Optical transfer function: Difference between revisions

Revision as of 17:31, 17 October 2021

Revision as of 22:28, 25 April 2022

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==The three-dimensional optical transfer function==

[[File:3DPSF_3DMTF_widefield_confocal3DPSF 3DMTF widefield confocal.png|right|thumb|600px|The three-dimensional point spread functions (a,c) and corresponding modulation transfer functions (b,d) of a wide-field microscope (a,b) and confocal microscope (c,d). In both cases the numerical aperture of the objective is 1.49 and the refractive index of the medium 1.52. The wavelength of the emitted light is assumed to be 600 nm and, in case of the confocal microscope, that of the excitation light 500 nm with circular polarization. A section is cut to visualize the internal intensity distribution. The colors as shown on the logarithmic color scale indicate the irradiance (a,c) and spectral density (b,d) normalized to the maximum value.]]

Although one typically thinks of an image as planar, or two-dimensional, the imaging system will produce a three-dimensional intensity distribution in image space that in principle can be measured. e.g. a two-dimensional sensor could be translated to capture a three-dimensional intensity distribution. The image of a point source is also a three dimensional (3D) intensity distribution which can be represented by a 3D point-spread function. As an example, the figure on the right shows the 3D point-spread function in object space of a wide-field microscope (a) alongside that of a confocal microscope (c). Although the same microscope objective with a numerical aperture of 1.49 is used, it is clear that the confocal point spread function is more compact both in the lateral dimensions (x,y) and the axial dimension (z). One could rightly conclude that the resolution of a confocal microscope is superior to that of a wide-field microscope in all three dimensions.

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where

* <math>Y_k\,</math> = the <math>k^\text{th}</math> value of the MTF

* <math>N\,</math> = number of data points

* <math>n\,</math> = index

* <math>k\,</math> = <math>k^\text{th}</math> term of the LSF data

* <math>y_n\,</math> = <math>n^\text{th}\,</math> pixel position

* <math>i=\sqrt{-1}</math>

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

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:<math>\operatorname{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

* <math>X\,</math> = the input array of pixel intensity data

* <math>x_i\,</math> = the ''i''^th element of <math>X\,</math>

* <math>\mu\,</math> = the average value of the pixel intensity data

* <math>\sigma\,</math> = the standard deviation of the pixel intensity data

* <math>n\,</math> = number of pixels used in average

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:

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

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

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==See also==

* [[Bokeh]]

* [[Gamma correction]]

* [[Minimum resolvable contrast]]

* [[Minimum resolvable temperature difference]]

* [[Optical resolution]]

* [[Signal-to-noise ratio]]

* [[Signal transfer function]]

* [[Strehl ratio]]

* [[Transfer function]]

* [[Wavefront coding]]

==References==

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==External links==

* [https://spie.org/publications/tt52_131_modulation_transfer_function "Modulation transfer function"], by Glenn D. Boreman on SPIE Optipedia.

* [https://www.optikos.com/wp-content/uploads/2013/11/How-to-Measure-MTF.pdf "How to Measure MTF and other Properties of Lenses"], by Optikos Corporation.

[[Category:Optics|Transfer function]]