zhang, G. Yang, Y. Lee, S. Chang, J. Lu, O. Zhou University of North Carolina at Chapel
Hill https://www.aapm.org/meetings/07AM/VirtualPressRoom/LayLanguage/IIMultiplexing.asp
http://www.sciencekids.co.nz/pictures/humanbody/braintomography.html

3. Different Modalities
CT : http://www.hiroshima-u.ac.jp/news/show/id/13475/dir_id/19
PET : http://www.hiroshima-u.ac.jp/hosp/hitokuchi/p_vdo8u6.html
MRI : http://www.hiroshima-u.ac.jp/news/show/id/14427/dir_id/19
CT
(X Ray)
MRI
(Magnetic field)
PET
(Positron) (infrared)

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

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

6. http://photosozai-database.com/modules/photo/photo.php?lid=664
Scattering

7. Volumetric scattering in CG
1987
1993
1994
Tomoyuki Nishita, Yasuhiro Miyawaki, and Eihachiro Nakamae.
1987. A shading model for atmospheric scattering considering
luminous intensity distribution of light sources. SIGGRAPH Comput.
Graph. 21, 4 (August 1987), 303-310. DOI=10.1145/37402.37437
http://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 scattering. In Proceedings of the 20th annual conference on Computer
graphics and interactive techniques (SIGGRAPH '93). ACM, New York, NY, USA, 175-182.
DOI=10.1145/166117.166140 http://doi.acm.org/10.1145/166117.166140
Tomoyuki Nishita and Eihachiro Nakamae. 1994. Method of displaying optical effects within water using
accumulation buffer. In Proceedings of the 21st annual conference on Computer graphics and interactive
techniques (SIGGRAPH '94). ACM, New York, NY, USA, 373-379. DOI=10.1145/192161.192261
http://doi.acm.org/10.1145/192161.192261
Copyright © 2011 by the Association for Computing Machinery, Inc. (ACM).

8. Tomoyuki Nishita, Yoshinori Dobashi, and Eihachiro Nakamae. 1996. Display of clouds taking into account
multiple anisotropic scattering and sky light. In Proceedings of the 23rd annual conference on Computer
graphics and interactive techniques (SIGGRAPH '96). ACM, New York, NY, USA, 379-386.
DOI=10.1145/237170.237277 http://doi.acm.org/10.1145/237170.237277
Yonghao Yue, Kei Iwasaki, Bing-Yu Chen, Yoshinori Dobashi, and Tomoyuki Nishita. 2010. Unbiased,
adaptive stochastic sampling for rendering inhomogeneous participating media. ACM Trans. Graph. 29, 6,
Article 177 (December 2010), 8 pages. In ACM SIGGRAPH Asia 2010 papers (SIGGRAPH ASIA '10).
DOI=10.1145/1882261.1866199 http://doi.acm.org/10.1145/1882261.1866199
Volumetric scattering in CG
1996
2010
Copyright © 2011 by the Association for Computing 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 observations and model prediction

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
900 30 60
input inputreconstructed
Gregson+ 2012TOG
Hasinoff and Kutulakos 2007PAMI
No scattering
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+ 2005TOGAn 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 estimation

11. INTRODUCTION TO
SCATTERING

12. Scattering model
• Volumetric scattering
Participating medium
(fog, cloud, smoke, etc)

13. Types of Phase function
Forward scattering
Backward scattering
Phase function
Isotropic scattering
input
Output distribution

14. Scattering model
Phase function

15. Rendering for CG
the line of sight
Numerical integration along

16. Light in participating medium
Attenuation
Absorption
Out-scattering
In-scattering
Multiple scattering
Emission
Usually ignored

17. Absorption
(Microscopic) Beer-Lambert Law
Absorption coefficient Infinitesimal
path length
(Macroscopic) Beer-Lambert Law
Integration
from 0 to d
or
Exponential attenuation
When σa is
not constant
optical depth
http://www.sciencekids.co.nz/pictures/humanbody/braintomography.html
CT: estimating
X-Ray absorption

18. Out-scattering
(Microscopic) Beer-Lambert Law
Scattering coefficient Infinitesimal
path length
(Macroscopic) Beer-Lambert Law
Integration
from 0 to d
or
Exponential attenuation

19. Attenuation = absorption + out-scattering
Absorption
Out-scattering
Attenuation
Extinction coefficient
Exponential attenuation

20. Emission
Emission
Absorption coefficient
(Emission of absorbed energy)
Emitted light

21. In-scattering
Phase function
(from ω’ to ω at x)
(usually common for all x)
Incident light
(from ω’ at x )
Integrating all incoming lights over the sphere
or

22. Rendering equation
Light transport equation
Volume rendering equation (differential form)
In-scattering termAttenuation term

23. DOT with Diffuse Approximation
In-scattering termAttenuation termTime-resolved
observation
PDE solved by FEM
No forward
scattering
Blurred results
Problem
Diffuse Approximation
(first order spherical
harmonics approximation)
http://commons.wikimedia.org/wiki/File:Harmoniki.png
Main assumption

24. Rendering equation
Light transport equation
Volume rendering equation (integral form)
In-scattering term
Attenuation
term
Attenuation of
scattered 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)
http://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
Integration

29. Path integral for participating media
introduced by Mark Pauly (1999 Master thesis)
http://graphics.stanford.edu/~mapauly/publications.html
Pixel value of pixel j
measure
Area measure if surface
Voxel measure if volume
Metropolis Light Transport for Participating 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
Attenuation
Geometric term
BRDF / scattering coeff, phase function
surface
inside
Cos term
All paths of length k
measure
Camera response
function
To be estimated!

