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1 Estimation of Shielding Object Distribution in Scattering Media by Analyzing Light Transport Shosei Moriguchi, 1 Yasuhiro Mukaigawa, 1 Yasuyuki Matsushita 2 and Yasushi Yagi 1 In this paper, we propose a new method to estimate the distribution of shielding objects in scattering media by analyzing light transport which is measured by a pair of illumination and camera. This estimation problem is regarded as an inverse problem compared to the regular problem. Many solutions for this problem usually have a limitation which can not be applied for strong scattering media because it is difficult to deal with light propagation in this environment. In our method, many light paths are generated by random sampling of scattering points. We estimate the distribution of shielding objects by voting the likelihoods of existing the occluding object on each light path. We have evaluated the proposed method using simulated scene data and confirmed that the location can be roughly estimated. 1. 1) Nayar 2) Narasimhan 3) Kim 4) CG CV 1 Institute of Scientific and Industrial Research, Osaka University 2 Microsoft Research Asia 1 c 2012 Information Processing Society of Japan

2 2. t L s 2.1 Chen 5) 6) 2.2 Visual Hull 7) Lanman 8) 9) 2.3 X CT 10) 11) v u O 1 Fig. 1 Light transport in the scene L O L (s, t)o (u, v) (s, t) (u, v) 2 (s, t) 2 (s, t, u, v) T (s, t, u, v) T (s, t, u, v) L O p 1 d 1 p 2 θ d 2 p 3 E p1,p 2,p 3 12) E p1,p 2,p 3 = σ s p(θ, g) exp( σ t (d 1 + d 2 )) (1) σ s σ t d 1d 2 p 1 p 2 p 2 p 3 2 c 2012 Information Processing Society of Japan

3 d1 p 1 p 2 θ p 3 t r 2 r 0 s t d 2 d 1 s Fig. 2 p(θ,g) d 2 2 Attenuation of the intensity. r m v r m+1 r m-1 r 1 u d m θ 1 θ m dm+1 v u s s s (a) (b) t t t 4 Fig. 4 Model of the light propagation. v u u u v v (a) (b) (c) 3 Fig. 3 Light paths in different distribution inside the scattering media. 3(a) 3(b) 3(b) 3(c) p(θ, g) 12) p(θ, g) = 1 4π 1 g 2 (1 + g 2 2g cos θ) 3 2 g, g (2) 4 4(a) j m r 1, r 2,..., r m 1, r m m λ m P oisson(λ, m) = λm exp( λ) m! r 1, r 2,..., r m 1, r m r 0 r m+1 R j (r 0 r 1,..., r m r m+1 ) 4(b) r k r k+1 θ k 1,k,k+1 d k,k+1 r k+1 Er k 1,r k,r k+1 r 0 r m+1 (3) 3 c 2012 Information Processing Society of Japan

4 E R j L s E R j = m (exp( σ td k 1,k )σ sp(θ k 1,k,k+1, g)) exp( σ td m,m+1) (4) t k=1 d k,k+1 θ k 1,k,k+1 d k,k+1 = r k r k+1 (5) θ k 1,k,k+1 = arccos( (r k+1 r k ) (r k r k 1 ) ) (6) r k+1 r k r k r k 1 v v u O u R j V (R j ) { Fig. 5 5 Illustration of the forward and inverse problems. V (R j) = 0 : if shielding objects exist on R j 1 : otherwise T R j R j E R j V (R j) T R j = V (R j )E R j (8) (7) t R j (s,t) s t R j V(R j ) (s,t) s R j T R j E R j j R j N T (s, t, u, v) j T R j (s, t, u, v) T (s, t, u, v) 1 N N T R j (s, t, u, v) (9) j=1 v u (u,v) v (u,v) E Rj (s,t,u,v) (a) (b) 6 Fig. 6 Solution of inverse problem. u T (s, t, u, v) V (R j ) T (s, t, u, v) 5 T (s, t, u, v) 4.2 6(a) 4 c 2012 Information Processing Society of Japan

5 j R j R j E R j R j T R j (s, t, u, v) R j V (R j) V (R j) = T R j (s, t, u, v) E R (10) j R j T R j (s, t, u, v) T R j (s, t, u, v) T (s, t, u, v) j T R j (s, t, u, v) D O L W L O T R j (s, t, u, v) T (s, t, u, v) (11) (a) 2 (b) R j V (R j ) T (s, t, u, v) V (R j) E R (12) j j R j N R j V (R j ) 6(b) j R j V (R j ) (a) W = 9 D = 9 σ s = 0.1 σ t = 0.15 g = Fig. 7 Scene of the simulation. W = 1 D = 1 7(b) W D 8 8(a) 8(b) T (s, u) u s 8(c) x y 5 c 2012 Information Processing Society of Japan

6 u x (x,y)=(1,5) (x,y)=(2,5) (x,y)=(3,5) (x,y)=(4,5) (x,y)=(5,5) (x,y)=(6,5) (x,y)=(7,5) (x,y)=(8,5) (x,y)=(9,5) s x s y y u (a) (b) (c) 8 Fig. 8 Visualization of the experimental results. 5.2 y = 5 x = 1 x = 9 9 s s u 5.3 Fig. 9 9 Estimated distribution by changing the horizontal locations of shielding object. (x,y)=(5,1) (x,y)=(5,2) (x,y)=(5,3) (x,y)=(5,4) (x,y)=(5,5) (x,y)=(5,6) (x,y)=(5,7) (x,y)=(5,8) (x,y)=(5,9) Fig Estimated distribution by changing the vertical locations of shielding object. 6 c 2012 Information Processing Society of Japan

7 x = 5 y = 1 y = 9 10 u u 6. 1) T. Treibitz and Y. Y. Schechner, Active Polarization Descattering, IEEE Trans. PAMI, Vol.31, No.3, pp , ) S. K. Nayar, G. Krishnan, M. D. Grossberg, and R. Raskar, Fast Separation of Direct and Global Components of a Scene using High Frequency Illumination, Proc. SIGGRAPH2006, pp , ) S. G. Narasimhan, S. K. Nayar, B. Sun, S. J. Koppal, Structured light in scattering media, Proc. ICCV2005, ) J. Kim, D. Lanman, Y. Mukaigawa, R. Raskar, Descattering transmission via angular filtering, Proc, ECCV2010, ) T. Chen, H. P. A. Lensch, C. Fuchs, H. P. Seidel, Polarization and Phase-Shifting for 3D Scanning of Translucent Objects, Proc. CVPR2007, ),,,, (MIRU2011), OS4-4, ) A. Laurentini, The Visual Hull Concept for Silhouette-Based Image Understanding, IEEE Trans. PAMI, Vol.16, No.2, pp , ) D. Lanman, R. Raskar, A. Agrawal, G. Taubin, Shield Fields: Modeling and Capturing 3D Occluders, Proc. SIGGRAPH Asia ) Kiriakos N. Kutulakos and Steven M. Seitz, A Theory of Shape by Space Carving, IJCV Vol.38, No.3, pp , ) D. A. Boas, D. H. Brooks, E. L. Miller, C. A. DiMarzio, M. Kilmer, R. J. Gaudette, and Q. Zhang, Imaging the body with diffuse optical tomography, IEEE Signal Processing Magazine, Vol.18, Issue 6, pp.57-75, ) J.Chen, V. Venugopal, X. Intes, Diffuse ptical tomography with Time-gated Perturbation Monte Carlo Method, Proc. SPIE Vol.7171, ) N. Kurachi, The Magic of Computer Graphics, CRC Press (ISBN ), c 2012 Information Processing Society of Japan

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