Many continuous wavelets are derived from a probability density (e.g. Sombrero). This approach also sets up a link among probability densities, wavelets and ‘’blur derivatives’’. To begin with, let P(.)- be a probability density, , the space of complex signals infinitely differentiable. In numerical analysis, continuous wavelets are functions used by the continuous wavelet transform. ... In quantum mechanics, a probability amplitude is a complex number-valued function which describes an uncertain or unknown quantity. ... Sombrero Sombrero means hat in spanish. ...

A blurred signal can be derived from f(.) by using the probability density P(.)- according to:

The classical derivative

of the blurred version is referred to as the blur derivative of f(.) through the density P(.)-.

Blur derivative and wavelets

If

then

is a wavelet engendered by P(.)-.

Given a mother wavelet ψ that holds the admissibility condition then the continuous wavelet transform is defined by // Formulation In mathematics and signal processing, the continuous wavelet transform (CWT) of a function is a wavelet transform defined by where represents translation, represents scale and is the mother wavelet. ...

, .

Continuous wavelets have often unbounded support, such as Morlet wavelet, Meyer, Mathieu wavelet, de Oliveira wavelet. The Morlet wavelet, named after Jean Morlet, was originally formulated by Goupillaud, Grossmann and Morlet in 1984 as a constant subtracted from a plane wave and then localised by a Gaussian: where is defined by the admissibility criterion and the normalisation constant is: The Fourier transform of the Morlet wavelet... // People Meyer, Adolf (1866–1950), Swiss-born U.S. psychiatrist Meyer, Adolf (1881-1921), architect Meyer, Adolf Bernard (1840-1911), German anthropologist and ornithologist Meyer, Albert Cardinal (1903–1965), Roman Catholic Archbishop of Chicago Meyer, Albert (1870–1953), Swiss politician Meyer, Alfred (1891–1945), German Nazi official Meyer, Alfred Richard...

In the case where the wavelet was generated from a probability density, one has

Now

so that

If the order of the integral and derivative can be permuted, it follows that

Defining the LPFed signal as theblur signal

an interesting interpretation can be made: set a scale a and take the average (smoothed) version of the original signal - the blur version . The blur derivative

is the n^th derivative regarding the shift b of the blur signal at the scale a.

The blur derivative coincide with the wavelet transform CWT(a,b)- at the corresponding scale. Details (high-frequency) are provided by the derivative of the low-pass (blur) version of the original signal.

Many continuous wavelets can be derived by this approach.

References

• [1] G. Kaiser, A Friendly Guide to Wavelets, Boston: Birkhauser, 1994.

• [2] H.M. de Oliveira, G.A.A. Araújo, Compactly Supported One-cyclic Wavelets Derived from Beta Distributions, Journal of Communication and Information Systems, (former Journal of the Brazilian Telecommunications Society), vol.20, n.3, pp.27-33, 2005.

• http://www.iecom.org.br/

• [3] M.M.S. Lira, H. M. de Oliveira and R.J.S. Cintra, Elliptic-Cylinder Wavelets: The Mathieu Wavelets, IEEE Signal Process. Letters, vol. 11, n.1, Jan., pp. 52 - 55, 2004.

• [4] H.M. de Oliveira, L.R. Soares and T.H. Falk, A Family of Wavelets and a New Orthogonal Multiresolution Analysis Based on the Nyquist Criterion, J. of the Brazilian Telecomm. Soc., Special issue, vol. 18, N.1, pp. 69-76, Jun., 2003.