Optogeometric factor

The optogeometric factor (symbol ${\displaystyle F_{\mathrm {opg} }}$) is a radiometric physical quantity that characterizes the geometric–optical throughput between a single detector pixel of an imaging system and its corresponding footprint in the observed scene. It is formally related to the concept of étendue^[1][2], but defined at the level of an individual pixel. The quantity was introduced in the context of quantitative thermography in order to establish a precise formulation of the thermography equation and to enable analysis of the angular dependence of the radiant flux received by a pixel.

In its initial scene–based formulation, published in Infrared Physics & Technology ^[3] the optogeometric factor was derived as an explicit surface–solid–angle integral. This allowed a rigorous description of how the geometry of the camera and the orientation of the observed surface influence the radiant flux incident on a pixel. The derivation was motivated by the need to evaluate the role of viewing angle in infrared temperature measurements, particularly to determine whether oblique observation introduces systematic errors. The analysis demonstrated that, for surfaces with homogeneous emissivity and temperature, the measured signal is effectively independent of the inclination angle within the validity of the approximation (typically up to 55–70°, depending on conditions). This invariance arises because the cosine decrease of radiance with angle, according to Lambert's cosine law, is compensated by the corresponding increase of the projected footprint area.

The optogeometric factor thus provides a pixel-level analogue of étendue that enables the quantitative thermography equation to be written in explicit radiometric form and to be applied consistently across different detector geometries and measurement conditions.

Definition

In the scene–based formulation, the optogeometric factor is defined as a surface–solid–angle integral:

${\displaystyle F_{\mathrm {opg} }=\iint _{A_{\mathrm {fp} }}\iint _{\Omega _{\mathrm {pix} }}\cos \theta \,d\Omega \,dA}$ 

where ${\displaystyle A_{\mathrm {fp} }}$  is the pixel footprint on the object surface, and ${\displaystyle \Omega _{\mathrm {pix} }}$  is the solid angle of the entrance pupil as seen from a point in the footprint. In the sensor–based formulation, the corresponding expression is

${\displaystyle F_{\mathrm {opg,s} }=\iint _{A_{\mathrm {pix} }}\iint _{\Omega _{\mathrm {ap} }}\cos \theta '\,d\Omega '\,dA'}$ 

where ${\displaystyle A_{\mathrm {pix} }}$  is the physical area of the pixel in the focal plane and ${\displaystyle \Omega _{\mathrm {ap} }}$  is the solid angle subtended by the aperture stop.

Both definitions are equivalent under conservation of étendue, and yield the same radiant flux incident on a pixel:

${\displaystyle \Phi _{\mathrm {pix} }\approx L\cdot F_{\mathrm {opg} }}$ 

where ${\displaystyle L}$  is the radiance of the observed scene. The optogeometric factor has the same physical units as étendue, square meter–steradian (${\displaystyle {\text{m}}^{2}\,{\text{sr}}}$ ). In the reduced forms below, the steradian factor is absorbed into the small-angle term (iFOV, in radians), so the expressions evaluate numerically in square meters.

Reduced forms

Under paraxial and idealized conditions, the factor admits compact reduced forms (with the factor of π eliminated):

• Scene–based form: ${\displaystyle {\tilde {\bar {F}}}_{\mathrm {opg} }^{(D,\varphi )}={\tfrac {1}{4}}(D\,\varphi _{\mathrm {iFOV} })^{2}}$ 
• Sensor–based form: ${\displaystyle {\tilde {\bar {F}}}_{\mathrm {opg,s} }^{(a,f\#)}={\tfrac {1}{4}}\left({\tfrac {a}{f\#}}\right)^{2}}$ 

where ${\displaystyle D}$  is the entrance pupil diameter, ${\displaystyle \varphi _{\mathrm {iFOV} }}$  the instantaneous field of view of a pixel, ${\displaystyle a}$  the pixel pitch, and ${\displaystyle f\#}$  the f-number of the camera lens.

Applications

The optogeometric factor (${\displaystyle F_{\mathrm {opg} }}$ ) represents the spatial–angular throughput of a single detector pixel. It is the direct analogue of étendue in geometrical optics, but reduced to the level of an individual detector element. This quantity provides a quantitative link between the projected footprint of a pixel in the scene and the radiant flux collected by the sensor.

In other words, ${\displaystyle F_{\mathrm {opg} }}$  specifies how many spatial–angular modes (i.e. phase–space cells defined by the étendue) a single pixel can accept from the observed scene. It therefore serves as a universal coupling term between the physical radiance of a surface and the electrical signal generated in the detector.

Thermography

The optogeometric factor appears in the quantitative thermography equation, where it explicitly couples the geometry of a thermal camera to the radiant flux received by a pixel:

${\displaystyle \Phi _{\mathrm {pix} }={\Bigl [}\varepsilon \,\sigma T_{\mathrm {obj} }^{4}+(1-\varepsilon )\,\sigma T_{\mathrm {ref} }^{4}{\Bigr ]}\,F_{\mathrm {opg} }}$ 

where ${\displaystyle \varepsilon }$  is the surface emissivity, ${\displaystyle \sigma }$  the Stefan–Boltzmann constant, ${\displaystyle T_{\mathrm {obj} }}$  the object temperature, and ${\displaystyle T_{\mathrm {ref} }}$  the reflected apparent temperature of the environment.

In the reduced scene-based form, the quantitative thermography equation can be written explicitly in terms of camera parameters as

${\displaystyle \Phi _{\mathrm {pix} }={\Bigl [}\varepsilon \,\sigma T_{\mathrm {obj} }^{4}+(1-\varepsilon )\,\sigma T_{\mathrm {ref} }^{4}{\Bigr ]}\,{\tfrac {1}{4}}\,{\bigl (}D\,\varphi _{\mathrm {iFOV} }{\bigr )}^{2}}$ 

where ${\displaystyle \varepsilon }$  is the surface emissivity, ${\displaystyle \sigma }$  the Stefan–Boltzmann constant, ${\displaystyle T_{\mathrm {obj} }}$  the object temperature, ${\displaystyle T_{\mathrm {ref} }}$  the reflected apparent temperature of the environment, ${\displaystyle D}$  the entrance pupil diameter, and ${\displaystyle \varphi _{\mathrm {iFOV} }}$  the instantaneous field of view of a pixel.

This reduced form is valid under paraxial and idealized conditions, assuming homogeneous temperature and emissivity of the observed surface and negligible atmospheric effects.

See also

• Étendue
• Radiometry
• Thermography
• Lambert's cosine law
• Field of view
• F-number

References

1. McCluney, William R. (1994). Introduction to Radiometry and Photometry. Academic Press.
2. Boyd, Robert W. (1983). Radiometry and the Detection of Optical Radiation. John Wiley & Sons.
3. Hofreiter, M.; Sova, J.; Kolaříková, M.; Kolařík, L.; Němec, T. (2025). "Thermography Equation for Non-Perpendicular Infrared Measurements: Derivation, Analysis, and Experimental Validation". Infrared Physics & Technology. doi:10.1016/j.infrared.2025.106010.