Wavelet-Based Sampling

Updated 30 June 2025

Wavelet-based sampling is a framework that employs wavelet transforms to capture multi-scale features for efficient signal and image reconstruction.
It utilizes generalized sampling, binary measurements, and compressed sensing to achieve stable, linear-rate recovery across diverse applications.
Applications in imaging, audio processing, and generative modeling demonstrate its practical benefits in reducing computational costs and enhancing reconstruction accuracy.

Wavelet-based sampling constitutes a broad framework in modern signal processing, imaging, generative modeling, and computational sensing, wherein sampling, analysis, or reconstruction is performed in, with respect to, or through wavelet bases or transforms. This methodology connects the multi-resolution, localization, and sparsity properties of wavelets with the needs of efficient, robust, and often adaptive data acquisition and reconstruction. The following sections provide a comprehensive account of the theory, algorithms, and practical significance of wavelet-based sampling across diverse domains.

1. Generalized Sampling Theory and Stable Sampling Rate

The generalized sampling framework formalizes reconstruction in one basis (e.g., wavelets) from samples taken in a different domain (e.g., Fourier samples). Let $\mathcal{S} = \text{span}\{s_j\}$ denote the sampling space (e.g., spanned by Fourier or Walsh functions) and $\mathcal{T} = \text{span}\{\varphi_j\}$ the reconstruction space (e.g., wavelets such as Daubechies). For $f \in \mathcal{H}$ , given $M$ samples $\langle f, s_j \rangle$ , the goal is to reconstruct the optimal $N$ -term wavelet approximation.

Generalized sampling computes the coefficients $\alpha^{[N,M]}$ by solving

$U^{[N,M]} \alpha^{[N,M]} = \hat{f}^{[M]},$

where $U^{[N,M]}_{ij} = \langle \varphi_j, s_i \rangle$ and $\hat{f}^{[M]} = (\langle f, s_1 \rangle, \ldots, \langle f, s_M \rangle)^T$ . The stability and accuracy of this procedure are governed by the stable sampling rate: $\Theta(N; \theta) = \min\left\{ M \in \mathbb{N} : C_{N,M} > \frac{1}{\theta} \right\},$ where $C_{N,M}$ is the subspace angle-based stability constant. The foundational result is that for a broad class of wavelets—including Daubechies, Haar, and their multidimensional tensor products—the stable sampling rate is linear: $M = \mathcal{O}(N)$ . This crucially means high-fidelity wavelet reconstruction is achievable from a number of samples proportional to the number of reconstructed wavelet coefficients, with the proportionality constant being precisely characterized for Daubechies wavelets as $\eta^* = 1/(\epsilon \lceil a \rceil)$ , where $a$ is the support of the wavelet and $\epsilon$ the sampling density (1208.5959).

Such generalized sampling is optimal up to a constant: no method satisfying exact recovery for functions in the reconstruction space (a "perfect method") can operate with a lower sampling ratio without causing exponential ill-conditioning and instability.

2. Binary and Structured Sampling for Wavelet Reconstruction

Beyond classical Fourier sampling, binary (e.g., Walsh-Hadamard) sampling permits similar guarantees for wavelet-based reconstruction. Binary measurements correspond to inner products against functions taking values in $\{-1,1\}$ , as realized by Walsh functions and Hadamard matrices. The critical theoretical result is that, just as with Fourier sampling, the stable sampling rate for recovering wavelet coefficients from binary samples is linear in the number of coefficients, across both one and higher spatial dimensions (1908.00185, 2106.00554). This equivalence makes binary mask-based modalities (e.g., single-pixel cameras, compressive holography, fluorescence microscopy using binary illuminations) highly practical.

Efficient recovery proceeds via either generalized sampling or compressed sensing approaches. In large-scale applications, fast algorithms leveraging the algebraic structure of the Walsh and wavelet bases are essential. For instance, the fast change-of-basis algorithm introduced by Antun achieves $\mathcal{O}(N\log N)$ cost and $\mathcal{O}(2^q)$ storage for computing matrix-vector products between the binary sampling and arbitrary wavelet reconstruction bases, thus facilitating practical deployment in medical and computational imaging systems (2106.00554).

3. Compressed Sensing, Adaptive, and Structured Sampling in the Wavelet Domain

The sparsity-inducing property of wavelets is exploited in compressed sensing (CS), where signals with sparse wavelet representations can be recovered from dramatically fewer measurements than dictated by the Nyquist rate. Measurement schemes are adapted to account for the coherence between measurement and sparsifying bases. For example, variable (VDS) and multilevel (MDS) density sampling strategies, which adapt sampling densities according to local and multi-scale coherence with wavelet atoms, enable successful recovery from Hadamard or other structured measurements even when classical random subsampling fails due to high coherence (1907.09795).

In sensor networks and applications such as early detection of vibrational rogue waves, random positioning of physical sensors directly implements such compressive sampling in the wavelet domain. The sparsity of characteristic event spectra (e.g., the triangular shape in Haar wavelet spectra for emerging rogue waves) allows accurate recovery and localization from a minimal number of samples, enabling reduced hardware, memory, and early warning capabilities (1706.01972).

