In thermometry, the displayed temperature value of an object depends on the size of the object. This behaviour, also known as the size-of-source effect (SSE), might be a major cause of measurement uncertainty in a thermoscene. The SSE is caused by diffraction, scattering, reflection, aberration and digitization in the optoelectronic propagation path. The influence of diffraction and digitization (sampling and pixelization) can be described
advantageously with the modulation transfer function MTF. The system MTF of an uncooled camera is determined by the diffraction in
the lens (optical MTF) and the averaging of the radiation over the pixel area (pixel MTF). If the system MTF is known,
the contrast reduction and, thus, the SSE can be calculated. Especially with very small objects, e.g. hotspots creating an image covering less than
4 pixel
The size-of-source effect (SSE) in pyrometry might be a major cause of temperature measurement uncertainty. It describes the effect by which the measurement signal and thus the temperature show a change in conjunction with changing size of the measured object. Its cause is diffraction, scattering, reflection and aberration of the infrared radiation to be measured in the optical channel. In pyrometry, this effect is well understood and extensively described in the literature, e.g. in Gutschwager et al. (2008) and Machin and Sergienko (2001).
This effect also occurs in thermography. The displayed temperature of an object in the thermoscene depends on the size of the object. Particularly for
very small objects (forming an image of less than 4 pixel
In the following, this contrast reduction is described for an uncooled thermal imaging camera using the MTF. With the aid of the MTF, the SSE can then be calculated for very small and very large objects and subsequently a scene (Siemens star) presented to measure the image quality.
Relative radiation flux vs. temperature for the uncooled infrared camera used.
Optical MTF for a wavelength
Pixel MTF. Parameter: pixel width
Optical, pixel and resulting system MTF for an uncooled thermal-imaging camera.
Comparison of the theoretical and measured MTF. QVGA microbolometer camera with 25
Measured values and regression curve of the MTF. Red crosses: measured values. Black line: regression polynomial third degree. QVGA microbolometer camera with 25
For optoelectronic systems, the optical transfer function OTF describes the image quality (DIN ISO 9335, 2015; DIN ISO 15529, 2010). The modulation transfer function MTF is the magnitude of the OTF (Budzier and Gerlach, 2011). In uncooled infrared cameras with microbolometers, the system MTF is obtained through multiplication of the optical MTF of the lens used and the pixel MTF.
At this point it must be explicitly stated that all considerations of the MTF always refer to the radiation flux and not to temperatures. Thus, an MTF of 0 means an averaging of the radiation in the image, which does not correspond to the average value of the temperatures in the thermal image due to the non-linear correlation of radiation flux and temperature (Planck's radiation law).
Temperatures of a hotspot shown in the thermal image. Dashed line: true hotspot temperature (75
Profile of the thermal image of an ideal blackbody radiator at 75
Temperature difference of the blackbody surface between the ideal blackbody and profile (Fig. 8); edge pixels 106 and 406 not shown (
Photo of the Siemens star screen, diameter 110
In addition to the thermal image (temperature in
In the far-infrared spectrum (wavelengths from 8 to 14
Figure 2 shows the optical MTF for ordinary f numbers. It has to be noted here that already at very low spatial frequencies a clear reduction in contrast occurs.
A pixel integrates the incident radiation over its area. From this is derived the pixel MTF, also called the geometric MTF or
detector MTF, by Daniels (2018):
For the sake of simplicity, in the following, the pixel width is set equal to the pixel raster size, in full cognizance of the fact that the fill factors of real microbolometers are less than 1.
The system MTF is the product of the various MTF components:
The system MTF is to be interpreted using an example (Fig. 4). In Fig. 4, three characteristic regions are labelled in colour. Region I
starts at spatial frequency 0 (uniform illumination of the thermal image) and ends at the Nyquist frequency (
Spatial frequencies above the Nyquist limit are incorrectly sampled (region II) because aliasing occurs. From this, the most varied artefacts can arise.
At spatial frequencies above the spatial frequency of the first zero of the pixel MTF (
Thermal image of the Siemens star in front of a surface blackbody radiator at 92
In this case the measurement of the MTF is carried out at an edge (Budzier and Gerlach, 2011). The image at the edge is spatially
differentiated, Fourier-transformed and then normalized. The result then forms the system MTF. Figure 5 shows a comparison of theoretical values (MTF given by the lens manufacturer multiplied by the pixel
For small objects, e.g. images of hotspots extending a few pixels in the image, the measured object temperature is always less than the actual temperature (Fig. 7). For very small objects, e.g. smaller than 4 pixel
When imaging large objects, such as area radiators, not only the edges are blurred, but the entire surface is also affected (Fig. 8). For example, the profile of the hot surface is rounded (Fig. 9). The measured temperature decreases from the centre of the image of the surface to the edge. Colder objects are accordingly measured warmer than their true temperature. Colder or hotter refers to the average temperature of the thermoscene. All
profiles have been calculated with the measured MTF from Fig. 5 (image centre) and for a QVGA microbolometer with a pixel size of 25
The Siemens star offers a good opportunity to test the quality of an optical imaging system. To this end, a suitable scene was created (Fig. 10). In
the thermal image (Fig. 11) it can clearly be seen how small structures are blurred out and approach the average temperature of approx.
62
If the distance between the camera and the scene is increased, the structures shown on the thermal image become smaller (Fig. 12). In the profile,
there is clearly an area at average temperature (MTF
Thermal image (section) of the Siemens star in front of a surface blackbody radiator at 92
Measured profile through the middle of the Siemens star taken from Fig. 12.
The modulation transfer function MTF is suitable for explaining the phenomena of contrast reduction and the size-of-source effect (SSE). Therefore, it provides a good and above all impartial description of the image quality. The MTF of uncooled infrared cameras is determined by the diffraction of the lens and the sampling caused by the pixel size. It also allows an estimate of how large an object needs to be that the displayed temperature shows a sufficient accuracy.
The SSE leads to blurring in the image. This is of particular importance for the imaging of small objects, where the displayed temperatures are particularly incorrect. Since the size of an object in the image depends on the measuring distance, there is an apparent distance dependence of the temperature measurement. When imaging large objects, not only the edges become blurred, but the entire image is also affected.
However, other image errors such as reflections in the optical channel or defects in sensor electronics cannot be explained by the MTF.
The underlying measurement data are not publicly available but can be requested from the authors if of interest.
HB conceptualized and administrated the project, investigated the research subject, visualized the data, and wrote the original draft of the manuscript. GG supervised the project and reviewed and edited the manuscript.
The authors declare that they have no conflict of interest.
Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
This open-access publication was funded by the Technische Universität Dresden (TUD).
This paper was edited by Thomas Fröhlich and reviewed by two anonymous referees.