Point spread function: Difference between revisions

Revision as of 00:00, 7 September 2022

Latest revision as of 15:47, 8 May 2025

(16 intermediate revisions by 14 users not shown)

Line 1:

{{Short description|Response in an optical imaging system}}

{{Refimprove|date=September 2010}}

[[Image:Convolution Illustrated eng.png|thumb|right|270x270px|Image formation in a [[confocal microscope]]: central longitudinal (XZ) slice. The 3D acquired distribution arises from the [[convolution]] of the real light sources with the PSF.]]

Line 4 ⟶ 5:

[[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 a focused optical imaging system to a [[point source]] or point object. A more general term for the PSF is the system's [[impulse response]]; the PSF is the impulse response or impulse response function (IRF) of a focused optical imaging system.
The PSF in many contexts can be thought of as the extendedshapeless blob in an image that representsshould represent a single point object,.
We thatcan isconsider consideredthis as a spatial [[impulse response function]].
In functional terms, it is the spatial ___domain version (i.e., the inverse Fourier transform) of the [[Optical transfer function|optical transfer function (OTF) of an 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) in the image of a point object for an imaging system is a measure of the quality of the imaging system. In [[coherence (physics)|non-coherent imaging]] 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 that 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|pagespage=355}}</ref>

==Introduction==

Line 13 ⟶ 17:

: ''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 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.<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 |pagespage=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/>

==Theory==

Line 26 ⟶ 30:

Mathematically, we may represent the object plane field as:

:<math> O(x_o,y_o) = \int\!\!\intiint O(u,v) ~ \delta(x_o-u,y_o-v) ~ du\, dv</math>

i.e., as a sum over weighted impulse functions, although this is also really just stating the siftingshifting property of 2D delta functions (discussed further below). Rewriting the object transmittance function in the form above allows us to calculate the image plane field as the superposition of the images of each of the individual impulse functions, i.e., as a superposition over weighted point spread functions in the image plane using the ''same'' weighting function as in the object plane, i.e., <math>O(x_o,y_o)</math>. Mathematically, the image is expressed as:

:<math>I(x_i,y_i) = \int\!\!\intiint O(u,v) ~ \mathrm{PSF}(x_i/M-u , y_i/M-v) \, du\, dv</math>

in which <math display="inline">\mbox{PSF}(x_i/M-u,y_i/M-v)</math> is the image of the impulse function <math> \delta(x_o-u,y_o-v)</math>.

Line 38 ⟶ 42:

[[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/w², 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 sifting property (which is implied in the equation above), which says that when the 2D impulse function, δ(''x''&nbsp;&minus;&nbsp;''u'',''y''&nbsp;&minus;&nbsp;''v''), is integrated against any other [[continuous function]], {{nowrap|''f''(''u'',''v'')}}, it "sifts out" the value of ''f'' at the ___location of the impulse, ''i''.''e''., at the point {{nowrap|(''x'',''y'')}}.

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,

:<math>\delta (x,y) \propto \int\!\!\intiint e^{j(k_x x + k_y y)} \, d k_x\, d k_y</math>

[[Image:PSF.svg|Truncation of Spherical Wave by Lens|right|thumb|400px]]

Line 54 ⟶ 58:

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|last1=Ahi|first1=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|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|s2cid=11781848}}</ref>

:<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>

where ''k-factor'' depends on the truncation ratio and level of the [[irradiance]], ''NA'' is numerical aperture, ''c'' is the [[speed of light]], ''f'' is the photon frequency of the imaging beam, ''I_r'' is the intensity of reference beam, ''a'' is an adjustment factor and <math>\rho</math> is the radial position from the center of the beam on the corresponding ''z-plane''.

==History and methods==

Line 80 ⟶ 84:

| publisher = Academic Press

| year = 2006

| isbn = 9780121828196978-0-12-182819-6

| pages = [https://archive.org/details/methodsinenzymol414ingl/page/223 223–224]

| chapter-url = https://books.google.com/books?id=5bczPeokiqAC&pg=PA223

Line 89 ⟶ 93:

| journal = Journal of Microscopy

| title = The point-spread function of a confocal microscope: its measurement and use in deconvolution of 3-D data

| author1=P. J. Shaw |author2=D. J. Rawlins

| |name-list-style=amp | volume = 163

| issue = 2

| pages = 151–165

Line 104 ⟶ 108:

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 [[AttitudeSpacecraft 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>

Line 114 ⟶ 118:

=== 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.<ref>{{Cite journal|last1=Roorda|first1=Austin|last2=Romero-Borja|first2=Fernando|last3=Iii|first3=William J. Donnelly|last4=Queener|first4=Hope|last5=Hebert|first5=Thomas J.|last6=Campbell|first6=Melanie C. W. |author6-link=Melanie Campbell|date=2002-05-06|title=Adaptive optics scanning laser ophthalmoscopy|journal=Optics Express|language=EN|volume=10|issue=9|pages=405–412|doi=10.1364/OE.10.000405|issn=1094-4087| bibcode=2002OExpr..10..405R |pmid=19436374|s2cid=21971504|doi-access=free}}</ref>

==See also==

* [[Airy disc]]

* [[Circle of confusion]], for the closely related topic in general photography.

* [[Airy discDeconvolution]]

* [[Encircled energy]]

* [[PSFImpulse Labresponse function]]

* [[Deconvolution]]

* [[Microscope]]

* [[Microsphere]]

* [[ImpulsePSF response functionLab]]

==References==

Line 132 ⟶ 135:

* {{cite journal

|title=3-D PSF Fitting for Fluorescence Microscopy: Implementation and Localization Application

|authorsauthor=Hagai Kirshner, |author2=François Aguet, |author3=Daniel Sage, |author4=Michael Unser

|journal=[[Journal of Microscopy]]

|volume=249

Line 143 ⟶ 146:

|s2cid=5318333

}}

* {{cite journal

|title=Multi-scale Optics for Enhanced Light Collection from a Point Source

|authorsauthor=Rachel Noek, |author2=Caleb Knoernschild, |author3=Justin Migacz, |author4=Taehyun Kim, |author5=Peter Maunz, |author6=True Merrill, |author7=Harley Hayden, |author8=C.S. Pai, and |author9=Jungsang Kim

|arxiv=1006.2188

|journal=[[Optics Letters]]