Completeness, stability, and provably accurate reconstruction for the wavelet transform modulus maxima representation of signals and images

Mallat and Hwang (1992) showed that the local regularity of a signal can be read off the wavelet transform modulus maxima - the points where the wavelet transform modulus is locally maximum across scales - and introduced an alternating-projection algorithm that reconstructs a signal or image from these maxima plus a low-frequency residue. Y. Meyer showed that the maxima do not characterize a function uniquely, and the accuracy of the reconstruction (empirically about 30 dB SNR) rests on numerical experiments, not theory. The open problem is to give this singularity-based representation a rigorous foundation: characterize the ambiguity class of signals sharing the same modulus maxima, decide whether the ambiguity is bounded (stability), and either prove error bounds for the minimal-Sobolev-norm reconstruction used in practice or produce a provably accurate algorithm; the source framework also leaves open the extension of the maxima-based Lipschitz-exponent characterization to integer and negative exponents (distributions) and statistical models for texture maxima. A later phase-retrieval literature has settled uniqueness and stability for the full wavelet transform modulus, but the modulus-maxima representation is a different, sparser data set that remains without a completeness or stability theory, and this is the surviving open core.

Background

The continuous wavelet transform of a signal $f$ convolves $f$ with dilated copies of a wavelet $\psi$, giving $Wf(s,x)=f*\psi_s(x)$, a function of position $x$ and scale $s$. When $\psi$ is the $n$-th derivative of a smoothing function, it has $n$ vanishing moments, meaning $\int x^k\psi(x)\,dx=0$ for $k<n$, and $Wf$ responds only to variations of $f$ at scales near $s$. The modulus maxima are the local maxima of $|Wf(s,x)|$ in $x$ at each scale; they form curves in the position-scale plane that converge, as $s\to 0$, to the singular points of $f$. The Lipschitz (Hölder) exponent $\alpha$ measures pointwise smoothness: $f$ is Lipschitz $\alpha$ at $x_0$ if $|f(x)-P(x)|\le C|x-x_0|^{\alpha}$ for some polynomial $P$; $\alpha=0$ is a jump, $\alpha<0$ (defined for distributions such as the Dirac delta or white noise) a stronger singularity. Mallat and Hwang proved, for non-integer $\alpha<n$ and isolated non-oscillating singularities satisfying a cone-of-influence condition (all maxima near $x_0$ lying inside $|x-x_0|\le Cs$) and a sign condition, that $\alpha$ is read off as the slope of $\log|Wf|$ versus $\log s$ along a maxima line. For processing, they used the dyadic wavelet transform ($s=2^j$), a redundant (frame-like) sampling in scale, and proposed reconstructing $f$ from the values of $Wf$ at the maxima points plus the low-frequency residue (the coarsest-scale component captured by the scaling function rather than by the wavelet) by alternating orthogonal projections - successively projecting onto each of two sets - between the space $V$ of all dyadic wavelet transforms (the functions of scale and position that actually arise as wavelet transforms of some signal; characterized by reproducing-kernel equations, i.e. fixed linear consistency relations) and the affine set $\Gamma$ of transforms matching the recorded maxima values; the algorithm converges to the element of $V\cap\Gamma$ of minimal Sobolev norm (the Sobolev norm measures a function together with finitely many of its derivatives in $L^2$), chosen to suppress oscillations. Reconstruction was reported at about 30 dB SNR (1D, 20 iterations; 31 dB for images), the error concentrated at high frequencies, with Gibbs-like spurious oscillations near sharp isolated singularities only partly removed by convex constraints. Y. Meyer proved that the modulus maxima do not characterize a function uniquely, so reconstruction accuracy cannot follow from injectivity; no bound on the approximation error for general signals was available. This motivated maxima-based denoising (noise maxima have negative exponents and increase toward fine scales, so they are removed by ad hoc propagation and chaining criteria), which systematically deletes image textures because texture maxima do not propagate to coarser scales, and which depends on manually chosen chain-length and scale-count thresholds. The question of what a maximally decimated wavelet-maxima representation retains is adjacent to a phase retrieval problem - recovering an element of a complex vector space from magnitudes of linear measurements - which uses the full wavelet modulus rather than the transform values at the maxima.

Problem Statement

Give the wavelet transform modulus maxima representation a rigorous foundation, with reconstruction accuracy guarantees, in the setting of Mallat and Hwang (1992): the dyadic wavelet transform $Wf(2^j,x)$ computed with a compactly supported wavelet with $n$ vanishing moments (the derivative of a smoothing function), the data being the dyadic transform values $Wf(2^j,x_j)$ at the modulus-maxima positions - equivalently the moduli $|Wf(2^j,x_j)|$ together with the sign of the transform at each maximum - together with the low-frequency residue. Meyer's result shows this data is not injective, so the central questions are: (1) characterize the ambiguity class - the set of signals, and distributions, whose dyadic wavelet transforms have modulus maxima at the same positions and take the same signed transform values there (and the same low-frequency residue); (2) decide whether the ambiguity is bounded, i.e. whether any two signals with the same maxima data are close in a norm stated explicitly by the solver, or construct unbounded ambiguity examples, thereby settling the stability of the representation; (3) either prove a quantitative error bound, valid for all signals in a stated class, for the minimal-Sobolev-norm reconstruction produced by the alternating-projection algorithm (including its 2D version), or exhibit a provably accurate reconstruction algorithm with comparable complexity (accuracy meaning that the algorithm's output satisfies an explicit quantitative error bound in a stated norm, and comparable complexity meaning on the order of the alternating-projection algorithm's); and, as linked components of the same objective, (4) extend the maxima-based Lipschitz-exponent characterization to integer exponents and to negative exponents for distributions, removing or justifying the cone-of-influence and sign conditions, and (5) formulate the statistical model of texture singularities (maxima that do not propagate to coarser scales) under which maxima-based denoising can preserve textures, together with an adaptive, data-driven choice of the chain-length and number-of-scales thresholds. The mathematical core comprises components (1)-(3), which together determine whether singularity-based, maximally decimated wavelet representations are a complete and stable signal representation or only an empirical approximation; components (4)-(5) are linked extensions of the same objective.

References

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