Multi-scale Frequency Extraction & Alignment (MFEA)

• Multi-scale Frequency Extraction and Alignment (MFEA) is a computational paradigm that decomposes signals into multi-scale spectral components, enabling precise extraction and alignment.
• It employs techniques such as FFT, wavelet decomposition, and Log-Gabor filtering to mitigate cross-scale noise and enhance feature discrimination across diverse domains.
• MFEA integrates multi-stage alignment and fusion methods in applications like time series forecasting, image segmentation, and adaptive DNN solvers to boost prediction accuracy.

Multi-scale Frequency Extraction and Alignment (MFEA) is a computational paradigm for explicitly decomposing data into multi-scale spectral components, extracting dominant or informative frequency bands, and synthesizing them via alignment mechanisms to facilitate robust prediction, segmentation, synchronization, or registration. Across domains—time series, images, deep neural representations, and signal processing—MFEA designs have been developed to overcome limitations in traditional multi-scale architectures: namely, the prevalence of cross-scale noise, spectral heterogeneity, and misaligned semantic features. Architectures employing MFEA integrate specialized modules (FFT-based selection, wavelet decomposition, Log-Gabor filtering, DCT/DFT spectral branches, frequency-guided attention, and learnable alignment units) to maximize energy capture, structural feature discrimination, and spatial/temporal consistency. MFEA is a central component in time series models ("KFS" (Wu et al., 1 Aug 2025)), medical and remote sensing image registration (Gao et al., 2023), frequency-adaptive DNN solvers (Huang et al., 2024), polyp segmentation (Xu et al., 2024), and ultrasound boundary refinement (Zhang et al., 12 Dec 2025).

1. Architectural Principles and Workflow

Canonical MFEA pipelines share a five-stage abstraction:

1. Multi-scale Decomposition: Input signals/images/features are down-sampled via pooling, pyramid generation, or encoder block-stacks (e.g., in KFS (Wu et al., 1 Aug 2025): average pooling; in PSTNet (Xu et al., 2024): multi-stage transformer outputs; in (Gao et al., 2023): Gaussian pyramids).
2. Spectral Extraction: At each scale, the signal is decomposed via FFT, DCT, wavelets (e.g., Haar), or Log-Gabor filters, yielding frequency coefficients. Dominant frequencies are identified by energy-based selection (see FreK module (Wu et al., 1 Aug 2025): $E(f)=|X(f)|^2$, select top K such that $\sum_{f\in\mathcal K}E(f)\ge\delta\sum_f E(f)$) or by orientation (WPMOM (Gao et al., 2023)).
3. Alignment Feature Embedding: Temporal/spatial indices (timestamps, semantic locations, keypoint maps) are aligned across scales using down-sampling/upsampling, embedding layers, and attention mechanisms (timestamp embedding alignment (Wu et al., 1 Aug 2025); bilinear interpolation and DCNv2 alignment (Xu et al., 2024)).
4. Representation Synthesis: Nonlinear, group-theoretic, or deep neural representations capture interactions between frequency components and aligned features (Group-Rational KAN (Wu et al., 1 Aug 2025); attention fusion (Zhang et al., 12 Dec 2025, Xu et al., 2024), spectral alignment for phase synchronization (Gao et al., 2019)).
5. Fusion and Prediction: Aligned, multi-scale features are fused (averaging, residual sum, skip connections) and projected to produce forecasts, segmentation masks, synchronized phases, or transformation matrices.

This workflow is explicitly reflected in the KFS pseudocode and block diagram (Wu et al., 1 Aug 2025), PSTNet architecture (Xu et al., 2024), FreqDINO fusion module (Zhang et al., 12 Dec 2025), invariant image matching (Gao et al., 2023), and MscaleDNN embedding adaptation (Huang et al., 2024).

