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convolution theorem (for images)

Convolving two images in space is exactly pointwise multiplying their DFTs in frequency.

The convolution theorem states \mathcal{F}\{f * g\} = F(u,v)\cdot G(u,v) -- spatial-domain convolution (Chapter 3) corresponds exactly to pointwise multiplication of the two operands' DFTs in the frequency domain, and back again (f\cdot g \Leftrightarrow \frac{1}{HW}F*G). This is what turns filter design into choosing a frequency-domain mask rather than hand-picking spatial weights.

See also

2D discrete Fourier transform frequency-domain filtering (low-pass and high-pass)
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