Point spread function: Difference between revisions

Revision as of 05:08, 29 March 2020

Revision as of 05:12, 29 March 2020

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[[Image:spherical-aberration-disk.jpg|thumb|269x269px|A [[point source]] as imaged by a system with negative (top), zero (center), and positive (bottom) [[spherical aberration]]. Images to the left are [[defocus]]ed toward the inside, images on the right toward the outside.]]

The '''point spread function''' ('''PSF''') describes the response of an imaging system to a [[point source]] or point object. A more general term for the PSF is a system's [[impulse response]], the PSF being the impulse response of a focused optical system. The PSF in many contexts can be thought of as the extended blob in an image that represents a single point object. In functional terms, it is the [[spatial ___domain]] version of the [[Optical transfer function|optical transfer function of the imaging system]]. It is a useful concept in [[Fourier optics]], [[astronomy|astronomical imaging]], [[medical imaging]], [[electron microscope|electron microscopy]] and other imaging techniques such as [[dimension|3D]] [[microscopy]] (like in [[confocal laser scanning microscopy]]) and [[fluorescence microscopy]].

The degree of spreading (blurring) of the point object is a measure for the quality of an imaging system. In [[coherence (physics)|non-coherent]] imaging systems, such as [[fluorescent]] [[microscopes]], [[telescopes]] or optical microscopes, the image formation process is linear in the image intensity and described by [[linear system]] theory. This means that when two objects A and B are imaged simultaneously, the resulting image is equal to the sum of the independently imaged objects. In other words: the imaging of A is unaffected by the imaging of B and ''vice versa'', owing to the non-interacting property of photons. In space-invariant system, i.e. the PSF is the same everywhere in the imaging space, the image of a complex object is then the [[convolution]] of the true object and the PSF.

==Introduction==

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The figure above illustrates the truncation of the incident spherical wave by the lens. In order to measure the point spread function — or impulse response function — of the lens, a perfect point source that radiates a perfect spherical wave in all directions of space is not needed. This is because the lens has only a finite (angular) bandwidth, or finite intercept angle. Therefore, any angular bandwidth contained in the source, which extends past the edge angle of the lens (i.e., lies outside the bandwidth of the system), is essentially wasted source bandwidth because the lens can't intercept it in order to process it. As a result, a perfect point source is not required in order to measure a perfect point spread function. All we need is a light source which has at least as much angular bandwidth as the lens being tested (and of course, is uniform over that angular sector). In other words, we only require a point source which is produced by a convergent (uniform) spherical wave whose half angle is greater than the edge angle of the lens.

Due to intrinsic limited resolution of the imaging systems, measured PSFs are not free of uncertainty<ref>{{Cite journal|last=Ahi|first=Kiarash|last2=Shahbazmohamadi|first2=Sina|last3=Asadizanjani|first3=Navid|date=July 2017|title=Quality control and authentication of packaged integrated circuits using enhanced-spatial-resolution terahertz time-___domain spectroscopy and imaging|url=https://www.researchgate.net/publication/318712771|journal=Optics and Lasers in Engineering|volume=104|pages=274–284|doi=10.1016/j.optlaseng.2017.07.007|via=|bibcode=2018OptLE.104..274A}}</ref>. In imaging, it is desired to suppress the side-lobes of the imaging beam by [[apodization]] techniques. In the case of transmission imaging systems with Gaussian beam distribution, the PSF is modeled by the following equation<ref>{{Cite journal|last=Ahi|first=K.|date=November 2017|title=Mathematical Modeling of THz Point Spread Function and Simulation of THz Imaging Systems|journal=IEEE Transactions on Terahertz Science and Technology|volume=7|issue=6|pages=747–754|doi=10.1109/tthz.2017.2750690|issn=2156-342X|bibcode=2017ITTST...7..747A}}</ref>:

<math>PSF(f,z)=I_r(0,z,f)\exp(-z\alpha(f))-\dfrac{2\rho^2}{0.36{\frac{cka}{\text{NA}f}}\sqrt{{1+\left ( \frac{2\ln 2}{c\pi}\left ( \frac{\text{NA}}{0.56k} \right )^2 fz\right )}^2}},</math>

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[[File:Hubble PSF with flawed optics.jpg|thumb|The point spread function of [[Hubble Space Telescope]]'s [[Wide Field and Planetary Camera|WFPC]] camera before corrections were applied to its optical system.]]

In [[observational astronomy]], the experimental determination of a PSF is often very straightforward due to the ample supply of point sources ([[star]]s or [[quasars]]). The form and source of the PSF may vary widely depending on the instrument and the context in which it is used.

For [[radio telescopes]] and [[Diffraction-limited system|diffraction-limited]] space [[telescopes]], the dominant terms in the PSF may be inferred from the configuration of the aperture in the [[Fourier ___domain]]. In practice, there may be multiple terms contributed by the various components in a complex optical system. A complete description of the PSF will also include diffusion of light (or photo-electrons) in the detector, as well as [[Attitude control|tracking]] errors in the spacecraft or telescope.

For ground-based optical telescopes, atmospheric turbulence (known as [[astronomical seeing]]) dominates the contribution to the PSF. In high-resolution ground-based imaging, the PSF is often found to vary with position in the image (an effect called anisoplanatism). In ground-based [[adaptive optics]] systems, the PSF is a combination of the aperture of the system with residual uncorrected atmospheric terms.<ref>{{Cite web|url=http://www.telescope-optics.net/diffraction_image.htm|title=POINT SPREAD FUNCTION (PSF)|website=www.telescope-optics.net|access-date=2017-12-30}}</ref>

=== Lithography ===