Wavelet-Based Targets

• Wavelet-based targets are feature representations using wavelet transforms to capture localized, multi-scale information for detecting and classifying patterns.
• They leverage properties like sparsity, translation invariance, and noise robustness to enhance performance in applications such as radar, biomedical monitoring, and imaging.
• Implementations utilize various transforms including DWT, CWT, and scattering methods to yield improved detection accuracy and computational efficiency.

Wavelet-based targets refer to the use of wavelet-domain feature representations or reconstruction objectives specifically designed for the detection, identification, classification, or analysis of localized patterns (“targets”) amid complex data and noise. These approaches leverage the time-scale or space-scale localization, invariance, sparsity, and multi-resolution decomposition properties of wavelet transforms to extract features that are simultaneously robust and interpretable. Wavelet-based targets are widely employed in radar/sonar detection, biomedical monitoring, financial signal analysis, imaging science, and machine learning. Their formal definition, structure, and implementation depend on task, modality, and specific wavelet basis and decomposition strategy.

1. Mathematical Formulation of Wavelet-Based Targets

Wavelet-based targets derive directly from the wavelet transform, either discrete (DWT), continuous (CWT), scattering, or dual-tree complex variants. The generic form involves dilated and shifted copies of a mother wavelet $\psi$. For a 1D signal $x(t)$, the CWT is

$$W_x(a, b) = \int_{-\infty}^{\infty} x(t)\, \frac{1}{\sqrt{|a|}}\, \psi^*\left(\frac{t-b}{a}\right)\, dt,$$

where $a$ is scale, $b$ is translation, and $\psi$ is the wavelet (Ni et al., 2023, Zavanelli, 2023, Medaiyese et al., 2021, Hirn et al., 2019). DWTs are constructed via inner products with dyadic translations and scales: $$d_{j, k} = \sum_n x[n]\, \psi_{j,k}[n],\quad \psi_{j,k}[n]=2^{-j/2}\psi(2^{-j} n - k).$$ Scattering transforms build from cascaded WTs and modulus nonlinearities to compute translation-invariant, deformation-stable targets: $$S_0 f = f * \phi_J;\quad S_1(\lambda_1) = |f * \psi_{\lambda_1}| * \phi_J; \quad S_2(\lambda_1, \lambda_2) = ||f * \psi_{\lambda_1}| * \psi_{\lambda_2}| * \phi_J$$ (Jia et al., 21 Oct 2025, Medaiyese et al., 2021). For image data, 2D tensor-product bases or dual-tree complex wavelets are used.

Target features may be defined as:

• Localized maxima in the WT modulus, corresponding to isolated events or generator functions (Element Analysis) (Zavanelli, 2023)
• Scale/band-specific energy or statistical moments, i.e., $\text{cont}_j = \sum_k |d_{j,k}|^2$ and $$\text{rel}_j = \text{cont}_j / \sum_i \text{cont}_i$$ (Wagner et al., 2016)
• Full space–frequency maps (scalograms, scattergrams) for input to deep neural networks (Medaiyese et al., 2021, Ni et al., 2023, Jing et al., 2 Aug 2025)
• Multi-scale multi-frequency coefficient tuples for reconstruction in self-supervised learning (Xiang et al., 2 Mar 2025)

2. Target Construction and Feature Extraction Pipelines

The operational definition of a wavelet-based target depends on application specifics and transform domain. Examples include:

