when one performs an FFT on a 2D signal, sometimes the axis of the signal on the spatial frequency plane would be shifted 90 degrees from the original axis of the input signal. what may be wide in the spatial plane may be narrow in the frequency plane. this is known as anamorphism. here are some figures that illustrate this phenomenon.
the next few slideshows demonstrate the FFT’s of different sinusoidal patterns. this one shows the FFT of sinusoidal patterns with different frequencies. the larger the frequency of the sine pattern, the larger the gap between the two dots in the frequency plane.
below is the effect of putting a bias on a sinusoidal pattern. the larger the bias, the more prominent the intensity of the center dot, and the less prominent the two dots on either side of the center.
this slideshow shows the FFT of rotated sine patterns. notice that the resulting FFT pattern is always shifted 90 degrees from the original sine pattern.
this slideshow shows the FFT of the addition of two sinusiodal patterns in different axes, with different frequencies.
this slideshow illustrate the FFT of the multiplication of two sine patterns with same frequencies, but with varying phase differences.
the next 6 slideshows display the FFT’s of the following:
- two dots with different gap distances
- two circles with different radii
- two squares with different sizes
- two gaussian shapes with different variances
- random dirac deltas convolved with different patterns
- equally spaced dirac deltas with increasing gap size
the following figures show how to remove the lines in a photograph of the lunar surface (1). first the image was converted to grayscale, and its FFT was generated (2). a filter was made to remove the horizontal and vertical line present in the fourier domain (3), since these lines contribute to the lines in the original image. note that there is a gap in the center of the filter. this is to preserve the information of that original image, which is located in the center of the fourier plane. the FFT and the filter was multiplied, and then shifted back to the spatial domain, resulting to the cleaned image (4).
the following figure used the same technique that was utilized in the previous exercise, except that the filter used was different (3). from the FFT (2) one can see the presence of big dots surrounding the center, which account for the noise present in the original image (1). the final image is found in (4).
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