Photon Mapping. Photon Mapping. Why Map Photons? Sources. What is a Photon? Refrac=on of a Caus=c. Jan Kautz

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1 Refrac=on of a Caus=c Photon Mapping Jan Kautz Monte Carlo ray tracing handles all paths of light: L(D S)*E, but not equally well Has difficulty sampling LS*DS*E paths, e.g. refrac=on of a caus=c Path tracing would need a very lucky first hit Bidirec=onal ray tracing can find caus=c, but reflec=on of caus=c s=ll needs lucky first hit during path tracing Featuring images swiped from Henrik Wann Jensen Photon Mapping Why Map Photons? Jensen EGRW 95, 96 Simulates the transport of individual Photons emixed from light sources Photons bounce off of specular surfaces Photons deposited on diffuse surfaces Held in a 3 D spa=al data structure Surfaces need not be parameterized Photons collected by path tracing from eye High variance in Monte Carlo renderings results in noise Collec=on of deposited into a photon map (a 3 D spa=al data structure) provides a flux density es=mate Flux samples filtered easier than path samples, resul=ng in error at lower frequencies Error is a result of bias, which decreases as the number of samples increase And, oh yeah, it s a lot faster The scene above contains glossy surfaces, and was rendered in 50 minutes using photon mapping. The same scene took 6 hours for render with Radiance, a rendering system that used radiosity for diffuse reflec=on and path tracing for glossy reflec=on. What is a Photon? Sources A photon p is a par=cle of light that carries flux ΔΦ p (x p, ω p ) Power: ΔΦ p magnitude (in WaXs) and color of the flux it carries, stored as an RGB triple Posi=on: x p loca=on of the photon Direc=on: ω p the incident direc=on ω i used to compute irradiance Photons vs. rays Photons propogate flux Rays gather radiance ω p ΔΦ p x p Point source Photons emixed uniformly in all direc=ons Power of source (W) distributed evenly among Flux of each photon equal to source power divided by total # of For example, a 60W light bulb would send out a total of 100K, each carrying a flux ΔΦ of 0.6 mw Photons sent out once per simula=on, not con=nuously as in radiosity 1

2 Mixed Surfaces Russian RouleXe? Surfaces have specular and diffuse components ρ d diffuse reflectance ρ s specular reflectance ρ d + ρ s < 1 (conserva=on of energy) Let ζ be a uniform random value from 0 to 1 If ζ < ρ d then reflect diffuse Else if ζ < ρ d + ρ s then reflect specular Otherwise absorb ρ d = 50% ρ s = 30% Arvo & Kirk, S90 Reflected flux only a frac=on of incident flux Aker several reflec=ons, spending a lot of =me keeping track of very lixle flux Instead, completely absorb some and completely reflect others at full power Spend =me tracing fewer full power Probability of reflectance is the reflectance ρ. Probability of absorp=on is 1 ρ. ρ = 60% Storing Photons Uses a kd tree a sequence of axisaligned par==ons 2 D par==ons are lines 3 D par==ons are planes Axis of par==ons alternates wrt depth of the tree Average access =me is O(log n) Worst case O(n) when tree is severely lopsided Need to maintain a balanced tree, which can be done in O(n log n) Can find k nearest neighbors in O(k + log n) =me using a heap Given a large number of points p 1,,p n in 3D space we want to classify them and be able to make fast queries: Find all the points within a cuboid Find all the points within a neighbourhood of a given point These points are photon posi=ons on surfaces. A K D tree is just an axis aligned BSP tree. Each node of the tree stores a separa=ng plane, defined by the median value along one of the coordinates. The leaves of the tree contain the original data points. Write p i = (x 1i, x 2i, x 3i ) Let x j * be the median of the values of the jth coordinate (j=1,2,3). Start with x 1 * which will par==on the original set of points into two sets (divided along the X axis). Now apply the same procedure to each of the lek and rightsets, except now subdivide on x 2 * Now apply the same procedure recursively to each of these subsets except now subdivide on x 3 * Keep applying this recursively un=l each leaf of the tree contains a data point. 2

3 2D Example 2D Example 2D Example Code KDTree makekdtree(int n, point_kd p[], int depth) /*returns a kd-tree for the n 3-dimensional points in p - assume all indices start from 1*/ { if n==1 return a leaf containing p[1]; /*base case*/ x = median of values of (depth mod 3) coordinate in the points; pleft is the set of points to the left of x and pright is the set to the right; leftnode = makekdtree(n/2,pleft,depth+1); rightnode = makekdtree(n/2,pright,depth+1); /*assumes that the median splits the points exactly in two*/ /*compose a new node and return*/ return compose(leftnode,x,rightnode); } Advantages Using a K D tree solu=on does not depend on the providing a mesh on the surfaces the distribu=on of the is maintained independently of the representa=on of the surfaces The method does not rely on regular surfaces such as polygons, but could equally well apply to fractal type surfaces. Reflected Radiance Recall the reflected radiance equa=on Convert incident radiance into incident flux Reflected radiance in terms of incident flux Numerically ΔA = πr 2 3

4 How Many Photons? Filtering How big is the disk radius r? Large enough that the disk surrounds the n nearest. The number of used for a radiance es=mate n is usually between 50 and 500. Too few cause blurry results Simple averaging produces a box filtering of Photons nearer to the sample should be weighted more heavily Results in a cone filtering of ΔA = πr2 Radiance es=mate using 50 Radiance es=mate using 500 Mul=ple Photon Maps Rendering Global L(S D)*D photon map Rendered by glossy surface distributed ray tracing When ray hits first diffuse surface Photon s=cks to diffuse surface and bounces to next surface (if it survives Russian roulexe) Photons don t s=ck to specular surfaces Caustic map Global map Caus=c LSS*D photon map High resolu=on Light source usually emits only in direc=ons that hit the thing crea=ng the caus=c Direct Illumina=on Compute direct illumina=on Compute reflected radiance of caus=c map Ignore global map Importance sample BRDF fr as usual Use global photon map to importance sample incident radiance func=on Li Evaluate reflectance integral by cas=ng rays and accumula=ng radiances from global photon map First diffuse intersec=on. Return radiance of caus=c map here, but ignore global map Use global map to return radiance when evalua=ng Li at first diffuse intersec=on. Global Photon Map 4

5 Indirect Illumina=on Photon Map Example Final Rendering Photon Map Example Photon Mapping One of the recent of the established global illumina=on solu=ons Has a pre rendering phase to es=mate global illumina=on A path tracing phase for the final rendering (final gathering) It is significantly faster than path tracing (minutes compared to hours), and the pre rendering phase only has to be done once. Most =me goes into the final rendering phase, and in par=cular in es=ma=ng the radiance from nearby. 5