Rendering equation
- 2. Empirical Model Phong Shading Ambient + Diffuse + Specular = Phong Reflection Parameters are chosen arbitrary by the user
- 3. Realistic Model Radiosity Radiosity (radiometry), the total radiation (emitted plus reflected) leaving a surface, certainly including the reflected radiation and the emitted radiation. BRDF Equation A BRDF, bi-directional reflectance distribution function, is a tool for describing the distribution of reflected light at a surface. Light Transport Equation The light transport equation (LTE) is the governing equation that describes the equilibrium distribution of radiance in a scene.
- 4. Spectrum and Spectral Power Distribution For each surface, it reflects lights with different wavelengths. The power of different lights can be plot as Spectral Power Distribution graph. A remarkable property of the human visual system makes it possible to represent colors for human perception with just three floating-point numbers. The tristimulus theory of color perception says that all visible SPDs can be accurately represented for human observers with three values.
- 5. Spectrum and Spectral Power Distribution When we display an RGB color on a display, the spectrum that is displayed is basically determined by the weighted sum of three spectral response curves, one for each of red, green, and blue, as emitted by the display’s phosphors, LED or LCD elements, or plasma cells. Figure plots the red, green, and blue distributions emitted by a LED display and an LCD display; note that they are remarkably different. Given (X, Y, Z) representation of an SPD, we can convert it to corresponding RGB coefficients, given the choice of a set of SPDs that define red, green, and blue for a display of interest.
- 6. Angle and Solid Angle In order to define light intensity, we first need to define the notion of a solid angle. Solid angles are just the extension of 2D angles in a plane to an angle on a sphere. 2D angles are defined by the length of the arc on the unit circle. Solid Angle (3D Angle) are defined by the area on the unit sphere. So for a 2D angle s, if we calculate 360 degree, then it is equal to ! 𝑑𝑠 = 2𝜋 While for solid angle 𝜔 ! 𝑑𝜔 = 4𝜋
- 7. Radiometry Quantities Definition Energy (Single Photon) 𝑄 = ℎ𝑐 𝜆 𝑐 is the speed of light, ℎ is the Planck’s constant, and 𝜆 is the wavelength. Flux (Power) 𝜙 = 𝑑𝑄 𝑑𝑡 , Q = + ∅𝑑𝑡 Energy per second Irradiance and Radiant Exitance 𝐸 = 𝑑𝜙 𝑑𝐴 , ∅ = + 𝐸𝑑𝐴 Flux (Power) per area Intensity 𝐼 = 𝑑𝜙 𝑑𝜔 , 𝜙 = + ! 𝐼𝑑𝜔 Flux (Power) per solid angle Radiance (Brightness) 𝐿 𝑝, 𝜔 = 𝑑𝜙 𝑑𝜔 ⋅ 𝑑𝐴 ⋅ 𝑐𝑜𝑠𝜃 Flux (Power) per area per solid angle
- 8. Radiometry By knowing Radiance, then all other quantities can be calculated by integrating Radiance over solid angle or over area. Consider point P, then the received Irradiance is 𝐸! 𝑝, 𝑛 = ! " 𝐿! 𝑝, 𝜔 |𝑐𝑜𝑠𝜃|𝑑𝜔 Most importantly, we need to know 𝐿# 𝑝, 𝑤# , as it is the power our eye receives.
- 9. Linearity Assumption If the direction 𝜔! is considered as a differential cone of directions, the differential irradiance at 𝑝 is 𝑑𝐸 𝑝, 𝜔! = 𝐿! 𝑝, 𝜔! |𝑐𝑜𝑠𝜃!|𝑑𝜔! A differential amount of radiance will be reflected in the direction 𝜔" due to this irradiance. Because of the linearity assumption from geometric optics, the reflected differential radiance is proportional to the irradiance 𝑑𝐿"(𝑝, 𝜔") ∝ 𝑑𝐸(𝑝, 𝑤!) A good way to understand is that the outgoing radiance has a constant ratio against with the incoming irradiance. When the irradiance increase the outgoing radiance increase as well. Considering different directions, when the irradiance increases, it results in increasing different amounts of the outgoing radiance. The ratio between the outgoing radiance and incoming irradiance is like the probability density function between probability and random variables.
- 10. BRDF Equation Because we have 𝑑𝐿!(𝑝, 𝜔!) ∝ 𝑑𝐸(𝑝, 𝑤") The constant of proportionality defines the surface’s BRDF for the pair of directions 𝜔" and 𝜔!: 𝑓# 𝑝, 𝜔", 𝜔! = 𝑑𝐿!(𝑝, 𝜔!) 𝑑𝐸(𝑝, 𝜔") = 𝑑𝐿!(𝑝, 𝜔!) 𝐿" 𝑝, 𝜔" |𝑐𝑜𝑠𝜃"|𝑑𝜔" So we have 𝐿! 𝑝, 𝜔! = 2 $ 𝑓# 𝑝, 𝜔", 𝜔! 𝐿" 𝑝, 𝜔" |𝑐𝑜𝑠𝜃"|𝑑𝜔" Physically based BRDFs have two important qualities: 1. Reciprocity: For all pairs of directions: 𝑓# 𝑝, 𝜔", 𝜔! = 𝑓# 𝑝, 𝜔!, 𝜔" 2. Energy conservation: The total energy of light reflected is less than or equal to the energy of incident light. For all directions: 2 $ 𝑓# 𝑝, 𝜔, 𝜔! 𝑐𝑜𝑠𝜃𝑑𝜔 ≤ 1
- 11. Rendering Equation 𝐿!$ = 𝐿#% = ! 𝐿!% …