2/1/10 Outline The Radiance Equation Basic terms in radiometry - PDF document

2/1/10 Outline The Radiance Equation • Basic terms in radiometry Jan Kautz • Radiance • Reflectance • The Radiance Equation • The operator form of the radiance equation • Meaning of the operator form • Approximations to the radiance equation 2005 Mel Slater, 2006 Céline Loscos, 2007–2010 Jan Kautz Light: Radiant Power Light: Flux Equilibrium • Φ denotes the radiant energy or flux in a volume V . • Total flux in a volume in dynamic equilibrium – Particles are flowing • The flux is the rate of energy flowing through a surface per unit time (watts). – Distribution is constant • Conservation of energy • The energy is proportional to the particle flow, since each photon carries energy. – Total energy input into the volume = total energy that is output by or absorbed by matter within the volume. • The flux may be thought of as the flow of photons per unit time. Light: Equation Radiance • Radiance ( L ) is the flux that leaves a surface, per unit projected area of the surface, per unit solid angle of direction. • Φ (p, ω ) denotes flux at p ∈ V, in direction ω d Φ = [ ∫ L cos θ d ω ] dA • It is possible to write down an integral equation for Φ (p, ω ) based on: L = d 2 Φ / [ cos θ d ω dA ] – Emission+Inscattering = Streaming+Outscattering + Absorption n • Complete knowledge of Φ (p, ω ) provides a complete solution to the graphics rendering problem. • Rendering is about solving for Φ (p, ω ). L θ dA 1

2/1/10 Radiance Solid angle dB cos θ B d ω b = • For computer graphics the basic particle is not the r 2 photon and the energy it carries but the ray and its dB associated radiance. n A θ B n n B θ A r d ω L dA θ dA Radiance is constant along a ray. Radiance: Radiosity, Irradiance Radiosity and Irradiance • Radiosity - is the flux per unit area that radiates • L(p, ω ) is radiance at p in direction ω from a surface, denoted by B . • E(p) is irradiance at p – d Φ = B dA • E(p) = (d Φ /dA) = ∫ L(p, ω ) cos θ d ω • Irradiance is the flux per unit area that arrives at a surface, denoted by E. • (or: L = dE/dA) – d Φ = E dA Light Sources – Point Light Light Sources • Point light with isotropic radiance • Other types of light sources – Spot-lights – Power (total flux) of a point light source • Cone of light • Φ s = Power of the light source [Watt] • Radiation characteristic of cos n θ – Intensity of a light source – Area light sources • I= Φ s /( 4 π sr) [Watt/sr] – Point light sources with non-uniform – Irradiance on a sphere with radius r around light source: directional power distribution • E r = Φ s /( 4 π r 2 ) [Watt/m 2 ] • Other parameter – Irradiance on a surface A – Atmospheric attenuation with distance (r) for point light sources • 1/(ar 2 +br+c) • Physically correct would be 1/r 2 • Correction of missing ambient light 2

2/1/10 Reflectance BRDF • BRDF • Boils down to: How much light is reflected for a given light/ f(p, ω i , ω r ) Unit: 1/sr – Bi-directional view direction at a point? – Reflectance ω i ω r – Distribution • Defines the "look" of the surface – Function • Important part for realistic surfaces: Reflected ray • Relates – Variation (in texture, gloss, …) Incident ray – Reflected radiance to incoming irradiance Illumination hemisphere How to compute reflected light? Properties of BRDFs • Integrate all incident light * BRDF • Non-negativity • Energy Conservation • Reciprocity – Aside: actually does often not hold for real materials [see Eric Veach's PhD Thesis!] Reflectance: BRDF The Radiance Equation • Reflected Radiance = • Radiance L(p, ω ) at a point p in direction ω is the L(p, ω r ) = ∫ f(p, ω i , ω r ) L(p, ω i ) cos θ i d ω I sum of • In practice BRDF’s are hard to specify – Emitted radiance L e (p, ω ) • Commonly rely on ideal types – Total reflected radiance – Perfectly diffuse reflection Radiance = Emitted Radiance + Total Reflected Radiance – Perfectly specular reflection – Glossy reflection • BRDFs taken as additive mixture of these 3

