Point spread function

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The point spread function (PSF) defines the propagation of electromagnetic radiation or other imaging waves from a point source or point object. It is defined in spherical coordinates for a Lambert-type radiator. It has units of <math>m^{-2}</math>. It is a useful concept in Fourier optics, 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. (The sum is of the light waves which may result in destructive and constructive interference at non-image planes.)

1 Introduction
2 History and methods
3 PSF in microscopy
4 Point spread functions in ophthalmology
5 External links

[edit] Introduction

As a result of the linearity property, the image of any object in a microscope or telescope can be computed by treating the object in parts, imaging each of these, and summing the results. When one divides the object 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 determinely 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.

[edit] 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.

[edit] PSF in microscopy

In microscopy, experimental determination of a PSF is usually tricky, due to the difficulty of finding sub-resolution (point-like) radiating sources. Quantum dots and fluorescent beads are usually considered for this purpose. In observational astronomy the experimental determination of a PSF is often very straightforward due to the ample supply of point sources (stars or quasars). In high-resolution ground-based imaging, the PSF is often found to vary with position in the image (an effect called anisoplanatism).

[edit] Point spread functions in ophthalmology

PSF's have recently become a useful diagnostic tool in clinical ophthalmology. Patients are measured with a wavefront sensor, and special software calculates the PSF for that patient’s eye. In this manner a physician can "see" what the patient sees. This method also allows a physician to simulate potential treatments on a patient, and see how those treatments would alter the patients PSF.