Approximate Program Smoothing Using Mean-Variance Statistics, with Application to Procedural Shader Bandlimiting

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1 EUROGRAPHICS 28 / D. Gutierrez and A. Sheffer (Guest Editors Volume 37 (28, Number 2 Approximate Program Smoothing Using Mean-Variance Statistics, with Applicati to Procedural Shader Bandlimiting Y. Yang and C. Barnes,2 University of Virginia, USA 2 Adobe Research Abstract his is a supplemental document to the paper Approximate Program Smoothing Using Mean-Variance Statistics,with Applicati to Procedural Shader Bandlimiting." We present in this supplemental document additial plots of time and error for shaders, formulas, derivatis, and proofs. Note that the main paper and supplemental use a single csistent set of references. 8. ime and Error Plots for Planar Geometry Please see Figure 7 the next page, where we present time and error plots for all 2 shaders that were defined in the main paper, applied to planar geometry. 9. able of Smoothed Formulas In able 3, we show a table of functis and their correspding cvolutis with box and Gaussian kernels. hese are needed for the approximatis we developed in the main paper. his table can be viewed as an extensi of the table presented in Dorn et al. [DBLW5], with some errors fixed. Note that in particular, for each functi f (x, we report not ly the result of smoothing f (x but also smoothing f 2 (x (e.g. if we report cos(x then we also report cos 2 (x. his is needed to determine the standard deviatis output by a given compute stage for the adaptive Gaussian approximati rule. In able 3, we give that bandlimiting x n by a Gaussian is a generalized Hermite polynomial He [α] n (x. his can be derived from a property of generalized Hermite polynomials: the nth ncentral moment of a Gaussian distributi X with expected value µ and variance σ is a generalized Hermite polynomial [Wik7]: He [α] n 2 n! n (x = (n 2k!k! ( 2 k x n 2k α k (2 k=. Multivariate Smoothed Functis In this secti, we extend the analysis for the adaptive Gaussian rule from the main paper Secti 4.2 to some additial multivariate functis. We explain how to derive formulas for modulo, compariss, and a ternary selecti functi. he modulo functi, f mod (a,b = a%b, can be rewritten as f mod (a,b = b fract( b a. Here, fract(x is the fractial part of x. We make the simplifying assumpti that the secd argument b of mod is an ordinary (n-random variable (so σ B =, to obtain: µ 2 mod =µ B fract( µ A, σ2 A µ B µ 2 B σ mod =µ 2 B fract 2 ( µ A, σ2 A µ B µ 2 µ 2 mod B (3 Comparis functis (>,, <, are approximated by cverting them to univariate functis including the Heaviside step functi H(x. As an example, the functi greater than (> can be rewritten as f >(a,b = H(a b. One other important multi-variate functi we approximated is the ternary select(a,b,c functi, which returns b if a is nzero, otherwise c. We approximated this in the same manner as Dorn et al. [DBLW5] as a linear interpolati based binary operators: select(a,b,c = a b + ( a c.. Correlati Coefficients for Multivariate Functis In this secti, we describe rules to compute the correlati coefficient ρ, which is briefly discussed in Secti 4.2. Specifically, we are given a binary functi f (a,b, which receives two inputs a and b, with associated random variables A and B, respectively. We discuss the following two rules: a assume ρ is cstant and estimate by sampling and b compute ρ under the assumpti that computatis are affine. Estimate ρ by sampling. In a training stage, we use n samples to approximate ρ of two random variables A and B. he samples drawn from these two distributis are represented as a i and b i with correspding sample mean a and b. hus, ρ can be estimated by: ρ = n i= (a i a(b i b n i= (a i a 2 n i= (b i b 2 (4 c 28 he Author(s Computer Graphics Forum c 28 he Eurographics Associati and John Wiley & Ss Ltd. Published by John Wiley & Ss Ltd.

