Threshold-Based Noise Filtering Framework

• Threshold-based noise filtering frameworks are defined by using adaptive or fixed thresholds to distinguish signal-dominated components from noise.
• They employ statistical models, wavelet/frame decompositions, and probabilistic risk controls to optimize noise suppression in tasks like denoising, anomaly detection, and robust estimation.
• Adaptive and learning-based approaches iteratively refine threshold estimates to enhance performance across diverse, high-dimensional data environments.

A threshold-based noise filtering framework is a broad class of signal and data processing architectures that leverages adaptive or fixed thresholds to discriminate between signal-dominated and noise-dominated components. These frameworks enable selective suppression of unwanted noise while preserving relevant information, using principled statistical, optimization, or learning-driven rules. The threshold determination is typically grounded in probabilistic modeling, distributional analysis, risk control (e.g., Type I error), or task-driven criteria, and the resulting algorithms are deployed in denoising, robust estimation, anomaly detection, and signal separation tasks.

1. Statistical Foundations of Thresholding in Noise Filtering

Threshold-based noise filtering exploits the statistical distribution of observed data under a noise model, setting a threshold to reject data contributions likely arising from noise while retaining those likely corresponding to signal. For additive white Gaussian noise (WGN), which underpins canonical denoising scenarios, the fundamental principle is as follows:

Let $y[n] = x[n] + n[n]$, with $n[n] \sim \mathcal{N}(0,\sigma^2)$. After a (possibly redundant) linear transform (such as a wavelet or frame expansion), the transformed coefficients $w_k = c_k + z_k$ can be statistically tested:

• Under the null hypothesis $\mathcal{H}_0$, $c_k = 0$ (noise only),
• Under the alternative $\mathcal{H}_1$, $c_k \neq 0$ (signal).

The optimal threshold differentiates the two by controlling, for instance, the probability of Type I error (false positive), or by reference to an extreme-value characterization of the maximum expected noise coefficient (Bayer et al., 2023, Haltmeier et al., 2012). In redundant representations, the threshold may be universal (extreme-value-based) or adaptively local, and typically relates to the standard deviation $\sigma$ and the number of coefficients.

2. Wavelet- and Frame-Based Thresholding Frameworks

In wavelet shrinkage, threshold selection can be universal or adaptive. Universal thresholds arise via extreme-value theory, exploiting the asymptotic Gumbel law for the maximum coefficient under Gaussian noise. For a set of $N$ analysis coefficients, the threshold

$$\tau = \sigma \left( \sqrt{2 \ln N} - \frac{\ln\ln N + \ln\pi}{2\sqrt{2\ln N}} + a_{\chi,n} t_\alpha \right)$$

with $t_\alpha = -\ln\bigl[ -\ln(1-\alpha) \bigr]$, controls the exceedance probability at level $\alpha$ (Haltmeier et al., 2012). Subband-dependent and signal-adaptive rules refine this, as in SpcShrink (Bayer et al., 2023) which iteratively determines per-scale thresholds using control limits inspired by statistical process control. This method sets the threshold at each resolution as $\lambda_j = d_j s_j$, with $s_j$ as the sample standard deviation of survivors and $d_j$ tuned for a desired Type I error rate, iteratively updating thresholds until convergence. The ultimate coefficient shrinkage is applied via hard or soft rules.

The table below summarizes core elements:

Approach	Threshold Selection	Shrinkage Rule
Universal/Gumbel	Global via max noise	Soft: $S(y;\tau)$
Empirical-subband	Iterative, SPC-based	Hard/Soft, adaptive $\lambda_j$

Frame-based denoising generalizes this to redundant dictionaries, leveraging the asymptotic distribution of frame coefficients and adjusting the threshold to maintain prescribed error or confidence bounds (Haltmeier et al., 2012).

3. Adaptive and Learning-Based Threshold Estimation

Learning-based approaches can integrate threshold selection into a neural network or data-driven pipeline, estimating thresholds as latent variables conditioned on input features or task context.

• Adaptive channel-wise thresholding: FN-Net (Lv, 2022) learns per-channel thresholds $t_s$ via a side network that aggregates global or local feature statistics, using either linear or quadratic soft-thresholding functions. The threshold parameters are updated end-to-end during training via loss minimization, ensuring instance-level adaptivity.
• Descriptor-weighted adaptive filtering: The WATF framework (Yan, 2024) computes descriptor importance scores based on prototype similarity, models their distribution (empirically Gaussian), and sets the threshold as $\tau = \mu - \sigma$ (mean minus standard deviation), discarding descriptors below this value. This enables context- and task-aware filtering in few-shot learning.

These paradigms require no manual tuning and enable differentiation between sample-varying noise regimes, offering improved robustness across different data distributions and tasks.

