By J. C. van den Berg
This booklet surveys the applying of the lately built means of the wavelet rework to quite a lot of actual fields, together with astrophysics, turbulence, meteorology, plasma physics, atomic and strong kingdom physics, multifractals happening in physics, biophysics (in drugs and body structure) and mathematical physics. The wavelet remodel can study scale-dependent features of a sign (or photograph) in the neighborhood, not like the Fourier remodel, and extra flexibly than the windowed Fourier remodel built by way of Gabor fifty years in the past. the continual wavelet remodel is used normally for research, however the discrete wavelet rework permits very quick compression and transmission of information and hurries up numerical calculation, and is utilized, for instance, within the resolution of partial differential equations in physics. This ebook could be of curiosity to graduate scholars and researchers in lots of fields of physics, and to utilized mathematicians and engineers attracted to actual software.
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Extra resources for Wavelets in Physics
P. 2 The continuous WT Actually one should distinguish two different versions of the wavelet transform, the continuous WT (CWT) and the discrete (or more properly, discrete time) WT (DWT) [10,14]. The CWT plays the same roˆle as the Fourier transform and is mostly used for analysis and feature detection in signals, whereas the DWT is the analogue of the Discrete Fourier Transform (see for instance  or ) and is more appropriate for data compression and signal reconstruction. The situation may be caricatured by saying that the CWT is more natural to the physicist, while the DWT is more congenial to the signal analyst and the numericist.
There, indeed, a 2-D multiresolution is simply the tensor product of two 1-D schemes, one for the horizontal direction and one for the vertical direction (in technical terms, one uses only separable ﬁlters). However the 2-D continuous WT, including the orientation parameter , may be used for detecting oriented features of the signal, that is, regions where the amplitude is regular along one direction and has a sharp variation along the perpendicular direction, for instance, in the classical problem of edge detection.
Suppose the signal sðx~Þ is strongly oriented, for instance a long segment along the xaxis. Then its Fourier transform b s ðk~Þ is a long segment along the ky -axis. In order to detect such a signal, with a good directional selectivity, one needs a wavelet supported in a narrow cone in k~-space. Then the WT is negligible unless b ðk~Þ is essentially aligned onto b s ðk~Þ: directional selectivity demands to restrict the support of b, not . The corresponding standard practice in signal processing is to design an adequate ﬁlter in the frequency domain (high pass, band pass, .