Point spread function: Difference between revisions

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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 (OTF) 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) ofin the image of a 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 a [[linear system]] theory. This means that when two objects A and B are imaged simultaneously by a non-coherent imaging system, 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 systems, i.e. those in which the PSF is the same everywhere in the imaging space, the image of a complex object is then the [[convolution]] of the truethat object and the PSF. The PSF can be derived from diffraction integrals.<ref>{{Cite book|url=https://books.google.com/books?id=lCm9Q18P8cMC&q=diffraction+integral+point+spread+function&pg=PA355|title=Progress in Optics|date=2008-01-25|publisher=Elsevier|isbn=978-0-08-055768-7|language=en|pages=355}}</ref>

The PSF can be derived from diffraction integrals<ref>{{Cite book|url=https://books.google.com/books?id=lCm9Q18P8cMC&q=diffraction+integral+point+spread+function&pg=PA355|title=Progress in Optics|date=2008-01-25|publisher=Elsevier|isbn=978-0-08-055768-7|language=en|pages=355}}</ref>

==Introduction==

By virtue of the linearity property of optical ''non-coherent'' imaging systems, i.e.,

: ''Image''(''Object''₁ + ''Object''₂) = ''Image''(''Object''₁) + ''Image''(''Object''₂)

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 overof 2D impulse functions, and then expressing the image plane field as thea weighted sum overof 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. (inIn 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.<ref name=Kiarash2>{{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=Modeling of terahertz images based on x-ray images: a novel approach for verification of terahertz images and identification of objects with fine details beyond terahertz resolution |url=https://www.researchgate.net/publication/303563365 |journal=Proc. SPIE 9856, Terahertz Physics, Devices, and Systems X: Advanced Applications in Industry and Defense, 98560N |volume=9856 |pages=985610 |doi=10.1117/12.2228685|series=Terahertz Physics, Devices, and Systems X: Advanced Applications in Industry and Defense |bibcode=2016SPIE.9856E..10A |s2cid=124315172 }}</ref> For instance, deconvolution of the mathematically modeled PSF and the image, improves visibility of features and removes imaging noise.<ref name=Kiarash1/>