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Digital Image Processing (2000)

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8.3 Power Spectrum

In Section 8.2 we computed the magnitude and phase spectra of 1D and 2D sequences. The power spectrum is a measure of the distribution of signal energy as a function of energy and graphs of the spectrum are commonly used to visually analyze the frequency content of a signal or the frequency response of a system. The power spectrum is defined as the squared magnitude of the discrete Fourier transform of a signal. Therefore, for the case of 2D signals, we get
    P[k1,k2] = X[k1,k2] |2
where X[k1,k2] is the discrete Fourier transform of the 2D signal x[n1,n2], as defined in Equation (8.2.4).
Several techniques, now considered classical, for estimating the power spectrum of a finite length sequence were developed in the 1940s, 50s, and 60s [Pro96]. All these methods use the DFT and assume no prior knowledge of the signal.

Power spectrum function.

Function PowerSpectrum, in addition to the direct realization of Equations (8.2.1) and (8.2.4) , implements three more methods for calculating the power spectrum of a signal or image. These are known as the Bartlett, Welch, and Blackman and Tukey methods [Opp89, Pro96, Lim90].

Options for PowerSpectrum.

Bartlett's method returns a power spectrum that represents the average of the power spectra evaluated over non-overlapping blocks of the signal. Welch extended this method by smoothing each block with a suitable window sequence and allowing for overlapping blocks. Therefore, Bartlett's method is effectively equivalent to Welch's method with no overlap and a rectangular window. Blackman and Tukey used an indirect approach in calculating the power spectrum. They proposed a method based on smoothing the auto-correlation coefficients of a signal with a suitable window in calculating the power spectrum. Here is a comparison of the methods for a 2D sinusoidal signal. This loads the ImageProcessing package.
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Here we define a sinusoidal signal with noise.
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Here is the power spectrum of the example sequence using the default method.
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Here the Bartlett, Welch, and Blackman-Tukey spectra of the example signal are shown.
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This loads a fragment of the example books image.
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This shows the resulting power spectra when using each of the three nondefault evaluation methods. The matrices have also been rotated following a common convention of displaying the low-frequency spectral coefficients in the center.
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The power spectra of images have very large dynamic ranges making them difficult to plot and visualize. Note the washed-out rendering of the sample values in the center of each spectrum shown above. Logarithmic scaling is therefore frequently used, with the so-called deciBel (dB) scaling, named for Alexander Graham Bell, being the most widely encountered. Here again are the power spectra of the example books image, but now using dB scaling.
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