A Pixel-Level Thermographic Equation for Emissivity Estimation Without Temperature Inversion Based on the Optogeometric Factor

Emissivity is a key parameter in quantitative thermography, but it is in practice usually obtained indirectly by first inverting Planck’s law to recover temperature and then applying approximate Stefan–Boltzmann relations. This procedure introduces additional uncertainty, combines radiance and radiant flux terms inconsistently, and neglects the geometric factors of the imaging system — aperture, distance, pixel field of view, and pixel–surface orientation — that determine how much radiation is collected by a pixel. This study develops an equation that yields emissivity directly from the measured flux, without requiring prior temperature inversion. The formulation is based on the optogeometric factor, which in its general form is defined through the radiometric integral, ensuring that the pixel-level emissivity equation remains dimensionally consistent and geometrically explicit by linking collected flux to surface radiance. For practical use, however, the optogeometric factor can be introduced in an approximate form, where it is expressed in terms of the entrance pupil diameter D and the instantaneous field of view (iFOV). This reduced expression condenses aperture size, distance, and angular subtense into a single multiplicative term. In contrast to classical relations, the proposed pixel-level emissivity equation is dimensionally consistent and geometrically explicit, linking the flux collected by a detector element rigorously to the radiance balance of the observed surface patch. The resulting framework provides a physically consistent foundation for emissivity retrieval at the pixel scale using thermal cameras. It remains directly compatible with established calibration procedures, such as the reflector method (ISO 18434), while offering improved transparency and consistency for quantitative thermographic practice.