Radiometry - the study of the propagation of electromagnetic radiation in an environment
There are four radiometric quantities that are central to rendering
They can each be derived from energy by successively taking limits over time, area, and directions
Thus all of the radiometric quantities are in general wavelength dependent
Energy is measured in joules (\( J \)). Sources of illumination emit photons, each of which is at a particular wavelength and carries a particular amount of energy. All of the basic radiometric quantities are effectively different ways of measuring photons.
A photon at wavelength \( \lambda \) carries energy
$$ Q= \frac{hc}{\lambda},$$
where \( c \approx 3\times 10^8 m/s\) is the speed of light and \( h\approx 6,626\times 10^{-34} m^2kg/s \) is Planck's constant
There are four radiometric quantities that are central to rendering
$$ Q= \frac{hc}{\lambda},$$
Radiant power (Flux) is the total amount of energy passing through a surface or region of space per unit time. Radiant power can be found by taking the limit of differential energy per differential time:
$$ \Phi = \frac{dQ}{dt} $$
Its units are joules/second (\( J/s \)), or more commonly, watts (\( W \)). Total emission from light sources is generally described in terms of flux.
Note that the total amount of flux measured on either of the two spheres is the same - although less energy is passing through any local part of the large sphere than the small sphere, the greater area of the large sphere means that the total flux is the same.
Flux from a point light source measured by the total amount of energy passing through imaginary spheres around the light
There are four radiometric quantities that are central to rendering
$$ \Phi = \frac{dQ}{dt} $$
$$ Q= \frac{hc}{\lambda},$$
Irradiance (the area density of flux arriving at a surface), or radiant exitance (the area density of flux leaving a surface) is the average density of power over the area over which photons per time is being measured:
$$ E=\frac{\Phi}{A} $$
Its units are \( W/m^2 \).
Flux from a point light source measured by the total amount of energy passing through imaginary spheres around the light
More generally, we can define irradiance by taking the limit of differential power per differential area at a point \( p \):
$$ E(p) = \lim_{\Delta A \rightarrow 0} \frac{\Delta\Phi(p)}{\Delta A} = \frac{d\Phi(p)}{dA}$$
There are four radiometric quantities that are central to rendering
$$ \Phi = \frac{dQ}{dt} $$
$$ Q= \frac{hc}{\lambda},$$
Lambert’s Law: Irradiance arriving at a surface varies according to the cosine of the angle of incidence of illumination, since illumination is over a larger area at larger incident angles.
Consider a light source with area \(A\) and flux \(\Phi\) that is illuminating a surface. If the light is shining directly down on the surface, then the area on the surface receiving light \(A_1\) is equal to \(A\). Irradiance at any point inside \(A_1\) is then \(E_1=\frac{\Phi}{A}\)
However, if the light is at an angle to the surface, the area on the surface receiving light is larger. If \(A\) is small, then the area receiving flux, \(A_2\) , is roughly \(A/\cos\Theta\). For points inside \(A_2\), the irradiance is therefore \(E_2=\frac{\Phi\cos\Theta}{A}\)
Irradiance (the area density of flux arriving at a surface), or radiant exitance (the area density of flux leaving a surface) is the average density of power over the area over which photons per time is being measured:
$$ E=\frac{\Phi}{A} $$
Its units are \( W/m^2 \).
There are four radiometric quantities that are central to rendering
$$ \Phi = \frac{dQ}{dt} $$
$$ E=\frac{\Phi}{A} $$
$$ Q= \frac{hc}{\lambda},$$
The planar angle \(\theta\) is the total angle subtended by some object with respect to some position.
If we project a shaded object onto a unit circle, some length of the circle \(l\) will be covered by its projection. The arc length (\(l=\theta\)) is the angle subtended by the object.
Planar angles are measured in radians.
The solid angle \(\omega\) extends the 2D unit circle to a 3D unit sphere. The total area \(s\) is the solid angle subtended by the object.
Solid angles are measured in steradians [\(sr\)].
The entire sphere subtends a solid angle of \(4\pi sr\), and a hemisphere subtends \(2\pi sr\).