Methods such as WTDUN (Wavelet Tree-Structured Sampling and Deep Unfolding Network) combine wavelet decompositions with adaptive measurement allocation. In these, the sampling budget is not uniformly assigned, but distributed across wavelet subbands according to per-band energy and sparsity metrics. Furthermore, the tree-structured sparsity among wavelet coefficients is enforced both in sampling and in group-sparsity regularization, leading to highly efficient and detail-preserving image reconstructions (2411.16336).

4. Wavelet-based Sampling in Generative Modeling and Fast Diffusion

State-of-the-art generative models—including score-based diffusion models (SGMs) and GANs—inherit practical limitations when operating in the spatial domain, such as ill-conditioning due to power-law covariance spectra, and slow convergence due to the need for many sampling (discretization) steps as image resolution increases. Recent theoretical and algorithmic work demonstrates that, by transitioning to the wavelet domain, these conditioning problems are ameliorated: the low-frequency (LL) wavelet coefficients are nearly Gaussian and well-conditioned at coarse scales, so SGMs are highly efficient; high-frequency coefficients, being sparse and non-Gaussian, are better handled by conditional adversarial models (GANs) or via hierarchical, multi-scale factorization (2208.05003, 2411.09356).

Practical frameworks such as WaveDM (Wavelet-Based Diffusion Model) and WMGM (Wavelet-based Multi-scale Generative Model) thus operate by performing diffusion sampling on only the low-frequency wavelet bands and reconstructing high-frequencies through lightweight refinement or conditional GANs. This hybrid strategy reduces sampling steps by orders of magnitude, achieving state-of-the-art image restoration or synthesis even at high resolutions and under severe computational constraints (2305.13819, 2411.09356). Similar principles underpin rapid generative models for super-resolution (e.g., nsb-GAN (2009.04433)) and efficient, generalizable NeRFs for 3D rendering (e.g., WaveNeRF (2308.04826)).

The central mathematical insight is that, in the wavelet domain, multi-scale factorization allows the score-based model to be computed efficiently with the same number of steps at each scale, making sampling complexity linear in image size rather than superlinear (2208.05003).

5. Grid-Based, Non-Uniform, and Advanced Sampling Strategies

In time-frequency analysis, notably for audio, classic wavelet sampling leads to irregular data shell structures which are not compatible with efficient matrix-style block processing. Recent approaches propose uniform, quasi-random grid-based decimation of wavelet coefficients, resulting in time-frequency coefficients arranged in block-structured matrices (2301.01640). This allows the application of standard methods (NMF, DNNs, blockwise onset detection) prevalent in STFT-based pipelines, but with the perceptual and frequency advantages of wavelets. Mathematical analysis confirms that such uniformly decimated systems serve as frames for the signal space, with bounded frame ratios ensuring invertibility and stability at moderate oversampling rates.

Continuous wavelet systems can also be stably sampled using rotated and scaled phase-space lattices, rather than canonical dyadic grids. By constructing sampling sets via rotated lattices generated by badly approximable numbers (e.g., inverse golden ratio), frame property and local covering guarantees are obtained via counting arguments and oscillation method results from coorbit theory (2307.13481). This extends the design flexibility of sampling schemes in analog or continuous applications.

6. Extensions: Scattered Data, Samplets, and Multiresolution Operators

Classical wavelet sampling presumes grid-structured data, but real-world datasets are often scattered or unstructured. Samplets generalize wavelets for scattered data by building localized, signed-measure, multiresolution bases (via cluster trees) adapted to arbitrary data sites (2503.17487). These support fast basis transforms, adaptive subsampling, matrix compression for nonlocal kernel operators, and meaningful application of sparsity-based methods (e.g., LASSO) in high-dimensional, manifold, or hybrid measurement settings.

Wavelet-based sampling Kantorovich operators and associated error analysis provide a rigorous theoretical framework for wavelet sampling and interpolation in multiresolution spaces. By employing cardinal orthogonal scaling functions and making precise use of the modulus of continuity and vanishing moment properties, they yield explicit rates of convergence for sampling operators, with proven error decay for smooth functions and quantitative assessments near sharp transitions (2506.18912).

Summary Table: Major Wavelet-Based Sampling Strategies

Application Domain	Methodology/Algorithm	Key Characteristics / Rates
Imaging (Fourier samples)	Generalized Sampling, Stable Sampling	$M = \mathcal{O}(N)$ , explicit constants
Binary/Single pixel cam	Walsh/Hadamard + Fast Wavelet Recovery	Linear rate, fast transforms, low memory
Compressed Sensing	Adaptive/Variable Density in Wavelet	$M \sim K \log N$ , coherence adapted
Generative Modeling	Multi-scale (wavelet) hybrid SGM+GAN	Fast sampling, low steps, model compression
Audio Time-Freq	Uniform grid, quasi-random decimation	Matrix-structured, stable, invertible
Scattered data	Samplets, cluster trees	Fast transforms, compression, sparsity

Concluding Remarks

Wavelet-based sampling harmonizes multi-resolution, localization, and sparsity with contemporary requirements in data acquisition, computational imaging, inverse problems, and machine learning. The theoretical framework provides nearly optimal guarantees for stability and accuracy, while practical algorithms deploy the approach in compressed sensing, robust generative modeling, and even non-grid (scattered) data contexts. Current and future work focuses on advancing adaptive, data-dependent, and high-dimensional sampling schemes that exploit the full power of the wavelet transform in an era of data-driven, computationally constrained, and structurally diverse applications.