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Global Illumination using Photon Ray Splatting

EG 2007, 3.9.-7.9. Prague. Global Illumination using Photon Ray Splatting. Robert Herzog * Vlastimil Havran ‡ S. Kinuwaki * K. Myszkowski * H.-P. Seidel *. * MPI Informatik ‡ Czech Technical University Prague. Germany Czech Republic. Motivation:.

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Global Illumination using Photon Ray Splatting

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  1. EG 2007, 3.9.-7.9. Prague Global Illumination using Photon Ray Splatting Robert Herzog* Vlastimil Havran‡ S. Kinuwaki* K. Myszkowski* H.-P. Seidel* *MPI Informatik ‡Czech Technical University Prague Germany Czech Republic

  2. Motivation: Filling the gap between photon density estimation and final gathering Photon Density Estimation Photon Ray Splatting Low-frequency noise Overestimation at corners Darkening near boundaries Direction undersampling for glossy BRDFs Washed out illumination for high-frequency geometry

  3. Related Work • Ray Maps for Global Illumination, [Havran et al., EGSR 2005] • Gathering of photon rays for a disc in the tangent plane • Eliminates boundary bias • Complex and memory demanding search data structures • Scalable Photon Splatting for Global Illumination, [Lavignotte el al., Graphite 2003] • Efficient image based photon splatting approach with adaptive bandwidth selection • Restricted to triangle meshes and only diffuse light transport towards camera • Path Differentials and Applications, [Suykens et al., EGSR 2001] • Designed for antialiasing and adaptive radiosity tessellation • Radiance Caching for Efficient Global Illumination Computation, [Křivánek et al., IEEE TVCG 2005] • State-of-the-art for (ir)radiance caching on diffuse and glossy surface

  4. Density Estimation Metrics Standard photon density estimation on surfaces [Photon-maps, H.W. Jensen ‘01] Photon density estimation for a disc in the tangent plane [Ray-maps, Havran et al. EGSR ’05] Cosine-weighted photon density estimation in ray-space [Eurographics ’07]

  5. E(x) = L(x,ω) cosθx dσ(ω) E(x) = L(x,ω) cosθx dσ(ω) d2Φ(x,ω) dΦ(x,ω) cosθxdσ(ω) = = d2Φ(x,ω) dA(x) cosθxdσ(ω) dA(x) cosθx = dA┴(x,ω) ΔΦi (xi ,ωi) ΔΦi (xi ,ωi) cosθi Kh(ri) Kh(ri) ΔA┴(x,ωi) ΔA(x) Basic Derivation Irradiance from incident photon rays: ( i.e. use directional information of incident flux ) Irradiance from incident photon hit points:

  6. Ray Splatting instead of Gathering • Efficient gathering of photon rays intersecting a sphere needs complex search data structures (e.g. ray maps [Havran et al. 05]) • Our solution: photon ray splatting instead of gathering • Boils down to search over point data Kh(r)

  7. Choosing the Bandwidth (Splat-radius) • Photon density estimation needs adaptive bandwidth selection • Various statistical methods for bandwidth selection exist – though often too complex to be used for efficient photon density estimation • Common k-nearest-neighbor (k-nn) method is computational intensive • Either costly search for each density estimation query (gathering methods) • Or costly precomputation of the k-nn radius per photon (splatting methods) • Remedy: exploit knowledge from photon sampling, i.e. choose a photon’s splatting radius according to photon path probability density • We get this almost for free !!!

