High-Order Similarity Relations in Radiative Transfer: Supplementary Document

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1 High-Order Similarity Relations in Radiative Transfer: Supplementary Document Shuang hao Ravi Ramamoorthi Kavita Bala Cornell University University of California, Berkeley Cornell University Overview This is the supplementary document for our SIGGRAPH 204 paper titled High-Order Similarity Relations in Radiative Transfer. Section describes the similarity relations described in Section 4. of the paper. Section 2 shows the connection between similarity theory and classical diffusion theory. We present an alternative derivation of the diffusion equation (DE) based on the assumption that the radiance field is linearly anisotropic, which is shared by the order- similarity relation. Based on Section 2, we show in Section 3 a generalization of the order- similarity relation, following the derivation introduced by Wyman et al. [989b]. Section 4 provides background on a function s monomial and Legendre moments. The rest of this document shows extra results. Section 5 contains the rendered images used to create the plots in Figure 6 of the paper. This figure discusses how the user-specified parameter α affects performance and accuracy. Section 6 provides two examples of overfitting. This issue has been discussed in Section 7.3 of the paper. Section 7 shows the rendered images and the corresponding phase functions used to create the three 2D embeddings in Figure 9 of the paper. These results demonstrate that to accurately capture important visual cues, higher-order analysis is crucial. Recap: Similarity Relations For completeness of this document, this section restates the similarity relations introduced by Wyman et al. [989a] and presented in Section 4. of the paper. We are given a radiative transfer equation (RTE) (ω )L(ω) = σ t L(ω) + σ s 2 f (ω ω)l(ω ) dω () and an altered version (ω )L(ω) = σ t L(ω) + σ s 2 f (ω ω)l(ω ) dω (2) with identical boundary conditions. Assume that the solution radiance field L of () has limited angular frequency, namely a mn = 0 for n > N, n m n (3) for some fixed N. Then (2) has the same solution L as () if the similarity relation of order N: σ a = σ, a σ s ( f n ) = σ ( f ) for n N (4) s n We assume that the medium has no internal source, so Q vanishes. holds where f n, f denote the n-th Legendre moments of phase n functions f ( ), f ( ), respectively. See Section 4 for more information on function moments. 2 Derivation of the Diffusion Equation In this section, we present an alternative derivation of the diffusion equation (DE) based on the same assumption taken by the order- similarity relation: the radiance field L is linearly anisotropic, namely a mn = 0 for all n >. (5) We first rewrite the RTE () in a series of integrated forms (Section 2.). Then, we present in Section 2.2 the order-0 and order- versions of the integrated RTE and simplify them based on the assumption (5). Finally, we combine the results from Section 2.2 to obtain the DE (Section 2.3). A similar derivation has been proposed by Wyman et al. [989b] to obtain the generalized similarity relations described in Section Integrated RTE Let ω i i 2...i n := where ω i j denotes the i j -th component of ω, which is a 3-vector (so i j {, 2, 3} for all j). In addition, we define the integrated forms of the three terms (including the source term Q) appearing in the RHS of the RTE (): φ i i 2...i n := χ i i 2...i n := Q i i 2...i n := n j= ω i j ω i i 2...i n L(ω) dω, (6) ω i i 2...i n f (ω ω)l(ω ) dω dω, (7) 2 ω i i 2...i n Q(ω) dω. (8) An integrated version of the RTE () can then be obtained by multiplying ω i i 2...i n and integrating over 2 on both sides, yielding j φ ji...i n = σ t φ i...i n + σ s χ i...i n + Q i...i n (9) j= where i denotes ω i. 2.2 Order-0 and Order- Versions We now write down the order-0 and order- versions of the integrated RTE (9) and simplify them based on (5). Order-0. The order-0 version of (9) is j φ j = σ t φ + σ s χ + Q (0) (0) j=

