Eurographics Symposium on Rendering 2019T. Boubekeur and P. Sen(Guest Editors)

Volume 38 (2019), Number 4

Orthogonal Array Sampling for Monte Carlo Rendering:supplemental document

Wojciech Jarosz Afnan Enayet Andrew Kensler Charlie Kilpatrick Per Christensen

Dartmouth College Pixar Animation Studios

1. Further reading list

Below is our suggested sequence through a subset of the OA litera-ture for graphics researchers to most quickly get ramped up on thesetopics.

Graphics

1. First become familiar with Chiu, Shirley, and Wang’s originalmulti-jittered sampling paper [CSW94] as it is the conceptualprecursor to orthogonal arrays.

2. Then read Kensler’s correlated multi-jittered tech report [Ken13].While the title suggests the main contribution is to incorporatefavorable correlations, more importantly, this paper shows howto construct multi-jittered samples in-place, one sample at a timewithout requiring any storage.

Orthogonal array basics

3. The preface and first chapter of Hedayat, Sloane, and Stufken[HSS99] provides a great non-technical introduction to orthogo-nal arrays from the experimental design perspective.

4. Sections 10.3 and 10.4 of Owen [Owe13] provide a great MonteCarlo integration-centric introduction to Latin hypercubes andorthogonal arrays. We suggest reading it before diving into moredetail in the original publications.

5. [Tan93] first introduced the idea of enforcing both OA stratifica-tion and Latin hypercube stratification with “orthogonal array-based Latin hypercube designs” also called “U-designs” (similarto Chiu, Shirley, and Wang’s multi-jittered stratification, but forpoints of arbitrary dimension).

6. Many more general OA construction routines require arithmeticon finite/Galois fields. This is more involved than our routinesusing modular arithmetic, but many open-source libraries exist(particularly for binary finite fields) since these are used in cryp-tography. Hedayat, Sloane, and Stufken [HSS99] provides a goodintroduction to Galois fields within the context of orthogonalarrays.

Additional avenues

After becoming familiar with the basics from the main paper and thereferences above, there are several promising off-shoots that may beuseful for graphics:

7. Strong orthogonal arrays: These enforce stratification in alldimensional projections, and are strongly related to (t,m,s)-netsfrom QMC. [HT12] originally proposed them, and there has beenactive work in this area since then [HCT18; HT14; LL15; Wen14].

8. Nested orthogonal arrays/Latin hypercube designs: Theseare a stepping stone to progressive sample sequences. They al-low creating two or more layers of nested point sets [HHQ16;AZ16; HQ10; HQ11; QA10; WL13; ZWD19; RHVD10; Qia09;QAW09; YLL14; SLQ14].

9. Correlation-controlled Latin hypercubes/orthogonal arrays:Instead of independently randomizing the factors and levels ofan orthogonal array, variance can be reduced by controlling thecorrelations [Owe94; CQ14; Tan98]. This is the same idea usedby correlated multi-jittered sampling [Ken13].

2. Additional and alternate code listings

Here we include alternate code listings and additional visualizationsand variance graphs.

Listing 1: A modified version of Listing 1 from the main documentwhich computes all dimensions (instead of just one) of an arbitrarysample from a Bose OA pattern.

1 vector<float> boseOA(unsigned i, // sample index2 unsigned d, // number of dimensions (<= s+1)3 unsigned s, // number of levels/strata4 unsigned p[], // array of d pseudo-random seeds5 OffsetType ot) { // J, MJ, or CMJ6 vector<float> Xi(d);7 i = permute(i % (s*s), s*s, p[0]);8 unsigned Ai0 = i % s;9 unsigned Ai1 = i / s;

10 unsigned stratumX = permute(Ai0, s, p[0] * 0x51633e2d);11 unsigned stratumY = permute(Ai1, s, p[1] * 0x51633e2d);12 float sstratX = offset(Ai0, Ai1, s, p[0] * 0x68bc21eb, ot);13 float sstratY = offset(Ai1, Ai0, s, p[1] * 0x68bc21eb, ot);14 float jitterX = randfloat(i, p[0] * 0x02e5be93);15 float jitterY = randfloat(i, p[1] * 0x02e5be93);16 Xi[0] = (stratumX + (sstratumX + jitterX) / s) / s;17 Xi[1] = (stratumY + (sstratumY + jitterY) / s) / s;18 for (unsigned j = 2; j < d; ++j) {19 unsigned Aij = (Ai0 + (j-1) * Ai1) % s;20 unsigned j2 = (j % 2) ? j-1 : j+1;21 unsigned Aij2 = (Ai0 + (j2-1) * Ai1) % s;22 unsigned stratumJ = permute(Aij, s, p[j] * 0x51633e2d);23 unsigned sstratJ = offset(Aij, Aij2, s, p[j] * 0x68bc21eb, ot);24 float jitterJ = randfloat(i, p[j] * 0x02e5be93);25 Xi[j] = (stratumJ + (sstratJ + jitterJ) / s) / s;26 }27 return Xi;28 }

submitted to Eurographics Symposium on Rendering (2019)

2 Jarosz, Enayet, Kensler, Kilpatrick, and Christensen / Orthogonal Array Sampling for Monte Carlo Rendering:supplemental document

Listing 2: A modified version of Listing 2 from the main documentwhich computes all dimensions (instead of just one) of an arbitrarysample from a Bush OA pattern.

