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arxiv: 1907.09795 · v1 · pith:FKR4MS7Enew · submitted 2019-07-23 · 💻 cs.IT · math.IT

Close Encounters of the Binary Kind: Signal Reconstruction Guarantees for Compressive Hadamard Sampling with Haar Wavelet Basis

Amirafshar Moshtaghpour , Jos\'e M. Bioucas Dias , Laurent Jacques This is my paper

Pith reviewed 2026-05-24 17:08 UTC · model grok-4.3

classification 💻 cs.IT math.IT
keywords compressive sensingHadamard samplingHaar waveletsrecovery guaranteessample complexityvariable density samplingsingle-pixel imaging
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The pith

Density-matched sampling yields explicit recovery guarantees for Hadamard measurements of Haar-sparse signals

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes explicit sample-complexity bounds plus uniform and non-uniform recovery guarantees for recovering signals that are sparse in the Haar wavelet basis from subsampled Hadamard measurements. Variable and multilevel density sampling strategies overcome the coherence obstacle between the two bases by tailoring the subsampling pattern to local and multilevel coherence levels. This result supplies the theoretical support that had been missing for practical compressive imaging systems such as single-pixel cameras. A sympathetic reader would care because the bounds make concrete how many binary measurements suffice for faithful reconstruction in both one and two dimensions.

Core claim

By adjusting the subsampling process to the local and multilevel coherence between the Hadamard and Haar bases, the variable and multilevel density sampling strategies enable successful signal recovery, yielding explicit sample-complexity bounds together with uniform and non-uniform recovery guarantees for one- and two-dimensional signals.

What carries the argument

Variable and multilevel density sampling strategies that adjust the subsampling process to the local and multilevel coherence between the Hadamard and Haar bases

Load-bearing premise

The premise that sampling densities can be chosen to match the coherence structure between the two bases closely enough to guarantee recovery without other error sources dominating.

What would settle it

A concrete Haar-sparse test signal that remains unrecoverable from the number of Hadamard samples prescribed by the derived bound when the density strategies are applied.

Figures

Figures reproduced from arXiv: 1907.09795 by Amirafshar Moshtaghpour, Jos\'e M. Bioucas Dias, Laurent Jacques.

Figure 1
Figure 1. Figure 1: An example of the 2-D isotropic wavelet levels [PITH_FULL_IMAGE:figures/full_fig_p011_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Block structure of the matrices HrW(1) r (left) and HrW(0) r (right) where Tl = T 1d l for l ∈ JrK0. Gray color represents zero value. 3.2 Main Results Equipped with the definitions above, we are now ready to develop our main results. To do so, we need to calculate the local coherence (5), multilevel coherence (10), and relative sparsity (9) for the Hadamard-Haar systems in one and two dimensions. Note tha… view at source ↗
Figure 3
Figure 3. Figure 3: Structure of the matrix |(Φ> hadΨdhw)k,k0 | (left), |(Φ> 2hadΨadhw)k,k0 | (middle), and |(Φ> 2hadΨidhw)k,k0 | (right) for N = 8. We observe that the figure in the middle is the Kronecker product of the matrix on the left with itself. This is actually the consequence of the construction of the 2-D Hadamard matrix and ADHW basis using the Kronecker product. where (u)+ := max(u, 0). Noting that |(Hr)k,k0| = 2… view at source ↗
Figure 4
Figure 4. Figure 4: The exact local coherence values for µ loc l (Φ> hadΨdhw) (left), µ loc l (Φ> 2hadΨadhw) (middle), and µ loc l (Φ> 2hadΨidhw) (right) for N = 8, with l1 and l2 defined in Prop. 3. The values shown here are equal to the estimated values in Prop. 3. The block structure of these figures, as represented by the constant color areas fits the definition of the wavelet levels, i.e., with 1-D dyadic (left), 2-D ani… view at source ↗
Figure 5
Figure 5. Figure 5: Rearrangement of the rows and columns of the matrices shown in Fig. 3 with respect to the sampling and [PITH_FULL_IMAGE:figures/full_fig_p017_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: The exact multilevel coherence values for Hadamard-Haar systems with [PITH_FULL_IMAGE:figures/full_fig_p017_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: The reconstruction performance comparison of the proposed MDS and VDS schemes with the traditional [PITH_FULL_IMAGE:figures/full_fig_p020_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Recovering four special 1-D signals from 20% Hadamard measurements. [PITH_FULL_IMAGE:figures/full_fig_p021_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: The SRE of phantom image recovery from subsampled Hadamard measurements. [PITH_FULL_IMAGE:figures/full_fig_p022_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Global (left) and local (right) sparsity of the phantom image in 2-D Haar wavelet basis. On the right [PITH_FULL_IMAGE:figures/full_fig_p023_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: An example of the reconstructed images from 10% of the Hadamard measurements. These images [PITH_FULL_IMAGE:figures/full_fig_p024_11.png] view at source ↗
read the original abstract