2 where Q (0) := 2 Q(ω) dω. Since χ = 2 L(ω ) (0) simplifies to Order-. 2 f (ω ω) dω dω = 2 L(ω ) dω = φ, j φ j = σ a φ + Q (0). () j= The order- version of (9) is j φ ji = σ t φ i + σ s χ i + Q i. (2) j= Due to (5), it holds that χ i = = = 2 ω i f (ω ω)l(ω ) dω dω 2 n i f n a ji n=0 m= n i=0 j= i 2 ω i Y m n (ω) n n=0 m= n f n a mn 2 Ȳ m n (ω )Y j i (ω ) dω }{{} = δ ni δ mj ω i Y m (ω) dω n 2 = f a m ω i Y m (ω) dω 2 m= dω (3) where the last equality follows the fact that ω 2 i Y 0 (ω) dω = 0 0 for all i {, 2, 3}. Similarly, we have φ i = 2 ω i L(ω) dω = a m ω i Y m (ω) dω. (4) 2 m= From (3) and (4), we know that χ i = f φ i. (5) It is easy to verify that for all i and i 2, ω 2 i i 2 Y 0 (ω) dω = 2 π 3 δ i i 2 and ω 2 i i 2 Y m (ω) dω = 0. Thus, 2 π φ i i 2 = ω i i 2 L(ω) dω = a 00 3 δ i i 2. 2 Note that a 00 = Y 0 φ (ω)l(ω) dω = 2 0 2, we have π φ i i 2 = φ 3 δ i i 2. (6) Substituting (5) and (6) into (2) leads to 3 iφ = σ t r, φ i + Q i (7) where σ t r, = σ t f σ s follows the definition in () of the paper Diffusion Equation In this subsection, we show that the DE can be obtained easily from the integrated RTE (9) in its simplified order-0 () and order- (7) forms. Let E := 2 ωl(ω) dω = φ φ 2 φ 3, Q () := Q Q 2 Q 3 and κ := (3σ t r, ). Then () and (7) can be rewritten as E = σ a φ + Q (0), (8) E = 3κ Q () κ φ (9) where is the divergence operator and is the gradient operator. Substituting (9) into (8) yields 3κ Q () κ φ = 3κ Q () (κ φ) = σ a φ + Q (0). Namely, (κ φ) + σ a φ = Q (0) 3κ Q (), (20) and (20) is known as the diffusion equation. 3 Generalized Order- Similarity Relation The similarity relations (4) requires σ a = σ a to ensure that the altered RTE (2) has the same solution radiance field L as the (). However, if we relax this equal-radiance constraint and only ask for the same fluence φ = 2 L(ω) dω, generalized versions of the similarity relations can be derived [Wyman et al. 989b]. Unlike (4), unfortunately, these relations do not easily generalize to higher orders. Assuming (5) holds, namely the radiance field is linearly anisotropic, () and (7) can be used to derive the generalized similarity relation of order- as follows. Applying i and summing over to 3 on both sides of (7) and substituting () into the resulting equation yields i= 2 i φ = σ a σ t r, φ σ t r, Q (0) + i Q i. For participating media with no internal source, all moments of Q vanish, giving σ a σ t r, φ i 2 φ = 0. Similarly, it holds that i σ a σ φ t r, i 2 φ = 0. Equating these two equations yields i i= σ a σ t r, = σ a σ t r,, (2) which is the order- generalized similarity relation. In addition, it has been shown in [Wyman et al. 989b] that the solution radiance field L of the RTE and the solution L of the altered one are related as follows: L(ω) = σ a L (ω) + σ a σ a φ σ 4π. (22) a Note that when σ a = σ a, (22) collapses to L(ω) = L (ω) and (2) becomes the order- similarity relation, namely (4) with N =. 4 Phase Function Moments Given a phase function f ( ), its monomial moments and Legendre moments are defined as follows., 2

