INFOMAGR – Advanced Graphics Jacco Bikker - November 2016 - February 2017 Lecture 14 - “Grand Recap” Welcome! 𝑱 𝒚, 𝒚 ′ = 𝒉(𝒚, 𝒚 ′ ) 𝝑 𝒚, 𝒚 ′ + 𝝇 𝒚, 𝒚 ′ , 𝒚 ′′ 𝑱 𝒚 ′ , 𝒚 ′′ 𝒆𝒚′′ 𝑻
Today’s Agenda: Monte Carlo Sampling an Area Light with One Ray Sampling Multiple Area Lights with One Ray Difficult Cases: Spherical Lights, Occluded Lights The Random Walk Random Walk with Next Event Estimation Russian Roulette Example Exam Questions
Advanced Graphics – Grand Recap 3 Lights Light arriving at 𝑞 from a point light at distance 𝑒 : 𝑂∙𝑀 Case 1: Point Light 𝐽 𝑒 2 per unit surface area. This is the irradiance from the 𝒇 Situation: light at 𝑓 arriving at point 𝑞 . surface point: location 𝒒 , normal 𝑶 ; point light: location 𝒇 , intensity 𝑱 (in Watt, or joule per second); distance between 𝒒 and 𝒇 : 𝑒 . The contribution unit vector from 𝒒 to 𝑓 : 𝑴 . of multiple lights is summed. Flux leaving 𝒇 : 𝑱 joules per second. Flux arriving at a sphere, radius 𝑠 , surface 4𝜌𝑠 2 around 𝒇 : 𝑱 𝑱 4𝜌𝑠 2 ( 𝑋/𝑛 2 ) Irradiance arriving on that sphere: 𝑶 𝑴 𝑱 Flux arriving per steradian: 4𝜌 𝑂∙𝑀 Steradians for a unit area surface patch at location 𝒒 : 𝑒 2 This is the solid angle of the unit area surface patch as seen from 𝒇 , or: The area of the patch projected on the unit sphere around 𝒇 . 𝒒
Advanced Graphics – Grand Recap 4 Lights Case 2: Area Light Situation: 𝒇 surface point, location 𝒒 , normal 𝑶 𝒒 ; single-sided area light, intensity 𝑱 , area A, normal 𝑶 𝒇 . 𝐵 𝑂 𝑓 ∙−𝑀 Steradians for the area light, as seen from 𝒒 : (approximately) . 𝑒 2 𝑶 𝒇 The radiance (joules per second per unit area per unit solid angle) 𝐵 𝑂 𝑓 ∙−𝑀 arriving at 𝒒 is thus: 𝑱 𝑒 2 𝑶 𝒒 𝑴 𝐵 𝑂 𝑓 ∙−𝑀 𝑂 𝑞 ∙𝑀 The irradiance (joules per second per unit area) is: 𝑱 . 𝑒 2 𝒒
Advanced Graphics – Grand Recap 5 Lights Sampling an Area Light 𝒇 𝐵 𝑤𝑗𝑡𝑗𝑐𝑚𝑓 𝑂 𝑓 ∙−𝑀 𝑂 𝑞 ∙𝑀 The irradiance (joules per second per unit area) is: 𝑱 . 𝑒 2 Here, 𝐵 𝑤𝑗𝑡𝑗𝑐𝑚𝑓 is the visible area. 𝐵 𝑤𝑗𝑡𝑗𝑐𝑚𝑓 may be smaller than 𝐵 in the presence of occluders. 𝑶 𝒇 We send 1 million rays to the light source. 𝑂 rays reach the light source. 𝑂 The visible area is estimated as 𝐵 𝑤𝑗𝑡𝑗𝑐𝑚𝑓 = 𝐵 1000000 . Now, we send a single ray to the light source. The probability of a ray 𝑶 𝒒 𝑴 reaching the light source is 𝜍 . Now, 𝐵 𝑤𝑗𝑡𝑗𝑐𝑚𝑓 = 𝐵 𝜍 . For this single ray, the answer is usually wrong. However, on average the answer is correct. 𝒒
Today’s Agenda: Monte Carlo Sampling an Area Light with One Ray Sampling Multiple Area Lights with One Ray Difficult Cases: Spherical Lights, Occluded Lights The Random Walk Random Walk with Next Event Estimation Russian Roulette Example Exam Questions
Advanced Graphics – Grand Recap 7 Lights Generalized: If we have 𝑂 lights, and we sample each with a 1 probability 𝜍 𝑗 , we scale the contribution by 𝜍 𝑗 to Sampling Multiple Lights get an unbiased sample of the set of 𝑂 lights. “To sample 𝑂 lights with a single ray, chose a random Any 𝜍 𝑗 is valid, as long as 𝜍 𝑗 = 1 and 𝜍 𝑗 > 0 light, and multiply whatever the ray returns by 𝑂 ”. unless we know the sample will yield 0. Situation: two lights . Mental steps: If the lights would have been point lights, we would have sampled both and summed the results. We can sample an area light with a single ray. So, we sample both using a single ray, and sum the results. Using one ray, we could sample alternating lights. Since each light is now sampled in half the cases, we should increase the result we get each time by 2. Or, we can sample a randomly selected light. On average, each light is again sampled in half of the cases, so we scale by 2. 𝒒 In other words, we scale by 1/50%=2, where 50% is the probability of selecting a light.
