Lecture 6 - Light Transport Welcome! , = (, ) , - PowerPoint PPT Presentation

INFOMAGR – Advanced Graphics Jacco Bikker - November 2017 - February 2018 Lecture 6 - “Light Transport” Welcome! 𝑱 𝒚, 𝒚 ′ = 𝒉(𝒚, 𝒚 ′ ) 𝝑 𝒚, 𝒚 ′ + න 𝝇 𝒚, 𝒚 ′ , 𝒚 ′′ 𝑱 𝒚 ′ , 𝒚 ′′ 𝒆𝒚′′ 𝑻

Today’s Agenda:  Introduction  The Rendering Equation  Light Transport

Advanced Graphics – Light Transport 3 Introduction Whitted

Advanced Graphics – Light Transport 4 Introduction Whitted

Advanced Graphics – Light Transport 5 Introduction Whitted Missing:  Area lights  Glossy reflections  Caustics  Diffuse interreflections  Diffraction  Polarization  Phosphorescence  Temporal effects  Motion blur  Depth of field  Anti-aliasing

Advanced Graphics – Light Transport 6 Introduction Anti-aliasing Adding anti-aliasing to a Whitted-style ray tracer: Send multiple primary rays through each pixel, and average their result. Problem:  How do we aim those rays?  What if all rays return the same color?

Advanced Graphics – Light Transport 7 Introduction Anti-aliasing – Sampling Patterns Adding anti-aliasing to a Whitted-style ray tracer: Send multiple primary rays through each pixel, and average their result. Problem:  How do we aim those rays?  What if all rays return the same color?

Advanced Graphics – Light Transport 8 Introduction Anti-aliasing – Sampling Patterns

Advanced Graphics – Light Transport 9 Introduction Anti-aliasing – Sampling Patterns

Advanced Graphics – Light Transport 10 Introduction Anti-aliasing – Sampling Patterns Adding anti-aliasing to a Whitted-style ray tracer: Send multiple primary rays through each pixel, and average their result. Problem:  How do we aim those rays?  What if all rays return the same color?

Advanced Graphics – Light Transport 11 Introduction Whitted Missing:  Area lights  Glossy reflections  Caustics  Diffuse interreflections  Diffraction  Polarization  Phosphorescence  Temporal effects  Motion blur  Depth of field  Anti-aliasing

Advanced Graphics – Light Transport 12 Introduction Distribution Ray Tracing* Soft shadows *: Distributed Ray Tracing, Cook et al., 1984

Advanced Graphics – Light Transport 13 Introduction Distribution Ray Tracing* Glossy reflections *: Distributed Ray Tracing, Cook et al., 1984

Advanced Graphics – Light Transport 14 Introduction Distribution Ray Tracing* ? *: Distributed Ray Tracing, Cook et al., 1984

Advanced Graphics – Light Transport 15 Introduction Distribution Ray Tracing* *: Distributed Ray Tracing, Cook et al., 1984

Advanced Graphics – Light Transport 16 Introduction Distribution Ray Tracing Whitted-style ray tracing is a point sampling algorithm:  We may miss small features  We cannot sample areas Area sampling:  Anti-aliasing: one pixel  Soft shadows: one area light source  Glossy reflection: directions in a cone  Diffuse reflection: directions on the hemisphere

Advanced Graphics – Light Transport 17 Introduction Area Lights Visibility of an area light source: 𝑊 𝐵 = න 𝑊 𝑦, ꙍ 𝑗 𝑒ꙍ 𝑗 𝐵 Analytical solution case 1: 𝑊 𝐵 = 𝐵 𝑚𝑗𝑕ℎ𝑢 − 𝐵 𝑚𝑗𝑕ℎ𝑢⋂𝑡𝑞ℎ𝑓𝑠𝑓 Analytical solution case 2: 𝑊 𝐵 = ?

Advanced Graphics – Light Transport 18 Introduction V A Approximating Integrals An integral can be approximated as a Riemann sum: 𝑂 𝑂 𝐶 A B 𝑊 𝐵 = න 𝑔(𝑦) 𝑒𝑦 ≈ ෍ 𝑔 𝑢 𝑗 𝛦 𝑗 , where ෍ 𝛦 𝑗 = 𝐶 − 𝐵 Image from Wikipedia 𝐵 𝑗=1 𝑗=1 Note that the intervals do not need to be uniform, as long as we sample the full interval. If the intervals are uniform, then 𝑂 𝑂 𝑂 𝑔 𝑢 𝑗 = 𝐶 − 𝐵 ෍ 𝑔 𝑢 𝑗 𝛦 𝑗 = 𝛦 𝑗 ෍ ෍ 𝑔 𝑢 𝑗 . 𝑂 𝑗=1 𝑗=1 𝑗=1 Regardless of uniformity, restrictions apply to 𝑂 when sampling multi-dimensional functions (ideally, 𝑂 = 𝑁 𝑒 ). Also note that aliasing may occur if the intervals are uniform.

Advanced Graphics – Light Transport 19 Introduction Monte Carlo Integration Alternatively, we can approximate an integral by taking random samples: 𝑂 𝐶 𝑔(𝑦) 𝑒𝑦 ≈ 𝐶 − 𝐵 𝑊 𝐵 = න ෍ 𝑔 𝑌 𝑗 𝑂 𝐵 𝑗=1 Here, 𝑌 1 . . 𝑌 𝑂 ∈ [𝐵, 𝐶] . As 𝑂 approaches infinity, 𝑊 𝐵 approaches the expected value of 𝑔 . Unlike in Riemann sums, we can use arbitrary 𝑂 for Monte Carlo integration, regardless of dimension.