  8. direct light (1 bounce) indirect light (>1 bounces) x0 Photon Path Probability Density Photon path probability: ·g(x0, x1) ·g(xi-1, xi) pe(x0, x1) ps┴(xi-1 , xi , xi+1) p( X ) = pe(x0, x1) ~ Le ·cos(θ0 →1) g(xi, xj) ~ cos(θj←i)/||xi-xj||2 ps┴(xi, xj, xk) ~ fbrdf (xi, xj, xk)·cos(θj→k) θ1 x1 θ2 x2

  9. Adaptive Bandwidth Selection • Choose the photon bandwidth inversely proportional to its path probability density • Reliable estimate only for a small number of bounces (1 to 2) • Need to slightly modify formula to become robust • See paper for details! Color-coded visualization of the bandwidth (shown as photon splats) 2. bounce 4. bounce 1-6 bounces Indirect Illumination

  10. Efficient Nearest Neighbor Search • Nearest neighbor search for ray splatting is simpler than for ray gathering (e.g. ray maps [Havran et al. EGSR05]) • Boils down to a search for point samples in a cylindrical bounding volume • We use a standard kd-tree constructed on top of the eye samples with axis-aligned spatial splitting planes

  11. Search in the Spatial kD-Tree tb R t+ t- ta tb > t- ta< t+ !tb > t- ta< t+ t- leaf R R tb ta Eye sample array Search buffer ,(ta ,t+) stack

  12. Moderately Glossy Surfaces • With ray density estimation we gain more information about the direction of incident flux • since more photons contribute to a surface point • no matter what is the size and orientation of the surface • Means: less noise in the directional domain at grazing angles • However, sampling is view-independent!!! • View-dependent importance sampling of BRDF (future work)

  13. dΦ(x,ω) cosθx Ylm(ω) λlm(x)≈ dA┴(x,ω) Glossy Light Field Representation • We tested 2 different methods for capturing the incident radiance field • Accumulating flux in discrete directional strata (uniformly distributed) • Projecting the incident flux onto a spherical basis (spherical harmonics) (all BRDF data is precomputed in spherical harmonics basis) ΔΦ(θ0,φ0) ΔΦ(θ0,φ0) ΔΦ(θ1,φ1) ΔΦ(θ1,φ1) λlm(x)+= Ylm (θ0,φ0) *Kh(r1) cos(θ1) *Kh(r0) cos(θ0) (θ1,φ1) *Kh(r1) cos(θ1) *Kh(r0) cos(θ0) * s(θ0,φ0) →S0(x) += s(θ1,φ1) →S3(x)+= ΔA┴ ΔA┴ ΔA┴ ΔΦ(θ1,φ1) ΔΦ(θ1,φ1) ΔΦ(θ0,φ0) ΔΦ(θ0,φ0) S3 r0 r1 S0

  14. (Ir)radiance Caching • Computation of (ir)radiance for each pixel is still too slow • However, irradiance function is really smooth due to low-pass filtering via density estimation •  sparse sampling of the (ir)radiance is sufficient • We propose an algorithm similar to cache splatting[Gautron et al., EGSR ’05]in the spherical harmonics basis • Cache interpolation gives additional filtering of noisy photon splats  reduce overall splat radius  more efficient ray splatting

  15. Radiance Cache Splatting Noisy input from ray splatting Adaptive radiance cache splatting BRDF modulated and filtered result

  16. Changes to (Ir)radiance Caching • Due to view-independent ray splatting, we require a multi-pass cache splatting algorithm • For filtering purposes use smooth weighting function (e.g. bi-weight)  More than one cache records must contribute to a pixel • Cache density and thus the filtering depends on the geometric gradient estimated using the split-sphere model [Ward ’88] Ward ’88 Bi-weight Estimating harmonic mean distance Rc at cachesr1and r2from projected ray distances (z1,2) to the ray origins (red circle). a·Rc -a·Rc Cache weighting functions

  17. Multi-Pass Radiance Cache Splatting Cache locations (red) + cache weights (~grey shade) Cache splatting result Pass 2 Pass 1 Pass 3

  18. Final gathering (1200 rays per pixel) Photon density estimation (500 k-nn) Ray splatting with caching and filtering Photon ray splatting ~4000 sec 29 sec 35 sec 20 sec Results: Diffuse Sibenik Scene • 500,000 indirect diffuse photons • 512 x 512 image resolution • Compared against reference (photon mapping + final gathering)