There are four radiometric quantities that are central to rendering
$$ \Phi = \frac{dQ}{dt} $$
$$ E=\frac{\Phi}{A} $$
$$ Q= \frac{hc}{\lambda},$$
Intensity is the angular density of emitted power. For an infinitesimal light source surrounded with a unit sphere, we can compute the angular density of emitted power over the entire sphere of directions:
$$ I=\frac{\Phi}{4\pi} $$
Its units are \( W/sr \). But more generally we are interested in taking the limit of a differential cone of directions:
$$ I=\lim_{\Delta\omega\rightarrow 0}\frac{\Delta\Phi}{\Delta\omega}=\frac{d\Phi}{d\omega}$$
Intensity describes the directional distribution of light, but it is only meaningful for point light sources
There are four radiometric quantities that are central to rendering
$$ \Phi = \frac{dQ}{dt} $$
$$ E=\frac{\Phi}{A}, E(p)=\frac{d\Phi(p)}{dA} $$
$$ Q= \frac{hc}{\lambda},$$
$$I=\frac{d\Phi}{d\omega}$$
Radiance is the flux density per unit area and per unit solid angle. In terms of flux, it is defined by:
$$ L=\frac{d\Phi}{d\omega dA^{\bot}}, $$
where \(dA^\bot\) is the projected area of \(dA\) on a hypothetical surface perpendicular to \(\omega\).
Radiance \(L\) is defined as flux per unit solid angle \(d\omega\) per unit projected area \(dA^\bot\)
In terms of irradiance, radiance measures irradiance or radiant exitance with respect to solid angles:
$$L(p,\omega)=\lim_{\Delta\omega\rightarrow 0}\frac{\Delta E_\omega(p)}{\Delta\omega} = \frac{dE_\omega(p)}{d\omega},$$
where \(E_\omega\) denotes irradiance at the surface that is perpendicular to the direction \(\omega\).
We make a distinction between radiance arriving at the point (e.g., due to illumination from a light source) and radiance leaving that point (e.g., due to reflection from a surface)
The incident radiance function \(L_i(p, \omega)\) describes the distribution of radiance arriving at a point as a function of position and direction
The exitant radiance function \(L_o(p, \omega)\) gives the distribution of radiance leaving the point
Note that for both functions, \(\omega\) is oriented to point away from the surface, and, thus, for example, \(L_i(p,-\omega)\) gives the radiance arriving on the other side of the surface than the one where \(\omega\) lies
Photometry is the study of visible electromagnetic radiation in terms of its perception by the human visual system
All of the radiometric measurements like flux, radiance, etc. have corresponding photometric measurements
Our visual system responds differently to different wavelengths
Each radiometric quantity can be converted to its corresponding photometric quantity by integrating against the luminosity function.
Radiance \(L(\lambda)\) can be converted to luminance \(Y\) as
$$ Y = \int_\lambda L(\lambda)V(\lambda)d\lambda$$
Photometry is the study of visible electromagnetic radiation in terms of its perception by the human visual system
All of the radiometric measurements like flux, radiance, etc. have corresponding photometric measurements
Radiometric | Unit | Photometric | Unit | |
---|---|---|---|---|
Radiant energy | joule (Q) | Luminous energy | talbot (T) | |
Radiant flux | watt (W) | Luminous flux | lumen (lm) | |
Intensity | W / sr | Luminous flux | lm / sr = candela (cd) | |
Irradiance | W / m^2 | Illuminance | lm / m^2 = lux (lx) | |
Radiance | W / (m^2 sr) | Luminance | lm / (m^2 sr) = cd / m^2 = nit |
In Physics called radiative transport equation and expresses energy equilibrium in scene:
Exitant radiance = emitted radiance + reflected radiance
Emitted radiance (\(L_e\)) describes the emissivity of the surface and is non-zero only for light sources
Reflected radiance (\(L_r\)) is in general an integral over all possible incoming directions of radiance times angle-dependent surface reflection function.