We investigate the problems of 1-D and 2-D signal recovery from subsampled Hadamard measurements using Haar wavelet sparsity prior. These problems are of interest in, e.g., computational imaging applications relying on optical multiplexing or single-pixel imaging. However, the realization of such modalities is often hindered by the coherence between the Hadamard and Haar bases. The variable and multilevel density sampling strategies solve this issue by adjusting the subsampling process to the local and multilevel coherence, respectively, between the two bases; hence enabling successful signal recovery. In this work, we compute an explicit sample-complexity bound for Hadamard-Haar systems as well as uniform and non-uniform recovery guarantees; a seemingly missing result in the related literature. We explore the faithfulness of the numerical simulations to the theoretical results and show in a practically relevant instance, e.g., single-pixel camera, that the target signal can be obtained from a few Hadamard measurements.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript investigates 1-D and 2-D signal recovery from subsampled Hadamard measurements under Haar wavelet sparsity priors, motivated by applications such as single-pixel imaging. It identifies coherence between the Hadamard and Haar bases as a barrier to recovery and proposes variable-density and multilevel-density sampling strategies that adapt to local and multilevel coherence. The central contribution is the derivation of explicit sample-complexity bounds together with uniform and non-uniform recovery guarantees for the Hadamard-Haar pair; these are validated through numerical experiments that compare theoretical predictions with observed reconstruction performance.

Significance. If the coherence calculations and application of standard CS recovery theorems are rigorous, the explicit bounds constitute a concrete, previously missing result that directly informs sampling design in optical multiplexing systems. The work supplies parameter-free theoretical guidance rather than purely empirical tuning, which strengthens its utility for practical compressive imaging.

minor comments (3)
  1. The abstract states that bounds and guarantees are computed but does not indicate the precise form of the sample-complexity expression (e.g., dependence on sparsity level, dimension, or coherence parameters); placing the leading-order bound in the abstract would improve immediate accessibility.
  2. Notation for the variable-density and multilevel-density sampling operators is introduced without an explicit comparison table; a short table listing the sampling probabilities per wavelet scale would clarify the distinction between the two strategies.
  3. Several numerical figures compare empirical phase-transition curves to the derived bounds, yet the caption does not state the number of Monte-Carlo trials or the precise recovery algorithm (e.g., basis pursuit or iterative hard thresholding) used; this information belongs in the figure captions.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their supportive summary of the manuscript, recognition of its significance for compressive imaging applications, and recommendation of minor revision. No major comments appear in the report, so we have nothing to address point by point.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper derives explicit sample-complexity bounds and recovery guarantees for the Hadamard-Haar pair by computing local and multilevel coherences between the two fixed bases and applying standard compressive sensing theorems (e.g., RIP or NSP) under variable/multilevel density sampling. These steps use the known matrix properties of the Hadamard and Haar transforms without any reduction to fitted parameters, self-referential definitions, or load-bearing self-citations. The sampling densities are defined directly from the coherence quantities rather than being tuned to the target recovery result, and the bounds are presented as new explicit expressions rather than renamings or imported uniqueness results. The derivation chain remains independent of the paper's own outputs and is self-contained against external CS theory.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard compressive-sensing sparsity and coherence assumptions applied to the specific Hadamard-Haar pair; no free parameters or new entities are introduced in the abstract.

axioms (1)
  • domain assumption Signals of interest are sparse in the Haar wavelet basis
    This sparsity prior underpins both the recovery problem and the sample-complexity analysis.

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