Point spread function

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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.^[3] For instance, deconvolution of the mathematically modeled PSF and the image, improves visibility of features and removes imaging noise.^[2]

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:

${\displaystyle (x_{i},y_{i})=(Mx_{o},My_{o})}$.

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:

${\displaystyle O(x_{o},y_{o})=\iint O(u,v)~\delta (x_{o}-u,y_{o}-v)~du\,dv}$

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

${\displaystyle I(x_{i},y_{i})=\iint O(u,v)~\mathrm {PSF} (x_{i}/M-u,y_{i}/M-v)\,du\,dv}$

in which ${\textstyle {\mbox{PSF}}(x_{i}/M-u,y_{i}/M-v)}$ is the image of the impulse function ${\displaystyle \delta (x_{o}-u,y_{o}-v)}$.

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.

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).

The concept of a perfect point source object is central to the idea of PSF. However, there is no such thing in nature as a perfect mathematical point source radiator; the concept is completely non-physical and is rather 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

The quadratic

Therefore, the converging (partial) spherical wave shown in the figure above produces an

Due to intrinsic limited resolution of the imaging systems, measured PSFs are not free of uncertainty.[4] In imaging, it is desired to suppress the side-lobes of the imaging beam by apodization techniques. In the case of transmission imaging systems with Gaussian beam distribution, the PSF is modeled by the following equation:^[5]

${\displaystyle \mathrm {PSF} (f,z)=I_{r}(0,z,f)\exp \left[-z\alpha (f)-{\dfrac {2\rho ^{2}}{0.36{\frac {cka}{{\text{NA}}f}}{\sqrt {{1+\left({\frac {2\ln 2}{c\pi }}\left({\frac {\text{NA}}{0.56k}}\right)^{2}fz\right)}^{2}}}}}\right],}$

where k-factor depends on the truncation ratio and level of the irradiance, NA is numerical aperture, c is the speed of light, f is the photon frequency of the imaging beam, I_r is the intensity of reference beam, a is an adjustment factor and ${\displaystyle \rho }$ is the radial position from the center of the beam on the corresponding z-plane.

History and methods

The diffraction theory of point spread functions was first studied by

In microscopy, experimental determination of PSF requires sub-resolution (point-like) radiating sources.

In

For

For ground-based optical telescopes, atmospheric turbulence (known as astronomical seeing) dominates the contribution to the PSF. In high-resolution ground-based imaging, the PSF is often found to vary with position in the image (an effect called anisoplanatism). In ground-based adaptive optics systems, the PSF is a combination of the aperture of the system with residual uncorrected atmospheric terms.^[8]

Lithography

The PSF is also a fundamental limit to the conventional focused imaging of a hole,^[9] with the minimum printed size being in the range of 0.6-0.7 wavelength/NA, with NA being the numerical aperture of the imaging system.^[10][11] For example, in the case of an EUV system with wavelength of 13.5 nm and NA=0.33, the minimum individual hole size that can be imaged is in the range of 25-29 nm. A phase-shift mask has 180-degree phase edges which allow finer resolution.^[9]

Ophthalmology

Point spread functions have recently become a useful diagnostic tool in clinical

• Airy disc
• Circle of confusion, for the closely related topic in general photography.
• Deconvolution
• Encircled energy
• Impulse response function
• Microscope
• Microsphere
• PSF Lab

References