Wavelet

A wavelet is a localized oscillating function used to represent signals at different scales and positions. Its finite extent or rapid decay makes it useful for analyzing local structure.

Continuous wavelet transform: A transform that measures a signal’s similarity to translated and dilated copies of a wavelet. It applies a wavelet across continuous positions and scales to reveal local signal features.
Admissibility condition: A condition on a wavelet that permits reconstruction through the continuous wavelet transform. It ensures the wavelet transform retains enough information to recover the original signal.
Image compression: The reduction of image data size through coding that preserves selected visual information. Wavelet coefficients can concentrate image information, as in JPEG 2000 compression.
Haar wavelet: The simplest orthogonal wavelet, formed from a step function with compact support. It gives an intuitive two-scale decomposition but represents smooth signals with visible discontinuities.
Discrete wavelet transform: A transform that represents a signal with wavelet coefficients at discrete positions and scales. It samples wavelet scales and translations efficiently, often producing a compact representation.
vanishing moment: A wavelet property in which its integrals against polynomials up to a given degree are zero. Vanishing moments make wavelet coefficients insensitive to smooth polynomial trends.
Denoising: The removal or reduction of noise from measured or recorded data. Thresholding small wavelet coefficients can suppress noise while retaining localized features.
Daubechies wavelet: A family of compactly supported orthogonal wavelets with a specified number of vanishing moments. These wavelets improve smoothness and polynomial cancellation while retaining finite support.
Wavelet coefficient: A scalar measuring a signal’s contribution at a particular wavelet position and scale. Each coefficient records how strongly a translated, dilated wavelet matches the signal locally.
Orthogonal wavelet: A wavelet whose translated and dilated basis functions are mutually orthogonal. Orthogonality supports stable, nonredundant decompositions and straightforward coefficient recovery.

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