1. Tuesday February 19 th, 2002 Deep Shadow Maps Tom Lokovic & Eric Veach Pixar Animation Studios Presented by Tom Lechner

2. Tuesday February 19 th, 2002Deep Shadow Maps Outline Traditional Shadow Maps (TSMs) Other Shadowing Techniques Deep Shadow Maps (DSMs) GenerationSampling The Transmittance Function The Visibility Function CompressionLookups Comparing DSMs to TSMs ImplementationsExamples

3. Tuesday February 19 th, 2002Deep Shadow Maps So why are Shadow Maps Important?

4. Tuesday February 19 th, 2002Deep Shadow Maps Traditional Shadow Maps (TSMs) Generation Shadow Camera (SC) Rectangular array of depths to the closest surface of a given pixel Sampling Transform P into SC coordinate system Compare point depth to shadow depth Higher quality images require Percentage closer filtering - examines depth samples within a given filter region and computes the fraction that are closer than a given depth z Stratified sampling in both the original shadow map and filtering subset

5. Tuesday February 19 th, 2002Deep Shadow Maps Traditional Shadow Maps (cont’d) Advantages Renders large objects well Stores only one depth value per pixel Disadvantages Poorly Renders highly detailed geometries (e.g. fur, hair) Produces artifacts, esp. in animation (e.g. “sparkling”) Rendering time and memory allocation for detailed renders rapidly increase due to super sampling

6. Tuesday February 19 th, 2002Deep Shadow Maps Other Shadowing Techniques Ray Casting Fuzzy objects (potentially millions of hairs) too expensive! No soft shadows (unless using an expensive area light source) Smoke and fog require ray marching for each hair 3D Texturing Has had some success for volume datasets (clouds, fog, medical imaging) Relatively course resolution Low accuracy in z (creates bias problems) Become prohibitively large as detail increases Multi-layer Z-buffers Renders opaque surfaces from differing viewpoints Similar drawbacks to TSMs

7. Tuesday February 19 th, 2002Deep Shadow Maps Deep Shadow Maps (DSMs) Generation Rectangular array of pixels in which every pixel stores a visibility function Every value is a function of transmittance - the fraction of light that penetrates to a given depth z The original beam can be shaped and filtered to any particular filter A visibility function is calculated by filtering the nearby transmittance functions and re-sampling at the pixel center = transmittance function = desired band limiting pixel filter (centered around the origin) = filter radius

8. Tuesday February 19 th, 2002Deep Shadow Maps Deep Shadow Maps (cont’d) Generation (cont’d) Visibility functions are closely related to alpha channels Thus a DSM is equivalent to computing the approximate value ‘1 -  ’ at all depths, and storing result as a function of z Contains the combined attenuation and coverage information for every depth A stack of semitransparent objectsPartial coverage of opaque blockersVolume attenuation due to smoke

9. Tuesday February 19 th, 2002Deep Shadow Maps Deep Shadow Maps (cont’d) Sampling Select a set of sample points across the shadow camera’s image plane For each sample point, determine the corresponding transmittance function Given an image point (x,y) compute the surfaces and volume elements intersected by the corresponding primary ray surface transmission function * volume transmission function = transmission function of (x,y) For each pixel, compute its visibility function by taking a weighted combination of the transmittance function at nearby sample points.

10. Tuesday February 19 th, 2002Deep Shadow Maps Deep Shadow Maps (cont’d) The Transmittance Function Surface Transmittance Each surface hit has a depth value, z i s, and an opacity, O i Start with a transparency of 1 and then multiply by 1 – O i at every surface hit (resulting in piecewise constant function T s ) Volumetric Transmittance Sample atmospheric density at regular intervals along primary ray Each volume sample has a depth value, z i v, and an extinction coefficient, k i, that measures light falloff per unit distance Linearly interpolate between the samples to get the extinction function, k Since not piecewise linear, approximate by evaluating transmittance at each vertex of the extinction function and linearly interpolating Composite like surface transparencies to find T v, except this time interpolate between vertices rather than forcing discrete steps

