Digital Image Processing: Transforms, Compression, and Filters
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Fourier Transform
1. Definition
The Fourier Transform (FT) is used to convert an image from the spatial domain to the frequency domain. It expresses the image as a sum of sinusoidal functions of varying frequencies, amplitudes, and phases.
2. Intuition
In the spatial domain, we deal with pixel intensity at each location. In the frequency domain, we analyze how intensity varies—i.e., how fast brightness changes across pixels. This helps in analyzing patterns, removing noise, and performing filtering.
3. Mathematical Forms
- A. 1D Continuous FT (for signals):
- B. 2D Continuous FT (for images):
- C. 2D Discrete FT (DFT):
Used in DIP because images are digital:
- f(x,y): Input image
- F(u,v): Frequency domain representation
- M×N: Size of image