Scattering optical tomography with discretized path integral

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 scanning 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, cylindrical 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 scattered 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 function 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 extinction 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 displaying realistic clouds. However, the realism of the resulting images 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; Î(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 proposed 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 coefficients 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 extinction 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 coefficients 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 proposed 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 coefficients 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 extinction 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 coefficients 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 proposed 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 coefficients 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 extinction 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 coefficients 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 absorb 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 object 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 object 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