Point Spread Function - Theory

Theory

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:

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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 PSF(x_i − Mu,y_i − Mv) is the image of the impulse function δ(x_o − u,y_o − 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 (click to enlarge).

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).

Since the concept of a perfect point source object is so central to the idea of PSF, it's worth spending some time on that before proceeding further. First of all, there is no such thing in nature as a perfect mathematical point source radiator; the concept is completely non-physical and is nothing more than 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 (click to enlarge). 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. This comes about in the following way. 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 (click to enlarge).

Therefore, the converging (partial) spherical wave shown in the figure above produces an Airy disc in the image plane. The argument of the Airy function is important, because this determines the scaling of the Airy disc (in other words, how big the disc is in the image plane). If Θ_max is the maximum angle that the converging waves make with the lens axis, r is radial distance in the image plane, and wavenumber k = 2π/λ where λ = wavelength, then the argument of the Airy function is: kr tan(Θ_max). If Θ_max is small (only a small portion of the converging spherical wave is available to form the image), then radial distance, r, has to be very large before the total argument of the Airy function moves away from the central spot. In other words, if Θ_max is small, the Airy disc is large (which is just another statement of Heisenberg's uncertainty principle for FT pairs, namely that small extent in one domain corresponds to wide extent in the other domain, and the two are related via the space-bandwidth product. By virtue of this, high magnification systems, which typically have small values of Θ_max (by the Abbe sine condition), can have more blur in the image, owing to the broader PSF. The size of the PSF is proportional to the magnification, so that the blur is no worse in a relative sense, but it is definitely worse in an absolute sense.

In the figure above, illustrating the truncation of the incident spherical wave by the lens, we may note one very significant fact. In order to measure the point spread function — or impulse response function — of the lens, we do not need a perfect point source that radiates a perfect spherical wave in all directions of space. This is because our 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.