2.
Probabilistic Approaches to Shadow Maps Filtering <ul><li>Marco Salvi </li></ul><ul><li>PS3 Engineer </li></ul><ul><li>LucasArts </li></ul>v1.0

3.
Talk Breakdown <ul><li>Motivations and previous work </li></ul><ul><li>VSM extensions to many moments </li></ul><ul><li>PDFs models </li></ul><ul><li>Higher order moments </li></ul><ul><li>Exponential shadow maps </li></ul><ul><li>Conclusions and Q&A </li></ul>

4.
Motivations <ul><li>PCF is very popular, but.. </li></ul><ul><ul><ul><li>..need many samples to fight aliasing </li></ul></ul></ul><ul><ul><ul><li>..only scales linearly with sample count </li></ul></ul></ul><ul><ul><ul><li>..difficult to re-use previous computations on neighbouring pixels </li></ul></ul></ul><ul><li>Can we do any better than PCF ? </li></ul><ul><ul><ul><li>ultimately inspired by VSM paper </li></ul></ul></ul>

5.
Statistics 101 (in a slide..  ) <ul><li>Probability Density Function ( PDF ) of a random variable </li></ul><ul><ul><ul><li>Describe the probability that a random event can take place. </li></ul></ul></ul><ul><li>Raw nth moment </li></ul>

6.
Previous Work Variance Shadow Maps (1/2) <ul><li>Depth samples are seen as an (unknown a priori) PDF </li></ul><ul><ul><ul><li>PDF represented by first two raw moments, pre-filterable with separable filters! </li></ul></ul></ul><ul><ul><ul><li>moments can be hw filtered and mip-mapped </li></ul></ul></ul><ul><li>Occlusion as an upper bound for probability of being in shadow </li></ul><ul><ul><ul><li>Chebyshev ’s inequality ( CI ) </li></ul></ul></ul>

7.
Previous Work Variance Shadow Maps (2/2) <ul><li>Light bleeding </li></ul><ul><li>Only two moments are not enough to represent any PDF </li></ul>

8.
Many Moments (1/2) <ul><li>Natural extension to VSM </li></ul><ul><ul><ul><li>to compute extra moments is easy </li></ul></ul></ul><ul><li>Generalize CI to higher moments </li></ul><ul><ul><ul><li>a very tough problem </li></ul></ul></ul><ul><li>First 3 rd and 4 th order moments inequalities exist.. </li></ul><ul><ul><ul><li>..expensive to evaluate </li></ul></ul></ul>

9.
Many Moments (2/2) <ul><li>Numerical accuracy issues with computations involving higher order moments </li></ul><ul><ul><ul><li>Would need double precision math </li></ul></ul></ul><ul><li>VSM still more robust/looks better </li></ul><ul><ul><ul><li>These naïve extensions to VSM simply don’t work very well </li></ul></ul></ul>

10.
PDFs Models (1/5) <ul><li>We can adopt an a priori distribution model for our PDFs </li></ul><ul><ul><ul><li>Abandon inequalities and compute occlusion evaluating model’s CDF </li></ul></ul></ul><ul><li>Many distribution types available </li></ul><ul><ul><ul><li>Dirac, triangular, gauss , etc.. </li></ul></ul></ul><ul><ul><ul><li>N ‘free’ parameters: mean, variance, skewness, kurtosis , etc.. </li></ul></ul></ul>

11.
PDFs Models (2/5) Rendering steps <ul><li>Render N moments </li></ul><ul><ul><ul><li>Mip map and filter them! </li></ul></ul></ul><ul><li>Use method of moments to fit a distribution to its moments </li></ul><ul><ul><ul><li>Solve a (sometimes non) linear NxN system of equations </li></ul></ul></ul><ul><li>Evaluate CDF to compute the occlusion term </li></ul>

12.
PDFs Models (3/5) A simple example <ul><li>Normal distribution as PDF model </li></ul><ul><ul><ul><li>A function of mean and variance </li></ul></ul></ul><ul><li>Fit it to first two moments </li></ul>

13.
PDFs Models (4/5) A simple example <ul><li>Occlusion (CDF) is a function of mean, variance and receiver depth </li></ul><ul><li>Image quality comparable to VSM </li></ul><ul><ul><ul><li>Same light bleeding issues </li></ul></ul></ul>

