# Path tracing everything. D.A. Forsyth

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1 Path tracing everything D.A. Forsyth

2 The Rendering Equation- 1 We can now write L o (x, ω o )=L e (x, ω o )+ Ω Angle between normal and incoming direction ρ bd (x, ω o, ω i )L i (x, ω i ) cos θ i dω i BRDF Incoming irradiance Average over hemisphere Radiance emitted from surface at that point in that direction Radiance leaving a point in a direction

3 The Rendering Equation - II This balance works for each wavelength, at any time, so So L o (x, ω o, λ,t) = L e (x, ω o, λ,t)+ Ω ρ bd(x, ω o, ω i, λ,t)l i (x, ω i, λ,t) cos θ i dω i

4 Detectors respond to irradiance Report D Ω λ T σ(x, λ)l(x, ω, λ,t) cos θdxdωdλdt sigma is sensitivity to wavelength typically: shutter is open for short time (T) particular detector is tiny so integral over D isn t significant Omega is set of directions through lens usually close to normal to device, so cos theta doesn t vary much

5 Strategy: evaluate this integral At detector, average many samples of incoming radiance at different times, directions, wavelengths, perhaps locations value? rendering equation Evaluating the rendering equation at each point, compute L_e (usually zero) Integral fire off some sample rays, evaluate at the far end very like diffuse path tracing improvements russian roulette (seen this) importance sampling

6 Importance sampling - 1 We have N samples x_i from probability distribution p(x) Then Generally, we ve assumed p is uniform not a great idea - what if f is large in some places, small in others? variance in the estimate we can shape p to get a better estimate of the integral 1 N f(x i ) i f(x)p(x)dx

7 Importance sampling - 2 As long as p(x) is non-zero when f(x) is non-zero We could use p(x) to improve our estimate of integral 1 N i f(x i ) p(x i ) f(x)dx

8 Path tracing To avoid variance problems, choose good P for surfaces with any diffuse component, P should have strong bias to light for specular surfaces, P should heavily emphasize the specular direction

9

10 Example pathtracer

11 Variance problems Paths may not find the light often this could be fixed by clever choice of P to heavily emphasize directions toward the source Caustics will be poorly rendered, because the path to the source is obscure

12 Bidirectional path tracing Start paths at both eye and light and join them Notice: a pair of eye-light paths generates many possible transfer paths we can use each of these, if we compute weights correctly to get integral estimate right Figure from Bidirectional path tracing. Lafortune and Willems, 1993

13 From the eye From the source Bidirectional Figure from Dutre, Bekaert, Bala 03; rendered by Suykens-De Laet

14 Photon maps Drop the requirement of an unbiased estimate of illumination accept some bias for better variance properties Propagate photons from source, cache when they arrive at surfaces Interpolate illumination value by averaging over k-nearest neighbours Caustic variance use two classes of photon: sample specular, refractive directions separately How many photons? keep trying till it looks good

15 Photon propagation Photons carry Power scale photons from source by number emitted reflected diffuse store in map when it arrives, propagate prob proportional to cos theta power scaled by albedo or use russian roulette specular do not store in map, propagate along specular direction power scaled by reflectivity or use russian roulette arbitrary BRDF inportance sample outgoing direction

16 Photon propagation When a photon arrives at complex surface, multiple photons could be generated eg specular + diffuse russian roulette to decide whether specular reflected/absorbed diffuse reflected/absorbed Photons are stored at diffuse (non-specular!) surfaces only Stored as: Power, Location, Normal

17 Photon storage and querying Store in k-d tree to look up r closest photons tree represents free space close to surfaces

18 Evaluating Radiance Reflected radiance is: L r L(x, r (x, ω) ω) = = Ω Ω Each photon carries known power, in known direction assume the relevant photons all arrive at x each contributes radiance (power/da) assume surface is flat around x, build a circle photons in this circle contribute area is known ρ bd ρ bd (ω, (ω, ω i ω)l i )L i (x, i (x, ω i ω) i cos ) cos θdω θdω i L r (x, ω) 1 L r (x, ω) = 1 ρbd πr 2 (ω, ω ρbd j )P πr 2 (ω, j (x, ωω j )P j ) j (x, Figure from Jensen s book

19 Figure from Jensen s book

20 A two pass renderer Propagate photons two classes caustic photons toward specular/glossy, refractive objects large numbers caustic map global illumination photons toward diffuse objects small numbers Gather render using direct term by area source sampling specular term by ray-tracing caustic term by direct query to photon map global illumination term by gathering the photon map

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