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Notation

Type Font Examples
Variables (scalars) italics $a, b, x, y$
Functions upright $\mathrm{f}, \mathrm{g}(x), \mathrm{max}(x)$
Vectors bold, elements row-wise $\mathbf{a}, \mathbf{b}= \begin{pmatrix}x\\y\end{pmatrix} = (x, y)^\top,$ $\mathbf{B}=(x, y, z)^\top$
Matrices Typewriter $\mathtt{A}, \mathtt{B}= \begin{bmatrix}a & b\\c & d\end{bmatrix}$
Sets calligraphic $\mathcal{A}, B=\{a, b\}, b \in \mathcal{B}$
Number systems, Coordinate spaces double-struck $\mathbb{N}, \mathbb{Z}, \mathbb{R}^2, \mathbb{R}^3$

Light

  • Light is a quantized, electromagnetic wave
  • The visible range is between 380 and 770nm
  • The color of visible light corresponds to its wavelength
  • Light sources often emit a wide spectrum. Due to the superposition of many frequencies, the light appears white (e.g., daylight)
light_wavelength

Perception of Light by the Human Eye

lightpaths
  • Light is emitted from a light source
  • Some rays of light may hit the eye directly; others are reflected from an object's surface towards the eye
  • When it is reflected on the object surface, part of the light is typically absorbed, i.e. the light changes its color
  • If the light rays hit the eye, receptors on the retina are activated and an image is formed in the brain

Color Perception: Trichromatic Theory

There are two systems of light sensory cells in humans:

  • System 1: rods that only react to light/dark contrasts
  • System 2: Three types of color receptors
    • L-cones
      wavelength_human_eye
      Wavelength
      Normalized absorption
    • M-cones
    • S-cones
Data source for figure: J. K. Bowmaker, H. J. A. Dartnall: Visual pigments of rods and cones in a human retina., The Journal of Physiology, Volume 298, Issue 1, Jan. 1980

Color Spaces

rgb_add
RGB color model
  • A color space comprises all colors that can be represented within the color model
  • In this lecture the additive RGB color space (red, green, blue) is used exclusively
  • But there are also numerous other models, such as

Rendering Equation Fundamentals: Solid Angle

steradian
$r$
$r^2$
$\Omega=1\,\mathrm{sr}$
solidangle
  • The solid angle $\Omega$ of an arbitrary area $A$ corresponds to the quotient of the area $S$, which results when $A$ is projected onto a sphere of radius $r$, and $r^2$:

    $\Omega = \int\limits_{\omega}d\omega = \frac{S}{r^2}$

  • Although the solid angle is a dimensionless quantity, it is specified in the unit "steradian" $\mathrm{sr}$
  • The solid angle of a surface of constant size decreases quadratically with the distance from the center of the sphere
  • Example: A solid angle of $\Omega=1\,\mathrm{sr}$ encloses ​​on a sphere with radius $1\,\mathrm{m}$ an area of $1\,\mathrm{m}^2$. The same area located at twice the radius results in $\Omega=\frac{1}{4}\,\mathrm{sr}$
  • The following relationship exists between the differential solid angle $d\omega$ and the spherical coordinates $\theta$ and $\phi$:

    $\Omega = \int\limits_{\omega}d\omega = \int\limits_{\omega}d\theta(\sin \theta \, d\phi)= \int\limits_{\omega}\sin \theta \, d\theta \, d\phi$

Rendering Equation Fundamentals: Radiant Flux

luminous_flux
  • Radiant flux $\Phi$
  • Radiant flux = Radiant energy per time

    $\Phi = \frac{dQ}{dt} \approx \frac{\Delta Q}{\Delta t}$

  • Simplified view:
    • Every photon has the energy $E_{\small\mathrm{photon}}=\frac{h \, c}{\lambda}$ with
      $h$: Planck constant
      $c$: Speed ​​of light
      $\lambda$: Wavelength
    • Radiant flux = Sum of the photon energies that are emitted per time $\Delta t$
  • Unit: Watt $[\mathrm{W}] = [\frac{\mathrm{J}}{\mathrm{s}}]$.

