Radiometry

• Measurement system and units for illumination
• Accurately measure the spatial properties of light: Radiant flux, intensity, irradiance, radiance
• Perform lighting calculations in a physically correct manner

$[*]$ is the unit

Radiant energy is the energy of electromagnetic radiation. It is measured in units of joules, and denoted by the symbol: $\Large Q[J=Joule]$

Radiant flux (power) is the energy emitted, reflected, transmitted or received, per unit time. $\Large\Phi\equiv\frac{dQ}{dt}[W=Watt][lm=lumen]^*$

Radiant (Luminous) intensity is the power per unit solid angle(立体角) emitted by a point light source. $\Large I(\omega)\equiv\frac{d\Phi}{d\omega}$ $\Large[\frac{W}{sr}][\frac{lm}{sr}=cd=candela]$

The candela is one of the seven SI base units.

Angle: ratio of subtended arc length on circle to radius $\large\theta=\frac{l}{r}$, Circle has $2\pi$ radians. $l$ is the arc length, $r$ is the radius.

Solid angle: ratio of subtended area on sphere to radius squared $\large\Omega=\frac{A}{r^2}$. Sphere has $4\pi$ steradians. $A$ is the area on sphere, $r$ is the radius.

Differential Solid Angles $d\omega$

Sphere: $S^2$

$\omega$ as a direction vector (unit length)

Isotropic Point Source: $\Phi=\int_{S^2}Id\omega=4\pi I$, $I=\Large\frac{\Phi}{4\pi}$

Irradiance is the power per (perpendicular/projected) unit area incident on a surface point. $\Large E(x)\equiv\frac{d\Phi(x)}{dA}$ $\Large[\frac{W}{m^2}][\frac{lm}{m^2}=lux]$

Irradiance at surface is proportional to cosine of angle between light direction and surface normal.

Radiance is the fundamental field quantity that describes the distribution of light in an environment. Radiance is the quantity associated with a ray. Rendering is all about computing radiance.

The radiance (luminance) is the power emitted reflected, transmitted or received by a surface, per unit solid angle, per projected unit area.

$\cos\theta$ accounts for projected surface area.

Incident radiance is the irradiance per unit solid angle arriving at the surface. $L(p,\omega)=\large\frac{dE(p)}{d\omega\cos\theta}$

Exiting surface radiance is the intensity per unit projected area leaving the surface. $L(p,\omega)=\large\frac{dI(p,\omega)}{dA\cos\theta}$

Irradiance vs. Radiance

• Irradiance: total power received by area dA

• Radiance: power received by area dA from "direction" $d\omega$

• $dE(p,\omega)=L_i(p,\omega)\cos\theta d\omega$

  $E(p)=\int_{H^2}L_i(p,\omega)\cos\theta d\omega$

  Unit Hemisphere: $H^2$

  

Bidirectional Reflectance Distribution Function (BRDF)

Reflection at a Point: Radiance from direction $\omega_i$ turns into the power E that dA receives. Then power E will become the radiance to any other direction $\omega$.

Differential irradiance incoming: $dE(\omega_i)=L(\omega_i)\cos\theta_id\omega_i$

Differential radiance exiting (due to $dE(\omega_i)$): $dL_r(\omega_r)$

BRDF: The Bidirectional Reflectance Distribution Function (BRDF) represents how much light is reflected into each outgoing direction $\omega_r$ from each incoming direction.

The Reflection Equation: $\Large L_r(p,\omega_r)=\int_{H^2}f_r(p,\omega_i\to\omega_r)L_i(p,\omega_i)\cos\theta_id\omega_i$

The Rendering Equation: Re-write the reflection equation by adding an Emission term to make it general.

assume that all directions are pointing outwards

Understanding the rendering equation:

• Reflection Equation

  

  

• Rendering Equation

  

  Rendering Equation as Integral Equation

  Is a Fredholm Integral Equation of second kind [extensively studied numerically] with canonical form. $K(u,v)dv$ is the Kernel of equation

  Linear Operator Equation

  Can be discretized to a simple matrix equation [or system of simultaneous linear equations] (L, E are vectors, K is the light transport matrix)

  • $L=E+KL$
  • $IL-KL=E$
  • $(I-K)L=E$
  • $L=(I-K)^{-1}E$
  • Binomial Theorem $L=(I+K+K^2+K^3+...)E$
  • $L=E+KE+K^2E+K^3E+...$

• Ray Tracing: $L=E+KE+K^2E+K^3E+...$

  • $E$: Emission directly From light sources

  • $KE$: Direct illumination on surfaces

    The following belong to Global illumination

  • $K^2E$: Indirect illumination (One bounce indirect) [Mirrors, Refraction]

  • $K^3E$: Two bounce indirect illumination

  • ...