Point spread function

The point spread function (PSF) describes the response of a focused optical imaging system to an idealized point source of light. In casual terms, for a given camera, it is the blurry blob image captured from pointing that camera at a single speck of light. More technically, a PSF is a form of impulse response function (IRF) for a focused optical imaging system, in spatial terms (as opposed to temporal terms). In functional terms, it is the spatial domain version (i.e., the inverse Fourier transform) of the optical transfer function (OTF) of an imaging system. It is a useful concept in Fourier optics, astronomical imaging, medical 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) in the image of a point object for an imaging system is a measure of the quality of the 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 a linear system theory. This means that when two objects A and B are imaged simultaneously by a non-coherent imaging system, the resulting image 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. In space-invariant systems, i.e. those in which the PSF is the same everywhere in the imaging space, the image of a complex object is then the convolution of that object and the PSF. The PSF can be derived from diffraction integrals.

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

Image(Object1 + Object2) = Image(Object1) + Image(Object2) the image of an object in a microscope or telescope as a non-coherent imaging system can be computed by expressing the object-plane field as a weighted sum of 2D impulse functions, and then expressing the image plane field as a weighted sum of 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 (PSF), 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 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:

( x i , y i ) = ( M x o , M y o ) {\displaystyle (x_{i},y_{i})=(Mx_{o},My_{o})} . 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 ) = ∬ O ( u , v ) δ ( x o − u , y o − v ) d u d v {\displaystyle O(x_{o},y_{o})=\iint O(u,v)~\delta (x_{o}-u,y_{o}-v)~du\,dv}

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., O ( x o , y o ) {\displaystyle O(x_{o},y_{o})} . Mathematically, the image is expressed as:

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