[Paper Review] Wavelets with the Translation Invariance Property of Order N

S. Schäffer, Eric Weber|ArXiv.org|Feb 15, 2000

Image and Signal Denoising Methods4 references3 citations

TL;DR

This paper introduces and constructs wavelets with a novel translation invariance property of order $ n $, where the generalized multiresolution analysis (GMRA) subspaces $ V_j $ are invariant under integer translations only for $ j \geq n $. The authors prove the existence of such wavelets for any dilation factor $ d \geq 2 $, using frequency-domain interpolation of wavelet sets, and show that these wavelets form nested, non-closed collections $ \mathcal{M}_n $, generalizing MSF wavelets.

ABSTRACT

All wavelets can be associated to a multiresolution like structure, i.e. an incr easing sequence of subspaces of L^2(R). We consider the interaction of a wavel et and the translation operator in terms of which of the subspaces in this multi resolution like structure are invariant under the translation operator. This ac tion defines the notion of the translation invariance property of order n. In this paper we show that wavelets of all levels of translation invariance exist, first for the classic case of dilation by 2, and then for arbitrary integral di lation factors.

Motivation & Objective

• To define and characterize wavelets with translation invariance of order $ n $, where only subspaces $ V_j $ with $ j \geq n $ are invariant under integer translations.
• To extend the concept of translation invariance beyond MSF wavelets, which have invariance for all $ j \in \mathbb{Z} $, by introducing a hierarchy of invariance levels.
• To demonstrate the existence of such wavelets for arbitrary integer dilation factors $ d \geq 2 $, generalizing prior results for $ d=2 $.
• To analyze the topological structure of the collections $ \mathcal{L}_n $ and $ \mathcal{M}_n $, showing $ \mathcal{L}_n $ is relatively closed while $ \mathcal{M}_n $ is not.

Proposed method

• Define the translation invariance property of order $ n $ via the invariance of $ V_{-n} $ under $ T^{m/d^n} $, linking it to invariance of $ W_0 $ under the same operators.
• Use the Fourier transform to characterize wavelets by the support of $ \hat{\psi} $, focusing on sets that are partially self-similar under translation by $ 2\pi/d^n $.
• Construct wavelets via interpolation pairs of wavelet sets $ W $ and $ W_n $, where $ W_n \to W $ in symmetric difference, ensuring convergence in $ L^2 $.
• Apply Merrill's theorem to verify that the resulting generalized scaling sets $ E_n $ satisfy the necessary conditions for generating wavelets.
• Use measure-theoretic arguments involving $ L^2 $ convergence and uniform bounds on Fourier transforms to show that $ \mathcal{L}_j $ is closed under limits.

Experimental results

Research questions

• RQ1 Do wavelets exist for which the GMRA subspaces $ V_j $ are invariant under integer translations only for $ j \geq n $, for any given $ n \in \mathbb{N} $?
• RQ2 Can such wavelets be constructed for arbitrary integer dilation factors $ d \geq 2 $, not just $ d=2 $?
• RQ3 Are the collections $ \mathcal{M}_n $ of wavelets with translation invariance of order $ n $ closed in the $ L^2 $ topology?
• RQ4 How do these wavelets relate to MSF wavelets, which correspond to $ \mathcal{M}_\infty $?

Key findings

• For every positive integer $ n $, there exists a wavelet with translation invariance property of order $ n $, i.e., $ \mathcal{M}_n \neq \emptyset $, for any integer dilation factor $ d \geq 2 $.
• The collections $ \mathcal{L}_n $, consisting of wavelets for which $ V_{-n} $ is translation invariant, are relatively closed in the $ L^2 $ norm.
• The collections $ \mathcal{M}_n $, which are the differences $ \mathcal{L}_n \setminus \mathcal{L}_{n+1} $, are not closed, as demonstrated by a sequence in $ \mathcal{M}_1 $ converging to an MSF wavelet in $ \mathcal{M}_\infty $.
• Wavelets with higher-order translation invariance can be constructed via interpolation of wavelet sets in the frequency domain, using a sequence of generalized scaling sets $ E_n $ converging to a limiting set.
• The construction relies on frequency-domain sets that are partially self-similar under translation by $ 2\pi/d^n $, ensuring the required invariance in $ W_0 $.