Point spread function

The PSF is defined as the image of a mathematical point source, such as a distant star, through an optical system. In a perfect, diffraction-limited system, the PSF is the Airy pattern, described by the squared Fourier transform of the aperture function. The central peak's angular radius is approximately 1.22λ/D, where λ is wavelength and D is aperture diameter, setting the Rayleigh resolution limit.

In the presence of aberrations or scattering, the PSF broadens and distorts, degrading resolution. The PSF is also central to image restoration: deconvolution algorithms use the PSF to reverse blurring, as in Hubble Space Telescope's early corrective optics.²

PSFs are measured using point-like sources, such as stars or sub-resolution fluorescent beads in microscopy. In astronomy, the PSF is often estimated from field stars, but its shape varies across the field and with time due to atmospheric turbulence, which is why adaptive optics systems measure and correct the PSF in real time.³

Common metrics include full width at half maximum (FWHM) and the Strehl ratio, which compares the peak intensity to the ideal diffraction-limited value. A Strehl ratio above 0.8 is often considered diffraction-limited. In fluorescence microscopy, the PSF determines the resolution limit of ~200 nm laterally, but super-resolution techniques like STED and STORM exploit PSF engineering to break this barrier.⁴

In astronomy, the PSF is crucial for photometry and astrometry, as it affects the apparent brightness and position of stars. In medical imaging, the PSF of a PET scanner or ultrasound transducer affects image quality and quantitative accuracy. In computer vision, PSF models are used for motion deblurring and depth estimation from defocus.

In optical tweezers and holography, the PSF determines the trapping force and reconstruction fidelity. In X-ray and electron microscopy, the PSF is often called the point response function, and its aberrations limit atomic-scale resolution.

Beyond the standard Airy pattern, the PSF can exhibit exotic shapes: in vortex coronagraphs, the PSF is engineered to have a dark center for exoplanet detection.⁵ In gravitational lensing, the PSF of a telescope is used to separate lensed images from the background. The PSF also varies with wavelength, which is why chromatic aberrations are modeled as a wavelength-dependent PSF.

In radio astronomy, the PSF is called the synthesized beam, and its shape depends on the array configuration. A notable edge case: in confocal microscopy, the PSF is the product of the illumination and detection PSFs, which improves resolution. Historically, the PSF concept was formalized by Lord Rayleigh in the 19th century, but the term 'point spread function' came into common use in the 1960s with the rise of Fourier optics.

Airy pattern

The diffraction pattern of a circular aperture, consisting of a central bright spot and concentric rings.

Strehl ratio

Ratio of the peak intensity of the actual PSF to that of the ideal diffraction-limited PSF.

Deconvolution

A computational process that reverses the blurring effect of the PSF to restore the original image.