the image plane. (In some branches of mathematics and physics, these might be referred to as Green's functions or impulse response functions. PSFs are considered impulse response functions for imaging systems. ) 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 imaging 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) object with deconvolution. For the case of laser beams, the PSF can be mathematically modeled using the concepts of Gaussian beams. For instance, deconvolution of the mathematically modeled PSF and the image, improves visibility of features and removes imaging noise. 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: . 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 sifting 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 . 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