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Chapter 7 ■ Image Restoration 253

constant over a small region of the image, and to restore the image in pieces
using very similar techniques to those that will be discussed.
The first thing to do is to see if a blurred image can be created artificially,
by convolving a known PSF with a created image. Then methods of reducing
the blur can be applied to these known images, and the results can be assessed
objectively. Since it is important to know the PSF, methods of estimating it from
a distorted image must also be discussed, as will certain special cases (such as
motion blur) for which specific restoration schemes have been devised.
A key tool in the analysis and restoration of images is the Fourier transform.
This is a mathematical tool devised in the mid-twentieth century and based
on the Fourier series, which was itself devised more than 200 years ago. Its
goal is to determine how much of each possible frequency occurs in a specific
signal. It was first used to analyze sound waves and like signals that were
specifically composed of the sum of many sine wave-like signals, but the
use has been expanded to include other kinds of signals and other kinds of
frequencies. Because the Fourier transform is so crucial to further work on
image restoration, it needs to be examined in more detail.

7.2 The Frequency Domain

A convolution can be carried out directly on an image by moving the convolu-
tion matrix (image) so that it is centered on each pixel of the image in turn, then
multiplying the corresponding elements and summing the products. This was
described in Equation 2.13, for example. This is a time-consuming process for
large images, and one that can be speeded up by using the Fourier transform.
A transform is simply a mapping from one set of coordinates to another. For
example, a rotation is a transform; the rotated coordinate system is different
from the original, but each coordinate in the original image has a corresponding
coordinate in the rotated image. The Hough transform is another example, in
which pixel coordinates (i,j) are converted into coordinates (m,b)representing
the slope and intercept of the straight lines that pass through the pixel.
The Fourier transform converts spatial coordinates into frequencies. Any
curve or surface can be expressed as the sum of some number (perhaps
infinitely many) of sine and cosine curves. In the Fourier domain (called
the frequency domain as well) the image is represented as the parameters of
these sine and cosine functions. The Fourier transform is the mathematical
mechanism for moving into and out of the frequency domain.
The frequency domain is so named because the two parameters of a sine
curve are the amplitude and the frequency. The fact that an image can be
converted into a frequency domain representation implies that the image can
contain high-frequency or low-frequency information; this is true. If the grey
level of some portion of the image changes slowly across the columns, then