# Stochastic Transparency

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3 on modern hardware. Throughout the discussion we refer to triangles, but the ideas extend naturally to other primitives. We also often use fragment which refers to the part of the triangle inside a given pixel. 3.1 Sampling Transparency We assume a collection of triangles where the ith triangle has color c i and opacity α i, so that its correct contribution to a pixel is α ic i modulated by the transparency of the triangles in front of it. We introduce two running examples. Example 1. Consider a pure red (c 0 = (1, 0, 0)) triangle in front of a pure green (c 1 = (0, 1, 0)) triangle, each with an opacity of 0.45 (α 0 = α 1 = 0.45). If the background is pure blue (c b = (0, 0, 1)) then the resulting pixel color is: c = αc 0 + (1 α)αc 1 + (1 α) 2 c b = (0.45, , ), which corresponds to the over operator as defined by Porter and Duff [1984]. This simple artificial example isolates contributions from each triangle and the background into separate color channels. Example 2. Consider N triangles each with α = 1/N and color c f. The resulting pixel color is: N 1 X c = (1 α) N c b + (1 α) i αc f. Here we consider a 25% grey background (c b = (0.25, 0.25, 0.25)) with N = 4 triangles. This example resembles the situation when multiple thin, softly lit, partially transparent smoke blobbies overlap in screen space, and helps illustrate the inherent complexity of transparency: the color contributed by a triangle depends on α of all closer triangles at that pixel. Consider an ordinary z-buffer where each pixel in the frame buffer consists of a number of samples S. During rasterization, a fragment covers some or all of those samples. Each sample retains the depth and color of the front-most fragment that has covered it. For our purposes we want to encode transparency for a fragment using the percentage of samples we enable during rasterization, so we begin by considering all S samples to be located at the center of the pixel. In this case, an opaque fragment will cover all the samples, or none. A transparent fragment will cover a stochastic subset of R samples, resulting in an effective α = R/S. Each sample does ordinary z-buffer comparisons, retaining the front-most fragment it sees, so this is a winner take all process. The expected value of the alpha and color for the fragment is correct provided the probability a sample is covered is equal to α. For example, if we make these opacity choices independently for S = 4, then for one of the fragments in Example 1 with α = 0.45, R might take on any value of 0, 1, 2, 3, 4 but on average it should be 0.45S = 1.8. For a fragment that is not in front, its effective α is the product of its own α and (1 α i) for all triangles i in front of this fragment. The expected product of two random variables is the product of the expected values, if the variables are uncorrelated. Thus using stochastic subsets for each triangle works correctly, provided that our method for selecting those subsets ensures no correlation. There will of course be some variance, i.e., random noise, around the correct value. Reducing that noise can be achieved by increasing the number of samples S, but for a given triangle this will have the classic Monte Carlo problem of diminishing returns where halving the average error requires quadrupling S. i=0 The standard way to attack diminishing returns is to use stratified sampling. That can be accomplished here by setting the R samples as a group rather than independently. One approach is to first set R: R i = α is + ξ where ξ is a canonical random number (uniform in [0, 1)). For example, with S = 4 and α 0 = 0.45, we have a 20% chance of using R 0 = 1 and an 80% chance of using R 0 = 2. (Or R could be dithered across a tile, for further stratification.) We then choose a random subset of R samples. This allows the error for one fragment to decrease linearly, thus avoiding diminishing returns. All the images in this paper use stratified sampling. See Figure 2. Put differently, naive sampling flips an α-weighted coin for each sample, while stratified sampling randomly selects one of the S- choose-r possible subsets of samples. We illustrate the practical impact of this difference with the two examples at the beginning of the section with S = 4 samples per pixel. For Example 1, the RMS error (standard deviation) rounded to hundredths per RGB channel is: σ naive = (0.25, 0.21, 0.23). When stratified sampling is used, each of the triangles will have 1 or 2 samples representing them. For the front triangle all samples will count, while for the back triangle they will count only if not also covered by the front triangle. This introduces more stability: σ stratified = (0.10, 0.17, 0.18). For Example 2 with S = 4, we have for each RGB channel: σ naive = 0.35, σ stratified = This example shows that although we do not have stratification between the fragments, we do still benefit significantly from the stratification within a pixel. The actual number is also interesting because the example resembles cloudy sprites on a dark background. An RMS error of 0.12 in this situation, while somewhat higher than practical, is approaching a usable level. The benefits of the stratification in the one opaque sample per fragment example raises the question of whether in all cases stochastic transparency has error that decreases with 1/S and thus avoids the law of diminishing returns. Unfortunately this is not the case. Our example with S fragments each with 1 opaque sample becomes a version of the balls and bins problem (dropping M balls at random into S bins), where the number of empty bins determines alpha. The asymptotic variance in the number of empty bins increases linearly with M [Flajolet and Sedgewick 2009]. When M = S (the thin media case) and we normalize by S to get our opacity, this implies variance decreases as 1/S and thus the RMS error decreases as 1/ S. So the benefits of stratification for the thin media case comes in the constant factor. This behavior is similar in spirit to jittered sampling for opaque surfaces where the error varies between O(1/ S) and O(1/S) depending on the details of the fragments in the pixel [Mitchell 1996]. A transparent shadow map can also be rendered by stochastically discarding fragments and storing only depth. The simplest case is one sample per pixel, at a high resolution. Percentage closer filtering then approximates the transparent shadow map value. Alternatively, several depths can be stored at each pixel. This can be viewed as a Deep Shadow Map [Lokovic and Veach 2000] that encodes each ray s visibility function using S depth values z 1, z 2,...z S. Visibility is approximated by vis(z) count(z z i)/s. (1) This is crude but compact and regular, and the z s need never be sorted.

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