1. Photo-realistic Rendering and Global Illumination in Computer Graphics Spring Material Representation
K. H. Ko
School of Mechatronics
Gwangju Institute of Science and Technology

2. Hemispherical Coordinates
In photorealistic rendering, one often wants to work with functions defined over a hemisphere (one-half of a sphere) centered around a surface point.
A hemisphere consists of all the directions in which one can look when standing at the surface point.
A hemisphere is a two-dimensional space, in which each point on the hemisphere defines a direction.
Spherical coordinates are a useful way of parameterizing the hemisphere.

3. Hemispherical Coordinates

4. Hemispherical Coordinates
Each direction is characterized by two angles.
: the azimuth. It is measured with regard to an arbitrary axis located in the tangent plane at x.
Θ: the elevation, measured from the normal vector Nx at surface point x.
The direction can be represented as the pair of those two angles.

5. Hemispherical Coordinates
The values for the angles belong to the intervals
Any three-dimensional point is then defined by three coordinates including the distance from the center, r.

6. Hemispherical Coordinates
The transformation between Cartesian coordinates and spherical coordinates is given by

7. Hemispherical Coordinates
In most rendering algorithms, only hemispherical coordinates without the distance parameter r are used.

8. Solid Angle
In order to integrate functions over the hemisphere, a measure on the hemisphere is needed.
This measure is the solid angle.
A finite solid angle Ω subtended by an area on the hemisphere is defined as the total area divided by the squared radius of the hemisphere.

9. Solid Angle

10. Solid Angle
If the radius r = 1, the solid angle is simply the area on the hemisphere.
Since the area of the hemisphere equals 2πr2 the solid angle covered by the entire hemisphere equals 2π.
The solid angle covered by a complete sphere equals 4π.
Solid angles are dimensionless but are expressed in steradians (sr).

11. Solid Angle
Note that the solid angle is not dependent on the shape of surface A, but is only dependent on the total area.
To compute the solid angle subtended by an arbitrary surface or object in space,
First we project the surface or object on the hemisphere.
Next compute the solid angle of the projection.

12. Solid Angle
Two objects different in shape can still subtend the same solid angle.

13. Solid Angle
For small surfaces, the following approximation can be used to compute the solid angle subtended by a surface or object.

14. Integrating over the Hemisphere
Define differential solid angles to integrate functions in hemispherical space.
Similar to differential surface areas or differential volumes to integrate function sin Cartesian XY or XYZ.
Compared to Cartesian spaces, there is a difference:
The area on the hemisphere “swept” out by a differential dΘ is larger near the horizon than near the pole.
The differential solid angle takes this into account by using a sin(θ) factor.

15. Integrating over the Hemisphere
A differential solid angle is given by
Integrating a function over the hemisphere is then

16. Hemisphere-Area Transformation
In rendering algorithms, it is sometimes more convenient to express an integral over the hemisphere as an integral over visible surfaces seen from x.
To transform a hemispherical integral into an area integral, the relationship between a differential surface and a differential solid angle must be used:

17. Hemisphere-Area Transformation

18. Hemisphere-Area Transformation
Any integral over the hemisphere can also be written as an integral over each visible differential surface dAy in each direction Θ

19. The Physics of Light Transport
The goal of rendering algorithms is to create images that accurately represent the appearance of objects in scenes.
For every pixel in an image, these algorithms must find the objects that are visible at that pixel and then display their “appearance” to the user.
Questions that need to be answered.
What does the term “appearance” mean?
What quantity of light energy must be measured to capture “appearance”?
How is this energy computed?

20. Radiometry
The goal of a global illumination is to compute the steady-state distribution of light energy in a scene.
To compute this distribution, an understanding of the physical quantities that represent light energy is required.
Radiometry is the area of study involved in the physical measurement of light.
Radiometric Quantities
Radiant Power (flux), Irradiance, Radiant Exitance (Radiosity), Radiance.

21. Radiometry Radiant Power or Flux
The fundamental radiometric quantity, Φ
Unit: watts (W) (joules/sec)
This quantity expresses how much total energy flows from/to/through a surface per unit time.
A light source emits 50 watts of radiant power.
20 watts of radiant power is incident on a table.
Note that flux does not specify the size of the light source or the receiver, nor does it include a specification of the distance between the light source and the receiver.

22. Radiometry Irradiance (E)
It is the incident radiant power on a surface, per unit surface area.
Unit: watts/m2
E = dΦ/dA
If 50 watts of radiant power is incident on a surface that has an area of 1.25m2, the irradiance at each surface point is 40 watts/m2.

