Introduction

Polyphase decompositions, filter banks, lifting schemes, and wavelet packets are standard parts of multirate signal processing and harmonic analysis. Their classical development is described in [CrochiereRabiner1983], [Vaidyanathan1993], [VetterliKovacevic1995], [StrangNguyen1996], [Daubechies1992], [Mallat2009]. These theories are often introduced through several parallel languages. Residue classes appear in polyphase analysis. Matrix symbols appear in filter banks. Trees appear in wavelet packets. Singularities are handled through a different approximation or detection method. The first aim of this article is to place these constructions in one exact coordinate system.

The second aim is to separate representation from recovery. If a signal has already been written as

with a smooth part \(s\) and a singular packet \(j\), then

satisfies

This identity is exact, but it does not discover \(j\). A useful signal theory must therefore distinguish three errors. The first is the approximation error of the smooth component. The second is the error in the estimated singular packet. The third is the amplification caused by applying a reconstruction operator to that packet error. The middle sections of the paper isolate all three terms.

The singular packets considered here range from indicator functions of regions to truncated powers

The latter encode jumps in derivatives of order \(m\). They permit a piecewise smooth function to be reduced to a globally Sobolev remainder under explicit trace assumptions. This gives a precise version of the informal statement that a discontinuity should be represented directly rather than approximated by a global smooth basis.

The recovery theory begins with one dimensional separated jumps. For bounded deterministic noise, thresholding adjacent sample contrasts recovers the exact grid cells containing the jumps whenever the jump size exceeds the combined smooth variation and noise by an explicit margin. The same proof gives location and amplitude bounds. A sub Gaussian version follows from a union bound. Averaging repeated measurements reduces the required threshold in the expected way. Haar coefficients provide a second detection mechanism. Two grids shifted by half a cell guarantee that every jump is well placed in at least one grid, which removes the most elementary dyadic alignment failure.

The literature on singularity detection is much broader than the results proved here. Wavelet modulus maxima are developed in [MallatHwang1992]. Best basis selection in packet libraries was introduced in [CoifmanWickerhauser1992]. Nonlinear approximation is surveyed in [DeVore1998]. Geometric image models, wedgelets, and curvelets appear in [Donoho1999], [CandesDonoho2004]. The present article does not replace those theories. Its narrower purpose is to give a common exact calculus in which coordinate reconstruction, adaptive basis selection, singular packet estimation, and smooth approximation remain logically separate.

The article is organized as follows. Section E2 and Section E3 develop lattice addresses and exact residue coordinates in any dimension. Section E4 and Section E5 express convolution and filter banks in those coordinates, including stability under imperfect reconstruction and filter perturbation. Section E6 treats arbitrary pruned packet trees and stability of the best basis decision. Section E7 and Section E8 introduce step and higher order singular packets, exact reconstruction identities, and error formulas. Section E9 and Section E10 prove deterministic and probabilistic recovery results. Section E11 and Section E12 treat Gibbs oscillation and nonlinear interactions. The final sections record finite numerical illustrations and a secondary cyclic Cayley realization.

Several conclusions are deliberately excluded. The paper does not prove a universal image segmentation theorem. Exact recovery of separated jumps in one dimension does not imply stable recovery of arbitrary boundaries in higher dimensions. Retaining a correct singular packet does not remove noise or model error. An adaptive tree selected by an additive objective need not remain optimal when global hardware costs or cross packet interactions are introduced. The Cayley coordinate section is a secondary conformal realization and is not used in the recovery theorems.

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The first result is an exact coordinate theorem. Let \(L\) be a finite index sublattice of \({\mathbb Z}^d\), and let \(R\) contain one representative from every coset. The analysis map

is an isometric isomorphism on every \(\ell^p\) space. This statement includes the usual one dimensional polyphase decomposition, but it also applies to arbitrary finite index lattice packets in several dimensions. If \(L'\leq L\), the coarse packets split exactly into their fine descendants. A convolution operator becomes a matrix of convolution operators on \(L\), with entries

The matrix is therefore a coordinate representation of the original operator, not a reduced model.

The filter bank results distinguish exact reconstruction from quantitative near reconstruction. Exact recovery with delay \(d\) is the pointwise matrix identity

When this identity fails, the reconstruction error is controlled by the essential supremum of the delay compensated multiplier defect. If the filters themselves are perturbed, Theorem E5.4 separates the analysis error, the synthesis error, and their product. The frame bound in Theorem E5.7 shows that a lower singular value gap survives every sufficiently small perturbation. This makes the distinction between algebraic invertibility and stable numerical reconstruction explicit.

The packet tree results treat every admissible pruning, not only complete trees of a fixed depth. Unitary local splits give a unitary global transform. Local frame bounds propagate recursively in the nonunitary case. If the cost of a pruning is additive over retained nodes, then

gives the exact optimal tree. The cost and the selected basis are stable when the node costs change by less than the gap separating the optimal pruning from its competitors.

The singular packet results begin with trace data. If a piecewise smooth function has a jump in its derivative of order \(m\) at \(\tau_q\), then the packet

removes exactly that trace jump. Subtracting all packets through order \(r-1\) leaves a global \(W^{r,p}\) remainder. The packet coefficients are uniquely determined by the one sided trace jumps. This gives a finite and explicit singular part for the class covered by Theorem E7.2.

The reconstruction formula with an estimated packet is

This identity identifies the two terms that must be controlled. The exact packet formulas in Section E8 then evaluate or bound \(\|j-\widehat j\|\) from amplitude error, region overlap, boundary displacement, and the order of a derivative jump. A value jump has location modulus \(|\tau-\widehat\tau|^{1/p}\) in finite \(L^p\), while every positive order truncated power packet is Lipschitz in its location.

The principal recovery theorem uses adjacent sample contrasts. Let

be the largest contrast that can be produced by a Lipschitz background and bounded noise in one grid cell. If

then every threshold between \(b_h\) and \(a_{\min}-b_h\) detects exactly the cells containing jumps. The jump locations are recovered within \(h/2\), and each amplitude error is at most \(b_h\). The probability theorem replaces the deterministic noise term by

Two half shifted Haar grids provide a second recovery mechanism. Every point is at least one quarter of a cell away from the boundary of one of the grids, so the Haar coefficient gap can be made independent of dyadic alignment.

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The following symbols recur throughout the article.

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