Today’s Agenda: Monte Carlo Sampling an Area Light with One Ray Sampling Multiple Area Lights with One Ray Difficult Cases: Spherical Lights, Occluded Lights The Random Walk Random Walk with Next Event Estimation Russian Roulette Example Exam Questions
Advanced Graphics – Grand Recap 9 Lights Sampling a Spherical Light “Any 𝜍 𝑗 is valid, as long as 𝜍 𝑗 = 1 and 𝜍 𝑗 > 0 unless we know that the sample will yield 0.” Situation: spherical light source . Selecting points on the sphere: We can skip points on one hemisphere (i.e., 𝜍 = 0) This does not affect 𝜍 𝑗 Therefore it doesn’t affect the other probabilities Similar situation: when evaluating the Lambertian BRDF, we the hemisphere below the surface without accounting for the omission in any way. 𝒒
Advanced Graphics – Grand Recap 10 Lights The only difference between situation 1 and 2 is variance: in situation 1, we get twice the energy each time we sample light 1, but it gets sampled in only Sampling Occluded Lights 50% of the cases. In situation 2, we get a much more even amount of energy for each sample. Situation 1: We have no information about occlusion. NEE probes each light with 50% probability. 1 2 samples are scaled up by 1/50%=2; rays to light 2 always yields 0; point int 𝒒 receives en energy fro rom light 1 1 in 50 50% % of of the he ca cases ses, but ut the he light is s mult multip iplied by 2. 2. Situation 2: We know light 2 is occluded. NEE probes light 1 with 100% probability. samples are scaled by 1; point int 𝒒 receives en energy fro rom light 1 1 in 𝒒 10 100% of of the cas ases, mu multipl plie ier is 1. 1.
Today’s Agenda: Monte Carlo Sampling an Area Light with One Ray Sampling Multiple Area Lights with One Ray Difficult Cases: Spherical Lights, Occluded Lights The Random Walk Random Walk with Next Event Estimation Russian Roulette Example Exam Questions
Advanced Graphics – Grand Recap 12 Walk The Random Walk How much light gets transported to the eye? 1. The light that 𝒒 emits towards the eye (typically: nothing); 2. The light that 𝒒 reflects towards the eye. The answer to 2: Light coming from all directions, reflected towards a single point; i.e: BRDF: 𝑔 𝑠 𝑞, 𝜄 𝑝 , 𝜄 𝑗 𝑔 𝑠 (𝑞, 𝑀, 𝑊) 𝛻 𝒒
Advanced Graphics – Grand Recap 13 Walk Regarding 2b: The further away a point, the The Random Walk lower the probability that a random ray from 𝑞 strikes it. The probability is also How much light gets transported to the eye? proportional to 𝑂 𝑡,𝑞,𝑟 ∙ −𝑀 . At 𝑞 , we scale by 𝑂 𝑞 ∙ 𝑀 to 1. The light that 𝒒 emits towards the eye (typically: nothing); compensate for the fact that we 2. The light that 𝒒 reflects towards the eye: sample radiance, while in fact we a) That is: the light that q, r, s emit towards p, plus gather irradiance. b) The light that q, r, s (and all other scene surface points) reflects towards p. 𝒔 𝒕 𝒓 𝒒
Advanced Graphics – Grand Recap 14 Walk Sampling the Hemisphere using a Single Ray The light being reflected towards the eye is the light arriving from all directions over the hemisphere, scaled by the BRDF: 𝛻 𝑔 𝑠 𝑞, 𝜄 𝑝 , 𝜄 𝑗 . Sampling the integral using a single random ray: Scale up by 2𝜌 . 𝒒
Advanced Graphics – Grand Recap 15 Walk Random Walk 𝒇 Point 𝒒 reflects what point 𝒓 𝒔 reflects, which is what point 𝒔 emits. 𝒓 𝑶 𝒇 𝒒
Advanced Graphics – Grand Recap 16 Walk Random Walk 𝒇 If we leave the scene, the path returns no energy. 𝑶 𝒇 𝒒
Today’s Agenda: Monte Carlo Sampling an Area Light with One Ray Sampling Multiple Area Lights with One Ray Difficult Cases: Spherical Lights, Occluded Lights The Random Walk Random Walk with Next Event Estimation Russian Roulette Example Exam Questions
Advanced Graphics – Grand Recap 18 NEE Next Event Estimation 𝒇 At each vertex, we sample the light source using an explicit light ray. 𝑶 𝒇 𝒒
Advanced Graphics – Grand Recap 19 NEE Next Event Estimation Why does this work? “The light arriving via point 𝒒 is the light reflected by point 𝒒 , plus the light emitted by point 𝒒.” And thus: The light reflected by point 𝒒 is the light arriving at 𝒒 originating from light sources (1), plus the light reflected towards 𝒒 (2). 1: Direct light at point 𝒒 . 2: Indirect light at point 𝒒 .
Recommend
More recommend