2 Y. Yang & C. Barnes / Approximate Program Smoothing Using Mean-Variance Statistics, with Applicati to Procedural Shader Bandlimiting 3 Ours Dorn et al Supersampling 2 No Antialiasing Bricks Checkerboard Circles Color Circles Fire Quadratic Sine Zigzag Bricks Checkerboard Circles Color Circles Fire Quadratic Sine Zigzag with Bumps with Bumps with Bumps with Bumps with Bumps with Bumps with Bumps Bricks Checkerboard Circles Color Circles Fire Quadratic Sine Zigzag with Ripples with Ripples with Ripples with Ripples with Ripples with Ripples with Ripples Figure 7: ime versus error plots for the 2 shaders defined in the body of the paper, and applied to planar geometry. Here we show the Pareto frtier of program variants that optimally trade off running time and L 2 error. We show results for our method, Dorn et al [DBLW5], supersampling with varying numbers of samples, and the input shader without antialiasing. Note that our approach typically has significantly less error than Dorn et al [DBLW5] and is frequently an order of magnitude faster than supersampling for comparable error. Estimate ρ by an affine assumpti. When we calculate ρ under this rule, we assume the variables a and b input to f are affine transformatis of the variables x,..., x n which are input to the program. Under this assumpti, a and b can be expressed as: n n a = a c + a i x i, b = b c + b i x i (5 i= i= In Equati (5, a i and b i are coefficients of the affine transformati, and a c and b c end up not mattering for the ρ computati, so we ignore these cstants. In our implementati, we find a i and b i by taking the gradient, via the automatic differentiati of the expressi nodes a and b with respect to the inputs x i. Here ρ is computed as: ρ = n i= a ib i n i= a2 i n i= b2 i (6 Specifically, we assume the periodic functi f (x has period and its first and secd integrals within e period are also known. hese are denoted as F p(x and F p2 (x, respectively. F p(x = f (udu F p2 (x = F p(udu x [, (7 Using Equati (7, we derive the first and secd integral of f (x as follows. 2. Smoothing Result for Periodic Functis In this secti, we derive a cvenient formula that gives the bandlimited result for any periodic functi if its integral within a single period is known. We extend the analysis of fract( made by Dorn et al. [DBLW5] to any periodic functi. We use Heckbert s technique of repeated integrati [Hec86] to derive the cvoluti of a periodic functi with a box kernel. F(x = f (udu ( x = + F p( ( x = = x x f (udu ( x + F p( F p( + F p x x x F p( + F p ( x (8 c 28 he Author(s Computer Graphics Forum c 28 he Eurographics Associati and John Wiley & Ss Ltd.

3 Y. Yang & C. Barnes / Approximate Program Smoothing Using Mean-Variance Statistics, with Applicati to Procedural Shader Bandlimiting F 2 (x = F(udu u u = F p( du + F p(u du x ( x x =F p( i + x F p( + i= x ( x F p2 ( + F p2 x ( (q q =F p( + (x q q + 2 F p2 ( q + F p2 (x q x Here, q =. (9 Using Heckbert s result, the cvoluti of the periodic functi f (x with a box kernel that has support [ 3σ, 3σ] (correspding to a uniform kernel with standard deviati σ is: ˆf (x,σ = F(x + 3σ F(x + 3σ 2 3σ (2 he cvoluti of the periodic functi f (x with a tent kernel that has support [ 6σ, 6σ] (correspding to a uniform kernel with standard deviati σ is: ˆf (x,σ = F 2(x + 6σ 2 F 2 (x + F 2 (x 6σ 6σ 2 (2 3. Proof of Secd Order Approximati for a Single Compositi Here we show for a univariate functi, applying functi compositi using our adaptive Gaussian approximati from Secti 4.2 is accurate up to the secd order in standard deviati σ. Suppose we wish to approximate the compositi of two functis: f (x = f 2 ( f (x, where f, f 2 : R R. Assume the input random variable is X N (x,σ 2 : the Gaussian kernel centered at x. he output from f is an intermediate