User:CraigNGC/subpage PSF

The point spread function (PSF) describes the response of an imaging system to a point source or point object. Another commonly used term for the PSF is a system's impulse response. The PSF in many contexts can be thought of as the extended blob in an image that represents an unresolved object. In functional terms it is the spatial domain version of the modulation transfer function. It is a useful concept in Fourier optics, astronomical imaging, electron microscopy and other imaging techniques such as 3D microscopy (like in Confocal laser scanning microscopy) and fluorescence microscopy. The degree of spreading (blurring) of the point object is a measure for the quality of an imaging system. In incoherent imaging systems such as fluorescent microscopes, telescopes or optical microscopes, the image formation process is linear and described by linear system theory. This means that when two objects A and B are imaged simultaneously, the result 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. (The sum is of the light waves which may result in destructive and constructive interference at non-image planes.)

Introduction
By virtue of the linearity property of optical imaging systems, i.e.,

Image(Object1 + Object2) = Image(Object1) + Image(Object2)

the image of an object in a microscope or telescope can be computed by expressing the object-plane field as a weighted sum over 2D impulse functions, and then expressing the image plane field as the weighted sum over 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, 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 Greens functions or impulse response functions).

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 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) image with deconvolution.

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:

(xi, yi) = (M xo, M yo).

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:

$$O(x_o,y_o) = \int \int O(u,v) \delta(x_o-u,y_o-v) du dv$$

i.e., as a sum over weighted impusle functions, although this is also really just stating the sifting property of 2D delta functions (discussed further below). Rewriting the object transmitance 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., O(xo,yo). Mathematically, the image is expressed as:

$$I(x_i,y_i) = \int \int O(u,v) PSF(x_i - M u,y_i - M v) du dv$$

in which PSF(xi-Mu,yi-Mv) 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.

Therefore, the converging 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 Θmax is the maximum angle that the converging waves make with the lens axis, and r is radial distance in the image plane, and k=2π/λ where λ is the wavelength, then the argument of the Airy function is: kr sin(Θmax). If Θmax 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 Θmax is small, the Airy disc is large (which is just another statement of Heisenberg's 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 Θmax (by the Abbe sine condition), can have more blur in the image, owing to the broader PSF.

History and methods
The diffraction theory of point-spread functions was first studied by Airy in the nineteenth century. He developed an expression for the point-spread function amplitude and intensity of a perfect instrument, free of aberrations (the so-called Airy disc). The theory of aberrated point-spread functions close to the optimum focal plane was studied by the Dutch physicists Fritz Zernike and Nijboer in the 1930–40s. A central role in their analysis is played by Zernike’s circle polynomials that allow an efficient representation of the aberrations of any optical system with rotational symmetry. Recent analytic results have made it possible to extend Nijboer and Zernike’s approach for point-spread function evaluation to a large volume around the optimum focal point. This Extended Nijboer-Zernike (ENZ) theory is instrumental in studying the imperfect imaging of three-dimensional objects in confocal microscopy or astronomy under non-ideal imaging conditions. The ENZ-theory has also been applied to the characterization of optical instruments with respect to their aberration by measuring the through-focus intensity distribution and solving an appropriate inverse problem.