1
$\begingroup$

I am reading Albert Cohen, Ingrid Daubechies, Pierre Vial. Wavelets on the Interval and Fast Wavelet Transforms, 1993.

In this paper wavelet bases on an interval (for, example on space $L^2[0,1]$) are proposed. They propose the same number of scaling functions and mother wavelets (namely, $2^j$ for $j$-th scale).

Firstly, a half-line $[0,\infty)$ is considered. Suppose we are dealing with wavelets which have minimal support $[-N+1,N]$ for number $N$ of vanishing moments (like Daubechies wavelets). Then the construction of scaling functions, as I understand it, is following:

  1. For mother wavelets having $N$ vanishing moments, $\int_{-\infty}^{\infty}a \psi(x)dx = 0$ for $a = const$, so only scaling functions "generate" constants. This, however, is impossible on an (half-)interval (e.g. $[0,\infty)$), because a) $\phi(x)$ is continuous, b) $\phi_{0,k}(0) = \phi_{0,k}(-k) = 0$ for $k \ge N-1$ (here the first subscript index is scale index and the second is shift index). Thus, we add such a function $\phi^{0}(x)$ that it a) "generates" constants, b) is orthogonal to shifts of $\phi(x)$.
  2. Define $\phi^0(x)$ as $\phi^0(x) = 1 - \sum_{k=N-1}^{\infty} \phi(x-k)$. This can be treated as a "constant" minus projections of that constant to all shifts of $\phi(x)$ which reside completely in $[0, \infty)$. Using $\sum_{k=-\infty}^{\infty} \phi(x-k) = 1$ we get $\phi^{0}(x) = \sum_{k=-\infty}^{N-2} \phi(x-k) = \sum_{k=-N+1}^{N-2} \phi(x-k)$. In the last equality I dropped all shifts of $\phi(x)$ which do not "live" on $[0, \infty)$ even partially.
  3. Now we have an orthonormal set of scaling functions ($\phi^0(x)$ and $\phi_{0,k}(x)$ for $k \ge N-1$), but they still do not generate polynomials of the first degree (only constants). So we drop one (outermost) "internal" scaling function and add two new functions $\phi^{0}(x)$ and $\phi^{1}(x)$. I provide formulas for these functions from the article: $\phi^{k}(x) = \sum_{n=k}^{2N-2} \binom{n}{k}\phi(x+n-N+1)$ for $k = 0, 1, \cdots, N-1$

These functions, $\phi^{k}(x)$, $0 \le k \le N-1$ said to be independent, orthogonal to $\phi_{0,k}$, $k \ge N$ and have support $supp \phi^{k}(x) = [0, 2N-1-k]$.

I have the following questions:

  • For N=2 (For example, Daubechies D4 wavelet), $\phi^{0}(x) = \phi(x-1) + \phi(x) + \phi(x+1)$ (using the formula above). Isn't its support $[-2,3]$, not $[0, 3]$? Support of $\phi(x)$ is $[-1,2]$, supp of $\phi(x+1)$ is, obviously, $[-2, 1]$ and supp of $\phi(x-1)$ is $[0, 3]$, resulting in $[-2, 3]$. Why the authors claim it is $[0, 3]$?
  • Can you explain how $\phi^{1}(x)$, $\phi^{2}(x)$ and so on are constructed? $\phi^{0}(x)$ seems to be obvious to me, but not all others. As I understand $\phi^{1}(x)$ must be something like $\phi^{1}(x) = x - \sum_{k=N}^{\infty} \left<x, \phi(x-k)\right> \phi(x-k)$. How do we get the general formula for $\phi^{k}(x)$?
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.