page_title: Point spread function
section_title: Theory
hierarchical_section_title: Point spread function / Theory
context_page_description: The point spread function 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 of the imaging system. It is a useful concept in Fourier optics, astronomical imaging, medical imaging, electron microscopy and other imaging techniques such as 3D microscopy and fluorescence microscopy.
  The degree of spreading of the point object is a measure for the quality of an 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 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.
context_section_description: The point spread function may be independent of position in the object plane, in which case it is called shift invariant. In addition, if there is no distortion in the system, the image plane coordinates are linearly related to the object plane coordinates via the magnification M as:
  .
  If the imaging system produces an inverted image, we may simply regard the image plane coordinate axes as being reversed from the object plane axes. With these two assumptions, i.e., that the PSF is shift-invariant and that there is no distortion, calculating the image plane convolution integral is a straightforward process.
  Mathematically, we may represent the object plane field as:
  i.e., as a sum over weighted impulse functions, although this is also really just stating the shifting 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., .  Mathematically, the image is expressed as:
  in which  is the image of the impulse function δ(xₒ  −  u, yₒ  −  v).
  The 2D impulse function may be regarded as the limit (as side dimension w tends to zero) of the "square post" function, shown in the figure below.
  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 − u,y − v), is integrated against any other continuous function, f(u,v), it "sifts out" the value of f at the location of the impulse, i.e., at the point (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) 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,
  The quadratic lens intercepts a portion of this spherical wave, and refocuses it onto a blurred point in the image plane. For a single lens, an on-axis point source in the object plane produces an Airy disc PSF in the image plane. It can be shown (see Fourier optics, Huygens–Fresnel principle, Fraunhofer diffraction) that the field radiated by a planar object (or, by reciprocity, the field converging onto a planar image) is related to its corresponding source (or image) plane distribution via a Fourier transform (FT) relation. In addition, a uniform function over a circular area (in one FT domain) corresponds to the Airy function, J₁(x)/x in the other FT domain, where J₁(x) is the first-order Bessel function of the first kind. That is, a uniformly-illuminated circular aperture that passes a converging uniform spherical wave yields an Airy function image at the focal plane. A graph of a sample 2D Airy function is shown in the adjoining figure.
  Therefo
caption: Airy function