value in the computati: we can represent this with another random variable X = f (X. Similarly, the output random variable X 2 = f 2 (X. We cclude that f (X = f 2 ( f (X = f 2 (X = X 2. We apply Equati (6 and Equati (7 to f, and obtain the following mean and standard deviati. µ X = ˆf (x,σ 2 = f (x + 2 σ2 f (x + O(σ 4 f 2 (x,σ 2 = f 2 (x + 2 σ2 2 x 2 ( f 2 (x + O(σ 4 = f 2 (x + 2 σ2 (2 f f + 2( f 2 (x + O(σ 4 σ 2 X = f 2 (x,σ2 ˆf (x,σ 2 2 = σ 2 ( f 2 (x + O(σ 4 (22 Using our compositi rule, X is approximated as a normal distributi using the mean and standard deviati calculated from Equati (22. hat is, we approximate X as being distributed as N (µ X,σ 2 X. Similarly, µ X2, which is the output we care about, can be computed based Equati (7, Equati (22, and repeated aylor expansi in σ around σ =. µ X2 = ˆf 2 ( ˆf (x,σ 2,σ 2 X = f 2 ( f (x + 2 σ2 f (x + O(σ σ2 X f 2 ( ˆf (x,σ 2 + O(σ 4 X = f (x + 2 σ2 f 2( f (x f (x+ 2 σ2 f 2 ( f (x( f 2 (x + O(σ 4 = f (x + 2 σ2 f (x + O(σ 4 (23 Comparing Equati (23 with Equati (7, the functi compositi in our framework agrees up to the secd order term in the aylor expansi. We cclude that our approximati is accurate up to the secd order in standard deviati for a single compositi of univariate functis. he same property for additial compositis of univariate functis can be shown by inducti. 4. Short uning In Figure 8, we show 5 results for short tuning where the tuner has run for limited time. We log tuner output at the end of each generati, and use the first available results after the tuner has run for minutes. We compare the results of short tuning with full tuning: the tuner is run by default for 2 generatis. We described how we choose the result for our full tuning in Secti 7.. Similarly, we selected a result for our short tuning with sufficiently low error. 5. Geometry ransfer In Figure 9, we show 6 results for geometry transfer: we tune shaders e geometry, and use the Pareto frtier of tuned program variants to render the same shader a different geometry. We compare the results of geometry transfer with directly training the same geometry, as is described in Secti 7. We described how we choose the result of our method, Dorn et al. [DBLW5] and supersampling in Secti 7.. We then selected a result for geometry transfer that has time and error that is comparable to the directly trained result we previously selected. References [DBLW5] DORN J., BARNES C., LAWRENCE J., WEIMER W.: owards automatic band-limited procedural shaders. In Computer Graphics Forum (25, vol. 34, Wiley., 2, 3, 6 [Hec86] HECKBER P. S.: Filtering by repeated integrati. In ACM SIGGRAPH Computer Graphics (986, vol. 2, ACM, pp [Wik7] WIKIPEDIA: Hermite polynomials wikipedia, the free encyclopedia, 27. [Online; accessed 2-May-27]. URL: Hermite_polynomials&oldid= c 28 he Author(s Computer Graphics Forum c 28 he Eurographics Associati and John Wiley & Ss Ltd.

4 Y. Yang & C. Barnes / Approximate Program Smoothing Using Mean-Variance Statistics, with Applicati to Procedural Shader Bandlimiting No Antialiasing Our Short uning Our Full uning Bricks Hyperboloid Ground ruth 273 ms (x, L2 error: ms (x, L2 error:.25 2 ms, L2 error:.64 9 ms (x, L2 error: ms (2x, L2 error: ms, L2 error: ms (7x, L2 error: ms (28x, L2 error: ms, L2 error: ms (2x, L2 error: ms (2x, L2 error:.2 3 ms, L2 error: ms (2x, L2 error:.2 Zigzag Sphere Quadratic Sine Hyperboloid Circles with Ripples Plane Checkerboard Plane 48 ms, L2 error:.55 7 ms (2x, L2 error:.2 Figure 8: Results for short tuning. Shaders are tuned for about minutes ( Our Short uning and compared to tuning for 2 generatis ( Our Full uning. Note that many aliasing patterns can be reduced after short tuning. c 28 he Author(s Computer Graphics Forum c 28 he Eurographics Associati and John Wiley & Ss Ltd.

