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. What does the 2D Fourier transform decompose an image into?
The 2D DFT re-expresses the image as a sum of sinusoidal gratings, each with its own frequency, orientation, magnitude, and phase.
. Why does a real-valued sinusoidal grating produce two bright spots (not one) in its Fourier spectrum?
By Euler's formula, a real sinusoid decomposes into two complex exponentials at frequencies +f and -f, so its spectrum shows two symmetric spikes.
. What causes the visible ringing artifacts when applying an ideal (sharp-cutoff) low-pass filter in the frequency domain?
By the convolution theorem, an ideal sharp cutoff in frequency corresponds to a sinc function in space, whose infinite ripples cause the Gibbs phenomenon.
. According to the convolution theorem, multiplying two images' Fourier transforms elementwise and inverse-transforming is equivalent to what operation in the spatial domain?
The convolution theorem states that convolution in the spatial domain equals elementwise multiplication in the frequency domain, and vice versa.