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Author: Romain Vergne (website)
Please cite my name and add a link to my web page if you use this course

Image synthesis and OpenGL: lighting

Quick links to:

The problem
Color
The rendering equation
Local lighting
Types of lights
Types of reflections
Computing lighting on the GPU
Sources

The problem

The illumination on a given point of the surface depends on:

primary light sources

direct lighting

secondary light sources

all the objects in the scene
an illuminated object becomes a light source
indirect lighting

Complex phenomenon:

Color bleeding
Shadows

Cornel box (1984)

We will be mainly interested in direct lighting and local illumination in this course.

The illumination on a given point of the surface depends on:

The viewpoint
The surface properties

reflexion
absorption
diffusion

The light properties

direction
wavelenght
energy

Involve a lot of dimensions

Color

Physically, visible light is an electromagnetic wave


Physiologically, our eyes use 3 types of sensors to perceive light colors

In computer science


Additive models (lighting)	Soustractive models (painting)

RGB(A) for Red, Green, Blue (Alpha) is the most well known additive model

Easy to use in graphics applications
Difficult to ensure the same color on different screen... (gamut)

CIE 1931 XYZ color space

The tristimulus values can be conceptualized as amounts of three primary colors in a trichromatic additive color model.
The CIE (Comission Internationale de l'éclairage) was the first to mathematically define a trichromatic system to represent perceived colors.
It is based on the standard (colorimetric) observer and thus allows to uniquely represent colors in a 3D space.

Color matching functions: spectral sensitivity
curves of 3 linear light detectors

$I (\lambda)$ is the spectral distribution of a color, then:

$X = \int_{400}^{800} I(\lambda) \bar{x}(\lambda) d \lambda$

$Y = \int_{400}^{800} I(\lambda) \bar{y}(\lambda) d \lambda$

$Z = \int_{400}^{800} I(\lambda) \bar{z}(\lambda) d \lambda$

CIE Yxy color space

Normalize the XYZ color space and decompose it into Luminance (Y) and chrominance (xy)

CIE Lab color space

Perceptually uniform: distance between 2 colors = perceived distance between these colors
Used in most graphics applications
L = luminance
a = chrominance (red-green)
b = chrominance (blue-yellow)

HSV (Hue, Saturation, Value) color space

In most image processing software: Photoshop, Gimp, etc.
Intuitive for designers

CMY(K), for Cyan, Magenta, Yellow (and Key - Black)

Subtractive model
Mainly used for printing

ETC....: Adobe RGB, sRGB, CIELuv, CIEUvw, YIQ (NTSC), YUV (PAL), HSL, ....

Still, the perception of colors is not well understood and the perfect color space does not exist! (example of color contrast)
try this excellent demo

Coding colors

Binary representation: 0 or 1
8 bits: 0 to 255 grey levels (monochromatic)
24 bits: 8 bits per channel (polychromatic)

256 values per channel (usually RGB)
256*256*256 = 16 777 216 colors
limited: 8 orders of magnitude between the sun and stars

HDR (High Dynamic Range) images: float or double precision per channel

24 bits per channel = $256^9$ colors
Can be created using multiple LDR (Low Dynamic Range) images
Needs to be tonemapped to be displayed on a screen

The rendering equation [Kajiya 1986]

$L(\mathbf{p} \rightarrow \mathbf{e}) = L_e(\mathbf{p} \rightarrow \mathbf{e}) + \int_{\Omega_\mathbf{n}} \rho(\mathbf{p}, \mathbf{e}, \pmb{\ell}) (\mathbf{n}\cdot\pmb{\ell}) \ L(\mathbf{p} \leftarrow \pmb{\ell}) \ d\pmb{\ell}$

$\mathbf{p}$ is a point on the surface
$\mathbf{e}$ is the view direction
$\mathbf{n}$ is the normal of the surface at point $\mathbf{p}$
$\pmb{\ell}$ is the direction of a light in the hemisphere $\Omega_\mathbf{n}$

