Blur derivative

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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, $P\in \mathbb{C}^{\infty }$ , the space of complex signals $f:\mathbb{R}\rightarrow \mathbb{C}$ infinitely differentiable.

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

$\tilde{f}(t)=\int^{+\infty}_{-\infty} f(\tau) \cdot P({\tau- t})d \tau,$

The classical derivative

$\frac{\partial^n}{\partial t^n}\tilde{f}(t)$

of the blurred version $\tilde{f}(t)$ is referred to as the blur derivative of $f (.)$ through the density $P (.)$ .

[edit] Blur derivative and wavelets

$\lim {}_{t\rightarrow \infty} \frac{d^{n-1}P(t)}{dt^{n-1}}=0$

then

$\psi(t)= (-1)^n \frac{d^n P(t)}{dt^n}$ is a wavelet engendered by

P (.)

Given a mother wavelet $ψ$ that holds the admissibility condition then the continuous wavelet transform is defined by

$CWT(a,b) = \int^{+\infty}_{-\infty} f(t) \cdot \frac{1}{\sqrt{|a|}}\psi(\frac{t- b}{a})dt,$ $\forall a \in \mathbb{R}-\{0\}$ , $b\in \mathbb{R}$ .

Continuous wavelets have often unbounded support, such as Morlet wavelet, Meyer, Mathieu wavelet, de Oliveira wavelet.

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

$\frac{1}{\sqrt{|a|}}\psi_n(\frac{t-b}{a})=(-1)^n \frac{1}{\sqrt{|a|}}\frac{\partial^n P(\frac{t-b}{a})}{\partial t^n}.$

Now

$\frac{\partial^n P(\frac{t-b}{a})}{\partial b^n}= (-1)^n \frac{1}{a^n} P^{(n)}(\frac{t-b}{a}),$

so that

$CWT(a,b)=\frac{1}{\sqrt{|a|}} \int^{+\infty}_{-\infty} f(t) \cdot \frac{\partial^n P(\frac{t- b}{a})}{\partial b^n}dt.$

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

$CWT(a,b)= \frac{1}{\sqrt{|a|}} \frac{\partial^n }{\partial b^n}\int^{+\infty}_{-\infty} f(t) \cdot P(\frac{t- b}{a})dt.$

Defining the LPFed signal as theblur signal

$\tilde{f}(a,b)=\int^{+\infty}_{-\infty} f(t) \cdot \frac{1}{\sqrt{|a|}}P(\frac{t- b}{a})dt=\int^{+\infty}_{-\infty} f(t) \cdot P_{a,b}(t)dt,$

an interesting interpretation can be made: set a scale $a$ and take the average (smoothed) version of the original signal - the blur version $\tilde{f}(a,b)$ . The blur derivative

$\frac{\partial^n}{\partial b^n}\tilde{f}(a,b)$

is the $n t h$ derivative regarding the shift $b$ of the blur signal at the scale $a$ .

The blur derivative coincide with the wavelet transform $C W T (a, b)$ 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.

[edit] 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.

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Categories: Cleanup from October 2006 | Continuous wavelets

Blur derivative

From Wikipedia, the free encyclopedia

[edit] Blur derivative and wavelets

[edit] References

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