Chapter 30: Frequency-Domain Filtering
Weighting by frequency through the 2D Fourier transform. Low-pass, high-pass, and notch. Removing periodic noise.
Taking the 2D Fourier transform of an image separates gentle gradations into low frequencies (the center of the spectrum) and fine patterns, edges, and noise into high frequencies (the periphery). Weighting by frequency and transforming back performs the same smoothing or enhancement as a spatial convolution, but from the frequency side. Where the filters of the previous chapters worked in the spatial domain, computing in a pixel's neighborhood, this expresses the same thing in the language of frequency. It is the same stage as the ramp filter of CT reconstruction (Chapter 4) and MRI's -space (Chapter 17).
Low-pass, high-pass, band
The most basic filters decide what to pass by the radius in frequency.
- Low-pass: passes only low frequencies and removes high ones. Detail and noise disappear and the image smooths. It is equivalent to the spatial Gaussian (Chapter 28).
- High-pass: passes only high frequencies and removes the low ones (the gentle gradations). Edges and detail are emphasized and flat regions darken.
- Band-pass / band-stop: pass or remove a specific band of frequencies.
The frequency-domain filtering pipeline and three masks. Top: image → FFT → spectrum → apply filter → IFFT → result. Bottom: low-pass (passes the center), high-pass (passes the periphery), and notch (removes only the bright points in the spectrum).
Cutting the range abruptly makes the transformed image ripple (ringing, the Gibbs phenomenon), so in practice a gentle transition such as a Gaussian is used. Convolving in space and multiplying in Fourier space are mathematically equivalent (the convolution theorem), but convolution by a large kernel can be faster via the FFT.
The notch filter: targeting periodic noise
Where the frequency domain excels is removing periodic noise. Regular stripes and moiré, and periodic structures like the ring artifacts of CT, appear in the spectrum as bright points. A notch filter that zeros just those points with a Gaussian removes only the periodic noise, leaving the other frequencies almost untouched. Doing the same in the spatial domain is hard, and this is a distinctive strength of the frequency domain.
Simulation: look at the spectrum, then remove
A phantom with horizontal stripes is set up. In the middle spectrum, besides the central spread from the disk image, bright points corresponding to the stripes appear on the left and right. Switch the filter. Low-pass smooths the whole image; high-pass keeps edges and drops the flat regions. Choose notch and it removes only the stripe points, so the disk image is left nearly intact while the stripes are gone.
Original (with stripes)
Magnitude spectrum
Filtered
A phantom with horizontal stripes is 2D-FFT'd. In the middle spectrum, besides the central spread from the disk, bright points for the stripes appear left and right. Low-pass smooths the whole image; high-pass keeps edges and drops flat regions. Notch removes only the stripe points, leaving the disk nearly intact while the stripes go. Removing periodic noise is the frequency domain's forte.
Key points
The 2D Fourier transform separates an image into gentle components (low frequency, center) and fine components (high frequency, periphery). Weighting by frequency and transforming back gives smoothing by low-pass and edge enhancement by high-pass, equivalent to spatial convolution (the convolution theorem). Ideal filters cause ringing, so gentle transitions are used. The frequency domain is especially strong at removing periodic noise: zeroing the bright points in the spectrum with a notch removes stripes and moiré without harming the rest.
References
- Gonzalez RC, Woods RE. Digital Image Processing, 4th ed. Pearson (2018) — the standard chapter on frequency-domain filtering.
- Bracewell RN. The Fourier Transform and Its Applications, 3rd ed. McGraw-Hill (2000).
- Brigham EO. The Fast Fourier Transform and Its Applications. Prentice Hall (1988).
Chapter 29: Filtering and Denoising
Linear (Gaussian) and edge-preserving (median, bilateral, non-local means, anisotropic diffusion, TV) denoising. The trade-off between noise and detail.
Chapter 31: Edge Detection and Gradients
Capturing abrupt intensity changes with gradients (Sobel) and the Laplacian. Canny, which yields thin, connected contours via non-maximum suppression and hysteresis.