It leads to Fredholm integral equation of second kind: thus, numerical methods are necessary to compute an approximate solution
Computation of irradiance at a point
Irradiance at a point \(p\) with surface normal \(\vec{n}\) due to radiance over a set of directions \(\Omega\) is
$$E(p,\vec{n}) = \int_\Omega L_i(p, \omega_i)\left|\cos\theta_i\right|d\omega, $$
where \(L_i(p,\omega_i)\) is the incident radiance function and the \(\cos\theta_i\) term in this integral is due to the \(dA^\bot\) term in the definition of radiance
Computation of irradiance at a point
Irradiance at a point \(p\) with surface normal \(\vec{n}\) due to radiance over a set of directions \(\Omega\) is
$$E(p,\vec{n}) = \int_\Omega L_i(p, \omega)\left|\cos\theta\right|d\omega $$
Computation of irradiance at a point
Irradiance at a point \(p\) with surface normal \(\vec{n}\) due to radiance over a set of directions \(\Omega\) is
$$E(p,\vec{n}) = \int_\Omega L_i(p, \omega)\left|\cos\theta\right|d\omega $$
We can thus see that the irradiance integral over the hemisphere can equivalently be written as
$$E(p,\vec{n}) = \int^{2\pi}_{0}\int^{\pi/2}_{0} L_i(p, \theta,\phi)\cos\theta \sin\theta d\theta d\phi $$
Note: if the radiance is the same from all directions, the equation simplifies to
$$E=\pi L_i$$
Computation of irradiance at a point
Irradiance at a point \(p\) with surface normal \(\vec{n}\) due to radiance over a set of directions \(\Omega\) is
$$E(p,\vec{n}) = \int_\Omega L_i(p, \omega)\left|\cos\theta\right|d\omega $$
Computation of irradiance at a point
Irradiance at a point \(p\) with surface normal \(\vec{n}\) due to radiance over a set of directions \(\Omega\) is
$$E(p,\vec{n}) = \int_\Omega L_i(p, \omega)\left|\cos\theta\right|d\omega $$
Therefore, we can write the irradiance integral for the quadrilateral source as
$$E(p,\vec{n}) = \int_S L\cos\theta_i\frac{\cos\theta_o}{r^2}dS $$
To compute irradiance at a point \(p\) from a quadrilateral source, it’s easier to integrate over the surface area of the source than to integrate over the irregular set of directions that it subtends. Thus \(L\) here is the emitted radiance from the surface of the quadrilateral source.
We would like to know how much radiance is leaving the surface in the direction \(\omega_o\) toward the camera, \(L_o(p,\omega_o)\), as a result of incident radiance along the direction \(\omega_i\), \(L_i(p,\omega_i)\).
Because of the linearity assumption from geometric optics, the reflected differential radiance is proportional to the irradiance:
$$dL_r(p,\omega_o)\propto dE(p,\omega_i)$$
or
$$dL_r(p,\omega_o) = k\cdot dE(p,\omega_i)$$
or
$$dL_r(p,\omega_o) = f_r(p,\omega_o,\omega_i)\cdot dE(p,\omega_i)$$
Recall: Irradiance at a point \(p\) with surface normal \(\vec{n}\) due to radiance over a set of directions \(\Omega\) is
$$E(p,\omega_i) = \int_\Omega L_i(p, \omega_i)\left|\cos\theta_i\right|d\omega $$
We would like to know how much radiance is leaving the surface in the direction \(\omega_o\) toward the camera, \(L_o(p,\omega_o)\), as a result of incident radiance along the direction \(\omega_i\), \(L_i(p,\omega_i)\).
