1. Two numbers: average and difference
The first Haar coefficient measures shared level. The second measures local contrast. Large difference coefficients often indicate edges.
2. Multiscale Haar transform of a length-16 signal
The coefficient plot shows where the signal has energy in the Haar language. Coarse coefficients describe broad changes; fine coefficients describe local jumps.
3. Compression: keep only the largest Haar coefficients
Because the Haar matrix is orthonormal, the reconstruction error is exactly controlled by the discarded coefficients.
4. Threshold denoising
Small noisy wiggles often become small Haar coefficients. Removing them can reveal the larger piecewise structure.
5. Tiny image: edges through row and column Haar transforms
A 2D Haar transform uses C = H X H^T. It separates coarse brightness from horizontal, vertical, and diagonal changes.