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The Fourier transform gives us the ability to convert images into spatial frequency space and vice versa. This turns out to be a powerful tool for analyzing and enhancing astronomical images, because the spatial frequencies that make up the image are intimately mixed together throughout it. The Fourier transform separates the components, allowing us to understand and manipulate them individually. See Figures 17.10 and 17.11.

Figure 17.10 shows an image of a brick wall. The pattern formed by the bricks is obviously periodic, and therefore rich in well-defined spatial frequencies. The brick and mortar surfaces are very different, composed of small grains of clay and sand, and studded with tiny cracks, holes, and bubbles. We would therefore expect that the brick and mortar surfaces are rich in high-frequency components. Indeed, the Fourier transform shows this to be case. Spatial frequency bands extend along the u and v axes, strong at low frequencies and gradually tapering off

Cloud Belts Passed FFT Masked to Pass Belts

Figure 17.15 The intense low-frequency spike is blocked or passed in these images. In the upper one, the spike is blocked, leaving the disk of the planet without cloud belts; in the lower one, the spike is isolated, producing an image consisting of very little except the cloud belts.

Figure 17.15 The intense low-frequency spike is blocked or passed in these images. In the upper one, the spike is blocked, leaving the disk of the planet without cloud belts; in the lower one, the spike is isolated, producing an image consisting of very little except the cloud belts.

toward high ones. The strong pattern is enveloped in a halo of high-frequencies. Note the distinct weakening along an axis at the one o'clock position, in line with the shadow's orientation in the image.

Figure 17.11 shows the effect of masking the Fourier transform to isolate and separate these spatial frequency components. In the top Fourier pair, the transform was multiplied by a mask with pixel values of p vmax along the u and v axes and 0 everywhere else. After multiplying this mask times the transform, only the strongly aligned components remained. The inverse Fourier transform of the masked Fourier transform contains the brick-and-mortar pattern pretty much devoid of surface detail.

The bottom Fourier pair shows the effect of blocking the strong components along the u and v axes. The inverse Fourier transform now shows the wall with its semi-random high-frequency details of the clay and sand surfaces, without the brick-and-mortar pattern. A fairly faint checkerboard pattern remains, generated by the low frequencies in the mask itself. You can also see an outer region of smeary high frequency signals; these are due to the CCD characteristics and noise in the electronics of the digital camera used to make the picture of the wall.

â€¢ Tip: Efficient fast Fourier transform and inverse fast Fourier transform are built into AIP4Win. They work with any image with a power-of-two size. For other images, use the Float tool to put the image into the middle of an image with power-of-two dimensions. A!P4Win displays the logarithm of the modulus of the Fourier transform as an image that you can modify.

Fourier transforms make it possible to manipulate spatial frequencies directly in the frequency domain, then create a new image from the modified frequency spectrum. Perhaps the most flexible general-purpose filter for spatial frequencies is the Butterworth filter. By choosing several simple parameters, you can easily generate a Butterworth filter to pass low frequencies, to pass high ones, to pass a band of frequencies, or to block a band.

A spatial filter such as a Butterworth filter, is simply an image that will be multiplied with the Fourier transform. When a pixel value of zero in the filter is multiplied times the Fourier transform, it creates a zero pixel value in the new Fourier transformâ€”completely blocking that frequency. Other pixel values act to multiply the Fourier transform by the factor pv /pvmSLX. When the pixel value in the filter is /?vmax, the Fourier transform passes through the filter unchanged. The equation for the low-pass Butterworth filter is:

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