Definition
Wavelets are brief, wave-like oscillations with an amplitude that begins at zero, increases, and then decreases back to zero. They are used as mathematical bases to decompose data into different frequency components.
Why It Matters
Wavelets provide an efficient way to represent and compress complex data while preserving sharp transitions. They allow us to transmit high-resolution information across limited bandwidth without losing critical, localized details like edges in an image.
Core Concepts
- Multiresolution Analysis (MRA): Constructing an approximation by starting with a coarse average and adding increasingly fine layers of detail.
- Time-Frequency Localization: Unlike sine waves (which span all time but have one exact frequency), wavelets have “compact support” providing local information in both time and frequency.
- Daubechies wavelets: Advanced families of orthogonal wavelets that are continuous, improving upon the discontinuous Haar system.