Orthonormal Bases Of Compactly Supported Wavelets.pdf

Key Points:

Wavelet Basics: The document starts by defining wavelets, families of functions generated from a single mother function (h) through dilations and translations. It highlights their utility in various fields like signal and image processing.

Admissibility Condition: For a wavelet to be useful for representation and reconstruction, it must satisfy an "admissibility condition" ensuring sufficient decay and mean zero.

Continuous vs. Discrete Wavelets: The paper discusses both continuous wavelet transforms (using continuous dilations and translations) and discrete wavelet transforms (with discrete steps).

Frames and Redundancy:

A frame is a set of wavelets that allows for accurate reconstruction of signals from their wavelet coefficients.
Redundancy refers to the degree to which wavelets within a frame are linearly dependent on each other. Highly redundant frames use many redundant wavelets, leading to smaller representation spaces but potentially better signal understanding.

Orthonormal Bases: The focus shifts to orthonormal bases of wavelets - sets of wavelets that form an orthogonal basis for L2(R).

Haar Basis: A well-known example is the Haar basis, constructed with a simple step function. It serves as a starting point and benchmark.
Meyer Basis: Y. Meyer introduced a groundbreaking compactly supported C“-function leading to an orthonormal wavelet basis with powerful properties, including unconditionality for various Sobolev spaces.
Lemarié-B and Battle Bases: Independently, P. G. Lemarié-B and G. Battle constructed another orthonormal basis with exponential decay.

Conclusion: The paper emphasizes the significance of orthonormal wavelet bases in signal processing and analysis due to their ability to provide both accurate representation and efficient reconstruction.