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

If

\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 nth 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.

[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.
  • [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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