$L(\mathbf{p} \rightarrow \mathbf{e})$ :

outgoing radiance (in $Wm^{-2}sr^{-1})$
how much energy is arriving to the eye / camera

$L_e(\mathbf{p} \rightarrow \mathbf{e})$ :

emitted radiance
usually equal to 0 for object surfaces (they do not create energy)

$L(\mathbf{p} \leftarrow \pmb{\ell})$ :

incoming radiance
incident illumination leaving the light $\pmb{\ell}$ and arriving at the point $\mathbf{p}$ of the surface

$(\mathbf{n}\cdot\pmb{\ell})$ :

the orientation of the surface
dot product between $\mathbf{n}$ and $\pmb{\ell}$

$\rho(\mathbf{p}, \mathbf{e}, \pmb{\ell})$ :

material properties / BRDF (Bidirectional Reflectance Distribution Function)
how much energy the surface reflects in the viewing direction $\mathbf{e}$ given the incident light $\pmb{\ell}$

Local lighting

General equation

Empirical model for computing the outgoing radiance

$L(\mathbf{p} \rightarrow \mathbf{e}) = \rho_a L_a + \sum_{k} \rho(\mathbf{p}, \mathbf{e}, \pmb{\ell}_k) \ (\mathbf{n}\cdot\pmb{\ell}_k) \ L(\mathbf{p} \leftarrow \pmb{\ell}_k)$

$L(\mathbf{p} \rightarrow \mathbf{e})$ : outgoing radiance / light energy / color
$\rho_a L_a$ : ambient lighting (approximate indirect lighting)
$\sum_{k} \cdots$ : contribution of each light $\pmb{\ell}_k$
$\rho(\mathbf{p}, \mathbf{e}, \pmb{\ell}_k)$ : BRDF - how the light $\pmb{\ell}_k$ is reflected on top of the surface
$(\mathbf{n}\cdot\pmb{\ell}_k)$ : surface orientation (according to light $\pmb{\ell}_k$ )
$L(\mathbf{p} \leftarrow \pmb{\ell}_k)$ : incoming radiance for light $\pmb{\ell}_k$

The simplest model: assign a color at each point of the surface (albedo)

$L(\mathbf{p} \rightarrow \mathbf{e}) = color$

Reflecting power of the surface
Independent of the view direction
Independent of the light direction
Too simple:

adapted for static scenes (photos) but not for dynamic ones (image synthesis)
dynamic viewpoint?
dynamic lighting variations?

The second simplest model: consider a single dynamic light

$L(\mathbf{p} \rightarrow \mathbf{e}) = \rho(\mathbf{p}, \mathbf{e}, \pmb{\ell}) \ (\mathbf{n}\cdot\pmb{\ell}) \ L(\mathbf{p} \leftarrow \pmb{\ell})$

Which types of lights?
Which types of surface reflections?

Types of lights

Infinitesimal lights


Directionnal light Distant sources (sun) Environment maps (video-games) $L(\mathbf{p} \leftarrow \pmb{\ell}) = L$	Point light position in space $\mathbf{p}_{\pmb{\ell}}$ near, small sources $L(\mathbf{p} \leftarrow \pmb{\ell}) = L/r^2$ with $r = \|\| \mathbf{p} - \mathbf{p}_{\pmb{\ell}} \|\|$ and $\pmb{\ell} = \frac{\mathbf{p} - \mathbf{p}_{\pmb{\ell}} }{r}$	Spot light position in space near, small sources defined in a cone direction $\mathbf{s}_{\pmb{\ell}}$ exponent $e$ a cutoff $c$ $L(\mathbf{p} \leftarrow \pmb{\ell}) = \frac{(\mathbf{s}_{\pmb{\ell}} \cdot \pmb{\ell})^e L}{r^2}$ if $(\mathbf{s}_{\pmb{\ell}} \cdot \pmb{\ell})<c$ and 0 otherwise

Area lights


Defined on volumes / surfaces Soft shadows Expensive!

