Frequency-Domain Filtering

Updated 02 Oct 2026

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What it is
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  modifies selected frequency components before transforming back to the spatial domain
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Formula
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  $G[u,v]=H[u,v]F[u,v]$ , then $g=\mathcal F^{-1}{G}$ .
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Filter types
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  low-pass: retains low frequencies and attenuates higher ones.
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  high-pass: suppresses low frequencies while retaining higher ones.
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  band-pass: retains a selected frequency band while attenuating frequencies outside it.
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  notch-reject: suppresses localized frequency regions.
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Project correction
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  The project’s “band” branch computes low_pass + (image - low_pass), reconstructing the original image rather than performing band-pass filtering.
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Valid alternative
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  subtracting two differently smoothed images, $G_{\sigma_1}*f-G_{\sigma_2}*f$ with $\sigma_2>\sigma_1$ , gives a difference-of-Gaussians band-pass response.
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  Calculate the difference in a signed or floating-point type.
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Reconstruction
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  preserve the frequency symmetry needed for a real-valued inverse.
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  taking the absolute value of a signed inverse response changes that response; it is not equivalent to merely removing tiny imaginary roundoff.
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Related
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  Gaussian Blur
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  Periodic Noise

Linked references 4