Speaker
Description
The size-of-source effect (SSE) is among the largest contributors to measurement uncertainty in thermography. It describes a systematic deviation of the measured temperature of an object with respect to its size and results from a variety of effects, such as diffraction, detector discretization, aberrations, and scattering. According to theory, the SSE can be explained by the modulation transfer function (MTF), which describes the camera's system response in the spatial frequency domain. In practice, MTF measurements are commonly performed using the slanted edge method, which primarily captures the effects of diffraction and discretization in the vicinity of the slanted edge. In this case, broad scattering contributions are largely neglected, limiting the MTF's ability to fully describe the SSE.
This work addresses this limitation by modeling the SSE using a measured MTF complemented by a parametric scattering model.
To achieve this, a long-wave infrared (LWIR) camera equipped with a wide-angle optical lens was studied experimentally. The MTF and SSE were measured using a blackbody radiator with a slanted-edge target for the MTF measurement and two adjustable iris apertures for the SSE measurement to realize different target sizes. In the following, the system MTF
\begin{equation}
{MTF_\mathrm{sys}}({f}{\mathrm{x}})=MTF\mathrm{meas}({f}{\mathrm{x}})\cdot STF({f}{\mathrm{x}})
\end{equation}
is defined as the product of the measured MTF $MTF_\mathrm{meas}$ and a parametric ("scattering") transfer function
\begin{equation}
STF({f}{\mathrm{x}}) = A + B \cdot \exp{\left(-\pi{\left(C\cdot{f}{\mathrm{x}}\right)}^{2}\right)}
\end{equation}
with
\begin{equation}
A = 1-B,
\end{equation}
the model parameters $B$ and $C$ and the spatial frequency ${f}_{\mathrm{x}}$. The STF aims to approximate the scattering-based spatial frequency response neglected by the measured MTF and is strongly motivated by the Harvey-Shack surface scatter theory. The shape of the STF is based on the assumption that a small fraction $B$ of the radiation is scattered along its path, resulting in a Gaussian scattering pattern whose width is proportional to the parameter $C$.
In order to model the SSE, the camera is considered a black box receiving ideal rectangular input signals
\begin{equation}
{I}{d}(x) =
\begin{cases}
{L}{\mathrm{ref}} & \text{if } |x|\leq\frac{d}{2}\
{L}{\mathrm{bg}} & \text{if } |x|>\frac{d}{2}
\end{cases}
\end{equation}
of width $d$ and pixel coordinate $x$ along the horizontal detector axis, each representing a one dimensional slice of different-sized circular objects at constant radiance ${L}_{\mathrm{ref}}$ and background radiance ${L}_{\mathrm{bg}}$, where ${L}_{\mathrm{ref}}$ is introduced as an additional model parameter and ${L}_{\mathrm{bg}}$ is known. The ideal input signals are convolved with the inverse Fourier transform $\mathcal{F}^{-1}$ of the system MTF, producing the modeled output signals
\begin{equation}
{O}{d}(x)={I}{d}(x) \ast \mathcal{F}^{-1}\big{{MTF\mathrm{sys}}\big}
\end{equation}
in which the modeled SSE is evaluated for a given set of target diameters. The model parameters ${L}_{\mathrm{ref}}$, $B$ and $C$ are determined by minimizing a cost function between the modeled and the measured SSE.
The results show that the proposed model significantly improves the description of the SSE within a linear systems framework, particularly for large measurement objects, where the object's center is located far from the edges. In this case, the effects of diffraction and discretization are negligible with respect to the SSE, while the effect of scattering is more significant. The proposed model may provide a basis for SSE correction through the application of parametric compensation approaches, with parameters informed by physical knowledge.