14.
PDFs Models (5/5) <ul><li>Other models possible </li></ul><ul><ul><ul><li>For example mixtures of two Dirac or Normal distributions (3 and 5 free parameters) </li></ul></ul></ul><ul><li>Require to render more moments and to iteratively solve (per pixel) complex non linear systems </li></ul>

15.
Higher Order Moments (1/6) <ul><li>Markov’s Inequality ( MI ) </li></ul><ul><li>First order moment represents the whole depth distribution </li></ul>

16.
Higher Order Moments (2/5)

17.
Higher Order Moments (3/6) <ul><li>No VSM-like light bleeding </li></ul><ul><ul><ul><li>Because variance is not used </li></ul></ul></ul><ul><li>Markov’s upper bound is not good, it converges to zero too slowly </li></ul><ul><li>Light bleeds everywhere, two LB types : </li></ul><ul><ul><ul><li>Translation dependant (absolute LB) </li></ul></ul></ul><ul><ul><ul><li>Translation independent (relative LB) </li></ul></ul></ul>

18.
Higher Order Moments (4/6) <ul><li>MI still holds if its terms are replaced with f(x) , a positive and monotonically increasing function : </li></ul><ul><li>A good candidate for f(x) is </li></ul>

19.
Higher Order Moments (5/6) <ul><li>Varying n we get higher order moments : </li></ul><ul><li>No pre-filtering, one sample per pixel </li></ul>

20.
Higher Order Moments (6/6) <ul><li>Higher order moments reduce light bleeding with no over darkening </li></ul><ul><li>A very simple algorithm </li></ul><ul><ul><ul><li>Render only one moment </li></ul></ul></ul><ul><ul><ul><li>MI can be evaluated with a single filtered sample and one division </li></ul></ul></ul>

21.
Exponential Shadow Maps (1/7) <ul><li>How to remove absolute light bleeding? </li></ul><ul><ul><ul><li>Need a new f(x) that makes MI invariant under translations </li></ul></ul></ul><ul><li>It can be shown that such a function has to satisfy the following equation: </li></ul>

22.
Exponential Shadow Maps (2/7) <ul><li>The previous well known differential equation admits a simple solution: </li></ul><ul><li>Yes, we have to render </li></ul><ul><ul><ul><li>As with VSMs we can still pre-filter and sample the ( exponential ) shadow map in the usual ways </li></ul></ul></ul>

23.
Exponential Shadow Maps (3/7) 512 x 512 x 4 bytes (fp32) 5 x 5 box filter, 4 separable passes

24.
Exponential Shadow Maps (4/7) <ul><li>Runs out of range pretty soon </li></ul><ul><ul><ul><li>ln (max_fp32) ≈ 88 </li></ul></ul></ul><ul><li>Workaround: render linear depth and pre-filter in ln space: </li></ul><ul><ul><ul><li>Pre-filtering is still very fast and it also simplifies occlusion computations </li></ul></ul></ul>

25.
Exponential Shadow Maps (5/7) <ul><li>A detour: Convolution Shadow Maps </li></ul><ul><ul><ul><li>Assume planar receiver within a filtering region </li></ul></ul></ul><ul><ul><ul><li>Depth test approximated via Fourier expansion </li></ul></ul></ul><ul><ul><ul><li>Render N expansion terms </li></ul></ul></ul><ul><li>ESM is equivalent to CSM where depth test is approximated with exp(O-R) </li></ul><ul><ul><ul><li>Render only one term! </li></ul></ul></ul><ul><ul><ul><li>Potential light bleeding with no planar receiver </li></ul></ul></ul><ul><ul><ul><li>Can be reduced via over-darkening </li></ul></ul></ul>

26.
Exponential Shadow Maps (6/7) Overdarkening can reduce light bleeding

27.
Exponential Shadow Maps (7/7) <ul><li>Tweak amount of relative LB with a scale factor K : </li></ul><ul><li>Over darken shadows to control LB due to not planar receivers: </li></ul>occluder = K * depth occlusion = exp( Ko *(filtered_occluder-receiver))

