Wavelet vanishing moment

To measure the local regularity of a signal, it is not so important to use a wavelet with a narrow frequency support, but vanishing moments are crucial. If the wavelet has n vanishing moments, the wavelet transform can be interpreted as a multiscale differential operator of order n. This yields a first relation between the differentiability of f and its wavelet transform decay at fine scales.

Polynomial suppression

The Lipschitz property approximates f with a polynomial p_v in the neighborhood of v:

f(t) = p_v(t) + ε_v(t)   with  ε_v(t) ≦ K|t−v|^α

A wavelet transform estimates the exponent α by ignoring the polynomial p_v. For this purpose, we use a wavelet that has n > α vanishing moments:

∫_−∞^∞t^kψ(t)dt = 0  for  0 ≤ K < n

A wavelet with n vanishing moments is orthogonal to polynomials of degree n − 1. Since n > α, the polynomial p_v has degree of at most n − 1. With the change of variable t′ = (t−u)/s, we verify that:

$Wp_v(u,s)=\int_{-\infty}^{\infty} P_v( \frac{1}{\sqrt{s}}\psi\left ( \frac{t-u}{s} \right )dt=0$

Sincef = p_v + ε_v,

Wf(u,s) = Wε_v(u,s)

Multiscale differential operator

A wavelet with n vanishing moments can be written as the nth-order derivative of a function θ; the resulting wavelet transform is a multiscale differential operator. We suppose that ψ has a fast decay, which means that for any decay exponent m ∈ N there exists C_m such that:

$\forall t\in \mathbb{R}, \ |\psi(t)|\leq\frac{C_m}{1+|t|^m}$

If K = ∫_−∞^∞θ(t)dt ≠ 0, then the convolution$f*\bar{\theta_s(t)}$ can be interpreted as a weighted average of f with a kernel dilated by s. So Wf(u,s) = Wε_v(u,s) proves that Wf(u,s) is an nth-order derivative of an averaging of f over a domain proportional to s.

Since θ has a fast decay, one can verify that:

$\lim_{s \to 0} \frac{1}{\sqrt{s}}\bar{\theta}_s=K\delta$

in the sense of the weak convergence. This means that for any ϕ that is continuous at u:

$\lim_{s \to 0} \phi*\frac{1}{\sqrt{s}}\bar{\theta}_s(u)=K\phi(u)$

If f is n times continuously differentiable in the neighborhood of u, then Wf(u,s) = Wε_v(u,s) implies that:

$\lim_{s \to 0} \frac{Wf(u,s)}{s^{n+1/2}}=\lim_{s\to 0}\bar{\theta}_s(u)=Kf^{(n)}(u)$

In particular, if f is Cn with a bounded nth-order derivative, then |Wf(u,s)| = O(s^n + 1/2). This is a first relation between the decay of |Wf(u,s)| when s decreases and the uniform regularity of f.

Theorem

A wavelet ψ with a fast decay has n vanishing moments if and only if there exists θ with a fast decay such that:

$\psi(t)=(-1)^n \frac{d^n\theta(t)}{dt^n}$

As a consequence:

$Wf(u,s)=s^n\frac{d^n}{du^n}(f*\bar{\theta_s})(u)$

with $\bar{\theta_s}(t)=s^{\frac{-1}{2}}\theta(-t/s)$. Moreover, ψ has no more than n vanishing moments if and only if∫_−∞^∞t^kψ(t)dt ≠ 0.

References

1. Stéphane Mallat, A Wavelet Tour of Signal Processing, 3rd
2. Karlheinz Gröchenig, Foundations of Time-Frequency Analysis