By J. C. van den Berg
This booklet surveys the applying of the lately constructed means of the wavelet rework to quite a lot of actual fields, together with astrophysics, turbulence, meteorology, plasma physics, atomic and sturdy country physics, multifractals taking place in physics, biophysics (in medication and body structure) and mathematical physics. The wavelet remodel can learn scale-dependent features of a sign (or photograph) in the community, in contrast to the Fourier rework, and extra flexibly than the windowed Fourier remodel built by means of Gabor fifty years in the past. the continual wavelet rework is used typically for research, however the discrete wavelet remodel permits very speedy compression and transmission of knowledge and hurries up numerical calculation, and is utilized, for instance, within the resolution of partial differential equations in physics. This e-book should 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 info for Wavelets in Physics, 2nd Edition
5). 6), but our analysis will be rather brief, because a full treatment requires the language and techniques of signal processing (ﬁlter theory), which are in general more familiar to electrical engineers than to physicists. Moreover, the 2-D DWT is mostly used in image processing, which is not the main thrust of the present book. 4). We consider 2-D signals of ﬁnite energy, represented by complex-valued, square integrable functions s 2 L2 ðR2 ; d 2 x~Þ. This condition may be relaxed, to allow, for instance, a plane wave or a function.
As a matter of fact, all the approaches that we have mentioned above in the 1-D case have been extended to 2-D: orthonormal bases, biorthogonal bases, wavelet packets, lifting scheme. These topics will be discussed in detail in Chapter 2. Wavelet analysis: a new tool in physics 19 However, the continuous transform may also be extended to 2 (or more) dimensions, with exactly the same properties as in the 1-D case [2, 26]. 10)): Z ~ Sða; ; bÞ $ d 2 x~ " ðaÀ1 rÀ ðx~ À b~ÞÞsðx~Þ; a > 0; 0 < 2; b 2 R2 ; ð1:26Þ where s is the signal and is the analysing wavelet, which is translated by b~, dilated by a and rotated by an angle ðrÀ is the rotation operator).
For all these reasons, signal analysts turn to time-frequency (TF) representations. The idea is that one needs two parameters: one, called a, characterizes the frequency, the other one, b, indicates the position in the signal. This concept of a TF representation is in fact quite old and familiar. The most obvious example is simply a musical score! If one requires in addition the transform to be linear, a general TF transform will take the form: Z1 ð1:1Þ sðxÞ 7 ! Sða; bÞ ¼ ab ðxÞ sðxÞ dx; À1 where s is the signal and ab the analysing function.
Wavelets in Physics, 2nd Edition by J. C. van den Berg