Radiometry.txt

RADIOMETRY
==========

All of these quantities are generally wavelength (λ) dependent, but we
leave that off for brevity and simplicity. Just remember that these values
can be quite different at different wavelengths.

Energy (Q)
----------
  * Other Terms:  radiant energy, light, light energy, electromagnetic energy
  * Symbol:       Q
  * Units:        Joules

Flux (Φ)
--------
  * Other Terms   Power, radiant flux, radiant power
  * Described:    Radiant energy per unit time, passing through some given area.
  * Symbol:       Φ
  * Units:        Joules / Second   = Watts

  Φ = dQ/dt

Irradiance (E)
--------------
  * Other Terms:  Radiant Flux Density, Radiant Exitance (M) or Radiosity (M)
  * Described:    Flux per unit area.
		  or: Radiant energy per unit time, per unit area.
                  Either arriving or leaving (generally called Exitance if leaving)
                  Can be measured at any point in space.
  * Symbol:       E (or M if leaving)
  * Units:        Watts / meter²

    E = dΦ/dA     Φ = ∫ E·dA

    if uniform:  E = Φ/A
    if area is not perpendicular to light:  E = Φ·cosθ/A


Intensity (I)
-------------
  * Other Terms:  Radiant intensity
  * Described:    Angular density of emitted power
                  or: flux per unit solid angle
                  or: total flux output of a point source / 4π
  * Symbol        I
  * Units         W / sr

    I = dΦ/dω     Φ = ∫ I·dω

    ONLY MEANINGFUL FOR POINT SOURCES OF LIGHT.


Radiance (L)
------------
  * Symbol        L
  * Described:    flux density per unit area, per unit solid angle.
  * Units:        W/(m²·sr)

    L = dE(ω)/dω

    L = ∂²Φ / [∂ω·∂A┴]

    where E(w) is only the perpendicular component of the irradiance E (that way we
    eliminate the cosθ term).


  * "Infinitesimal amount of radiant flux in a single ray of light."
  * Function of the viewing direction.
  * ω is the solid angle

  * Emitted radiance:  A is the area on the surface of the source, ω is the
    solid angle into which the light is emitted (and spreads out).
  * Recevied radiance:  A is the area on the detector, ω is the solid angle
    subtended by the source as viewed from that detector.


It is useful to separate incoming radiance from outgoing radiance (rather than to
compute one-sided limits at discontinuities).


TERMINOLOGY GOTCHA:
  > Thermal engineers use the term "radiant intensity" for "radiance" (watts per unit
    area per unit solid angle).
  > This became an issue when Computer Graphics adopted concepts of radiative heat
    transfer to create radiosity theory, and adopted thermal engineering's terminology.
  > Best to stick to the above, with "radiant intensity" as "watts per unit solid angle"
    and "radiance" as "watts per unit area per unit solid angle", and ignore those
    thermal engineers.

Photometry deals with analogous quantities:
  Luminous Intensity
  Luminous Flux
  Luminous Energy
  Luminous Flux Density
  Luminance
  Illuminance (Luminous Exitance [sometimes Luminosity])

  Using luminous efficacy (maximum of 683 lumens/W), and a CIE luminosity
  function (luminous efficacy for a given wavelength)

=================================================================
Mike's examples one:  Surface of the sun
----------------------------------------

Sun provides some level of irradiance (E) here on Earth
   (maximum of 1361 W/m² above atmosphere, 1000 W/m² at surface)

Sun cuts some solid angle (ω) in the sky (6.87e-5 stradians)

  BTW: for a circle you can get steradians from linear radians with:
      steradians = 2 * pi * (1 - cos(radians))

L = dE(ω)/dω
Above atmosphere:
  L = 1361 W/m² / 6.87e-5 stradians
  L = 19,787,728 W/m²·sr above atmosphere
At Earth's surface:
  L = 1000 W/m² / 6.87e-5 stradians
  L = 14,539,110 W/m²·sr above atmosphere

Sun's luminous efficacy is 98

  Lum = 19,787,728 * 98 = 1,939,197,441 cd/m²
  Lum = 14,539,110 * 98 = 1,424,832,800 cd/m²

Let's check:
  According to this page:
    https://en.wikipedia.org/wiki/Orders_of_magnitude_(luminance)
  Solar disk at noon 1,600,000,000 cd/m², which lies between our
    above/below atmosphere numbers, and so is totally within reason.

=================================================================
Mike's example two:  Stars
--------------------------

Each star provides some level of irradiance (E) here on Earth.

Each star cuts some solid angle. Since they all cover less than one
pixel, but we cannot focus the light smaller than a pixel, we will
consider them to cover exactly one pixel.  One pixel covers an
arc of fov/width radians.  Making that into a circle, we use the
formula above:
	ω = 2 * pi * (1 - cos((fov/width)/2))
        (the linear angle is actually a bit more than half a pixel, partway between
         an inscribed circle and an circumscribed cirle. I haven't worked the formula
         for a square)

   star_radiance = star_irradiance / ω
                 = star_irradiance / 2 * pi * (1 - cos((fov/width)/2))

Each star has a luminous efficacy.  Multiply radiance by luminous efficacy
to get the luminance of the star.

Then take the star's RGB, convert to XYZ, scale Y to this luminance (considering the
luminance of the monitor -- e.g. Y=1 would be 80 cd/m² on an sRGB monitor),
then convert back to RGB.

This gives RGB values around 12000. There is no way....

=================================================================
Mike's recommendations
----------------------

Keep radiative light levels as irradiance values.
Keep ambient light levels as irradiance values.

Work out radiance of objects that are visible in the scene.

Convert that radiance into luminance near the end of processing.

=================================================================

----
http://www.helios32.com/Measuring%20Light.pdf

Spectral versions of the above

Spectral Radiant Energy (Q((λ))
  * Described:    Radiant Energy per unit wavelength
  * Symbol:       Q(λ)
  * Units:        Joules per nanometer

  Q(λ) = dQ/dλ

Spectral Radiant Flux (Φ(λ))
  * Other Terms:  Spectral Radiant Power
  * Symbol:       Φ(λ)
  * Units:        Joules / Second   = Watts

  Φ(λ) = dΦ/dλ

Spectral Radiant Flux Density (E(λ) or M(λ))
  * Other Terms:  Spectral Irradiance or Spectral Radiant Exitance
  * Symbol:       E(λ) or M(λ)
  * Units:        W/(m²·nm)

  E(λ) = dE/dλ
  M(λ) = dM/dλ

Spectral Radiance
  * Described:     Radiance per unit wavelength
  * Symbol         L(λ)
  * Units:         W/(m²·sr·nm)

   L(λ) = d³Φ / [dA·dω·cosθ·dλ]

Spectral Radiant Intensity
  * Symbol         I(λ)
  * Units          W/(sr·nm)

   I(λ) = dI/dλ