Optical transfer function: Difference between revisions

Revision as of 21:00, 10 March 2020

Revision as of 21:02, 10 March 2020

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The Fourier transform of the line spread function (LSF) can not be determined analytically by the following equations:

:<math>\textopertorname{MTF} = \mathcal{F} \left[ \textopertorname{LSF}\right] \qquad \qquad \textopertorname{MTF}= \int f(x) e^{-i 2 \pi\, x s}\, dx</math>

Therefore, the Fourier Transform is numerically approximated using the discrete Fourier transform <math>\mathcal{DFT}</math>.<ref>Chapra, S.C.; Canale, R.P. (2006). ''Numerical Methods for Engineers (5th ed.). New York, New York: McGraw-Hill</ref>

:<math>\textopertorname{MTF} = \mathcal{DFT}[\textopertorname{LSF}] = Y_k = \sum_{n=0}^{N-1} y_n e^{-ik \frac{2 \pi}{N} n} \qquad k\in [0, N-1] </math>

where

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:<math> e^{\pm ia} = \cos(a) \, \pm \, i \sin(a) </math>

:<math>\textopertorname{MTF}= \mathcal{DFT}[\textopertorname{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]</math>

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

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As shown in the right hand figure, an operator defines a box area encompassing the edge of a '''knife-edge test target''' image back-illuminated by a [[black body]]. The box area is defined to be approximately 10%{{citation needed|date=August 2013}} 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 (video)|line]] within the array is [[normalization (statistics)|normalized]] and averaged. This yields the edge spread function.

:<math>\textopertorname{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} </math>

where

*ESF = the output array of normalized pixel intensity data

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The line spread function is identical to the [[derivative|first derivative]] of the edge spread function,<ref name=Mazzetta2007>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</ref> which is differentiated using [[numerical analysis|numerical methods]]. In case it is more practical to measure the edge spread function, one can determine the line spread function as follows:

:<math>\textopertorname{LSF} = \frac{d}{dx} \textopertorname{ESF}(x)</math>

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

:<math> \textopertorname{LSF} = \frac{d}{dx}\textopertorname{ESF}(x) \approx \frac{\Delta \textopertorname{ESF}}{\Delta x}</math>

:<math> \textopertorname{LSF} \approx \frac{\textopertorname{ESF}_{i+1} - \textopertorname{ESF}_{i-1}}{2(x_{i+1} - x_i)}</math>

where:

*<math>i\,</math> = the index <math>i = 1,2,\dots,n-1</math>

*<math>x_i\,</math> = <math>i^\text{th}\,</math> position of the <math>i^\text{th}\,</math> pixel

*<math>\textopertorname{ESF}_i\,</math> = ESF of the <math>i^\text{th}\,</math> pixel

====Using a grid of black and white lines====