Radiometry, Surfaces and Rendering

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Radiometry, Surfaces and Rendering

• Anselmo Lastra
• COMP238

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Outline (only first part today)

• Radiometric Concepts
• BRDF and Reflectance
• Light Transport
• Radiosity
• Monte Carlo methods

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Radiometry

• Science of measuring light
• Analogous science called photometry is based on
  human perception.

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Radiometric Quantities

• Function of wavelength, time, position,
  direction, polarization.
• Add polarization to Plenoptic function
• Well make simplifying assumptions

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Wavelength

• Assume wavelengths independent
• No phosphorescence

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Time

• Equilibrium
• Light travels fast
• No luminescence

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Polarization

• Ignore it
• Would likely need wave optics to simulate

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Result five dimensions

• With little loss in usefulness
• Two quantities
• Position (3 components)
• Direction (2 components)

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Radiant Energy - Q

• Think of photon as carrying quantum of energy
  (hc/? where c is speed of light and h is Plancks
  constant)
• Total energy, Q, is then energy of the total
  number of photons.

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Power - ?

• Flow of energy (important for transport)
• Also called radiant flux.
• Energy per unit time (joules / s)
• Units watts
• ? dQ/dt

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Radiant Flux Area Density (1)

• This is a measure we need for power
  arriving/leaving a surface

dA
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Radiant Flux Area Density (2)

• Units of watts per meter squared
• Graphics doesnt use this term instead we use
  two terms

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Irradiance

• Power per unit area incident on a surface.
• E d ?/dA
• Units watts / m2

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Radiant Exitance

• Radiant flux area density leaving surface
• Also known as Radiosity
• B d ?/dA
• Same units as irradiance, of course.

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What about a point source?

• Not a lot of area

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Radiant Intensity

• Flux per unit solid angle
• Units watts per steradian
• Note term intensity is heavily overloaded.

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Solid Angle

• Size of a patch, dA, is
• Solid angle is

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Isotropic Point Source

• Even distribution over sphere
• Intensity is power over whole sphere

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Irradiance on Differential Patch

• Compute solid angle of surface element seen from
  light
• This is the Inverse Square Law

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Radiance

• Power per unit projected area per unit solid
  angle.
• Units watts per steradian m2
• We have now introduced projected area, a cosine
  term.

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Projected Area

• Ap A (N V) A cos ?

V
N
V
?
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Why the Cosine Term?

• Foreshortening is by cosine of angle.
• Radiance gives energy by effective surface area.

d cos?
?
d
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Irradiance from Radiance

• cos? d? is projection of a differential area

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Reciprocity

• Radiance from dS to dR

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• Radiance from dS to dR (l is distance)

Projected area
Solid angle
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Reciprocity

• Which is radiance from dS to dR

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Properties

• Whats Effect of Distance on Radiance?
• Lets look at thin pencil of light
• Whats radiance on a sensor?

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Total flux leaving one side flux arriving other
side, so
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therefore
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so

• Radiance doesnt change with distance!

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Radiance at a sensor

• Sensor of a fixed area sees more of a surface
  that is farther away.
• However, the solid angle is inversely
  proportional to distance.
• Response of a sensor is proportional to radiance.

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Radiance as unit of measure

• Radiance doesnt change with distance
• Therefore its the quantity we want to measure in
  a ray tracer.
• Radiance proportional to what a sensor (camera,
  eye) measures.
• Therefore its what we want to output.

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Photometry and Radiometry

• Photometry (begun 1700s by Bouguer) deals with
  how humans perceive light.
• All measurements relative to perception of
  illumination
• Units different from radiometric but conversion
  is scale factor -- weighted by spectral response
  of eye (over about 360 to 800 nm).

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CIE curve

• Response is the integral over all wavelengths

Violet
Green
Red
CIE, 1924, many more curves available, see
http//cvision.ucsd.edu/lumindex.htm
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Photometric Units

• Talbot ? Joules
• Lumens ? Watt
• Nit, Lux, Candela

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Recap Radiometry

• Energy
• Power energy/time
• Irradiance and Radiosity
• Power/area
• Intensity
• Power/solid angle
• Radiance
• Power/(projected-area solid-angle)

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Surface Properties

• Reflected radiance is proportional to incoming
  flux and to irradiance (incident power per unit
  area).

