ABSTRACT
The Fourier transform and the efficient algorithm for computing it, the fast Fourier transform, extend in a straightforward manner to two (or more) dimensions. The twodimensional version of the Fourier transform can be applied to images providing a spectral analysis of the image content. Of course, the resulting spectrum will be in two dimensions, and usually it is more difficult to interpret than a one-dimensional spectrum. Nonetheless, it can be a very useful analysis tool, both for describing the contents of an image and as an aid in the construction of imaging filters as described in the next section. When applied to images, the spatial directions are equivalent to the time variable in the one-dimensional Fourier transform, and this analogous spatial frequency is given in terms of cycles/unit length (i.e., cycles/cm or cycles/inch) or normalized to cycles per sample. Many of the concerns raised with sampled time data apply to sampled spatial data. For example, undersampling an image will lead to aliasing. In such cases, the spatial frequency content of the original image is greater than fs/2, where fs now is 1/(pixel size). Figure 11.1 shows an example of aliasing in the frequency domain. The upper left-hand upper image contains a chirp signal increasing in spatial frequency from left to right. The high frequency elements on the right side of this image are adequately sampled in the left-hand image. The same pattern is shown in the upper right-hand image except that the sampling frequency has been reduced by a factor of 6. The right side of this image also contains sinusoidally varying intensities, but at additional frequencies as
the aliasing folds other sinusoids on top of those in the original pattern. The lower figures show the influence of aliasing on a diagonal pattern. The jagged diagonals are characteristic of aliasing as are moire patterns seen in other images. The problem of determining an appropriate sampling size is even more acute in image acquisition since oversampling can quickly lead to excessive memory storage requirements.