2. Mathematical Formulation of Multi-scale Frequency Extraction

MFEA leverages foundational results from harmonic analysis and approximation theory to quantify energy, spectral dominance, and extraction efficacy:

• Dominant Energy Selection (Parseval-Guided):

$$E(f)=|X(f)|^2;\quad \mathcal K=\text{TopK}\{E(f)\};\; \sum_{f\in\mathcal K}E(f)\ge\delta\sum_f E(f)$$

Zero out non-dominant bins, reconstruct via inverse transform:

$$\tilde x(t)=\text{iFFT}(X(f)\cdot 1_{f\in\mathcal K})$$

Ensures most energy per Parseval’s theorem is preserved (Wu et al., 1 Aug 2025).

• Wavelet- and Log-Gabor Decomposition: At scale $s$,

$$G_{s,o}(r,\theta) = \exp\left[-\frac{(\ln(r/r_{0,s}))^2}{2\ln^2(\sigma_r)}\right] \exp\left[-\frac{(\theta-\theta_o)^2}{2\sigma_\theta^2}\right]$$

Wavelet transforms yield detail and structure coefficients per band (Gao et al., 2023, Zhang et al., 12 Dec 2025).

• Hybrid Feature Embedding in DNNs: For candidate frequency $\bm\alpha$,

$$\varphi(\bm x;\bm\alpha) = \begin{pmatrix} \bm\alpha \cdot \bm x \ \cos(\bm\alpha \cdot \bm x) \ \sin(\bm\alpha \cdot \bm x) \end{pmatrix}$$

MFEA constructs embeddings $\Phi(\bm x)$ via concatenation over scales, enabling frequency-adaptive approximation (Huang et al., 2024).

• Attention-Alignment (PSTNet, FreqDINO):

$$\mathcal{F}_{\mathrm{enh}} = \mathcal{F}_{\mathrm{spatial}} + \lambda(\mathcal{F}_{\mathrm{spatial}} \odot (\alpha\mathcal{A}_b + \beta\mathcal{A}_s))$$

$\mathcal{A}_b$: boundary attention from high-frequency detail, $\mathcal{A}_s$: structure attention from low-frequency (Zhang et al., 12 Dec 2025).

3. Alignment Strategies Across Scales and Modalities

Alignment within MFEA is domain-specific but uniformly addresses misregistration, temporal/semantic drift, and modal heterogeneity:

• Timestamp Embedding Alignment (Time Series): Downsampled timestamp vectors $T_i$ (via average pooling) are linearly embedded ($E_s^i = W_sT_i+b_s$), synchronizing every scale’s feature map indices (Wu et al., 1 Aug 2025).
• Spatial Alignment (Segmentation): PSTNet’s FSAM module applies upsampling, convolution, offset prediction (via DCNv2), and bilinear interpolation to spatially align deep features across multi-scale representations (Xu et al., 2024).
• Orientation Alignment (Image Matching): WPMOM computes the main orientation robustly via weighted fusion of Log-Gabor–based gradients across scales (Gao et al., 2023).
• Power Method Synchronization (Phase): In multi-frequency phase synchronization, multiple harmonics are aligned by harmonic retrieval (periodogram maximization) and joint spectral decomposition, ensuring consistent phase recovery even under severe noise (Gao et al., 2019).

In all cases, alignment is crucial for consistent fusion, accurate prediction, and improved interpretability.

4. Fusion, Representation, and Downstream Task Modules

The outputs of MFEA are typically subjected to further nonlinear transformations and aggregation:

• Group-Rational KAN (Time Series): Nonlinear pattern extraction performed per scale; fusion via $$FM^i = E_1^i + \text{KAN}_{\text{mix}}([E_1^i; E_s^i])$$ and inter-scale averaging (Wu et al., 1 Aug 2025).
• Boundary and Structure-Aware Fusion (Segmentation): In FreqDINO (Zhang et al., 12 Dec 2025), $\mathcal{F}_{\mathrm{enh}}$ is passed to the FGBR for boundary prototype distillation, then the MBGD decoder for multi-task mask prediction.
• Descriptor Construction (Image Matching): Combination of gradient-orientation histograms (GGLOH) and normalized, aligned descriptors is utilized for matching, consensus (FSC/RANSAC), and affine warping (Gao et al., 2023).
• Adaptive Fusion in MscaleDNNs: MFEA iteratively reconstructs input feature embeddings to target the posterior dominant frequencies, with adaptive subnetworks integrated according to empirical Fourier spectra (Huang et al., 2024).