28.
Conclusions (1/2) <ul><li>All techniques presented: </li></ul><ul><ul><ul><li>Allow to use separable filters and hardware filtering & mipmaps support </li></ul></ul></ul><ul><ul><ul><li>Reduce shadow maps aliasing </li></ul></ul></ul><ul><ul><ul><li>Are orthogonal to exotic shadow maps projection schemes ( LiSPSM, TSM, ... ) </li></ul></ul></ul><ul><ul><ul><li>Greatly reduce self-shadowing and shadow maps acne artifacts </li></ul></ul></ul>

29.
Conclusions (2/2) <ul><li>No presented technique is entirely satisfactory </li></ul><ul><li>Must get rid of light bleeding associated to non planar receivers </li></ul><ul><ul><ul><li>High order and exponential shadow maps </li></ul></ul></ul><ul><li>A lot of work still to be done! </li></ul><ul><ul><ul><li>Hope to have inspired you  </li></ul></ul></ul>

30.
Q&A <ul><li>If you have any question, please contact me: </li></ul><ul><li>email: [email_address] </li></ul><ul><li>blog: http://pixelstoomany.wordpress.com </li></ul><ul><li>For more details about Exponential Shadow Maps please refer to: </li></ul><ul><li>“ Rendering Filtered Shadows with Exponential Shadow Maps” </li></ul><ul><li>Marco Salvi, ShaderX6 , edited by W. Engel. Charles River Media, 2008 . </li></ul>

31.
Bibliography (1/2) [Williams83] Williams, Lance. “Pyramidal Parametrics”, SIGGRAPH83 Proceedings , Vol. 17, No. 3, July 1983&quot;. [Reeves87] William T. Reeves, David H. Salesin, Robert L. Cook “Rendering anti aliased shadows with depth maps”, SIGGRAPH87 Proceedings , (July, 1987), pp. 283-291[Lokovic00] Tom Lokovic and Eric Veach, “Deep Shadow Maps.” SIGGRAPH 2000 Proceedings (August 2000), Addison-Wesley. [Philips95] Thomas K. Philips, Randolph Nelson, “The Moment Bound Is Tighter Than Chernoff's Bound for Positive Tail Probabilities”, The American Statistician , Vol. 49, No. 2 (May, 1995), pp. 175-178 [Lokovic00] Tom Lokovic and Eric Veach, “Deep Shadow Maps.” SIGGRAPH 2000 Proceedings (August 2000), Addison-Wesley. [Amour04] Jean-Francois St. Amour, Eric Paquette, et al.”Real Time Soft Shadows Using the PDSM Technique.” ShaderX4, edited by W. Engel. Charles River Media, Hingham, MA, 2005. [Martin04] Tobias Martin and Tiow-Seng Tan ,”Anti-aliasing and Continuity with Trapezoidal Shadow Maps”, Proceedings of Eurographics Symposium on Rendering , 21-23 June 2004, Norrköping, Sweden, page 153-160 (text) and page 412 (color plate)

32.
Bibliography (2/2) [Wimmer04] Michael Wimmer, Daniel Scherzer, Werner Purgathofer, “Light Space Perspective Shadow Maps” , Eurographics Symposium on Rendering 2004 [Schueler05] Schueler, Christian. “Eliminating Surface Acne with Gradient Shadow Mapping.” ShaderX4 , edited by W. Engel. Charles River Media, Hingham, MA, 2005. [Donnelly06] William Donnelly, Andrew Lauritzen, “Variance Shadow Maps” Symposium on Interactive 3D Graphics and Games Proceedings (2006), ACM [Engel06] Wolfgang Engel, “Cascaded Shadow Maps”, ShaderX5, edited by W. Engel. Charles River Media, Hingham, MA, 2006. [Annen07] Thomas Annen, Tom Mertens, Philippe Bekaert, Hans-Peter Seidel, Jan Kautz, “Convolution Shadow Maps”, Rendering Techniques 2007: Eurographics Symposium on Rendering , June 2007. [Salvi08] Marco Salvi, “Rendering Filtered Shadows with Exponential Shadow Maps”, ShaderX6, edited by W. Engel. Charles River Media, Hingham, MA, 2008.

33.
Appendix (1/2) <ul><li>ESM: constant light bleeding </li></ul>

34.
Appendix (2/2) <ul><li>ESM, distance based light bleeding </li></ul>