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the image of an object in a microscope or telescope as a non-coherent imaging system can be computed by expressing the object-plane field as a weighted sum of 2D impulse functions, and then expressing the image plane field as a weighted sum of the ''images'' of these impulse functions. This is known as the ''superposition principle'', valid for [[linear systems]]. The images of the individual object-plane impulse functions are called point spread functions (PSF), reflecting the fact that a mathematical ''point'' of light in the object plane is ''spread'' out to form a finite area in the image plane. (In some branches of mathematics and physics, these might be referred to as [[Green's functions]] or [[impulse response]] functions. PSFs are considered impulse response functions for imaging systems.
[[File:PSF Deconvolution V.png|thumb|265x265px|Application of PSF: Deconvolution of the mathematically modeled PSF and the low-resolution image enhances the resolution.<ref name=Kiarash1>{{Cite journal |last1=Ahi |first1=Kiarash |first2=Mehdi |last2=Anwar |editor3-first=Tariq |editor3-last=Manzur |editor2-first=Thomas W |editor2-last=Crowe |editor1-first=Mehdi F |editor1-last=Anwar |date=May 26, 2016 |title=Developing terahertz imaging equation and enhancement of the resolution of terahertz images using deconvolution |url=https://www.researchgate.net/publication/303563271 |journal=Proc. SPIE 9856, Terahertz Physics, Devices, and Systems X: Advanced Applications in Industry and Defense, 98560N |volume=9856 |pages=98560N |doi=10.1117/12.2228680|series=Terahertz Physics, Devices, and Systems X: Advanced Applications in Industry and Defense |bibcode=2016SPIE.9856E..0NA |s2cid=114994724 }}</ref>]]
When the object is divided into discrete point objects of varying intensity, the image is computed as a sum of the PSF of each point. As the PSF is typically determined entirely by the imaging system (that is, microscope or telescope), the entire image can be described by knowing the optical properties of the system. This imaging process is usually formulated by a [[convolution]] equation. In [[microscope image processing]] and [[astronomy]], knowing the PSF of the measuring device is very important for restoring the (original) object with [[deconvolution]]. For the case of laser beams, the PSF can be mathematically modeled using the concepts of [[Gaussian beam]]s
==Theory==
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[[Image:SquarePost.svg|Square Post Function|right|thumb|220px]]
We imagine the object plane as being decomposed into square areas such as this, with each having its own associated square post function. If the height, ''h'', of the post is maintained at 1/w<sup>2</sup>, then as the side dimension ''w'' tends to zero, the height, ''h'', tends to infinity in such a way that the volume (integral) remains constant at 1. This gives the 2D impulse the
The concept of a perfect point source object is central to the idea of PSF. However, there is no such thing in nature as a perfect mathematical point source radiator; the concept is completely non-physical and is rather a mathematical construct used to model and understand optical imaging systems. The utility of the point source concept comes from the fact that a point source in the 2D object plane can only radiate a perfect uniform-amplitude, spherical wave — a wave having perfectly spherical, outward travelling phase fronts with uniform intensity everywhere on the spheres (see [[Huygens–Fresnel principle]]). Such a source of uniform spherical waves is shown in the figure below. We also note that a perfect point source radiator will not only radiate a uniform spectrum of propagating plane waves, but a uniform spectrum of exponentially decaying ([[Evanescent wave|evanescent]]) waves as well, and it is these which are responsible for resolution finer than one wavelength (see [[Fourier optics]]). This follows from the following [[Fourier transform]] expression for a 2D impulse function,
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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
:<math>\mathrm{PSF}(f, z) = I_r(0,z,f)\exp\left[-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}}\right],</math>
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=== Ophthalmology ===
Point spread functions have recently become a useful diagnostic tool in clinical [[ophthalmology]]. Patients are measured with a [[Shack–Hartmann wavefront sensor|Shack-Hartmann]] [[wavefront sensor]], and special software calculates the PSF for that patient's eye. This method allows a physician to simulate potential treatments on a patient, and estimate how those treatments would alter the patient's PSF. Additionally, once measured the PSF can be minimized using an adaptive optics system. This, in conjunction with a [[Charge-coupled device|CCD]] camera and an adaptive optics system, can be used to visualize anatomical structures not otherwise visible ''in vivo'', such as cone photoreceptors
==See also==
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