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Stability of the À Trous Algorithm Under Iteration

Brody Johnson, Simon McCreary-Ellis

TL;DR

It is shown that the stability of an infinitely iterated shift-invariant filter bank guarantees that of any associated finitely iterated shift-invariant filter bank, with uniform bounds, with uniform bounds.

Abstract

This paper examines the stability of the à trous algorithm under arbitrary iteration in the context of a more general study of shift-invariant filter banks. The main results describe sufficient conditions on the associated filters under which an infinitely iterated shift-invariant filter bank is stable. Moreover, it is shown that the stability of an infinitely iterated shift-invariant filter bank guarantees that of any associated finitely iterated shift-invariant filter bank, with uniform bounds. The reverse implication is shown to hold under an additional assumption on the low-pass filter. Finally, it is also shown that the separable product of stable one-dimensional shift-invariant filter banks produces a stable two-dimensional shift-invariant filter bank.

Stability of the À Trous Algorithm Under Iteration

TL;DR

It is shown that the stability of an infinitely iterated shift-invariant filter bank guarantees that of any associated finitely iterated shift-invariant filter bank, with uniform bounds, with uniform bounds.

Abstract

This paper examines the stability of the à trous algorithm under arbitrary iteration in the context of a more general study of shift-invariant filter banks. The main results describe sufficient conditions on the associated filters under which an infinitely iterated shift-invariant filter bank is stable. Moreover, it is shown that the stability of an infinitely iterated shift-invariant filter bank guarantees that of any associated finitely iterated shift-invariant filter bank, with uniform bounds. The reverse implication is shown to hold under an additional assumption on the low-pass filter. Finally, it is also shown that the separable product of stable one-dimensional shift-invariant filter banks produces a stable two-dimensional shift-invariant filter bank.
Paper Structure (12 sections, 12 theorems, 106 equations, 11 figures, 4 tables)

This paper contains 12 sections, 12 theorems, 106 equations, 11 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Mathematical Background and Notation
  4. Filter Bank Terminology
  5. Stability of Infinitely Iterated Shift-Invariant Filter Banks
  6. Sufficient Condition for a Bessel Bound
  7. Sufficient Condition for a Lower Frame Bound
  8. Stability of Finitely Iterated Shift-Invariant Filter Banks
  9. Examples
  10. Symmetric Filter Design
  11. Filter Banks with Two High-Pass Filters
  12. Separable Shift-Invariant Filter Banks in Two Dimensions

Key Result

Proposition 3.1

Let $h, g^{1}, g^{2}, \ldots, g^{L} \in \ell^{2}(\mathbb{Z})$ be low-pass and high-pass filters, respectively. Fix $0<A\le B<\infty$. The infinitely iterated shift-invariant filter bank associated with $h, g^{1}, g^{2}, \ldots, g^{L}$ is stable with frame bounds $A$ and $B$ if and only if Moreover, the infinitely iterated shift-invariant filter bank associated with $h, g^{1}, g^{2}, \ldots, g^{L}

Figures (11)

  • Figure 1.1: Analysis schematic for an infinitely iterated shift-invariant filter bank.
  • Figure 2.1: Synthesis schematic for an infinitely iterated shift-invariant filter bank.
  • Figure 4.1: Analysis schematic for a finitely iterated shift-invariant filter bank.
  • Figure 4.2: Analysis schematic for a $J$-upsampled infinitely iterated shift-invariant filter bank.
  • Figure 5.1: Frequency response for the filters $h$ and $g$ of Example \ref{['eg5.1']}.
  • ...and 6 more figures

Theorems & Definitions (30)

  • Definition 2.1
  • Proposition 3.1: Characterization of Stability
  • proof
  • Remark 3.2
  • Lemma 3.3
  • Lemma 3.4
  • Remark 3.5
  • Theorem 3.6
  • proof
  • Example 3.7
  • ...and 20 more