Jupiter Image Artifacts

FFT Masked to Pass Artifacts

Figure 17.16 The prominent broadband features in the Fourier transform turn out to have been generated from artifacts created when the original image (192 x 165 pixels) was centered and resampled to 256 x 256 pixels. Although it is normally not visible, resampling leaves traces in the Fourier spectrum.

Figure 17.16 The prominent broadband features in the Fourier transform turn out to have been generated from artifacts created when the original image (192 x 165 pixels) was centered and resampled to 256 x 256 pixels. Although it is normally not visible, resampling leaves traces in the Fourier spectrum.

In a computer, the Fourier transform is stored in two arrays, one containing the real components and the other the imaginary ones. The multiplication must be carried out for both components:

FOR v = 0 to vmax FOR u = 0 to umax filter = butterworth(u, v) real(u, v) = filter * real(u, v) imag(u, v) = filter * imag(u, v) NEXT u NEXT v where real () is the array of real numbers, imag () is the array of imaginary numbers, and butterworth () is the filter array.

Figure 17.13 shows the effect of low-pass filtering: it acts to blur the image. The high frequencies that create fine detail in the image are stopped, leaving only the low frequencies' contribution to it. Since this image is low in high frequencies to begin with, the blurring does not appear very dramatic. The equation for a high-pass Butterworth filter is:

where the variables all have the same meanings. The high-pass Butterworth filter transmits 0% at the center, 50% at distance d0 from the center, and 100% at the outer edge. The parameter y controls the abruptness of the transition from passing to blocking.

You can see the effect of a high-pass filter on an image of Jupiter in Figure 17.14. Because the zero-frequency component is absent, the center of the planet appears the same average brightness as the surrounding black sky. It is clear that high-frequency components define the edges of the belts, zones, and Great Red Spot.

To block a band of frequencies, use a Butterworth band-stop filter:

where the variables mean the same as above, and dw defines the band-width that the filter blocks. A Butterworth band-stop filter transmits 100% at the center and 100% at the edge, and 0% at frequency d0. The transmission at the edges of the blocked band of frequencies is 50%.

The complement of a band-stop filter is a band-pass filter:

A Butterworth pass-band filter blocks at the center and at the edge, but transmits 100% at frequency d0. The transmission at the edges of the transmission band of frequencies is 50%. Band-pass and band-stop filters are good for highlighting or blocking textures and patterns characterized by a well-defined size scale,

• Tip: AIP4Win has a built-in Butterworth filter tool. Select the size of the cutoff filters and the bandwidth of the band-pass and bandstop filters, and AIP4Win makes the filter to your specifications. The filter must have the same power-of-two dimensions as the image.

Image features sometimes have well-defined frequency characteristics, even when they are spread around an image. When this happens, an observer can create a special mask to either pass or block the defining frequencies. The mask must be designed to match the pattern in the Fourier transform, so it will have a size and shape determined by the image itself.

The images of Jupiter in Figures 17.14 and 17.15 show simple feature-passing and feature-blocking masks. Blocking areas are set to a pixel value of zero, and passing areas are set a pixel value of pvmdiX ; the Fourier transform is multiplied by the mask to remove the frequency bands; and finally the masked Fourier transform is inverse Fourier transformed back into an image.

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