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Bidirectional Reflection Distribution Function
(BRDF)
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Dimensionality

• Function of
• position,
• four angles (two incident, two reflected),
• wavelength and polarization (usually ignored).
• Material is often considered uniform, so position
  is ignored.
• If isotropic, one angle goes away.
• Result - 3 or 4 dimensional.

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Properties

• Reciprocity

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Lambertian (diffuse) Surface

• BRDF is a constant.
• Independent of direction of incoming light.
• Radiosity over irradiance is constant.

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Mirror (ideally specular) Surface

• Reflection on a plane perpendicular to surface.
• Angle of reflectance angle of incidence.
• BRDF cast as delta functions.

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Glossy

• Between lambertian and specular.

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Complex BRDF

• Combination of the three.
• An interesting BRDF is a retroreflector
• Whats range of values of BRDF?

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Representations

• 4D function, so awkward to represent directly.
• Most often represented as parametric equation
  (Phong, Cook-Torrance, etc.).
• Sometimes with basis functions (such as spherical
  harmonics, sum of cosines, etc.).

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Reflectance

• Ratio of reflected to incident flux
• Always 0 to 1 convenient
• Can be over part or all of incident and exitant
  hemispheres

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Rendering Equation

• Not exactly like Kajiya 86 (more like Radiosity
  equation).
• Often approximated by splitting diffuse,
  specular, and glossy.

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Transport

• Now we have models of reflection.
• How do we transfer energy?
• Approximations are used to make computation
  feasible
• Only certain paths accounted for

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Heckberts Notation

• For transport paths
• From Heckbert, SIGGRAPH 90
• L light
• E the eye
• S specular reflection
• D diffuse reflection
• Sometimes also G for glossy
• Example Path from light, to specular, to eye is
  LSE

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Regular Expressions

• (k) -- one or more
• (k) -- zero or more
• (k)? -- zero or one
• (kk) either one

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Possible Paths

• From Heckbert 90

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Transport Approximations

• Classical ray tracing
• LD?SE
• Direct lambertian
• Global specular
• Radiosity
• LDE
• Diffuse to diffuse global illumination
• View independent
• Bi-directional ray tracing
• Can be L(SD)E

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Next

• Formulation of Radiosity

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Rendering Equation

• Recall
• We want to simplify enough to solve

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Radiosity Assumptions

• 1.  Opaque surfaces
• 2.  Vacuum
• 3. Purely diffuse surfaces
• Solve in object space
• Solution represented in object space
• View independent render as tris w/ vertex color
  (or a radiosity texture)

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Other Surfaces

• Lets relate incoming radiance to other surfaces
• where
• and is 0 or 1.

X
X
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Radiance at x from x

• So now rendering equation is (w/o emitter)
• Next, lets make our integral over surfaces
  instead of solid angles

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Solid Angle to Area

• Recall that
• so

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Geometry Term

• For simplicity, define
• Therefore

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Diffuse Assumption

• All surfaces diffuse, so replace BRDF with a
  constant
• Also angles are now irrelevant, so

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Convert to Radiosities

• so L B / ?, and

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Radiosity Equation

• For convenience subsume the ? into G(). Also, add
  the emissive term back to get
• where

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Where are we?

• We have an expression relating radiosity at a
  point to radiosity at ALL other points
• But no method to solve for the values

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Next

• Practical aspects for computing a solution
• Later
• Monte Carlo methods
• Bi-directional ray tracing

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References

• Chapter 2 (by Hanrahan) in Cohen and Wallace,
  Radiosity and Realistic Image Synthesis.
• Glassner, Principles of Digital Image Synthesis,
  pp. 648 659 and Chapter 13.
• Radiometry FAQ
• http//www.optics.arizona.edu/Palmer/rpfaq/rpfaq.h
  tm

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References

• Geometrical Considerations and Nomenclature for
  Reflectance, F.E. Nicodemus, J.C. Richmond,
  J.J. Hsia, I.W. Ginsberg, and T. Limperis, Nat.
  Bureau Stand. (1977)
• Link to PDF is
• http//physics.nist.gov/Divisions/Div844/facilitie
  s/specphoto/pdf/geoConsid.pdf

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References

• Bastos dissertation, Chapter 3 in
  http//www.cs.unc.edu/bastos/PhD/2and3.pdf
• Heckbert, Adaptive radiosity textures for
  bidirectional ray tracing
• http//doi.acm.org/10.1145/97879.97895