radiances

Radiance is a fundamental radiometric quantity that describes the amount of radiant energy traveling through or emitted from a particular point in a specified direction. It is defined as the radiant flux dΦ per unit area dA, projected perpendicular to the direction of travel (dA cosθ), per unit solid angle dΩ: L = dΦ / (dA cosθ dΩ). Its SI units are watts per steradian per square meter (W·sr−1·m−2). Spectral radiance Lλ specifies the same measure for a given wavelength interval. In a lossless, homogeneous medium, radiance is invariant along a straight ray, and ideal optical elements cannot increase radiance; this invariance underpins étendue conservation in imaging systems.

Radiance is a directional quantity that relates to other radiometric measures through integration over directions. The irradiance E on a surface, for example, is the integral of radiance over the hemisphere of incoming directions weighted by cosθ: E = ∫∫ L cosθ dΩ. The radiant exitance M, the total emitted power per unit area from a surface, is similarly the integral of L over the outward hemisphere with cosθ. Radiance is distinct from radiant intensity (W·sr−1) and radiant flux (W), while its photometric counterpart is luminance, measured in candelas per square meter (cd/m^2), which incorporates the human eye’s spectral response.

Applications of radiance appear across computer graphics, remote sensing, astronomy, and photography. In rendering, radiance directly models light transport; in imaging, sensors respond to radiance, while luminance provides a perceptual measure of brightness. Spectral radiance is essential for color and wavelength-dependent analyses.