3.1.4 Monte Carlo Path Tracing Jacobs University Visualization and Computer Graphics Lab 320491: Advanced Graphics - Chapter 3 35 Monte-Carlo path tracing • Trace one secondary ray per recursion •Send manyprimaryrays Jacobs University Visualization and Computer Graphics Lab 320491: Advanced Graphics - Chapter 3 36 Monte-Carlo path tracing • To shade visible object, – trace random ray to sample incoming light (not just light ray!) –useBRDF forshading Jacobs University Visualization and Computer Graphics Lab 320491: Advanced Graphics - Chapter 3 37 Monte-Carlo path tracing •Result: Jacobs University Visualization and Computer Graphics Lab 320491: Advanced Graphics - Chapter 3 38 Monte-Carlo path tracing • 10 paths/pixel Jacobs University Visualization and Computer Graphics Lab 320491: Advanced Graphics - Chapter 3 39 Monte-Carlo path tracing • 10 paths/pixel Jacobs University Visualization and Computer Graphics Lab 320491: Advanced Graphics - Chapter 3 40 Monte-Carlo path tracing • 100 paths/pixel Jacobs University Visualization and Computer Graphics Lab 320491: Advanced Graphics - Chapter 3 41 Monte-Carlo path tracing • Why random scene? • Fixed random sequence produces structures in the error Jacobs University Visualization and Computer Graphics Lab 320491: Advanced Graphics - Chapter 3 42 3.1.5 Monte Carlo Integration Jacobs University Visualization and Computer Graphics Lab 320491: Advanced Graphics - Chapter 3 43 Monte-Carlo integration • The sampling for both Monte-Carlo distribution ray tracing and Monte-Carlo path tracing is based on Monte-Carlo integration. Jacobs University Visualization and Computer Graphics Lab 320491: Advanced Graphics - Chapter 3 44 Monte-Carlo computation of π • Take a unit square • Take a random point (x,y) in the square • Test whether (x,y) lies in the quarter disc x2+y2<1 • The probability is π/4 Jacobs University Visualization and Computer Graphics Lab 320491: Advanced Graphics - Chapter 3 45 Monte-Carlo computation of π • Iterate the process • π = 4 * (# inside trials) / (# total trials) • Error depends on the number of trials Jacobs University Visualization and Computer Graphics Lab 320491: Advanced Graphics - Chapter 3 46 Monte-Carlo computation of π • Link to ray tracing: – occluder is a sphere – somepixelshaveexactlyπ/4 visibility 1 shadow ray many shadow rays Jacobs University Visualization and Computer Graphics Lab 320491: Advanced Graphics - Chapter 3 47 Simplest Monte-Carlo integration • Compute 0.5 by flipping a coin •1 flip: –0 or1 – average error = 0.5 •2 flips – 0, 0.5, 0.5, 1 – average error = 0.25 •4 flips – 1*0, 4*0.25, 6*0.5, 4*0.75, 1*1 – average error = 0.1875 • does not converge very fast Jacobs University Visualization and Computer Graphics Lab 320491: Advanced Graphics - Chapter 3 48 Simplest Monte-Carlo integration •Variancedecreasesin 1/n •Convergenceis • Convergence is independent of dimension! • Good choice to integrate high-dimensional functions • Can integrate over any domain Jacobs University Visualization and Computer Graphics Lab 320491: Advanced Graphics - Chapter 3 49 Monte-Carlo integration •To evaluate use uniform random variable xi • better: use random variable xi with probability pi • The trick is to choose good xi and pi Jacobs University Visualization and Computer Graphics Lab 320491: Advanced Graphics - Chapter 3 50 Sampling strategies • Naive vs. optimal sampling Jacobs University Visualization and Computer Graphics Lab 320491: Advanced Graphics - Chapter 3 51 Sampling strategies • Sampling directions vs. sampling light sources Jacobs University Visualization and Computer Graphics Lab 320491: Advanced Graphics - Chapter 3 52 Sampling strategies • Sampling more the light vs. sampling more the BRDF Jacobs University Visualization and Computer Graphics Lab 320491: Advanced Graphics - Chapter 3 53 What can we integrate? • Pixel: anti-aliasing • Light sources: soft shadows • Hemisphere: indirect lighting • BRDF: glossy reflection •Lens: depthof field •Time: motionblur Jacobs University Visualization and Computer Graphics Lab 320491: Advanced Graphics - Chapter 3 54 Ambient occlusion • Illumination by an extremely large area light source • Outdoor light • Low and high ambient occlusion: Jacobs University Visualization and Computer Graphics Lab 320491: Advanced Graphics - Chapter 3 55 Ambient occlusion Jacobs University Visualization and Computer Graphics Lab 320491: Advanced Graphics - Chapter 3 56 Diffuse reflections • Indirect diffuse light • Cast rays in all directions of hemisphere with a cosine weight that gives more weight towards the pole Jacobs University Visualization and Computer Graphics Lab 320491: Advanced Graphics - Chapter 3 57 Diffuse reflections Jacobs University Visualization and Computer Graphics Lab 320491: Advanced Graphics - Chapter 3 58 Glossy reflection • Shoot rays in direction within range of glossy reflection distribution. Jacobs University Visualization and Computer Graphics Lab 320491: Advanced Graphics - Chapter 3 59 Glossy reflection Jacobs University Visualization and Computer Graphics Lab 320491: Advanced Graphics - Chapter 3 60 Depth of field • Real cameras have a finite aperture opening • This can be simulated by using slightly varying origin for the rays and using the following directions: Jacobs University Visualization and Computer Graphics Lab 320491: Advanced Graphics - Chapter 3 61 Depth of field Jacobs University Visualization and Computer Graphics Lab 320491: Advanced Graphics - Chapter 3 62 Motion blur • In real cameras the shutter has to open a finite amount of time to capture enough light on the film or CCD chip. • If the object is moving when the shutter is open, it will appear blurred. • Shoot rays at different times during shutter opening. • For ray-object intersection, take the object at the respective point in time. Jacobs University Visualization and Computer Graphics Lab 320491: Advanced Graphics - Chapter 3 63 Motion blur Jacobs University Visualization and Computer Graphics Lab 320491: Advanced Graphics - Chapter 3 64 3.1.6 Radiance and Radiosity Jacobs University Visualization and Computer Graphics Lab 320491: Advanced Graphics - Chapter 3 65 The rendering equation Jacobs University Visualization and Computer Graphics Lab 320491: Advanced Graphics - Chapter 3 66 The rendering equation Jacobs University Visualization and Computer Graphics Lab 320491: Advanced Graphics - Chapter 3 67 The rendering equation Jacobs University Visualization and Computer Graphics Lab 320491: Advanced Graphics - Chapter 3 68 The rendering equation Jacobs University Visualization and Computer Graphics Lab 320491: Advanced Graphics - Chapter 3 69 The rendering equation Jacobs University Visualization and Computer Graphics Lab 320491: Advanced Graphics - Chapter 3 70 The rendering equation Jacobs University Visualization and Computer Graphics Lab 320491: Advanced Graphics - Chapter 3 71 The rendering equation Jacobs University Visualization and Computer Graphics Lab 320491: Advanced Graphics - Chapter 3 72 Computation of radiosity •LetBi be the radiosity of surface patch i and Ei the amount of emitted light energy from surface patch i. • The emitted light energy Ei is non-zero, iff. surface patch i belongs to a light source. • Finally, let ρi be the (constant) diffuse reflection coefficient for surface patch i. • outgoing light emitted light incoming light reflected light Jacobs University Visualization and Computer Graphics Lab 320491: Advanced Graphics - Chapter 3 73 Form factors where Jacobs University Visualization and Computer Graphics Lab 320491: Advanced Graphics - Chapter 3 74 Solving radiosity equation • The equations for radiosity for all patches lead to a system of n linear equations with n unknown: Jacobs University Visualization and Computer Graphics Lab 320491: Advanced Graphics - Chapter 3 75 Radiosity vs. Monte-Carlo ray tracing • project onto finite • use probabilistic basis of functions sampling • linear system •view-independent •view-dependent • ideal diffuse surfaces •BRDFs Jacobs University Visualization and Computer Graphics Lab 320491: Advanced Graphics - Chapter 3 76 3.1.7 Irradiance Caching Jacobs University Visualization and Computer Graphics Lab 320491: Advanced Graphics - Chapter 3 77 Monte-Carlo ray tracing • Computational bottleneck: –Weneedtonsof raysper pixel. •Question: –Canweusespatialcoherence? Jacobs University Visualization and Computer Graphics Lab 320491: Advanced Graphics - Chapter 3 78 Direct illumination Jacobs University Visualization and Computer Graphics Lab 320491: Advanced Graphics - Chapter 3 79 Radiance Jacobs University Visualization and Computer Graphics Lab 320491: Advanced Graphics - Chapter 3 80 Irradiance Jacobs University Visualization and Computer Graphics Lab 320491: Advanced Graphics - Chapter 3 81 Irradiance • Observation: irradiance (= incident light) is smooth, i.e. changes slowly • Idea: interpolation is possible Jacobs University Visualization and Computer Graphics Lab 320491: Advanced Graphics - Chapter 3 82 Irradiance sampling Jacobs University Visualization and Computer Graphics Lab 320491: Advanced Graphics - Chapter 3 83 Irradiance change Jacobs University Visualization and Computer Graphics Lab 320491: Advanced Graphics - Chapter 3 84 Irradiance interpolation Jacobs University Visualization and Computer Graphics Lab 320491: Advanced Graphics - Chapter 3 85 Irradiance caching When computing the irradiance at a certain point x: • Search for all cached irradiance samples with w(x) > threshold. • If samples are found, interpolate irradiance E(x). • Otherwise,