1 vector<float> bushOA(unsigned i, // sample index2 unsigned d, // number of dimensions (<= s+1)3 unsigned s, // number of levels/strata4 unsigned t, // strength of OA (0 < t <= d)5 unsigned p[], // array of d pseudo-random seeds6 OffsetType ot) { // J or MJ7 vector<float> Xi(d);8 unsigned N = pow(s, t);9 i = permute(i, N, p[0]);

1011 // get digits of ‘i‘ in base ‘s‘12 auto iDigits = toBaseS(i, s, t);1314 unsigned stm = N / s; // pow(s, t-1)15 for (unsigned j = 0; j < s; ++j) {16 unsigned k = (j % 2) ? j - 1 : j + 1;17 unsigned phi = evalPoly(iDigits, j);18 unsigned stratum = permute(phi % s, s, p[j] * 0x51633e2d);19 unsigned sstratum = offset(i, s, stm, p[j] * 0x68bc21eb, ot);20 float jitter = randfloat(i, p[j] * 0x02e5be93);21 Xi[j] = (stratum + (sstratum + jitter) / stm) / s;22 }23 return Xi;24 }

Listing 3: A modified version of Listing 3 from the main documentwhich computes all dimensions (instead of just one) of a CMJ patterngeneralized to arbitrary dimensions d.

1 vector<float> cmjdD(unsigned i, // sample index2 unsigned s, // number of levels/strata3 unsigned t, // strength of OA (t=d)4 unsigned p[] { // array of d pseudo-random seeds5 vector<float> Xi(t);6 unsigned N = pow(s, t);7 i = permute(i, N, p[0]);89 // get digits of ‘i‘ in base ‘s‘

10 auto iDigits = toBaseS(i, s, t);1112 unsigned stm1 = N / s; // pow(s, t-1)13 for (unsigned j = 0; j < s; ++j) {14 unsigned stratum = permute(iDigits[j], s, p[j] * 0x51633e2d);15 auto pDigits = allButJ(iDigits, j);16 unsigned sstratum = evalPoly(pDigits, s);17 sstratum = permute(sstrata, stm1, p[j] * 0x68bc21eb);18 float jitter = randfloat(i, p[j] * 0x02e5be93);19 Xi[d] = (stratum + (sstratum + jitter) / stm1) / s;20 }21 return Xi;22 }

submitted to Eurographics Symposium on Rendering (2019)

Jarosz, Enayet, Kensler, Kilpatrick, and Christensen / Orthogonal Array Sampling for Monte Carlo Rendering:supplemental document 3

1

2

3

0 1 2 0 1 2 0 1 2

1

2

3

0 1 2 0 1 2 0 1 2

1

2

3

0 1 2 0 1 2 0 1 2(a) Jittered (b) Multi-Jittered (c) Correlated Multi-Jittered

Figure 1: All six bivariate projections of a 4D strength-2 Bush-OA with 121, 289 and 1369 samples using jittered, multi-jittered, and correlatedmulti-jittered offsets.

submitted to Eurographics Symposium on Rendering (2019)

4 Jarosz, Enayet, Kensler, Kilpatrick, and Christensen / Orthogonal Array Sampling for Monte Carlo Rendering:supplemental document

y

u

v

x y u x y u x y u

(a) XORed-padded 2D (0,2) sequence [KK02] (b) XORed+shuffle-padded 2D (0,2) sequence (c) Shuffle-padded 2D CMJ points [Ken13]

Figure 2: Alternate version of Fig. 2 from the main document with 3D projections of the 4D pointsets instead of power spectra. A commonway to create samples for higher-dimensional integration is to pad together high-quality 2D point sets. Kollig and Keller [KK02] proposedscrambling each dimension of a (0,2) sequence by XORing it with a different random bit vector. While odd-even dimension pairings producehigh-quality point sets, even-even (xu) or odd-odd (yv) dimensions are poorly distributed. These issues can be eliminated by additionallyrandomly shuffling the points across dimension pairs, but this decorrelates all cross dimensions, destroying 2D stratification.

No shear Slight shear Extreme shear

101 102 103 104 105 106

Samples

10−10

10−9

10−8

10−7

10−6

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−1.59, Sobol (2,14,15)

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101 102 103 104 105 106

Samples

10−10

10−9

10−8

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−1.43, Sobol (2,14,15)

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101 102 103 104 105 106

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Figure 3: Variance graphs for a cylinder sheared by different amounts. This integrand is defined by two circles that lie in the z = 0 andz = 1 planes with centers (x0,y0,0) and (x1,y1,1). The analytic integral is defined as

∫ 10 (x,y,z) = πr2√(x1− x0)2 +(y1− y0)2 +1 where the

radius r is an arbitrary constant. Since an extruded circle with no shear (left) is actually a two-dimensional integrand, samplers that have2D stratification perform best in this case. Once any shear is applied, however, the integrand is 3D and, asymptotically, 2D stratificationdegrades to O(N−1) convergence. With only slight shear (middle), 2D stratified samplers initially perform best, but their variance graphscurve and start to perform worse than 3D stratification at higher sampler counts. For larger shear angles the tipping point starts earlier, with3D stratification dominating even at lower sample counts.

submitted to Eurographics Symposium on Rendering (2019)

Jarosz, Enayet, Kensler, Kilpatrick, and Christensen / Orthogonal Array Sampling for Monte Carlo Rendering:supplemental document 51-

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submitted to Eurographics Symposium on Rendering (2019)