3 0% 0% 20% α = 5 Reference α = α = α = 0.75 γn := f (t) t n dt. f (t)pn (t) dt (24) f0 = 2π f (t) dt = and f = 2π t f (t) dt = f (cos θ ) dω (25) α = 0.75 i=0 S2 α = Pn Proof. Let Pn (t) = i=0 pn,i t i. According to (23) and (24), we know that for any n 0, n X f n = 2π pn,i f (t) t i dt. (27) where Pn denotes the degree-n Legendre polynomial and 2π is a normalization term. Since f ( ) is a phase function, it holds that α = In fact, monomial moments and Legendre moments are closely related. Precisely, for any N 0, the first (N + ) Legendre moments and the first (N + ) monomial moments uniquely determine each other. In other words, any phase function f ( ) with f0,..., f N fixed will have the same γ0,..., γn, and vice versa. α = 0.3 (23) Given f ( ), its n-th Legendre moment is f n := 2π α = Figure 2: Rendered images corresponding to the purple curve in Figure 6 of the paper. The relative error visualizations use the same color mapping as Figure. Legendre Moments. α = 0. α =.0 Given f ( ), its n-th monomial moment is α = 5 α =.0 Figure : Rendered images used to create the orange curve in Figure 6 of the paper. The relative error maps (using the color mapping shown at the top) are included. Monomial Moments. Reference α = 0. α = 0.3 α = 30% {z = γi cos θ f (cos θ ) dω } Let γ := (γ0 γ... γn ) T and f := ( f0 f... f N ) T, equation (27) can be rewritten in vector form as (26) f = 2π P γ. S2 where θ is the angle between ω and the incident direction. f is usually referred to as the average cosine of f ( ). For a HenyeyGreenstein (HG) phase function with parameter g, its n-th Legendre moment has been shown to equal g n (see Theorem 7. in [Van den Eynde 2005]). (28) where P is an (N + ) (N + ) matrix with P (i +, j + ) = pi, j for 0 i, j N. It is easy to verify that P is lower-triangular and non-singular. Thus, γ and f uniquely determine each other. Given f, γ can be computed by solving the linear system in (28). 3

4 f().0 Phase Function Plots Reference Order-2 Order-3 f() 0.6 Phase Function Plots Reference Order-2 Order-3 Figure 3: Two examples on overfitting. In both examples, the altered parameters satisfying the order-3 similarity relation overfit. 5 Results: Performance versus Accuracy Figures and 2 contain rendered images we used to study how α, the user-specified parameter, controls the balance between performance and accuracy. 5. Bust The renderings in Figure correspond to the orange curve in Figure 6 of the paper. The environment lighting used in the following rendering is Kitchen from [Debevec 998]. Note that The α = result is not identical to the reference since the latter was rendered with a much higher number of samples. The error in the results with high α values are mostly from Monte Carlo noise. 5.2 Dragon The renderings in Figure 2 correspond to the purple curve in Figure 6 of the paper. We used the environment lighting Ennis from [Debevec 998]. Similar to Figure, the reference image was rendered using a higher number of sample paths per pixel. 6 Results: Overfitting In our experiments, overfitting does not happen very often and normally causes only subtle differences. Figure 3 illustrates two examples where altered parameters adhering to the order-3 similarity relation overfit. Our method (Algorithm in the paper) successfully rejects these solutions (and returns the order-2 versions) since their coverages are too small (which is demonstrated by the phase functions plots). blue points in Figure 9-ab). The middle image is generated with the altered parameters provided by our algorithm (Algorithm in the paper). We use it for the higher-order embedding (the yellow points in Figure 9-a). The right image is created with the altered parameters satisfying only the order- similarity relation. The corresponding embedding (which collapses to a D line) is in Figure 9-b of the paper. The middle and right images share the same altered scattering coefficient σ and are rendered in similar s time. : Reference : Order-3 : Order- 2: Reference 2: Order-5 2: Order- 3: Reference 3: Order-5 3: Order- 7 Results: Spanning the Perceptual Space This section includes the images we used to create the embeddings in Figure 9 of the paper. The two examples in Figure 9-cde correspond to phase functions 8 and 35. 4: Reference 4: Order-5 4: Order- 7. Rendered Images This section illustrates 40 rows of images, and each row corresponds to a phase function that we picked from [Gkioulekas et al. 203]. In each row, the left image is rendered with the parameters and used for creating the reference embedding (the 5: Reference 5: Order-5 5: Order- 4