• Neurological Monitoring: Signal windows from wrist-worn IMUs are DWT-processed (Daubechies basis); energy distributions at each scale/axis form relative and absolute targets for SVM classifiers of tremor, dyskinesia, and bradykinesia (Wagner et al., 2016).
• Radar and RF Device Detection: CWT with Morse or Morlet wavelets maps time-frequency signatures of radar echoes or RF bursts; coefficients or scalogram images form the input for CNNs (e.g. RepVGG, SqueezeNet) or as compact feature vectors for ensemble classifiers; wavelet scattering features further enhance translational and deformation invariance (Ni et al., 2023, Medaiyese et al., 2021, Jia et al., 21 Oct 2025).
• Infrared Small Target Detection: HWConv uses two-level Haar DWTs inside convolutional blocks, extracting low-frequency (target energy) and directional high-frequency (edge) features; these are fused and enhanced with spatial/channel-wise attention mechanisms in end-to-end networks (Jing et al., 2 Aug 2025).
• Imaging and Masked Modeling: Multi-level DWT decomposes images for use as distinct reconstruction targets at aligned neural network layers (WaMIM), enabling frequency-localized, semantically meaningful learning objectives (Xiang et al., 2 Mar 2025).
• Finance and Adaptive Analysis: Element Analysis detects isolated wavelet maxima associated with statistically significant events, estimating amplitude, location, and scale parameters for direct, sparse reconstruction; Morse wavelet family is typically used (Zavanelli, 2023).
• Electro-sensing and Imaging: Measurements are projected onto multidimensional wavelet bases; sparse operator recovery via $\ell^1$ minimization identifies support regions strongly associated with target boundaries/shapes (Ammari et al., 2013, Afeyan et al., 2012).

A summary of deployment strategies is shown below:

Domain	Wavelet Target	Feature Structure
IMU symptom monitoring	DWT scale–axis energies	cont$_j$, rel$_j$, SVM input
Sea/airborne radar	CWT/CWS scalograms, coefficients	2D time–scale images, deep features
IR small target imaging	HWConv multi-scale convolution outputs	spatial + frequency attention fusion
Masked image modeling	Multi-level DWT coefficient targets	Layer-wise, scale-aligned objectives
RF device identification	CWT/WST scalograms & coefficients	Classic ML and CNN input features
Time series event analysis	Maxima of Morse CWT	Direct parameter extraction
ICF and medical imaging	(Undecimated) DWT, curvelets	Shell radii, denoised structure

3. Statistical and Computational Properties

Wavelet-based targets are often preferred for their invariance properties and noise robustness. First-order wavelet invariants,

$S[f](\lambda) = \|f*\psi_\lambda\|_{L^2}^2,$

are exactly translation-invariant (Hirn et al., 2019). Scattering transforms inherit stability to local translations and additive noise, with controlled Lipschitz constants (Jia et al., 21 Oct 2025, Medaiyese et al., 2021). Many target constructions are naturally sparse (e.g., element analysis, band-diagonal sparse shape matrices (Ammari et al., 2013)), facilitating robust recovery in high-noise or limited-data environments.

Bias correction—e.g., via moment expansion for random dilations—is feasible using higher wavelet-invariant derivatives (Hirn et al., 2019). Convex or $\ell^1$-regularized optimization (FISTA, projected gradient, or joint wavelet–curvelet functionals (Afeyan et al., 2012)) efficiently recovers interpretable sparse targets.

Empirical results underline substantial performance gains, e.g., improvements in detection probability $P_d$ from 0.772 (STFT) to 0.954 (CWT) in radar (Ni et al., 2023), nearly 99% multi-group identification accuracy for UAVs using scattering features (Medaiyese et al., 2021), and >5 points mIoU improvement in IR small target imaging versus conventional U-Nets (Jing et al., 2 Aug 2025). For masked image modeling, WaMIM matches or betters pixel- and Fourier-based objectives with up to 90% lower computational cost (Xiang et al., 2 Mar 2025).

4. Comparison to Alternative and Traditional Methods

Compared to Fourier or fixed-resolution windowed approaches:

• Wavelet-based targets provide true multi-resolution capability: broad support and fine time resolution at high-frequency—well-suited for transients, abrupt clutter, and local features (Ni et al., 2023, Medaiyese et al., 2021).
• Power spectrum invariants from wavelet transforms permit noise/dilation unbiasing not available to global power spectrum estimation (Hirn et al., 2019).
• Curvelets extend wavelet targets for imaging scenarios with curved, anisotropic edges, significantly outperforming decimated or undecimated wavelets in ICF implosion symmetry characterization (Afeyan et al., 2012).
• Wavelet scatterings, in the form used in AWSPNet, supply more noise-robust, translation-invariant descriptors than raw CNN features alone, particularly in low SNR jamming environments (Jia et al., 21 Oct 2025).