  19. 10001000 pixel, 200.000 photons 500500 pixel, 500.000 photons 10001000 pixel, 300.000 photons Results K-nn Photon Density Estimation 69 sec (400 k-nn) 66 sec (600 k-nn) 92 sec (400/ 250 k-nn) Photon Ray Splatting (with radiance caching) 64 sec (20 sec) 52 sec 63 sec (16 sec)

  20. Two-Pass Final Gathering Path-traced (2000 rays) Conference Scene 4700700 pixels, 160.000 photons Ray splatting (600 rays) ~ 64 min Photon mapping (70 k-nn 600 rays) ~ 85 min

  21. Conclusions • Ray density estimation as trade-off between photon density estimation and final gathering • No boundary bias, no topological bias • Works well with (ir)radiance caching • However, still problems with proximity bias (light leaking) • requires occlusion testing (future work) (see [Herzog, Pacific Graphics ‘07]) • only low-frequency (indirect) illumination can be rendered so far • More details provided in a technical report (coming soon)

  22. Future Work • Improving adaptive cache splat size computation • Make use of illumination gradients and density estimation variance (estimated from the number of photon splats per cache) to compute a better cache splat size • Temporal domain • Reuse caches for future frames and filter in temporal domain • Detect occlusions during photon ray traversal

  23. Occlusion Testing camera camera Current splatting neglects occlusion in the footprint Correctly masking the density estimation kernel

  24. Unpublished Extensions: What about Occlusion? Ray Splatting Visibility Ray Splatting Lightcuts [Walter 05] (reference) Direct light (50,000 photons/ VPLs) Conference Scene, (1000 x 1000 eye samples) 26 sec 30 sec 675 sec (~245 rays) Global Illumination (250,000 photons/ VPLs) 85 sec 104 sec 1070 sec (~339 rays)

  25. Thank you

  26. (Additional Material – Not presented at EG ’07)Bandwidth Selection: Practical Issues • Clamp squared distance term in path probability density • Two user parameters to control bandwidth: • Overall noise  bias level ( C ) • Variance in bandwidth ( S ), i.e. sensitivity to path probability density (S=0 : constant bandwidth) • For convenience S should be individually chosen for direct and indirect light • Direct light ( S ~ [ 0.5 .. 0.7 ] ) • Indirect light ( S ~ [ 0.1 .. 0.2 ] ) • Bandwidth automatically adapts when number of photons changes • (proportional to 6th root of photon number)

  27. (Additional Material – Not presented at EG ‘07)Cache Weighting Functions := := Ward ‘88 Clampled Ward := Smoothed Ward Biweight := wc(x):= cache weighting function Rc := harmonic mean distance a:= user defined cache-error nc:= cache surface normal n := surface normal at position x -a·Rc a·Rc

  28. (Additional Material – Not presented at EG ‘07)Irradiance Gradients for Ray Splatting Angle θi with surface normal does not change in our metric (ray’s direction is constant) (xi ,ωi) ray.dir r(x,xi,wi) Vx x Ux Kh is the Epanechnikov kernel function.

  29. (Additional Material – Not presented at EG ‘07)Ray Cache Splatting with Irradiance Gradients (i.e. sample denser and thus filter less where the gradient is high) Estimated irradiance gradient magnitude from ray splatting Caveat: use gradient only for determining sampling density, not for cache interpolation!!! (gradients are too noisy)  Sharpen shadow boundaries Cache splat result with gradients Cache splat result without gradients Difference image (color coded) Interpolated irradiance (cache splatting for direct + Indirect Light) (300.000 photons) 5.4 sec 5.7 sec

  30. (Additional Material – Not presented at EG ‘07)Scalability of Photon Ray Splatting • Ray splatting only (directly to pixels) (RS direct) • k-NN Photon density estimation (PM k-NN) • Ray splatting to the directional histogram (RS histogram) • Ray splatting + SH radiance caching (RS+SH caching) • Ray splatting time to SH radiance caches (RS only) 500.000 photons 500 x 500 pixels

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