Because of the linearity assumption from geometric optics, the reflected differential radiance is proportional to the irradiance:
$$dL_r(p,\omega_o)=f_r(p,\omega_o,\omega_i) L_i(p,\omega_i)\cos\theta_id\omega$$
Recall: Irradiance at a point \(p\) with surface normal \(\vec{n}\) due to radiance over a set of directions \(\Omega\) is
$$E(p,\omega_i) = \int_\Omega L_i(p, \omega_i)\left|\cos\theta_i\right|d\omega $$
The rendering Equation:
$$ L_o(p,\omega_o) = L_e(p,\omega_o) + \int_{H^2(\vec{n})} f_r(p,\omega_o,\omega_i)L_i(p,\omega_i)\cos\theta_i d\omega_i$$
The bidirectional reflectance distribution function (BRDF) gives a formalism for describing reflection from a surface
$$ f_r(p,\omega_o,\omega_i) = \frac{dL_o(p,\omega_o)}{dE(p,\omega_i)}$$
BRDF
is the key term of the rendering equation
BRDF
is the core of every software shader class
The bidirectional reflectance distribution function (BRDF) gives a formalism for describing reflection from a surface
$$ f_r(p,\omega_o,\omega_i) = \frac{dL_o(p,\omega_o)}{dE(p,\omega_i)}$$
Helmholtz reciprocity principle: BRDF remains unchanged if incident and reflected directions are interchanged
$$ f_r(p,\omega_o,\omega_i) = f_r(p,\omega_i,\omega_o) $$
The total energy of light reflected is less than or equal to the energy of incident light. For all directions \(\omega_o\)
$$ \int_{H^2(\vec{n})} f_r(p,\omega_o,\omega_i) \cos\theta_i d\omega_i \le 1$$
A device to measure BRDF is called Gonioreflectometer
4 directional degrees of freedom
BRDF representation
Aluminium; \(\lambda=2.0\mu m\)
Aluminium; \(\lambda=0.5\mu m\)
Magnesium; \(\lambda=0.5\mu m\)
In case when BRDF measurement is impossible, an approximated model may be used
Light equally likely to be reflected in any output direction (independent of input direction)
$$f_r(p,\omega_o,\omega_i)=k_d=const$$
$$L_{r,d}(p,\omega_o) = \int_{H^2(\vec{n})} k_dL_i(p,\cdot)\cos\theta_i d\omega_i = k_d L_i\cos\theta_i\int_{H^2(\vec{n})} d\omega_i = 2\pi k_d L_i \cos\theta_i \propto k_dL_i(\vec{n}\cdot\vec{I})$$
Light equally likely to be reflected in any output direction (independent of input direction)
$$f_r(p,\omega_o,\omega_i)=k_d=const$$
$$L_{r,d}(p,\omega_o) = k_dL_i(\vec{n}\cdot\vec{I})$$
$$E\propto L_o\cdot d\omega$$
$$E\propto L_o\cdot\cos\theta\cdot d\omega$$
CShaderFlat
CShaderEyelight
Absorption in photosphere
Path length through photosphere longer from the Sun’s rim
Surface covered with fine dust
Dust on TV visible best from slanted viewing angle
Neither the Sun nor the Moon are Lambertian
Angle of reflectance \(\theta_o\) is equal to angle of incidence \(\theta_i\)
Angle of reflectance \(\theta_o\) is equal to angle of incidence \(\theta_i\)
Dirac delta function \(\delta(x)\) equal to zero everywhere except at \(x=0\)
$$f_r(p,\omega_o,\omega_i)=\frac{\delta(\cos\theta_i-\cos\theta_o)}{\cos\theta_i}\cdot\delta(\phi_i-\phi_o\pm\pi)$$
$$L_{r,m}(p,\omega_o) = \int_{H^2(\vec{n})} \frac{\delta(\cos\theta_i-\cos\theta_o)}{\cos\theta_i}\delta(\phi_i-\phi_o\pm\pi)L_i(p,\theta_i,\phi_i)\cos\theta_i d\omega_i = L_i(p,\theta_o,\phi_o\pm\pi)$$
side view
top view
Glossy reflection appears due to surface roughness
$$f_r(p,\omega_o,\omega_i)=k_s\cos\theta^{k_e}_{\vec{r}\vec{v}}=k_s\left(\vec{r}\cdot\vec{v} \right)^{k_e}$$
$$L_{r,s}(p,\omega_o) = k_sL_i\left(\vec{r}\cdot\vec{v}\right)^{k_e}$$
$$L_{r,s}(p,\omega_o) = k_sL_i\left(\vec{r}\cdot\vec{v}\right)^{k_e}$$
Make use of a halfway vector \(\vec{h}=(\vec{I}+\vec{v})/|\vec{I}+\vec{v}|\) in order to approximate Phong reflectiion model.