Empirical studies generally report that the inclusion of wavelet-based targets or feature extraction front-ends yields higher accuracy, greater robustness, and more interpretable outputs compared to non-wavelet alternatives.

5. Applications Across Modalities

Biomedical and Wearable Sensing: Automated extraction of symptom signatures for Parkinson’s disease via DWT-based energy ratios per axis and scale allows for real-time, low-power patient monitoring (Wagner et al., 2016).

Radar, Sonar, and RF Sensing: CWT, wavelet scattering, and dual-tree complex wavelets serve as core feature extractors for robust target detection in sea clutter, airborne jamming, and UAV identification, leveraging their resilience to translation, wideband interference, and signal deformation (Ni et al., 2023, Medaiyese et al., 2021, Jia et al., 21 Oct 2025).

Computer Vision and Imaging: Wavelet decomposition provides the multiscale targets needed for efficient masked modeling, small target discrimination, and shape reconstruction; joint wavelet–curvelet denoising has been foundational for fusion and medical imaging (Xiang et al., 2 Mar 2025, Afeyan et al., 2012, Ammari et al., 2013).

Financial Time Series and Sparse Event Analysis: Element Analysis reconstructs signals from sparse maximal CWT responses, directly targeting events corresponding to real economic drivers and rejecting noise-induced fluctuations (Zavanelli, 2023).

Electro-sensing and Inverse Problems: Wavelet-based feature matrices and $\ell^1$ reconstruction exploit localization and sparsity to “read off” geometrical features (e.g., object boundaries) with superresolution even in low SNR regimes (Ammari et al., 2013).

6. Limitations and Future Directions

Limitations include sensitivity to wavelet basis selection (e.g., Haar vs. Morse for optimal trade-off between localization and shape matching (Zavanelli, 2023, Jing et al., 2 Aug 2025)), resolution dependence on discretization, and potential omission of features not well captured by separable wavelets (mitigated by directional or curvelet extensions) (Afeyan et al., 2012). Further, as noted in (Wagner et al., 2016), single-limb wearable wavelet-based sensing may miss symptoms localized elsewhere.

Future research directions encompass:

• Adaptive or learned wavelet dictionaries, hybridizing analytic and learned features (Medaiyese et al., 2021).
• Joint multi-domain objectives (e.g., cross-domain fusion with attention or masking (Jing et al., 2 Aug 2025, Xiang et al., 2 Mar 2025)).
• Enhanced real-time deployments leveraging re-parameterization and platform-efficient wavelet front-ends (Ni et al., 2023).
• Extension to graph and hypergraph domains, higher-order (beyond scattering order-2), and domain-adaptive transfer via fixed or recalibrated scattering layers (Jia et al., 21 Oct 2025).
• Robust phase retrieval and inverse modeling leveraging wavelet-invariant statistics, especially for non-stationary, non-Gaussian data (Hirn et al., 2019).

7. Core Theoretical Foundations and Statistical Guarantees

Wavelet-based targets inherit strong theoretical properties under mild conditions:

• Uniqueness: First-order wavelet invariants suffice to uniquely determine power spectra under linear independence of spectral envelopes (Hirn et al., 2019).
• Robustness: Statistically unbiased estimation is possible under translation, dilation, and additive noise with appropriate correction; sample complexity $M=O(\max\{\sigma^2, \eta^{-2}\})$ underpins high-SNR and moderate-warp applications.
• Feature localization: Band-diagonal structure and localization properties imply superresolution not achievable with direct pixel- or time-domain representations (Ammari et al., 2013, Zavanelli, 2023).
• Sparse representation: By mapping data to low-cardinality, interpretable generator sets or feature tuples, wavelet-based targets support efficient downstream learning, inference, and interpretability (Zavanelli, 2023, Xiang et al., 2 Mar 2025, Wagner et al., 2016).

Across domains, the formalism of wavelet-based targets underlies numerous advances in robust detection, compressive modeling, and interpretable machine intelligence.