The
Fourier transforms of typical images have been observed to have most of their
energy concentrated in a small region in the frequency domain, near the origin
and along the w_{1 }and w_{2 }axes. One reason for the energy concentration
near the origin is that images typically have large regions where
the’intensities changes slowly. Furthermore, sharp discontinuities such as
edges contribute to low-frequency as well as high-frequency components. The
energy concentration along the w_{1 } and w_{2 }axes is
in part due to a rectangular window used to obtain a finite-extent image. The
rectangular window creates artificial sharp discontinuities at the four boundaries.
Discontinuities at the top and bottom of the image contribute energy along the w_{2 }axis and discontinuities at the two sides
contribute energy along the w_{1 axis}. Figure 1.37 illustrates this
property. Figure 1.37(a) shows an original image of 512 x 512 pixels, and
Figure 1.37(b) shows of the image in Figure 1.37(a). The
operation has the effect of compressing
large amplitudes while expanding small amplitudes, and therefore shows _{} more clearly for higher frequency
regions. In this particular example, energy concentration along approximately
diagonal directions is also visible. This is because of the many sharp
discontinuities in the image along approximately diagonal directions. This
example shows that most of the energy is concentrated in a small region in the
frequency plane.

Since most of the signal energy is concentrated in a small frequency region, an image can be reconstructed without significant loss of quality and intelligibility from a small fraction of the transform coefficients. Figure 1.38 shows images that were obtained by inverse Fourier transforming the Fourier transform of the image in Figure 1.37(a) after setting most of the Fourier transform coefficients to zero. The percentages of the Fourier transform coefficients that have been preserved in Figures 1.38(a). (b). and (c) are 12.4%, 10%, and 4.8%, respectively. The frequency region that was preserved in each of the three cases has the shape (shaded region) shown in Figure 1.39.

The
notion that an image with good quality and intelligibility can be reconstructed
from a small fraction of transform coefficients for some transforms, for
instance the Fourier transform, is the basis of a class of image coding systems
known collectively as **transform coding techniques**. One objective of
image coding is to represent an image with as few bits as possible while
preserving a certain level of image quality and intelligibility. Reduction of
transmission channel or storage requirements is a typical application of image
coding. In transform coding, the transform coefficients of an image rather than
its intensities are coded. Since only a small fraction of the transform
coefficients need to be coded in typical applications, the bit rate required
in transform coding is often significantly lower than image coding techniques
that attempt to code image intensities.