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Frequency-domain SNR behaviour

PSNR reduces image quality to a single number. If the number is low, it offers no information about what parts of the signal have been lost. To analyze the restoration more carefully, it is useful to work in the frequency domain. Looking at the signal-to-noise ratio as a function of spatial frequency gives a breakdown of filter performance for features of various scales. One can immediately see whether a filter has trouble with fine or large-scale features.

The Discrete Fourier Transform (DFT) representation of an image g(n,m) is given by:

For the purpose of looking at frequency-domain SNR behaviour, it is useful to lump together the coefficients according to normalized spatial frequency, given by:

The range of is . The inverse of the normalized frequency ( ) gives the wavelength of the spatial frequency component. The DC component (or average) of an image corresponds to . Features on the scale of ten pixels would have ( ), and very fine detailed features (on the scale of 2 pixels) have . The highest possible frequency is - this corresponds to a checkerboard pattern. Figure 1.8 illustrates some spatial frequency components of various wavelengths and orientations.

  
Figure 1.8: Spatial frequency components of various wavelengths and orientations

Let s(n,m) be the original image and x(n,m) be a degraded version. To calculate SNR as a function of spatial frequency, the first step is to calculate the difference image:

Then Fourier transforms of s(n,m) and d(n,m) are taken, producing S(u,v) and D(u,v). The frequency domain is then divided into nonoverlapping frequency bands of the form . Figure 1.9 illustrates this for a sample image. The average band power is calculated by averaging over all (u,v) in the band. The average power for the difference image, is calculated similarly.

  
Figure 1.9: Illustration of how the frequency domain is divided into nonoverlapping bands

The signal-to-noise ratio for each band can then be calculated as:

  
Figure 1.10: Example signal-to-noise ratio plot

This SNR ratio can be plotted as a function of normalized spatial frequency. Figure 1.10 shows such a plot for the image of Figure 1.9 degraded by additive white Gaussian noise with . This plot illustrates that there is little difference between the Wiener and Lee filter result for low frequencies ( ). However, for higher frequencies ( ) the Lee filter has much better performance.

  
Figure 1.11: Example plot showing how MSE is distributed over frequency bands. The top figure illustrates that noise power is proportional to band area.

These SNR plots can be somewhat deceptive, since they do not convey how much noise is left in each band. From Figure 1.10, one might conclude that the very high frequency bands ( ) need improvement the most, since they have the lowest SNR. A better approach is to look at how the total MSE is distributed over frequency bands (Figure 1.11). The top plot shows how each band contributes to the total MSE for the raw image. Since additive white noise has a flat power spectrum, the contribution of each band is proportional to its area in the frequency domain. From the bottom plot, it is clear that comparatively little noise remains in the very high and very low frequency bands. The bands need the most improvement.

The frequency-domain SNR still does not give a complete picture of the image restoration result. For example, it does not distinguish between visually important and unimportant features. For this reason, it is important to look at the images themselves for a subjective estimate of image quality.


next up previous contents
Next: Are local filters good Up: Measures of image quality Previous: Measures of image quality

Todd Veldhuizen
Fri Jan 16 15:16:31 EST 1998