Radiometry

Outline

What is Radiometry? Quantities

Radiant energy, flux density Irradiance, Radiance

Spherical coordinates, foreshortening Modeling surface properties: the BRDF

Lambertian Specular

Radiometry

Geometry of perspective projection explains location of scene point in image, but what about its intensity and color?

Radiometry is the measurement of electromagnetic radiation, primarily optical Frequencies from infrared to visible to ultraviolet

Photometry quantifies camera/eye sensitivity

Measuring Light For now, ignore the fact that light has multiple

wavelengths; we’ll come back to this when we discuss color

Fundamental quantities Radiant energy Q (measured in joules J)

Proportional to number of photons (and photon frequency) Radiant flux/power © of a light source (watts W)

Joules per second emitted Radiant flux density (Wm-2)

Power per unit area through a surface (real or imaginary) from all directions

Incoming and Outgoing Light at a Surface

Irradiance E (Wm-2) Light arriving at a point on a surface from all

visible directions An image samples the irradiance

at the pinhole

Radiosity B (Wm-2) Light leaving a surface in all

directions

Radiance Radiance L (Wm-2sr-1)

Power at a point in space in a given direction, foreshortened

Can be incoming or outgoing Does not attenuate with distance in vacuum

What is stored in a pixel—the light energy arriving along a particular ray at a particular point.

The more the surface is tilted away, the larger an area the light energy is distributed over.

Foreshortening

Lines, patches tilted with respect to the viewing direction present smaller apparent lengths, areas, respectively, to the viewer

r

dl

d

E.g., angle subtended by differential line segment in 2-D follows from formula for arc length

Spherical Coordinate Terminology

Local coordinates for patch of surface Colatitude/polar angle ( = 0 at normal n)

Longitude/azimuth Solid angle (steradians sr): Area of surface patch

on unit sphere ( = 4 for entire sphere)

3-D Foreshortening

For patches, just extend 2-D argument to areas:

Foreshortening factor for light coming from

() is just cos

Solid Angle in Spherical Coordinates

Differential patch dd has smaller area

closer to pole due to shrinking width of d Circumference at a given polar angle is

2rsin, so the correct patch area is

d = sinddrsin

r

Computing Irradiance

Integrate radiance over the hemisphere

Thus, irradiance from a particular direction is

Describing an Environment’s Radiance

Radiance distribution 5-dimensional function (3 position + 2 direction

variables) Environment map: Radiance distribution at a given

point Plenoptic function: Time variable added

Light field Radiance distribution for an object 4-dimensional function: All positions on image planes of

all orthographic views of object

BRDF Bidirectional Reflectance Distribution

Function (BRDF): Ratio of outgoing radiance in one direction to incident irradiance

BRDF Properties

Energy leaving · energy arriving Symmetric in both directions (Helmholtz reciprocity) Generally, only difference between incident and emitted angle is

significant Dependence on absolute Anisotropy

Can view BRDF as probability that incoming photon will leave in a particular direction

courtesy of S. Rusinkiewicz

Things the BRDF neglects

Surface non-uniformity Subsurface scattering

Light moving under the surface and emerging elsewhere—e.g., marble, skin

Transmission Light going through to

other sidecourtesy of

H. JensenBSSRDF

BRDF

Reflectance Equation

Radiance for a viewing direction given all incoming light

This is proportional to the pixel brightness for that ray

Lambertian Surfaces

Diffuse/matte reflectance: Radiance leaving surface does not depend on angle

Albedo : Ratio of light reflected by an object to light received

BRDF: f() = /

Specular Surfaces

Mirrorlike: Radiance only leaves along specular direction (reflection of incoming direction)

BRDF:

Specular lobe models shiny surface that is not perfect mirror