Wavelet
A technique that decomposes images into multi-resolution representations preserving both spatial and frequency information, serving as the compression basis for JPEG 2000 and effective for denoising.
A wavelet decomposes an image into multi-resolution representations preserving both spatial and frequency information. Unlike the Fourier transform which provides only frequency content, wavelet transforms reveal where specific frequency components exist in the image.
The 2D Discrete Wavelet Transform (2D DWT) produces four subbands:
- LL (approximation): Low-frequency components, a downscaled version of the original
- LH (horizontal detail): Horizontal edge information
- HL (vertical detail): Vertical edge information
- HH (diagonal detail): Diagonal edge information
Common wavelet basis functions:
- Haar: A simple rectangular wave with fast computation
- Daubechies: Orthogonal with compact support
- CDF 9/7: JPEG 2000 lossy compression
- CDF 5/3: JPEG 2000 lossless compression, integer arithmetic
Applications include JPEG 2000 compression (block artifacts are less likely), denoising via thresholding detail coefficients (wavelet shrinkage), and texture analysis through energy distributions at each scale. Python's pywt library provides an implementation.