Fourier Transform
A mathematical technique that decomposes an image into spatial frequency components, where low frequencies represent smooth regions and high frequencies correspond to edges and noise.
The Fourier Transform decomposes an image's spatial pixel variations into frequency components. By converting to a frequency-domain representation, it enables processing that would be difficult in the spatial domain. Image processing uses the 2D Discrete Fourier Transform (2D DFT).
- Low-frequency components: Gradual brightness changes and smooth gradients
- High-frequency components: Edges, fine textures, and noise
- Magnitude spectrum: Strength of each frequency component, center is low frequency
- Phase spectrum: Positional information, critical for image structure
Frequency-domain filtering includes:
- Low-pass filter: Removes high frequencies for blur (ideal, Butterworth, Gaussian)
- High-pass filter: Removes low frequencies to emphasize edges
- Band-pass filter: Passes a specific frequency range, effective for periodic noise
- Notch filter: Removes specific frequencies for moire elimination
Computation uses the FFT algorithm at O(N² log N) complexity via np.fft.fft2() or cv2.dft(). The convolution theorem states that spatial convolution equals element-wise multiplication in the frequency domain, enabling speedup for large kernel operations.