23. Radiometry Radiant Exitance (M) or Radiosity (B)
It is the exitant radiant power per unit surface area and is also expressed in watts/m2.
M = B = dΦ/dA
Consider a light source, of area 0.1m2, that emits 100 watts. Assuming that the power is emitted uniformly over the area of the light source, the radiant exitance of the light is 1000w/m2 at each point of its surface.

24. Radiometry
Radiance
It is flux per unit projected area per unit solid angle.
Unit: watts/(steradian·m2)
It expresses how much power arrives at (or leave from) a certain point on a surface, per unit solid angle, and per unit projected area.
Radiance is a five-dimensional quantity that varies with position x and direction vector θ, and is expressed as L(x,θ)

25. Radiometry
Radiance
It is the most important quantity in global illumination algorithms because it is the quantity that captures the “appearance” of objects in the scene.

26. Radiometry
Radiance
The projected area Acosθ is the area of the surface projected perpendicular to the direction we are interested in.
This stems from the fact that power arriving at a grazing angle is “smeared out” over a larger surface.
Since we explicitly want to express power per projected area and per direction, we have to take the larger area into account.

27. Transport theory
It deals with the transport or flow of physical quantities such as energy, charge, and mass.
Assumption: consider the flow of light particles or photons.
It can explain relations between different radiometric terms.

28. Radiometric Quantities
Notations!!!!!

29. Radiometry Wavelength Dependency
The radiometric measures and quantities are dependent on the wave length of light energy.
When wavelength is explicitly specified for radiance, the corresponding radiometric quantity is called spectral radiance.
The units of spectral radiance are the units of radiance divided by meters (the unit of wavelength).
Radiance is computed by integrating spectral radiance over the wavelength domain covering visible light.
It is implicitly assumed to be part of the global illumination equations.

30. Radiometry Properties of Radiance
Radiance is invariant along straight paths.
L(x -> y) = L(y <- x)
It states that the radiance leaving point x directed towards point y is equal to the radiance arriving at point y from the point x.
This important property follows from the conservation of light energy in a small pencil of rays between two differential surfaces at x and y.

31. Radiometry Properties of Radiance
Radiance is invariant along straight paths.

32. Radiometry Properties of Radiance
Radiance is invariant along straight paths.

33. L(x->y) = L(y<-x)
Radiometry
Properties of Radiance
Radiance is invariant along straight paths.
L(x->y) = L(y<-x)

34. Radiometry Properties of Radiance
Radiance is invariant along straight paths.
It does not attenuate with distance.
Only valid in the absence of participating media.
Once incident or exitant radiance at all surface points is known, the radiance distribution for all points in a three-dimensional scene is also known.

35. Radiometry Properties of Radiance
Sensors, such as cameras and the human eye, are sensitive to radiance.
The response of sensors is proportional to the radiance incident upon them, where the constant of proportionality depends on the geometry of the sensor.

36. Light Emission
In most rendering algorithms, light is assumed to be emitted from light sources at a particular wavelength and with a particular intensity.
The computation of accurate global illumination requires the specification of the following three distributions for each light source:
Spatial, directional and spectral intensity distribution

37. Images obtained from SIGGRAPH 2005 Course Notes
Material Samples
Diffuse Material
Images obtained from SIGGRAPH 2005 Course Notes

38. Images obtained from SIGGRAPH 2005 Course Notes
Material Samples
Glossy Material
Images obtained from SIGGRAPH 2005 Course Notes

39. Images obtained from SIGGRAPH 2005 Course Notes
Material Samples
Mirror
Images obtained from SIGGRAPH 2005 Course Notes

40. Images obtained from SIGGRAPH 2005 Course Notes
Material Samples
Anisotropic Materials
Images obtained from SIGGRAPH 2005 Course Notes

41. Images obtained from SIGGRAPH 2005 Course Notes
Material Samples
Translucent Materials
Images obtained from SIGGRAPH 2005 Course Notes

42. Images obtained from SIGGRAPH 2005 Course Notes
Material Samples
Materials with Complex Surface Structure
Images obtained from SIGGRAPH 2005 Course Notes

43. Images obtained from SIGGRAPH 2005 Course Notes
Material Samples
Fibers
Images obtained from SIGGRAPH 2005 Course Notes

44. Interaction of Light with Surfaces
Materials interact with light in different ways.
The appearance of materials differs given the same lighting conditions.
The reflectance properties of a surface affect the appearance of the object.
The interaction of light with surfaces can be represented as a function of diverse quantities such as the incident light, exitant light, surface conditions, etc.

45. Interaction of Light with Surfaces
Illustration
Images obtained from SIGGRAPH 2005 Course Notes

46. Interaction of Light with Surfaces
Generalization – 12D
Images obtained from SIGGRAPH 2005 Course Notes