Wavelets

The wavelets are scaled and translated copies (known as "daughter wavelets") of a finite-length or fast-decaying oscillating waveform (known as the "mother wavelet").

Wavelets are more useful for describing these signals with discontinuities because of their time-localized behavior (both Fourier and wavelet transforms are frequency-localized, but wavelets have an additional time-localization property).

No wavelets running across our ankles at sunset?

A stone’s throw away is a small river and on this cool tropical evening, twilight rays rippled across its waters in a play of shimmering wavelets.

As a mathematical tool, wavelets can be used to extract information from many different kinds of data, including – but certainly not limited to – audio signals and images.

As wavelets have location as well as frequency, they are better able to represent location, at the expense of greater difficulty representing frequency.

A wavelet transform is the representation of a function by wavelets.

Daubechies and Symlet wavelets can be defined by the scaling filter.

Every point on the wave front acts as the source of secondary wavelets that spread out in the light cone with the same speed as the wave.

For analysis with orthogonal wavelets the high pass filter is calculated as the quadrature mirror filter of the low pass, and reconstruction filters are the time reverse of the decomposition filters.

For example, JPEG 2000 is an image compression standard that uses biorthogonal wavelets.

In the analysis of the single slit, the non-zero width of the slit is taken into account, and each point in the aperture is taken as the source of one contribution to the beam of light (Huygen's wavelets).

JPEG2000 uses wavelets, and these can also be used to encode motion without gaps between blocks in an adaptive way.

Later work by Dennis Gabor yielded Gabor atoms (1946), which are constructed similarly to wavelets, and applied to similar purposes.

Sets of wavelets are generally needed to analyze data fully.

The delta image can also be encoded as wavelets, so that the borders of the adaptive blocks match.

The expanding ring of waves is the wave group, within which one can discern individual wavelets of differing wavelengths traveling at different speeds.

These filter banks are called the wavelet and scaling coefficients in wavelets nomenclature.

Thus, sets of complementary wavelets are useful in wavelet based compression /decompression algorithms where it is desirable to recover the original information with minimal loss.