4. Thresholding in Robust Estimation and Anomaly Detection

Threshold-based filtering is central in robust estimation frameworks such as RANSAC, where a residual threshold determines set membership to the consensus. Recent advances in threshold-robust RANSAC estimate the inlier noise scale and convert it to a threshold via

$\tau^* = c \cdot \sigma^*,$

where $c$ is chosen from the inverse CDF of the appropriate error statistic to guarantee that a pre-specified fraction of inliers is retained (Edstedt, 17 Mar 2025). The SIMFIT++ estimator addresses bias due to model-fitting and sample truncation, employing validation splits and percentile corrections to refine noise-scale (and hence threshold) estimates.

In model-based anomaly detection (e.g., for cyber-physical systems), residual-based detectors form a $\chi^2$-statistic: $z_k = r_k' \Sigma_r^{-1} r_k,$ raising an alarm if $z_k > \alpha$, with $\alpha$ chosen from the $\chi^2$ distribution to control false alarm rates (Hashemi et al., 2019). Applying frequency-selective (e.g., low-pass) filters to the residuals and re-tuning the covariance and thresholds yields improved anomaly sensitivity in noisy environments, especially for slow or persistent attacks.

5. Random Matrix and Spectral Thresholding Methods

Thresholds derived from random matrix theory exploit the eigenvalue or singular-value spectrum of covariance matrices computed from noisy data:

• For Gaussian noise, the Marčenko–Pastur law delineates a noise bulk within $[\lambda_{\min},\lambda_{\max}]$; only eigenvalues/singular values exceeding $\lambda_{\max}$ are treated as signal-bearing (Basu et al., 2010, Staats et al., 2022). Filtering is then executed via projection onto the corresponding dominant eigenspaces and/or via spike shrinkage.
• In neural network weight matrices, singular values below the fitted MP threshold are set to zero (hard removal), and large spikes are shrunk towards their hypothetical “clean” values to counteract noise-induced level repulsion (Staats et al., 2022). This method robustifies generalization in the presence of label noise, without affecting performance on clean data.

6. Applications Across Domains

Threshold-based noise filtering is a central paradigm across a diverse range of signal processing, computer vision, control, and scientific computing tasks:

• Biomedical signal denoising: SpcShrink achieves superior SNR and RMSE on plethysmography and ECG data vis-à-vis classical and Bayesian shrinkage methods, preserving physiological features while suppressing high-frequency noise (Bayer et al., 2023).
• Few-shot learning: Weighted descriptor thresholding enhances class-discriminative embedding and mitigates background noise, yielding state-of-the-art performance on benchmark datasets (Yan, 2024).
• Event-based vision: In neuromorphic observation, threshold-driven filters (e.g., KNoise, TS, EvFlow) use spatial/temporal windows and correlation statistics to discriminate sparse signals from background and hot-pixel noise. Thresholds are tuned via ROC analysis for optimal trade-off between signal retention and noise rejection (Arja et al., 2024).
• Track reconstruction in particle physics: GNN-based edge-score filtering with tiered per-layer thresholds dramatically reduces fake track rates and recovers high efficiency in high-background regimes (Jia et al., 12 Jul 2025).
• Kalman filtering in non-Gaussian environments: Threshold-judgment partitions residuals into “small” and “impulsive,” with each group’s variance updated via variational Bayes, improving estimation under heavy-tailed and heterogeneous noise (Zhang et al., 2023).

7. Methodological Trade-Offs and Limitations

Threshold choice critically determines the trade-off between noise suppression and feature retention. Universal thresholds offer theoretical guarantees but may be conservative in practice. Adaptive, data-driven approaches provide flexibility but can introduce variance or bias if not properly regularized or validated.

Key limitations and practical considerations include:

• Threshold sensitivity: Overly aggressive thresholds risk signal loss; under-regularized adaptation may underfilter noise.
• Bias in threshold estimation: Fitting on the same data used for threshold determination can result in underestimated thresholds; splitting or cross-validation mitigates this effect (Edstedt, 17 Mar 2025).
• Noise model assumptions: Methods predicated on Gaussianity or independence may degrade under heavy-tailed or correlated noise. Extensions to robust, Bayesian, or distribution-free procedures are areas of ongoing research.
• Computational scaling: Spectral approaches require eigendecomposition or SVD, which can be costly for large datasets, though randomized or incremental variants exist (Basu et al., 2010, Staats et al., 2022).
• Applicability to high redundancy or strong correlation: Standard extreme-value thresholds may fail, necessitating alternate normalization (e.g., in translation-invariant frames) (Haltmeier et al., 2012).

In summary, threshold-based noise filtering provides a unifying and robust framework for denoising and noise rejection tasks, with a diverse toolkit ranging from analytically derived thresholds to fully learned adaptivity, and a broad spectrum of state-of-the-art applications in contemporary computational science and engineering.