Types of reflections

Lets consider a (white) directional light (

$L=cst=(1,1,1)$ ). The model is equal to:

$L(\mathbf{p} \rightarrow \mathbf{e}) = \rho(\mathbf{p}, \mathbf{e}, \pmb{\ell}) \ (\mathbf{n}\cdot\pmb{\ell})$

The lambertian model


Lambertian surface	$\rho(\mathbf{p}, \mathbf{e}, \pmb{\ell}) = cst$ Light is diffused in every direction Independant on the point of view

$L(\mathbf{p} \rightarrow \mathbf{e}) = \rho_d \ (\mathbf{n}\cdot\pmb{\ell}) L_d$	$L(\mathbf{p} \rightarrow \mathbf{e}) = \sum_k \rho_{kd} \ (\mathbf{n}\cdot\pmb{\ell}) L_{kd}$
One light: $\rho_d =$ constant diffuse material color $L_d =$ constant diffuse light color	Multiple light sum of contribution of each light may have different color coeficients

The mirror and transparent models


Mirror surface	$\rho(\mathbf{p}, \mathbf{e}, \pmb{\ell}) = cst\ if\ (\mathbf{r}\cdot\mathbf{e}=1),\ 0\ otherwise$ Reflected light vector $\mathbf{r}=2 \mathbf{n} (\mathbf{n}\cdot \pmb{\ell})-\pmb{\ell}$ Dependant on the point of view Usefull for environment maps compute the reflected view vector $\mathbf{r}=2 \mathbf{n} (\mathbf{n}\cdot \mathbf{e})-\mathbf{e}$ use it to fetch a color in the map


Transparent surface	$\rho(\mathbf{p}, \mathbf{e}, \pmb{\ell}) = cst\ if\ (\mathbf{r}\cdot\mathbf{e}=1),\ 0\ otherwise$ Refracted light vector $\mathbf{r}=e \pmb{\ell} - (e (\mathbf{n}\cdot \pmb{\ell}) + \sqrt{1-e^2 (1-(\mathbf{n}\cdot \pmb{\ell})^2)}) \mathbf{n}$ Dependant on the point of view Same principle for environment maps

$L(\mathbf{p} \rightarrow \mathbf{e}) = \rho_s(\mathbf{p}, \mathbf{e}, \pmb{\ell}) \ (\mathbf{n}\cdot\pmb{\ell}) L_s$	$L(\mathbf{p} \rightarrow \mathbf{e}) = \sum_k \rho_{ks} (\mathbf{p}, \mathbf{e}, \pmb{\ell})\ (\mathbf{n}\cdot\pmb{\ell}) L_{ks}$
One light: $\rho_s =$ constant specular material color $L_s =$ constant specular light color	Multiple light sum of contribution of each light may have different color coeficients

Glossy models


Glossy surface	$\rho(\mathbf{p}, \mathbf{e}, \pmb{\ell}) = \rho_d(\mathbf{p}, \mathbf{e}, \pmb{\ell}) + \rho_s(\mathbf{p}, \mathbf{e}, \pmb{\ell})$ Usually expressed as a sum of diffuse and specular terms $\rho_d = cst$ (as before) $\rho_s = (\mathbf{r}\cdot\mathbf{e})^e$ (for instance)

The Phong model

$\rho_s = (\mathbf{r}\cdot\mathbf{e})^e$

The general formulation of the Phong model is given by a weighted sum of an ambient, diffuse and specular term:

$L(\mathbf{p} \rightarrow \mathbf{e}) = \rho_a L_a + \sum_k \rho_{kd} L_{kd} (\mathbf{n}\cdot\pmb{\ell}) + \rho_{ks} L_{ks} (\mathbf{r}\cdot\mathbf{e})^e$
where

$\rho_a$ , $\rho_{kd}$ and $\rho_{ks}$ are the material colors for the ambient, diffuse and specular term, respectively.
$L_a$ , $L_{kd}$ and $L_{ks}$ are the light colors for the ambient, diffuse and specular term, respectively.