Refrac=on of a Caus=c • Monte‐Carlo ray tracing handles all paths of light: L(D|S)*E, but not equally well • Has difficulty sampling LS*DS*E paths, Photon Mapping e.g. refrac=on of a caus=c • Path tracing would need a very lucky first hit • Bidirec=onal ray tracing can find Jan Kautz 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 • High variance in Monte‐Carlo renderings results in noise • Simulates the transport of individual photons • Collec=on of deposited photons • Photons emiXed from light sources into a “photon map” (a 3‐D • Photons bounce off of specular spa=al data structure) provides a surfaces flux density es=mate The scene above contains glossy surfaces, and was rendered in 50 • Photons deposited on diffuse surfaces • Flux samples filtered easier than minutes using photon mapping. The – Held in a 3‐D spa=al data structure path samples, resul=ng in error at same scene took 6 hours for render – Surfaces need not be parameterized lower frequencies with Radiance, a rendering system that • Photons collected by path tracing from used radiosity for diffuse reflec=on and • Error is a result of bias, which eye path tracing for glossy reflec=on. decreases as the number of samples increase • And, oh yeah, it’s a lot faster What is a Photon? Sources ω p • A photon p is a par=cle of light • Point source that carries flux ΔΦ p ( x p , ω p ) – Photons emiXed uniformly in all direc=ons – Power: ΔΦ p – magnitude (in WaXs) and color of the flux it • Power of source (W) distributed ΔΦ p carries, stored as an RGB triple evenly among photons – Posi=on: x p – loca=on of the • Flux of each photon equal to source x p photon power divided by total # of photons – Direc=on: ω p – the incident • For example, a 60W light bulb would direc=on ω i used to compute send out a total of 100K photons, irradiance each carrying a flux ΔΦ of 0.6 mW • Photons vs. rays • Photons sent out once per simula=on, – Photons propogate flux not con=nuously as in radiosity – Rays gather radiance 1
Mixed Surfaces Russian RouleXe ? • Surfaces have specular and • Arvo & Kirk, S90 • Reflected flux only a frac=on of diffuse components ρ d = 50% incident flux – ρ d – diffuse reflectance ρ s = 30% • Aker several reflec=ons, spending – ρ s – specular reflectance a lot of =me keeping track of very – ρ d + ρ s < 1 (conserva=on of liXle flux energy) • Instead, completely absorb some photons and completely reflect • Let ζ be a uniform random value others at full power from 0 to 1 • Spend =me tracing fewer full • If ζ < ρ d then reflect diffuse power photons • Else if ζ < ρ d + ρ s then reflect • Probability of reflectance is the reflectance ρ . specular • Probability of absorp=on is 1 – ρ . ρ = 60% • Otherwise absorb Storing Photons K‐D Tree • Uses a kd‐tree – a sequence of axis‐ • Given a large number of points p 1 ,…,p n in 3D aligned par==ons – 2‐D par==ons are lines space we want to classify them and be able to – 3‐D par==ons are planes make fast queries: • Axis of par==ons alternates wrt depth of the tree – Find all the points within a cuboid • Average access =me is O (log n ) – Find all the points within a neighbourhood of a • Worst case O ( n ) when tree is severely lopsided given point • Need to maintain a balanced tree, which can be done in O ( n log n ) • These points are photon posi=ons on surfaces. • Can find k nearest neighbors in O ( k + log n ) =me using a heap K‐D Tree K‐D Tree • A K‐D tree is just an axis aligned BSP tree. • Write p i = (x 1i , x 2i , x 3i ) • Let x j * be the median of the values of the jth coordinate • Each node of the tree stores a separa=ng (j=1,2,3). • Start with x 1 * which will par==on the original set of points plane, defined by the median value along one into two sets (divided along the X axis). of the coordinates. • Now apply the same procedure to each of the lek‐ and right‐ sets, except now subdivide on x 2 * • The leaves of the tree contain the original data • Now apply the same procedure recursively to each of these points. subsets except now subdivide on x 3 * • Keep applying this recursively un=l each leaf of the tree contains a data point. 2
K‐D Tree 2D Example K‐D Tree 2D Example K‐D Tree 2D Example K‐D Tree 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); } K‐D Tree Advantages Reflected Radiance • Using a K‐D tree solu=on does not depend on • Recall the reflected radiance equa=on the providing a mesh on the surfaces – the Δ A = π r 2 distribu=on of the photons is maintained • Convert incident radiance into incident flux independently of the representa=on of the surfaces • The method does not rely on ‘regular’ surfaces • Reflected radiance in terms of incident flux such as polygons, but could equally well apply to fractal type surfaces. • Numerically 3
How Many Photons? Filtering • How big is the disk radius r ? • Too few photons cause blurry results • Large enough that the disk surrounds the n nearest • Simple averaging produces a box photons. filtering of photons Radiance es=mate using 50 photons • The number of photons used for • Photons nearer to the sample a radiance es=mate n is usually should be weighted more heavily between 50 and 500. • Results in a cone filtering of photons Radiance es=mate using 500 Δ A = π r 2 photons Mul=ple Photon Maps Rendering • Rendered by glossy‐surface • Global L(S|D)*D photon map distributed ray tracing – Photon s=cks to diffuse surface • When ray hits first diffuse and bounces to next surface (if surface… it survives Russian rouleXe) – Compute direct illumina=on – Photons don’t s=ck to specular – Compute reflected radiance Caustic map Global map surfaces of caus=c map photons First diffuse intersec=on. Use global photons photons Return radiance of caus=c map • Caus=c LSS*D photon map – Ignore global map photons map photons here, but photons to – Importance sample BRDF f r – High resolu=on ignore global map photons return as usual radiance – Light source usually emits – Use global photon map to when photons only in direc=ons that importance sample incident evalua=ng radiance func=on L i hit the thing crea=ng the caus=c Li at first – Evaluate reflectance integral diffuse by cas=ng rays and intersec=on accumula=ng radiances from . global photon map www.bennolan.com www.bennolan.com Examples Examples Direct Illumina=on Global Photon Map 4
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