5 6: Reference 6: Order-4 6: Order- 6: Reference 6: Order-2 6: Order- 7: Reference 7: Order-3 7: Order- 7: Reference 7: Order-2 7: Order- 8: Reference 8: Order-2 8: Order- 8: Reference 8: Order-3 8: Order- 9: Reference 9: Order-3 9: Order- 9: Reference 9: Order-3 9: Order- 0: Reference 0: Order-4 0: Order- 20: Reference 20: Order-5 20: Order- : Reference : Order-3 : Order- 2: Reference 2: Order-5 2: Order- 2: Reference 2: Order-3 2: Order- 22: Reference 22: Order-5 22: Order- 3: Reference 3: Order-2 3: Order- 23: Reference 23: Order-5 23: Order- 4: Reference 4: Order-3 4: Order- 24: Reference 24: Order-3 24: Order- 5: Reference 5: Order-2 5: Order- 25: Reference 25: Order-5 25: Order- 5

6 26: Reference 26: Order-4 26: Order- 36: Reference 36: Order-2 36: Order- 27: Reference 27: Order-4 27: Order- 37: Reference 37: Order-4 37: Order- 28: Reference 28: Order-5 28: Order- 38: Reference 38: Order-3 38: Order- 29: Reference 29: Order-5 29: Order- 39: Reference 39: Order-4 39: Order- 30: Reference 30: Order-4 30: Order- 40: Reference 40: Order-4 40: Order- 7.2 Phase Function Plots This section contains the plots of all phase functions used to create the renderings in Section 7.. 3: Reference 3: Order-4 3: Order- f().0 Phase Function f() Phase Function 2 32: Reference 32: Order-4 32: Order- 33: Reference 33: Order-4 33: Order- f() 0.6 Phase Function 3 f() Phase Function 4 34: Reference 34: Order-4 34: Order- f() Phase Function 5 f() Phase Function 6 35: Reference 35: Order- 35: Order- 6

7 f().0 Phase Function 7 f() Phase Function 8 f() Phase Function 2 f() Phase Function 22 f() Phase Function 9 f() Phase Function 0 f() Phase Function 23 f() 0. Phase Function 24 f() Phase Function f() Phase Function 2 f() Phase Function 25 f() Phase Function 26 f() Phase Function 3 f() Phase Function 4 f() Phase Function 27 f() Phase Function 28 f().0 Phase Function 5 f().0 Phase Function 6 f() Phase Function 29 f() Phase Function 30 f() Phase Function 7 f() Phase Function 8 f() Phase Function 3 f() Phase Function 32 f() Phase Function 9 f().0 Phase Function 20 f() Phase Function 33 f() Phase Function 34 7

8 f() Phase Function 35 f() Phase Function 36 f() Phase Function 37 f() Phase Function 38 f() 0.6 Phase Function 39 f() Phase Function 40 References D E B E V E C, P Rendering synthetic objects into real scenes: Bridging traditional and image-based graphics with global illumination and high dynamic range photography. In Proceedings of SIGGRAPH 998, G K I O U L E K A S, I., X I A O, B., H A O, S., A D E L S O N, E. H., I C K L E R, T., A N D B A L A, K Understanding the role of phase function in translucent appearance. ACM Trans. Graph. 32, 5, 47: 47:9. VA N D E N E Y N D E, G Neutron Transport with Anisotropic Scattering. PhD thesis, Ph. D. Thesis, Université Libre de Bruxelles. W Y M A N, D. R., PAT T E R S O N, M. S., A N D W I L S O N, B. C Similarity relations for anisotropic scattering in Monte Carlo simulations of deeply penetrating neutral particles. Journal of Computational Physics 8,, W Y M A N, D. R., PAT T E R S O N, M. S., A N D W I L S O N, B. C Similarity relations for the interaction parameters in radiation transport. Applied optics 28, 24,