2/1/10 The Radiance Equation: Reflection The Radiance Equation • p is considered to be on a surface, but can be anywhere, since radiance is constant along a ray, trace back until surface is reached at p’, then • Total reflected radiance in direction ω : – L(p, ω i ) = L(p’, ω i ) ∫ f(p, ω i , ω ) L(p*, - ω i ) cos θ i d ω I (p* is closest point L(p, ω ) depends on all in direction ω i ) L(p*, - ω i ) which in turn p* are recursively defined. • Full Radiance Equation: L(p, ω ) = L e (p, ω ) + ∫ f(p, ω i , ω ) L(p*, - ω i ) cos θ i d ω i ω i • L(p, ω ) p – (Integration over the illumination hemisphere) The radiance equation models global illumination. Operator form of the Radiance Equation Operator Form • Define the operator R to mean • Using this notation, the radiance equation can be rewritten as: • (RL)(p, ω ) = ∫ f(p, ω i , ω ) L(p*, - ω i ) cos θ i d ω I – L = L e + RL – Use the notation RL(p, ω ) = L 1 (p, ω ) • We can rearrange this as: – Repeated applications of R can be applied – (1-R)L = L e – R(RL(p, ω ))= R 2 L(p, ω ) = RL 1 (p, ω ) = L 2 (p, ω ) • Operator theory allows the normal algebraic operations: – … • The operator 1 means the identity: – L = (1-R) -1 L e – L = (1 + R + R 2 + R 3 + …) L e (Neumann series/expansion) – 1L(p, ω ) = L(p, ω ) Meaning of the Operator Meaning of the Operator e (p, ω i ) • R 2 L e (p, ω i ) = RL 1 is therefore light that is ‘twice removed’ from • L e (p, ω i ) is radiance corresponding to direct the light sources. lighting from a source (if • Similar meanings can any) from direction ω i at be attributed to point p. R 3 L e (p, ω i ), R 4 L e (p, ω i ) and • RL e (p, ω i ) is therefore the so on. radiance from point p in direction ω due to this direct lighting. In general R i L e (p, ω i ) is the contribution to radiance from p in direction ω from all light paths of length i+1 This is light that is ‘one step removed’ from the sources. back to the sources. 4

2/1/10 The Radiance Equation Truncating the Equation • Suppose the series is truncated after the first term (1)L e – Only objects that are emitters would be shown • In general the radiance equation in operator form • Suppose one more term is added (1+R)L e shows that L(p, ω ) may be decomposed into light – Only direct lighting (and shadows) are accounted for. due to • Suppose another term is added (1+R+R 2 )L e – The emissive properties of the surface at p – Additionally one level of reflection is accounted for. – Plus that due directly to sources • …and so on. – Plus that reflected once from sources • Each type of rendering method is a special case of – Plus that reflected twice this rendering equation, and computer graphics – … to infinity rendering consists of different types of approximation. Monte Carlo Methods Simple Example - Finding π • Choose a random sample of n uniformly distributed points in • The radiance equation is an integral equation. the square of side 2. • Count how many (r) in the circle. • Monte Carlo methods may be used to solve this. • r/n →π /4 with prob. 1 as n →∝ • Monte Carlo methods involve using a random sampling technique to solve deterministic problems. Variance Reduction Stratified Sampling • The convergence rate of this procedure will be • A uniform random sample is random ! quite slow. • Each type of pattern is • The standard error of the estimator ∝ 1/ √ n equally probable. • The problem in MC methods is to find ways to • A stratified sample, where reduce the variance. we sample randomly within • A standard technique is stratified sampling. strata significantly reduces the variance. stratified 5

2/1/10 Example Antialiasing in Ray Tracing • In order to reduce aliasing due to undersampling in ray tracing each pixel may be sampled and then the average radiance per pixel found. • A stratified sample over the pixel is preferable to a uniform sample – especially when the gradient within the pixel is sharply changing. Unstratified Stratified Slide borrowed from Henrik Wann Jensen Conclusion • Radiance equation formally revisited • And defined as an operator form • Introduction to Monte Carlo sampling • Applied to ray tracing 6