# Modulation transfer function (infrared imaging)

The modulation transfer function (MTF) is used to approximate the position of best focus of an infrared imaging system. In an imaging system, best focus is typically achieved when the MTF is between 0.4 and 0.6; most often at 0.5 (50% cutoff frequency of the MTF) MTF is inversely related to the minimum resolvable temperature difference (MRTD), which is a measure of an infrared sensor's ability to resolve temperature difference. MTF is defined as[citation needed] the discrete fourier transform of the line spread function (LSF). The LSF can be calculated by two different methods. One includes measuring the LSF directly from an idealized line approximation provided by an image of a slit target. The other involves differentiating the edge spread function (ESF).[1]

## ESF evaluation

In evaluating the ESF, an operator defines a box area equivalent to 10%[citation needed] of the total frame area of a knife-edge test target back-illuminated by a blackbody. The area is defined to encompass the edge of the target image.

An operator defines a box area encompassing the edge of a knife-edge test target image back-illuminated by a blackbody. The box area is defined to be approximately 10%[citation needed] of the total frame area. The image pixel data is translated into a two-dimensional array (pixel intensity and pixel position). The amplitude (pixel intensity) of each line within the array is normalized and averaged. This yields the edge spread function (ESF)[2][citation needed]

$\text{ESF} = \frac{X - \mu}{\sigma} \qquad \qquad \sigma\, = \sqrt{\frac{\sum_{i=0}^{n-1} (x_i-\mu\,)^2}{n}} \qquad \qquad \mu\, = \frac{\sum_{i=0}^{n-1} x_i}{n}$

where

• ESF = the output array of normalized pixel intensity data
• $X\,$ = the input array of pixel intensity data
• $x_i\,$ = the ith element of $X\,$
• $\mu\,$ = the average value of the pixel intensity data
• $\sigma\,$ = the standard deviation of the pixel intensity data
• $n\,$ = number of pixels used in average

## LSF evaluation

The line spread function (LSF) can be found using two different methods. It can be found directly from an ideal line approximation provided by a slit test target or it can be derived from the ESF. Using the latter method, the LSF is defined as the first derivative of the ESF,[3] which is differentiated using numerical methods.

$\text{LSF} = \frac{d}{dx} \text{ESF}(x)$

Typically the ESF is only known at discrete points, so the LSF is numerically approximated using the finite difference:

$\text{LSF} = \frac{d}{dx}\text{ESF}(x) \approx \frac{\Delta \text{ESF}}{\Delta x}$
$\text{LSF} \approx \frac{\text{ESF}_{i+1} - \text{ESF}_{i-1}}{2(x_{i+1} - x_i)}$

where

• $i\,$ = the index $i = 1,2,\dots,n-1$
• $x_i\,$ = $i^{th}\,$ position of the $i^{th}\,$ pixel
• $\text{ESF}_i\,$ = ESF of the $i^{th}\,$ pixel

## MTF evaluation

The modulation transfer function (MTF) is defined as the discrete Fourier transform of the line spread function. Thus, given the LSF, the MTF is approximated numerically. This data is graphed against the spatial frequency data. A sixth order polynomial is fitted to the MTF vs. spatial frequency curve to show the trend. The 50% cutoff frequency is determined to yield the corresponding spatial frequency. Thus, the approximate position of best focus of the unit under test is determined from this data.

### MTF calculations

The MTF data versus spatial frequency is normalized by fitting a sixth order polynomial to it, making a smooth curve. The 50% cut-off frequency is determined and the corresponding spatial frequency is found, yielding the approximate position of best focus.

The Fourier transform of the LSF can not be determined analytically by the following equations:

$\text{MTF} = \mathcal{F} \left[ \text{LSF}\right] \qquad \qquad \text{MTF}= \int f(x) e^{-i 2 \pi\, x s}\, dx$

Therefore, the Fourier Transform is numerically approximated using the discrete Fourier transform $\mathcal{DFT}$.[4]

$\text{MTF} = \mathcal{DFT}[\text{LSF}] = Y_k = \sum_{n=0}^{N-1} y_n e^{-ik \frac{2 \pi}{N} n} \qquad k\in [0, N-1]$

where

• $Y_k\,$ = the $k^{th}$ value of the MTF
• $N\,$ = number of data points
• $n\,$ = index
• $k\,$ = $k^{th}$ term of the LSF data
• $y_n\,$ = $n^{th}\,$ pixel position
• $i=\sqrt{-1}$
$e^{\pm ia} = \cos(a) \, \pm \, i \sin(a)$
$\text{MTF}= \mathcal{DFT}[\text{LSF}] = Y_k = \sum_{n=0}^{N-1} y_n \left[\cos\left(k\frac{2 \pi}{N} n\right) - i\sin\left(k \frac{2 \pi}{N} n\right)\right] \qquad k\in[0,N-1]$

The MTF is then plotted against spatial frequency and all relevant data concerning this test can be determined from that graph.

## References

1. ^ Holst, G.C. (1998). Testing and Evaluation of Infrared Imaging Systems (2nd ed.). Florida:JCD Publishing, Washington:SPIE.
2. ^ Electro Optical Industries, Inc.(2005). EO TestLab Methodology. In Education/Ref. http://www.electro-optical.com/html/toplevel/educationref.asp.
3. ^ Mazzetta, J.A.; Scopatz, S.D. (2007). Automated Testing of Ultraviolet, Visible, and Infrared Sensors Using Shared Optics. Infrared Imaging Systems: Design Analysis, Modeling, and Testing XVIII,Vol. 6543, pp. 654313-1 654313-14
4. ^ Chapra, S.C.; Canale, R.P. (2006). Numerical Methods for Engineers (5th ed.). New York, New York: McGraw-Hill