Point spread function: Difference between revisions

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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 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 [[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 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|page=355}}</ref>

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:<math> O(x_o,y_o) = \iint 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) = \iint O(u,v) ~ \mathrm{PSF}(x_i/M-u , y_i/M-v) \, du\, dv</math>

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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/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,

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:<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==

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==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==