11. Tuesday February 19 th, 2002Deep Shadow Maps Deep Shadow Maps (cont’d) The Transmittance Function (cont’d) Merge surface and volume transmission functions Since result is not piecewise linear, evaluate it at the combined vertices of T s and T v, interpolating linearly between them

12. Tuesday February 19 th, 2002Deep Shadow Maps Deep Shadow Maps (cont’d) The Visibility Function At each depth z, the nearby transmittance functions are filtered like ordinary image samples Results in a piecewise linear function with approx. n times as many vertices as the transmittance functions Takes into account the fractional coverage of semitransparent surfaces and fog as well as light attenuation = number of transmittance functions within the filter radius around (i + ½, j + ½) = normalized filter weight for each corresponding sample point (x k, y k )

13. Tuesday February 19 th, 2002Deep Shadow Maps Deep Shadow Maps (cont’d) Compression Visibility functions can have a large number of vertices depending on filter size and number of samples per shadow pixel Fortunately, functions tend to be very smooth Compression must preserve z values of important features, since simple errors in z can cause self-shadowing artifacts

14. Tuesday February 19 th, 2002Deep Shadow Maps Deep Shadow Maps (cont’d) Compression (cont’d)

15. Tuesday February 19 th, 2002Deep Shadow Maps Deep Shadow Maps (cont’d) Lookups Apply a reconstruction and re-sampling filter to a rectangular array of pixels (similar to textures) Given a point (x,y,z) at which to perform a lookup and a 2D filter kernel f, the filtered shadow value is Evaluating each visibility function requires searching through its data points to determine which segment contains the z-value May be implemented as a binary or linear search filter weight for pixel (i, j) = filter weight for pixel (i, j) sum over all pixels in filter radius = sum over all pixels in filter radius

16. Tuesday February 19 th, 2002Deep Shadow Maps Comparing DSMs to TSMs Support prefiltering Faster lookups Much smaller in comparison to an equivalent high resolution depth map (dependant upon compression) Fortunately, at any sampling rate there is an error tolerance that allows significant compression without compromising quality Shadows of detailed geometry have an expected error of about O(N -½ ) This error is a measure of the noise inherent in the sampled visibility function Actually implemented with a tolerance of 0.25*(N -½ ) – half of the maximum expected noise magnitude TSM uses O(N) storage, where DSM uses O(N -½ ), approaches O(N -¼ ) when functions are piecewise linear

17. Tuesday February 19 th, 2002Deep Shadow Maps Comparing DSMs to TSMs (cont’d) Significantly more expensive to compute than a regular shadow map at the same pixel resolution Bias artifacts possible, due to constant z depths Exacerbated by encouragement of large filter widths Might be useful, as it provides an extra degree of freedom Shadow resolution should be chosen according to minimum filter width desired (shadow detail) Number of samples per pixel should be determined by maximum acceptable noise Note: Bias artifacts and computational expense of DSMs are no worse off than TSMs that are equivalent in terms of pixel resolution or sample size

18. Tuesday February 19 th, 2002Deep Shadow Maps Comparing DSMs to TSMs (cont’d) 512x512 TSM 4Kx4K TSM 512x512 DSM

19. Tuesday February 19 th, 2002Deep Shadow Maps Comparing DSMs to TSMs (cont’d)

20. Tuesday February 19 th, 2002Deep Shadow Maps Implementations Incremental Updates Can be optimized to proceed in O(nlogn) Colored Shadows At expense of twice the storage Allows for some compression for gray shadows Mip-Mapping Can dramatically reduce lookup costs when objects are viewed over a wide range of scales Tiling and Caching Stored similar to textures and share in some of their advantages Tile directory Motion Blur

21. Tuesday February 19 th, 2002Deep Shadow Maps Examples With and without DSMs

22. Tuesday February 19 th, 2002Deep Shadow Maps Examples (cont’d)

23. Tuesday February 19 th, 2002Deep Shadow Maps Examples (cont’d)

24. Tuesday February 19 th, 2002Deep Shadow Maps Questions? E.G. Was there anything you didn’t understand about my explanation of the algorithms? Do you see the correlation between DSMs and Light Fields? Do you need clarification on the implementations? Thanks!