Jupiter High-Pass Enhanced 5x FFT High-Pass Enhanced

Figure 17.17 With its low frequencies passed intact and its high frequencies enhanced by a factor of five, Jupiter reveals cloud structures that are all but invisible to the eye in the original image. The artifact components have also been blocked, yielding an exceptionally "clean" image of the giant planet.

Jupiter High-Pass Enhanced 5x FFT High-Pass Enhanced

Figure 17.17 With its low frequencies passed intact and its high frequencies enhanced by a factor of five, Jupiter reveals cloud structures that are all but invisible to the eye in the original image. The artifact components have also been blocked, yielding an exceptionally "clean" image of the giant planet.

• Tip: To selectively attenuate spatial frequencies using AIP4Win, duplicate the modulus image of the Fourier transform. Use it to create a mask using the region-filling function of the Pixel Editor tool, and set the frequencies you want to block to zero.

Parseval's Theorem holds the key to practical image enhancement using Fourier transform techniques. The theorem states that the energy is conserved between an image and its Fourier transform. Blocking spatial frequencies removes energy from the image by reducing the brightness of the image at those frequencies. However, filters can do much more than block spatial frequencies.

Since the pixel values in a Butterworth filter range from 0 to 1, when a spectrum is multiplied by the filter, it is weakened. However, if a spatial frequency in the spectrum is multiplied by a number greater than 1, Parseval's theorem suggests that the corresponding spatial frequencies in the image will be boosted—and this can be used to enhance desirable frequencies.

For ease of application, the filter is scaled by adding a base transmission factor, b, and a contrast factor, c. The filter can thus be adjusted to pass any desired fraction of the original Fourier transform and boost the desired range of spatial frequencies:

In the above, B'(u, v) is the new value for the adjusted filter, b is the base transmission, and c is the contrast enhancement.

In computer code, the multiplication is carried out for both components:

FOR u = 0 to umax boost = b + c * filter(u, v) real(u, v) = boost * real(u, v) imag(u, v) = boost * real(u, v) NEXT u NEXT v

The image of Jupiter in Figure 17.17 is an enhanced one created by passing 100% of the original image and boosting high spatial frequencies by a contrast factor of five. The resampling and centering artifacts were blocked to produce a final clean image.

• Tip: When you apply a filter to a Fourier transform in AIP4Win, you can set the Base Value and Contrast. The default values are 0 and 1, values that filter without modification. To enhance images, the base should be between 0 and 1, and the contrast should be greater than 1.

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