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252 Chapter 7 ■ Image Restoration

The mathematics will be skimmed over, and readers interested in the details
will find references at the end of this chapter to explore further. However, not
all the math can be eliminated.
The example of the Hubble Space Telescope is very relevant, since it is an
ideal way to introduce a technique for characterizing the distortion inherent in
an image. A star, when viewed through a telescope, should be seen as a perfect
point of light. Ideally, all the light energy of the star would be focused on a
single pixel. In practice this is not so, because the distortions of the atmosphere
and the telescope optics will yield a slightly blurred image in which the central
pixel is brightest, and a small region around it is less bright, but brighter than
the background. The distortions that have been inflicted on the point image
of the star are reflected in the shape and intensity distribution of the star’s
image. All stars (for a reasonable optical system) will have the same distortions
imposed upon them; indeed, all points on the image have been replaced by
these small blobs, and the sum of all the blobs is the sampled image.
The effect that an image acquisition system has on a perfect point source
is called the point spread function (PSF). The sample image has been produced
by convolving the PSF with the perfect image, so that the same blur exists at
all points. Figure 7.1 shows a diagrammatic view of how distortion and noise
have been applied to the original image to give the sampled, observed image.
To obtain theperfect imagegiven thesampled oneisthe goal of restoration,
and it is not generally possible. We therefore wish to improve the image as
much as possible, and the PSF tells us what has been done to the image. The
ideal solution is to deconvolve the image and the PSF, but this can only be done
approximately and at some significant expense.

∗ +

h Add noise
Original scene

Perfect 2-D Convolve with Resulting image
Image point spread function f(i,j)
Figure 7.1: One model of how a perfect image becomes distorted by imperfect (real)
acquisition systems.

This discussion assumes that the PSF is the same at all points of the image,
in which case the system is said to be spatially invariant. This is the situation
generally assumed in the literature, but the Space Telescope was not spatially
invariant. In cases like this the solution is to assume that the PSF is almost