Some image operations do not fit into the well-defined categories; point, neighborhood, or morphological. They are nonetheless important and, in fact, include some of the most interesting image operations. One class of these are frequency domain operations.

Fast Fourier Transform (FFT)

Images can be thought of as a collection of pixels of varying brightness that are arranged in a fixed, two-dimensional pattern. It is this spatial positioning of pixel intensities that forms a digital image. Obviously images contain patterns of intensity change. These intensity changes correspond to spatial frequencies that define things such as the sharp edges of objects and slowly changing color hues.

An image also can be thought of as a collection of spatial signals. From signal processing, we know that any signal can be reduced, or decomposed, into a series of simple sinusoidal components, each of which has a frequency, amplitude, and phase. As such, it is possible to change, or transform, an image from the spatial domain into the frequency domain.  In the frequency domain image information is represented as signals having various amplitute, frequency and phase characteristics. Some frequency-based operations, such as high-pass, low-pass, and band-pass filtering, can be performed easily on a frequency domain image, while the equivalent operation in the spatial domain involves cumbersome and time-consuming convolutions. In addition, the accuracy of frequency oriented operations are often higher than if they were performed in the spatial domain.

Several mathematical transforms can be used to convert an image from the spatial to the frequency domain. The Discrete Fourier Transform, or DFT, is a well-known method. A related and more commonly used procedure, tailored to the needs of digital imagery, is the Fast Fourier Transform, or FFT. The inverse FFT is the corresponding procedure used to transform from the frequency domain back to the spatial domain.

The FFT is a sequential, one-dimensional process. It is performed first on each row of image pixels and then on the columns of image pixels. The result is a two-dimensional array of values, called the power spectra, that represents the frequency components of the original image. The power spectra can be displayed as a two-dimensional frequency distribution map. The frequencies will appear as mirror images of one another around the center point, since each positive frequency will be matched with a corresponding negative component.

The center of the transformed image is the zero frequency, also known as the dc offset (dc is a term used in signal processing to indicate the constant, non-varying component). As the distance from the center increases, the frequency becomes higher. The brightness of each point in the power spectra corresponds to the magnitude of that frequency in the signal.

As an example, consider the use of frequency domain processing for filtering. In the image first image shown below, there is a distinct repetitive diagonal noise pattern that runs primarily from the upper right to lower left. In addition, there is a more subtle pattern running from the upper left to lower right. The FFT of this image is shown next. This is a typical example of an image's power spectra. The four bright, off-axis spots represent the unwanted noise. (Remember, each frequency has an identical but negative component.)