5 Y. Yang & C. Barnes / Approximate Program Smoothing Using Mean-Variance Statistics, with Applicati to Procedural Shader Bandlimiting able 3: A table of univariate functis, and their correspding bandlimited result, using a box kernel B and a Gaussian G. he box kernel is the PDF of the uniform random variable U[ 3σ, 3σ]. he Gaussian kernel is the PDF of the random variable N (,σ 2. Each random variable has standard deviati σ. We define sinc(x = sin(x/x, and the Heaviside step functi H(x is for x and for x positive. Note that functis with undefined regis, such as x p for negative or fractial p have σ limited as described in Secti 4.4. Functi f (x Bandlimited with box kernel: ˆf B (x,σ 2 Bandlimited with Gaussian kernel: ˆf G (x,σ 2 [ x p, p (x + 3σ p+ (x 3σ p+] He [ σ2 ] 2σ(p+ p (x x 2 (x 2 3σ 2 x log 2σ x+ 3σ x 3σ x x x x 2 x 2 + σ 2 x 2 + σ 2 x 3 x 3 + 3xσ 2 x 3 + 3xσ 2 x 4 x 4 + 6x 2 σ σ4 x 4 + 6x 2 σ 2 + 3σ 4 x 5 x 5 + x 3 σ 2 + 9xσ 4 x 5 + x 3 σ 2 + 5xσ 4 x 6 x 6 + 5x 4 σ x 2 σ σ6 x 6 + 5x 4 σ x 2 σ 4 + 5σ 6 x 7 x 7 + 2x 5 σ x 3 σ xσ 6 x 7 + 2x 5 σ 2 + 5x 3 σ 4 + 5xσ 6 x 8 x x 6 σ x 4 σ 4 + 8x 2 σ 6 + 9σ 8 x x 6 σ 2 + 2x 4 σ x 2 σ 6 + 5σ 8 sin(x sin(xsinc( 3σ sin(xe σ2 2 cos(x cos(xsinc( 3σ cos(xe σ2 2 tan(x log 2σ cos(x+ 3σ cos(x 3σ sinh(x (cosh(x + 3σ cosh(x 3σ 2σ 2 (ex+ 2 σ2 e x+ 2 σ2 cosh(x (sinh(x + 3σ sinh(x 3σ 2σ 2 (ex+ 2 σ2 + e x+ 2 σ2 tanh(x (log(cosh(x + 3σ log(cosh(x 3σ 2σ sinh 2 (x 8 3σ ( 4 3σ + sinh(2 3σ 2x + sinh(2 3σ + 2x cosh 2 (x 8 3σ (4 3σ + sinh(2 3σ 2x + sinh(2 3σ + 2x tanh 2 (x 2 3σ (2 3σ tanh( 3σ x tanh( 3σ + x ( e x e x+ 3σ e x 3σ e x+ 2 σ2 2σ sin 2 (x cos 2 (x tan 2 (x H(x fract(x fract 2 (x x x 2 x x 2 x cos(2xsinc( 2σ cos(2xsinc( 2σ ( tan(x + 3σ tan(x 3σ 2σ x 3σ x 2 3σ + 2 3σ x 3σ x 3σ (fract 2 (x + 3σ + x + 3σ 48σ fract 2 (x 3σ x 3σ (fract 3 (x + 3σ + x + 3σ 8σ fract 3 (x 3σ x 3σ x fract(x x 2 + fract 2 (x F(x + 3σ + F(x 3σ where F(x = 2( x 3 + x( x 4 + x fract 2 (x fract (x 3 x + fract( x 2 2 cos(2xe 2σ cos(2xe 2σ2 2 ( + erf x 2σ c 28 he Author(s Computer Graphics Forum c 28 he Eurographics Associati and John Wiley & Ss Ltd.

6 Quadratic Sine Circles Color Circles Checkerboard Zigzag Checkerboard with Ripples with Ripples Sphere Hyperboloid Sphere Plane Sphere Plane (ransferred from (ransferred from (ransferred from (ransferred from (ransferred from (ransferred from Sphere Sphere Hyperboloid Sphere Hyperboloid Hyperboloid Y. Yang & C. Barnes / Approximate Program Smoothing Using Mean-Variance Statistics, with Applicati to Procedural Shader Bandlimiting Ground ruth Our Result Our ransfer Result Dorn et al. 25 Supersampling 3 ms (3x, L2 error: ms (3x, L2 error: ms (3x, L2 error:.46 8 ms (2x, L2 error: ms (2x, L2 error:.44 8 ms (4x, L2 error: ms (2x, L2 error: ms (3x, L2 error: ms (7x, L2 error: ms (6x, L2 error: ms (2x, L2 error: ms (8x, L2 error:.4 ms (3x, L2 error:.7 48 ms (5x, L2 error:.7 9 ms (3x, L2 error:.9 32 ms (4x, L2 error:.2 33 ms (4x, L2 error: ms (5x, L2 error: ms (7x, L2 error: ms (5x, L2 error: ms (2x, L2 error: ms (2x, L2 error: ms (2x, L2 error: ms (2x, L2 error:.44 Figure 9: Results for geometry transfer. Shaders trained a source geometry are applied to a target geometry ( Our ransfer Result" and compared to re-training the target geometry. Note that both transfer between curved geometries and from curved geometry to plane give good antialiasing results and have significant less error than Dorn et al. [DBLW5]. Note also that transfer results between curved geometries are typically competitive to direct training both in time and error. c 28 he Author(s Computer Graphics Forum c 28 he Eurographics Associati and John Wiley & Ss Ltd.