Radiometry

The most fundamental radiometric quantity we are concerned with is radiant flux, denoted with Φ and expressed in watts. This measure gives the energy transferred through a surface or a volume in unit time. The spectral radiant flux, distinguished by the subscript λ as Φ_λ, is defined by dividing the radiant flux with a unit wave-length interval:

Φ_λ = dΦ

dλ. (2.5)

This quantity tells the radiant flux falling within an infinitesimal wavelength interval dλ at the wavelength λ. The unit of spectral radiant flux is W/nm. [33]

Irradiance, denoted with E, describes the radiant flux through a unit area. The

unit of irradiance is W/m², and it can be described mathematically as E = dΦ

dA, (2.6)

where dA is an element of area. The corresponding spectral quantity, spectral irradiance, is given by

E_λ = dE

dλ = d²Φ

dAdλ. (2.7)

The unit of spectral irradiance is then W/(m² nm). When constraining the ir-radiance of an object, it is important to also specify the spatial point where the irradiance is considered. [33] This is especially true in imaging, where the spatial distributions of light in the scene are the object of interest.

Radiant intensity, I, describes the radiant flux per unit solid angle from or to a point in space in a specific direction. Radiant intensity has the unit W/sr, with sr denoting steradian, the basic unit of solid angles. The equation defining radiant intensity is

I = dΦ

dω, (2.8)

where dω is a differential element of the solid angle in the direction of the flux.

Spectral radiant intensity is defined similarly to spectral irradiance as I_λ = dI

dλ = d²Φ

dωdλ. (2.9)

While for irradiance the location of the point of interest was necessary, for radiant intensity both location and direction must be specified. [33]

Radiance, L, is a measure of the radiant flux density in unit area and unit solid angle. It is designed the unit W/(m²sr). Radiance can be described mathematically as

L= d²Φ

dωdA_proj = d²Φ

dωdAcosθ, (2.10)

where dA_proj = dAcosθ is the projected area on a surface, dependent of the angle θ between the direction of the flux and the surface normal. The relation between

radiance and spectral radiance is similar to that between irradiance and spectral irradiance, or radiant intensity and spectral radiant intesity:

L_λ = dL

dλ = d³Φ

dωdA_projdλ (2.11)

The unit of spectral radiance is then W/(m² sr nm). [33]

For the purposes of imaging, radiance has an interesting quality in its invariance:

if losses of scattering and absorption can be ignored, radiance remains the same when propagating through a medium. [33] The distance between an imager and its target will then make no difference in recorded radiance when the medium between the imager and its target is the vacuum of space. Further, radiance reflected from the target can be recorded accurately if the losses of reflection, absorption, and scattering in the imager are properly characterized.

The quantity of reflectance is a material property, that describes the ratio of reflected and incident light for a surface. For certain surfaces, reflectance can be strongly dependent on geometry, with the direction of incident light affecting the angular distribution of reflected light. In diffuse reflection the reflected light goes equally in all directions, while in specular reflection the reflected light propagates in only one direction determined by the incidence angle. For real–world objects, the reflection is typically a mix of both diffuse and specular. A complete characteriza-tion of reflectance requires determining the Bidireccharacteriza-tional Reflectance Distribucharacteriza-tion Function (BRDF), which takes into account these geometric considerations. [36,37]

In this work we will consider a quantity more readily measured than the BRDF:

the Bidirectional Reflectance Factor (BRF). This factor, denoted with R, is defined by comparing light reflected from a sample to that reflected from an ideal surface.

It can be calculated based on flux or radiance, as R= dΦ_r

dΦ_id = dL_r

dL_id, (2.12)

where Φ_r and L_r denote the flux and radiance reflected from the sample, and Φ_id and L_id denote those reflected from an ideal surface, respectively. [37]

In this case, the ideal reflector to which the sample is compared is completely re-flective and diffuse. The requirement of rere-flectiveness states that the surface reflects all light incident on it, for all wavelengths. A perfectly diffuse reflector reflects light uniformly in all directions. Such a surface obeys Lambert’s cosine law, which states

that the radiance reflected from the surface is proportional only to the cosine of the incidence angle, measured with respect to the surface normal. Radiance reflected from an ideal Lambertian surface can be expressed as

L_id= E_i

π , (2.13)

whereEiis the incident irradiance. [37] If the incidence angle measured with respect to the surface normal ϕ is not zero, this affects the projected area on the surface, modifying the incident irradiance:

L_id= E_icosϕ

π . (2.14)

Another concept directly related to reflectance is that of albedo. In astronomy, this term is connected to two quantities, the Bond albedo and the geometric albedo.

Bond albedo measures the total portion of light reflected by a body, integrated over wavelengths. Geometric albedo is a spectral quantity that compares the disc–

integrated reflected light from a body to that reflected from an ideal reflector disc of the same size as said body, in a measurement where light comes from directly behind the observer. [1]

While we have discussed quantities using differentials, the measuring techniques used for finding these always return discreet values. For example, the spectral chan-nels of an HSI are always finite in number and width.