The blurring characteristics of a shift-invariant linear system
are characterized by the point spread function (PSF) of the system.
The observed image, , is the result of convolving the PSF,
,
with the true image,
:
Since convolution in space is equivalent to multiplication in frequency, the blurred image can be also be obtained as follows:
where capitals and
denote the Fourier spectra of images
and
, respectively.
A diffraction-limited Hubble Space Telescope (HST) would have a PSF
just over 2.5 pixels in radius (0 1 at 633 nm) which would
include half of the second bright Airy ring. The actual PSF is over 68
pixels in radius (3
0) due to the spherical aberration
caused by malformation of the HST's primary mirror (Burrows 1991).
The PSF of an imaging system can be estimated mathematically or empirically. The computational construction of a PSF involves complicated and computationally demanding ray tracing methods based on a complete description of the optical system. Empirical estimates of the PSF can often be obtained by imaging a relatively bright, isolated point source. For example, an empirical estimate of the PSF of a fluorescence microscope can be obtained by imaging a tiny polystyrene bead coated with a fluorescent marker dye (Fay 1986). PSFs derived by either the computational or empirical approach can be used in iterative deblurring methods. Inaccuracies in the PSF estimate will cause some aspects of the true image data to be interpreted as image noise and limit the degree to which the image can be deblurred.
A PSF can be summarized compactly by encircled energy, the
integral of the PSF as a function of distance, , from the center
of the PSF. Indexing the PSF using polar coordinates
with the origin at the center of the PSF, the encircled energy is
Since optical systems do not add energy to a signal,
as
. Thus, a spatially
extensive PSF attenuates the signal at the center, in-focus location,
complicating detection of faint structures in complex images. In
crowded fields, the blurred images of nearby objects can interfere
with each other, requiring detection algorithms to distinguish
intensity peaks due to objects from intensity peaks of artifacts
arising from constructive interference between the blurred images of
separate objects. Nonlinear algorithms are prone to create false
objects at intensity peaks due to constructive interference or to
overfitting particularly bright noise peaks.
The effect of a deblurring algorithm can be summarized by plotting the
encircled energy of a point source as a function of radius. This
measure is computed in the same manner as above with the
polar coordinate system centered on the correct location of the point
source.