31. PROPOSED METHOD
TO THE INVERSE PROBLEM

32. Discretization. How?
Extinction coefficient
at voxel b
Length that
the path xi->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?
Extinction coefficient
at voxel b
Length that
the path xi->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
after

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_k-1
x_k
x1
x0
Source location s
Is,t =Le
X
⌦
Hk(s,t) e t·Dk(s,t)
Observation
location t
Observation at s,t
x_k-1
x_k
x1
x0
Source location s
Observation
location t
x_k-1
x_k
x1
x0
Source location s
Observation
location t

36. The inverse problem
t 0subject to
Coefficient must be positivemodelobservation
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 log-barrier interior point method
Cost function 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)
Initial value t 0
repeat
Solve
Log-barrier functions
by Quasi-Newton

37. Layer
2D3D
Target model
2D
Smallest cost
Rough approximation
Ideal
Costly in space and time
Second option
Less expensive
but still large cost

38. Proposed 2D layered model
• Light scatters:
– Layer by layer
– At voxel centers
– Forward only
• Subject to a simplified
phase function
– With uniform/known
scattering coefficients

39. Enumerating paths
Source location s
Observation
location t
Source location s
Observation
location t
Exhaustive (with thresholding) (Bi-directional) Monte Carlo-like

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
Vertical
estimates
Horizontal
EstimatesToru Tamaki, Bingzhi Yuan, Bisser Raytchev, Kazufumi Kaneda, Yasuhiro
Mukaigawa, "Multiple-scattering Optical Tomography with Layered Material,"
The 9th International Conference on Signal-Image Technology and Internet-
Based Systems (SITIS 2013), December 2-5, 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 optical tomography of multiple scattering media with combined constraint
optimization", in Proc. of FCV 2015 : 21st Korea-Japan joint Workshop on Frontiers of Computer Vision, January
28--30, 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]
• Extinction coefficients: between 1.05 and 1.55 [mm-1]
• Scattering coefficients (known): 1 [mm-1]
• Phase function (known): a simplified approximation
• Fixed 742 paths for each s,t
(exhaustive with th.)
• Joint optimization with
interior point method
Estimated 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
(Gauss-Newton)
DOT
(Primal-Dual 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 Optical Tomography
Reconstruction Software)
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 implementation: MATLAB

46. Computation cost NOW!

47. Devices for Scattering Tomography
Y.Ishii, T.Arai, Y.Mukaigawa, J.Tagawa, Y.Yagi, "Scattering Tomography by Monte Carlo Voting", Proc. IAPR International Conference on Machine Vision Applications
(MVA2013), May. 2013.
ubstances
ght scattering
Bread
ifficult to detect foreign substances
use of the influence of light scattering
),( xT
), x
x
+ … +
Needle
),( xE
Campus & Location
ロゴタイプ・ロゴマーク／学旗／同窓会
学 旗
ロゴタイプ・ロゴマーク
ロゴタイプ ロゴマーク
s & Location
イプ・ロゴマーク／学旗／同窓会
ロゴマーク
ロゴタイプ ロゴマーク

48. Devices for Scattering Tomography
Ryuichi Akashi, Hajime Nagahara, Yasuhiro Mukaigawa and Rin-ichiro Taniguchi, Scattering Tomography Using Ellipsoidal Mirror, FCV2015.
. 8. Experimental results for an object with two obstacles. (Left) actual
figuration; (Middle) image from wrap-around viewing; (Right) image from
gle-direction viewing.
Camera
Projector
Ellipsoidal mirror
Fig. 11. Projector-camera system
attered by and passes through the target object is reflected
ck and directed toward the camera lens at the other focal
int. In consequence of this setup, we can observe scattering
(a) Slit board (b) Grid pattern
Fig. 12. (a) Slit board and (b) projected grid pattern.
Camera
Projector
Screen
Slit
Beam splitter
Fig. 13. Optical layout during alignment
uring aligning, we replaced the ellipsoidal mirror with
een and placed the slit board in front of the screen.
13 shows the optical layout during the camera-projector
(a) Before alignment
Fig. 14. Alignm
Fig. 15. Cylindrical scattering ob-
ject made of silicon resin with a
single metal obstacle.
Fig
wr
and metal wires as obstacles. Th
in shape with diameter of 27 mm
as the darkened filament inside o
We give an image for the estim
bution for this real object (Fig. 16
Fig. 10. Configuration of the optical system for wrap-around viewing
pattern
grid pattern.
Screen
Slit
(a) Before alignment (b) After alignment
Fig. 14. Alignment result
Fig. 15. Cylindrical scattering ob-
ject made of silicon resin with a Fig. 16. Image obtained using
Real object
Estimation

49. Summary
• Optical tomography for scattering media
• Observation model: path integral
• Least-square approach with constraints
• Solver: interior point method
• Experiments: numerical simulation
• Future work: a lotGT
GT
2 = 0.2