Rendering Equation Fundamentals: Radiant Intensity

luminous_intensity
  • Radiant intensity $I$
  • Radiant intensity = Radiant flux per solid angle
    $I = \frac{d\Phi}{d\omega} \approx \frac{\Delta \Phi}{\Delta \omega}$
  • Radiant intensity = Sum of the photon energies that are emitted per time and solid angle
  • Unit: Watt per steradian $[\frac{\mathrm{W}}{\mathrm{sr}}]$
  • Required if the light source does not radiate equally in all directions

Rendering Equation Fundamentals: Irradiance

illuminance
  • Irradiance $E$
  • Irradiance = Radiant flux per area

    $E = \frac{d\Phi}{dA} \approx \frac{\Delta \Phi}{\Delta A}$

  • The radiant flux comes from all directions within the hemisphere above the surface
  • Unit: Watt per square metre $[\frac{\mathrm{W}}{\mathrm{m}^2}]$
  • Simplified view: Sum of the photon energies that are received per time and area

Rendering Equation Fundamentals: Radiance

luminance
  • Radiance $L$
  • Radiance = Radiant flux per solid angle and visble area

    $L = \frac{d^2\Phi}{d\omega\, \cos(\theta) \,dA} \approx \frac{\Delta \Phi}{\Delta \omega\, \cos(\theta) \,\Delta A}$

  • The area viewed from direction $\theta$ appears shortened by the factor $\cos(\theta)$
  • Application:
    • Corresponds to the observed brightness/color
    • Radiant flux in the direction of the eye
    shorter
    $\theta$
    $\theta$
    $\Delta A$
    $\cos(\theta) \,dA$
  • Simplified view: Sum of the photon energies that are emitted per time, solid angle, and visible area
  • Unit: Watt per steradian per square metre $[\frac{\mathrm{W}}{\mathrm{sr}\, \mathrm{m}^2}]$

Rendering Equation

  • The rendering equation calculates the outgoing radiance $L_o(\mathbf{v})$ for the surface point $\mathbf{x}$ with normal $\mathbf{n}$ in the direction $\mathbf{v}$ by integrating over the contributions of all incoming radiances $L_i(\mathbf{l})$ of the hemisphere above the surface

    $L_o(\mathbf{v}) = L_e(\mathbf{v}) + \int\limits_\Omega \mathrm{f}_r(\mathbf{v}, \mathbf{l})\, \, \underbrace{L_i(\mathbf{l}) \cos(\theta) \, d\omega}_{dE(\mathbf{l})}$

render_eqn
$\mathbf{x}$
$\theta$
$L_i(\mathbf{l})$
$L_o(\mathbf{v})$
$\Omega$
$\mathbf{n}$
  • $L_o(\mathbf{v})$ outgoing radiance
  • $L_e(\mathbf{v})$ is the radiance emitted by the surface itself
  • $L_i(\mathbf{l})$ incoming radiance
  • $E(\mathbf{l})$ irradiance
  • $\mathrm{f}_r(\mathbf{v}, \mathbf{l})$ is the so-called "Bidirectional Reflection Distribution Function" (BRDF)
  • $\Omega$ is the totality of all angles of the hemisphere above the surface

BRDF

BRDF (Bidirectional Reflection Distribution Function)

  • The BRDF describes the angle-dependent spectral reflection factor of a surface by the ratio of the outgoing radiance $L_o$ and the incomming irradiance $E$

    $\mathrm{f}_r(\mathbf{v}, \mathbf{l}) = \frac{ dL_o(\mathbf{v})} { dE(\mathbf{l})} $

  • Incoming direction $\mathbf{l}$ and outgoing direction $\mathbf{v}$ can be parameterized with polar angles $\theta$ und $\phi$. Consequently, the BRFD is a 4-dimensional function.
  • By specifying this 4D function, the reflection properties of a surface are described precisely

BRDF (Bidirectional Reflection Distribution Function)

  • The "shape" of the BRDF determines the reflective properties of a surface
glossyTodiffuse
glossyTodiffuse

BRDF (Bidirectional Reflection Distribution Function)

  • The BRDF of a particular material can be measured and stored in a 4D table
  • Such tables can be found in material databases
  • 4D tables require a lot of memory and the materials cannot be edited directly
  • Therefore, for most cases, parametric BRDF models are used (Phong, Blinn-Phong, Cook-Torrance etc.)

BRDF Properties

  • Positivity:
    $\mathrm{f}_r(\mathbf{v}, \mathbf{l}) \ge 0 \quad \forall \, \mathbf{v}, \mathbf{l} \in \Omega$
  • Helmholtz reciprocity:
    $\mathrm{f}_r(\mathbf{v}, \mathbf{l}) = \mathrm{f}_r(\mathbf{l}, \mathbf{v})$
    i.e., incoming and outgoing directions can be swapped
  • Energy conservation:
    $\int\limits_\Omega \mathrm{f}_r(\mathbf{v}, \mathbf{l})\, \cos(\theta) d\omega \le 1$

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questions

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