Introduction to the Frequency Domain

Overview

Much signal processing is done in a mathematical space known as the frequency domain. In order to represent data in the frequency domain, some transform is necessary. Perhaps the most studied one is the Fourier transform.

In 1807, Jean Baptiste Joseph Fourier presented the results of his study of heat propagation and diffusion to the Institut de France. In his presentation, he claimed that any periodic signal could be represented by a series of sinusoids. Though this concept was initially met with resistance, it has since been used in numerous developments in mathematics, science, and engineering. This concept is the basis for what we know today as the Fourier series. Figure 7.1 shows how a square wave can be created by a composition of sinusoids. These sinusoids vary in frequency and amplitude.

What this means to us is that any signal is composed of different frequencies, This applies to 1-dimensional signals such as an audio signal going to a speaker or a 2-dimensional signal such as an image.

A prism is a common example of how a signal is a composition of signals of varying frequencies. As white light passes through a prism, the prism breaks the light into its component frequencies revealing a full color spectrum.

The spatial frequency of an image refers to the rate at which the pixel intensities change. Figure 7.2 shows an image consisting of different frequencies. The high frequencies are concentrated around the axes dividing the image into quadrants. High frequencies are noted by concentrations of large amplitude swings in the small checkerboard pattern. The corners have lower frequencies. Low spatial frequencies are noted by large areas of nearly constant values.


FIGURE 7.1 (a) Fundamental frequency: sine(x); (b) Fundamental plus one harmonic: sine(x) + sine(3x)/3; (c) Fundamental plus 16 harmonics: sine(x) + sine(3x)/3 + sine(5x)/5. ...

The easiest way to determine the frequency composition of signals is to inspect that signal in the frequency domain. The frequency domain shows the magnitude of different frequency components. A simple example of a Fourier transform is a cosine wave. Figure 7.3 shows a simple 1-dimensional cosine wave and its Fourier transform. Since there is only one sinusoidal component in the cosine wave, one component is displayed in the frequency domain. You will notice that the frequency domain represents data as both positive and negative frequencies. Expanding that idea further, Figure 7.4 shows images of a cosine wave and its 2-dimensional discrete Fourier transform.

Notice the two dominant points in 7.4(b). These points represent a and -a along the vertical axis. This corresponds to the cosine wave in Figure 7.4(a) changing in the y direction. The two dominant points in Figure 7.4(d) show the a and -a points in the horizontal direction. As expected, Figure 7.4(f) shows all four points since we are using a summation of the two images.

Many different transforms are used in image processing (far too many begin with the letter H: Hilbert, Hartley, Hough, Hotelling, Hadamard, and Haar). Due to its wide range of applications in image processing, the Fourier transform is one of the most popular (Figure 7.5). It operates on a continuous function of infinite length. The Fourier transform of a 2-dimensional function is shown mathematically as

Where


It is also possible to transform image data from the frequency domain back to the spatial domain. This is done with an inverse Fourier transform:




It quickly becomes evident that the two operations are very similar with a minus sign in the exponent being the only difference. Of course, the functions being operated on are different, one being a spatial function, the other being a function of frequency. There is also a corresponding change in variables. In the frequency domain, u represents the spatial frequency along the original image's x axis and v represents the spatial frequency along the y axis. In the center of the image u and v have their origin. The Fourier transform deals with complex numbers (Figure 7.6), It is not immediately obvious what the real and imaginary parts represent. Another way to represent the data is with its sign and magnitude. The magnitude is expressed as
and phase as

where R(u,v) is the real part and I(u,v) is the imaginary. The magnitude is the amplitude of sine and cosine waves in the Fourier transform formula. As expected, 0 is the phase of the sine and cosine waves. This information along with the frequency, allows us to fully specify the sine and cosine components of an image. Remember that the frequency is dependent on the pixel location in the transform. The further from the origin it is, the higher the spatial frequency it represents

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Related Links:
Discrete Fourier Transform
Fast Fourier Transform
Fast Fourier Transform (FFT) Examples
IMAQ Automatic Focus Example

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