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Scattering tomography with path integral

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Abstract: Tomography, an inverse problem to recover inside an object by probing light and observing its output, is an important issue in physics, medical imaging, computer vision and related research fields. In this talk, we will present an approach to optical scattering tomography that uses the path integral, which has been developed for volume rendering, to model the light transport. We make assumptions on a specific scattering model to facilitate the computation, then formulate an inverse problem to be solved by an interior point method. We will show some simulation results, comparisons with diffusion optical tomography (DOT), and conclude with future directions.

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Scattering tomography with path integral

  1. 1. Scattering tomography with path integral Toru%Tamaki%(Hiroshima%University)% % % Joint%work%with% Bingzhi%Yuan,%Yasuhiro%Mukaigawa,% and%many%others%
  2. 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. 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. 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. 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. 6. h6p://photosozaiVdatabase.com/modules/photo/photo.php?lid=664% Sca6ering
  7. 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. 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. 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. 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. 11. INTRODUCTION TO SCATTERING
  12. 12. Scattering model •  Volumetric%sca6ering Par`cipa`ng%medium% (fog,%cloud,%smoke,%etc)%
  13. 13. Types of Phase function Forward%sca6ering Backward%sca6ering Phase%func`on Isotropic%sca6ering% input Output%distribu`on
  14. 14. Scattering model Phase%func`on
  15. 15. Rendering for CG the%line%of%sight screen Numerical%integra`on%along
  16. 16. Light in participating medium A6enua`on Absorp`on OutVsca6ering InVsca6ering Mul`ple%sca6ering Emission Usually%ignored
  17. 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. 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. 19. Attenuation = absorption + out-scattering Absorp`on OutVsca6ering A6enua`on Ex`nc`on%coefficient% Exponen`al%a6enua`on
  20. 20. Emission Emission Absorp`on%coefficient% (Emission%of%absorbed%energy)% Emi6ed%light%
  21. 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. 22. Rendering equation Light%transport%equa`on% Volume%rendering%equa`on%(differen`al%form) InVsca6ering%termA6enua`on%term
  23. 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. 24. Rendering equation Light%transport%equa`on% Volume%rendering%equa`on%(integral%form) InVsca6ering%term A6enua`on% term A6enua`on%of% sca6ered%light
  25. 25. Light Paths: from light sources to cameras Light%source Camera%/%eye
  26. 26. Light Paths: from light sources to cameras Light%source Camera%/%eye
  27. 27. Path (Ray) tracing: from cameras to light source Light%source Camera%/%eye Paths
  28. 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. 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. 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. 31. PROPOSED METHOD TO THE INVERSE PROBLEM
  32. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 43. Shepp-Logan phantom 20%x%20
  44. 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. 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. 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

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