Domain-Aware Multi-Threshold Filtering

• Domain-aware multi-threshold filtering is an adaptive method that adjusts local thresholds using domain-specific features and expert insights to improve segmentation accuracy.
• The approach integrates feature-adaptive interactive thresholding with spectral graph techniques, employing elastic-net regularization to compute local threshold corrections efficiently.
• Empirical results highlight enhanced noise reduction and segmentation performance across large-scale datasets, with scalability achieved through localized computations and optimized hyperparameters.

Domain-aware multi-threshold filtering refers to adaptive thresholding strategies that account for heterogeneity and local structure in high-dimensional data domains, including volumetric images and graph-structured signals. Unlike classical global thresholding—which fails in the presence of domain artifacts, noise, or fine-scale variations—domain-aware techniques leverage local features, domain knowledge, or multi-scale decompositions to learn or select threshold functions that vary spatially or across representation domains. This approach enhances sensitivity and accuracy in critical regions while maintaining computational tractability, addressing fundamental challenges in large-scale segmentation, denoising, and signal recovery.

1. Feature-Adaptive Interactive Thresholding for Large 3D Volumes

Feature-Adaptive Interactive Thresholding (FAITH) constitutes a paradigm for domain-aware multi-threshold filtering in volumetric image segmentation (Lang et al., 2022). FAITH augments classical global thresholding ($\theta_g$) with local adaptivity through supervised expert input and geometric feature extraction. A global threshold $\theta_g$, chosen by the user as suitable for most of the volume, serves as the baseline. In regions where $\theta_g$ underperforms (due to artifacts or intensity fluctuations), experts mark seed voxels. Around each seed, local subvolumes $U_j$ are extracted, and feature vectors $F(U_j) \in \mathbb{R}^d$ are computed based on geometric and intensity statistics: planeness, lineness, local mean, and local standard deviation.

The optimal local threshold $\theta^*(U_j)$ for each neighborhood is computed (e.g., via Minimum Cross-Entropy Thresholding). The desired threshold offset $\Delta\theta_j = \theta^*(U_j) - \theta_g$ becomes the target for learning. The optimization seeks a weight vector $w^* \in \mathbb{R}^d$ that linearly maps local features to offsets, yielding the adaptive threshold formula

$\theta(U) = \theta_g + w^{*\top} F(U)$

This construction keeps $\theta_g$ unchanged in well-behaved regions while adaptively modifying it in user-identified critical regions.

2. Mathematical Optimization and Algorithmic Implementations

The model fits the threshold-correction weights $w$ by minimizing an elastic-net-regularized least-squares loss subject to box constraints, ensuring validity:

$$\min_{w \in \mathbb{R}^d} \; \frac{1}{2} \|F w - \Delta\Theta\|_2^2 + \lambda \left( \frac{1-\mu}{2} \|w\|_2^2 + \mu \|w\|_1 \right) \quad \text{s.t.} \quad 0 \leq \theta_g + F_j^\top w \leq W, \; j=1\ldots M$$

where $F$ is the training feature matrix, $\Delta\Theta$ the threshold offsets, $\lambda$ controls regularization, $\mu$ the $\ell_1/\ell_2$ trade-off, and $W$ the maximum gray level.

The practical realization decomposes into two main routines:

• FAITH_Training: Given training seed neighborhoods, features, $\theta_g$, and hyperparameters, solve the box-constrained elastic-net QP for $w^*$ using a forward–backward (proximal gradient) scheme with projection (e.g., Hildreth’s method) and soft-thresholding for the $\ell_1$ penalty.
• FAITH_Segment: For all voxels, extract the local neighborhood, compute features, evaluate $\theta(U)$ as above, and binarize accordingly. The operation is embarrassingly parallel over voxels and only requires local memory.

Empirically, FAITH exhibits linear scaling in the number of voxels $N$ and small memory footprint, as it avoids global graph constructions or dense matrices.

3. Local Feature Choices and Hyperparameter Effects

The local feature extractor $F$ can combine arbitrary geometric and intensity features. In published exemplars, a low-dimensional subset—planeness and lineness derived from the structure tensor’s eigenvalues—sufficed to capture critical local structure, but the method supports any $d$-vector. Neighborhood size $K$ tunes the scale of context (larger $K$ smooths thresholds, smaller $K$ captures finer detail). The number of features $d$ and the number of seeds $M$ control, respectively, modeling power and training constraint diversity; both factors trade off accuracy, overfitting, and computational burden. Regularization parameters $\lambda$ (strength) and $\mu$ ($\ell_1/\ell_2$ balance) directly affect sparsity and generalization.