Advanced Graphics – Light Transport 20 Introduction Monte Carlo Integration of Area Light Visibility To estimate the visibility of an area light source, we take 𝑂 random point samples. In this case, 5 out of 6 samples are unoccluded: 𝑊 ≈ 1 6 1 + 1 + 1 + 0 + 1 + 1 = 5 6 In terms of Monte Carlo integration: 𝑂 𝒯 2 𝑊(𝑞) 𝑒𝑞 ≈ 1 𝑊 = න 𝑂 ෍ 𝑊 𝑄 𝑗=1 With a small number of samples, the variance in the estimate shows up as noise in the image.

Advanced Graphics – Light Transport 21 Introduction Monte Carlo Integration of Area Light Visibility We can also use Monte Carlo to estimate the contribution of multiple lights: 1. Take the average of N samples from each light source; 2. Sum the averages. 2 𝐹 𝑦 ← = ෍ 𝑀 𝑗 𝑊(𝑦 ↔ 𝑚 𝑗 ) 𝑗=1 𝑦

Advanced Graphics – Light Transport 22 Introduction Monte Carlo Integration of Area Light Visibility Alternatively, we can just take 𝑂 samples, and pick a random light source for each sample. 𝑂 𝐹 𝑦 ← = 2 𝑂 ෍ 𝑀 𝑅 𝑊 𝑅 𝑄 , 𝑅 ∈ {1,2} 𝑗=1 𝑂 𝑀 𝑅 𝑊 𝑅 𝑄 Probability of = 1 𝑂 ෍ sampling light 𝑀 𝑅 0.5 𝑗=1 𝑦

Advanced Graphics – Light Transport 23 Introduction Monte Carlo Integration of Area Light Visibility We obtain a better estimate with fewer samples if we do not treat each light equally. In the previous example, each light had a 50% probability of being sampled. We can use an arbitrary probability, by dividing the sample by this probability. 𝑂 𝑀 𝑅 𝑊 𝑅 𝑄 𝐹 𝑦 ← = 1 𝑂 ෍ , ෍ 𝜍 𝑅 = 1, 𝜍 𝑅 > 0 𝜍 𝑅 𝑗=1 𝑦

Advanced Graphics – Light Transport 24 Introduction Distribution Ray Tracing Key concept of distribution ray tracing: We estimate integrals using Monte Carlo integration. Integrals in rendering:  Area of a pixel  Lens area (aperture)  Frame time  Light source area  Cones for glossy reflections  Wavelengths  …

Advanced Graphics – Light Transport 25 Introduction Open Issues Remaining issues:  Energy distribution in the ray tree / efficiency  Diffuse interreflections

Today’s Agenda:  Introduction  The Rendering Equation  Light Transport

Advanced Graphics – Light Transport 27 Rendering Equation Whitted, Cook & Beyond Missing in Whitted: Cook:  Area lights  Area lights  Glossy reflections  Glossy reflections  Caustics × Caustics  Diffuse interreflections × Diffuse interreflections  Diffraction × Diffraction  Polarization × Polarization  Phosphorescence × Phosphorescence  Temporal effects × Temporal effects  Motion blur  Motion blur  Depth of field  Depth of field  Anti-aliasing  Anti-aliasing

Advanced Graphics – Light Transport 28 Rendering Equation Whitted, Cook & Beyond Cook’s solution to rendering: Sample the many-dimensional integral using Monte Carlo integration. න න න න න … 𝐵 𝑞𝑗𝑦𝑓𝑚 𝐵 𝑚𝑓𝑜𝑡 𝑈 𝑔𝑠𝑏𝑛𝑓 𝛻 𝑕𝑚𝑝𝑡𝑡𝑧 𝐵 𝑚𝑗𝑕ℎ𝑢 Ray optics are still used for specular reflections and refractions: The ray tree is not eliminated. (In fact: for each light, one or more shadow rays are produced)

Advanced Graphics – Light Transport 29 Rendering Equation God’s Algorithm 1 room 1 bulb 100 watts 10 20 photons per second Photon behavior:  Travel in straight lines  Get absorbed, or change direction:  Bounce (random / deterministic)  Get transmitted  Leave into the void  Get detected

Advanced Graphics – Light Transport 30 Light Transport

Advanced Graphics – Light Transport 31 Rendering Equation God’s Algorithm - Mathematically A photon may arrive at a sensor after travelling in a straight line from a light source to the sensor: 𝑀 𝑡 ← 𝑦 = 𝑀 𝐹 (𝑡 ← 𝑦) Or, it may be reflected by a surface towards the sensor: 𝑠 𝑡 ← 𝑦 ← 𝑦 ′ 𝑀 𝑦 ← 𝑦 ′ 𝐻 𝑦 ↔ 𝑦 ′ 𝑒𝐵(𝑦 ′ ) 𝑀 𝑡 ← 𝑦 = න 𝑔 𝐵 Those are the options. Adding direct and indirect illumination together: 𝑠 𝑡 ← 𝑦 ← 𝑦′ 𝑀 𝑦 ← 𝑦′ 𝐻 𝑦 ↔ 𝑦 ′ 𝑒𝐵(𝑦 ′ ) 𝑀 𝑡 ← 𝑦 = 𝑀 𝐹 𝑡 ← 𝑦 + න 𝑔 𝐵

Advanced Graphics – Light Transport 32 Rendering Equation God’s Algorithm - Mathematically 𝑠 𝑡 ← 𝑦 ← 𝑦′ 𝑀 𝑦 ← 𝑦′ 𝐻 𝑦 ↔ 𝑦 ′ 𝑒𝐵(𝑦 ′ ) 𝑀 𝑡 ← 𝑦 = 𝑀 𝐹 𝑡 ← 𝑦 + න 𝑔 𝐵 Geometry factor Indirect Reflection Hemisphere Emission