$$f_r(p,\omega_o,\omega_i)=k_s\cos\theta^{k_e}_{\vec{h}\vec{n}}=k_s\left(\vec{h}\cdot\vec{n} \right)^{k_e}$$
$$L_{r,s}(p,\omega_o) = k_sL_i\left(\vec{h}\cdot\vec{n}\right)^{k_e}$$
We have considered 3 elemental components that can be used to model a variety of light-surface interactions
(approximate indirect illumination)
(only possible with recursive raytracing)
$$L_r=k_aL_{i,a}+k_d\sum_lL_i(\vec{I}_l\cdot\vec{n})+k_s\sum_lL_i(\vec{r}(\vec{I}_l)\cdot\vec{v})^{k_e}$$
$$L_r=k_aL_{i,a}+k_d\sum_lL_i(\vec{I}_l\cdot\vec{n})+k_s\sum_lL_i(\vec{h}\cdot\vec{n})^{k_e}$$
Here index \(l\) represents point light sources and the color of specular reflection equals to the light source
The greater the variation of microfacet normals, the rougher the surface is
Smooth surfaces have relatively little variation of microfacet normals
Light bounces among the microfacets before reaching the viewer
Analogously, light does not reach the microfacet
The microfacet of interest is not visible to the viewer due to occlusion by another microfacet
For directions \(\omega_i\) and \(\omega_o\), only microfacets with normal \(\omega_h\) reflect light (i.e. \(\vec{n}_f\equiv\vec{h}\))
$$f_r(p,\omega_o,\omega_i)=\frac{D(\omega_h)G(\omega_i, \omega_o)F_r(\omega_o)}{4\cos\theta_i\cos\theta_i}$$
As an approximation we may assume isotropic microfacet distributions: \(D(\omega_h)\equiv D(\theta_h)\), where \(\theta_h=\vec{n}\cdot\vec{h}\)
$$f_r(p,\omega_o,\omega_i)=\frac{D(\omega_h)G(\omega_i, \omega_o)F_r(\omega_o)}{4\cos\theta_i\cos\theta_i}$$
Blinn: $$ D(\theta_h)=\frac{e+2}{2\pi}\cos^e\theta_h $$
Torrance - Sparrow:
$$D(\theta_h)=e^{-\left(\frac{\theta_h}{\sqrt{2}\alpha}\right)^2}$$
Beckmann - Spizzichino:
$$D(\theta_h)=\frac{1}{4\sigma^2\cos^4\theta_h}e^{-\tan^2\theta_h/\sigma^2}$$
Fully illuminated and visible: $$ G(\omega_i,\omega_o)=1 $$
Partial masking of reflected light:
$$G(\omega_i,\omega_o)=\frac{2\cos\theta_h\cos\theta_o}{(\vec{v}\cdot\vec{h})}$$
Partial shadowing of incident light:
$$G(\omega_i,\omega_o)=\frac{2\cos\theta_h\cos\theta_i}{(\vec{v}\cdot\vec{h})}$$
The resulting geometric attenuation factor:
$$G(\omega_i,\omega_o)=\min\left(1,\frac{2(\vec{n}\cdot\vec{h})(\vec{n}\cdot\vec{v})}{(\vec{v}\cdot\vec{h})},\frac{2(\vec{n}\cdot\vec{h})(\vec{n}\cdot\vec{I})}{(\vec{v}\cdot\vec{h})}\right)$$
Directional light
Diffuse emitters
Ambient light
Volume
Area
Line
“Point”
Power (total flux)
Active emission size
Spectral distribution
Directional distribution
Power (total flux) of a point light source
$$\Phi$$
Intensity of a light source (radiance cannot be defined, no area)
$$I=\frac{\Phi}{4\pi}$$
Irradiance on a sphere with radius \(r\) around light source:
$$E=\frac{\Phi}{4\pi r^2}$$
Irradiance on some other surface \(S\)
$$E(p)=\frac{d\Phi}{dS}=I\frac{d\omega}{dS}=\frac{\Phi}{4\pi}\frac{dS\cos\theta}{r^2dS}=\frac{\Phi\cos\theta}{4\pi r^2}$$
Flux from a point light source measured by the total amount of energy passing through imaginary spheres around the light