Frequency Domain Filtering

Implement image filtering in the frequency domain using FFT.

Convolution in spatial domain = multiplication in frequency domain: $\mathcal{F}(f * g) = \mathcal{F}(f) \cdot \mathcal{F}(g)$

Algorithm:

1. Compute FFT of image: $F = \mathcal{F}(I)$
2. Create frequency domain filter $H$ (same size as image)
3. Multiply: $G = F \cdot H$
4. Inverse FFT: $I_{filtered} = \mathcal{F}^{-1}(G)$

Common filters:

• Low-pass: Keep center (low frequencies) → blur
• High-pass: Keep edges (high frequencies) → sharpen
• Band-pass: Keep ring of frequencies