1.4.2 The Fourier Transform of Typical Images

    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 w₁ and w₂ 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 w₁  and w₂ 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 w₂ 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.