Definition
The Haar system is the simplest wavelet basis, consisting of mathematically orthogonal “box” and “step” functions that capture signal information at various resolutions.
Why It Matters
As the foundation of wavelet theory, the Haar system provides an intuitive framework for understanding how continuous signals can be perfectly decomposed into discrete blocks, forming the basis for image compression.
Core Concepts
- Scaling Function (): The “box” function, representing the average value at a specific resolution.
- How to read: “The scaling function phi.”
- Meaning: Captures the coarse, averaged signal at a given scale — the “smooth” component.
- Wavelet (): The “step” function, representing the change or detail at that resolution.
- How to read: “The wavelet function psi.”
- Meaning: Captures local differences/detail at a scale — what the scaling function misses.
- Compact Support: Haar functions are zero outside a small interval, preventing distant anomalies from corrupting local signal reconstruction.