Grid-search over hyperparameters with cross-validation on the annotated seed regions is recommended, with regularization sweeping $\lambda$ on $[10^{-3}\ldots10^3] \cdot \lambda_{\max}(\mu)$.

4. Graph Signal Processing: Data-Driven Adaptive Multi-Thresholding

In signal processing on graphs, domain-aware multi-threshold filtering is instantiated by data-driven threshold selection in spectral graph wavelet domains (Loynes et al., 2019). The semi-orthogonal Spectral Graph Wavelet Transform (SGWT) provides multi-scale representations of signals on graphs $G = (V, E, W)$. Denoising is formulated as multivariate thresholding in the (redundant) SGWT domain according to the mean squared error (MSE), for which Stein’s unbiased risk estimator (SURE) is derived, taking into account redundancy-induced noise correlations.

Thresholds can be optimized globally, per scale, or in blocks. For coordinatewise thresholding, parameterized threshold functions $\tau_\beta(x; t)$ encompassing soft-threshold (LASSO), James–Stein, and hard-threshold families are used. SURE is evaluated in closed form for each, enabling selection of optimal thresholds at each scale by minimization. Block-thresholding, where coefficient blocks (by scale or spatial partition) share thresholds, offers further flexibility but requires more complex optimization.

Key empirical findings include:

• Level-dependent, James–Stein-style coordinatewise thresholds selected by SURE outperform global or non-adaptive schemes by $\sim$1–4 dB SNR, depending on task and graph.
• Block-thresholding captures clustered structure but does not surpass finely scale-dependent coordinatewise schemes unless block geometry is well-matched to signal features.
• Extensions exist to correlated noise models and scale with the SGWT redundancy.

5. Computational Scalability and Practical Implementations

Both the FAITH and SGWT+SURE frameworks prioritize scalability. FAITH’s training cost is $O(Md)$ per iteration (with $M$ seeds, $d$ features), independent of data volume, and segmentation is $O(Nd)$ after preprocessing features, where $N$ is the total voxel count. Memory requirements during inference are minimal: only $w^*$ and local features need to be stored.

SGWT+SURE’s main bottleneck is the computation of spectral filters; scalable implementations leverage fast Chebyshev polynomial approximations of $g_j(L)$ for large graphs, alleviating the need for full eigendecomposition.

6. Application Domains and Representative Results

FAITH is well-suited for industrial computed tomography and other large-scale 3D imaging, demonstrated on 200 MiB and 4 GiB datasets with 50–160 seeds and $d=2$ features. Application to CT scans of a Peruvian mummy head and wolf jaw showed that FAITH filled in missing thin bone structures undetected by global thresholding, without introducing noise in well-controlled regions. For these datasets, run-times were $\sim$1800 s (200 MiB) and $\sim$3150 s (4 GiB) on a commodity Intel i7 CPU, confirming linear scaling.

SGWT+SURE has been benchmarked on the Minnesota roads, Facebook, and Pittsburgh Census-Tract graphs, with gains of up to 10–15 dB SNR over classical methods in high-noise or structured-signal regimes, and practical run-times orders of magnitude faster than graph trend-filtering for large graphs.

7. Insights, Limitations, and Future Perspectives

Domain-aware multi-threshold filtering leverages local domain structure and/or learned adaptation to overcome the limitations of global rules in heterogeneous data environments. Insights include:

• Local expert knowledge (e.g., seed voxels) and geometric feature descriptors are critical for robust segmentation in FAITH (Lang et al., 2022).
• Redundancy-aware threshold selection (via SURE) in SGWT enables practical multi-thresholding for signals on graphs, with robust gains confirmed across datasets (Loynes et al., 2019).
• Level-dependent (multi-scale) thresholds outperform single global thresholds at minimal extra cost.
• Over-parameterization (excess features or insufficient regularization) risks noisy or unstable thresholds, underscoring the need for targeted hyperparameter selection and validation.
• Scalability is achieved through strictly local computation: neither approach relies on global graphs or dense matrix manipulation in runtime segmentation.

A plausible implication is that further cross-fertilization between the volumetric and graph-based domains—for example, incorporating graph representations of 3D volumes or spatially-coupled thresholding strategies—could extend the reach and robustness of domain-aware threshold filtering. Block and scale-dependent thresholding, as well as extensions to correlated noise and block-sparsity, remain active areas of methodological development.