Fusion mechanisms may include linear projections, residual connections, skip summation, attention-weighted averaging, or spectral aggregation.

5. Objective Functions, Adaptation, and Supervision

MFEA frameworks employ task-specific compound objectives, with frequency-sensitive terms to regularize energy preservation, alignment quality, and discriminative performance:

• Combined Loss (Time Series Forecasting):

$$\mathcal{L} = \alpha \mathcal{L}_F + (1-\alpha) \mathcal{L}_{\mathrm{MSE}}$$

Where $\mathcal{L}_F$ penalizes misalignment at the top-K frequencies, enforcing spectral consistency with ground truth (Wu et al., 1 Aug 2025).

• Multi-task Losses (Segmentation): Weighted sum of binary cross-entropy, Dice, and focal losses, targeting both auxiliary outputs (frequency-infused features) and primary segmentation masks (Xu et al., 2024, Zhang et al., 12 Dec 2025).
• Error-Driven Frequency Adaptation: MscaleDNNs leverage empirical Fourier analysis of predictions to update the active frequency embedding set, refining the spectral receptive fields and subnetworks (Huang et al., 2024).

Optimization is end-to-end in deep architectures; MFEA modules are trained either jointly or with frozen feature extractors, depending on the domain.

6. Domain-Specific Applications and Experimental Performance

MFEA architectures are demonstrated in diverse contexts:

• Time Series: KFS (Wu et al., 1 Aug 2025) achieves SOTA forecasting accuracy by preserving cross-scale signal energy and aligning periodic/heterogeneous patterns.
• Medical/Remote Sensing Image Registration: The PC + Log-Gabor + WPMOM pipeline (Gao et al., 2023) yields highest correct match rates and sub-pixel spatial alignment, robust to intensity, rotation, and scale variation.
• Adaptive Solvers: Frequency-adaptive MscaleDNNs (Huang et al., 2024) provide $10^2$–$10^3$–fold accuracy improvements on PDEs after iterated frequency extraction and embedding realignment.
• Semantic Segmentation: PSTNet (Xu et al., 2024) and FreqDINO (Zhang et al., 12 Dec 2025) produce significant improvements in Dice and IoU scores for polyp and ultrasound segmentation, driven by frequency-domain fusion and boundary-aligned features.

The following table summarizes representative architectures:

Domain	Spectral Extraction	Alignment Mechanism	Reference
Time Series Forecasting	FFT-based energy selection	Timestamp embedding	(Wu et al., 1 Aug 2025)
Image Registration	Log-Gabor filters, PC	WPMOM orientation fusion	(Gao et al., 2023)
Semantic Segmentation	2D-DCT, Haar wavelets	DCN/Bilinear interpolation	(Xu et al., 2024, Zhang et al., 12 Dec 2025)
DNN-based PDE Solvers	DFT/sinusoidal embedding	Frequency-adaptive update	(Huang et al., 2024)

7. Theoretical Foundations and Generalizations

MFEA is theoretically grounded in classic results from harmonic analysis (Parseval’s theorem, energy preservation), group representation theory (Peter–Weyl theorem for generalized phase synchronization (Gao et al., 2019)), and approximation theory (error bounds for high-frequency functions via multi-scale embeddings (Huang et al., 2024)). The rigorous decoupling of network size from maximal frequency (as in MscaleDNNs) and the provable energy capture in selection-based extraction (FreK, PC, Log-Gabor) are central.

In generalized synchronization, multi-frequency harmonics are leveraged simultaneously, allowing robust estimation under high noise, extending to compact Lie groups (SO(3), U(1)) (Gao et al., 2019). In segmentation and matching, the fusion of orthogonal spectral and spatial cues directly improves discriminative and localization accuracy.

Conclusion

Multi-scale Frequency Extraction and Alignment is a methodological paradigm for constructing computational architectures capable of robust spectral signal decomposition, selective feature extraction, and precise multi-scale alignment. Its instantiations span time series prediction, medical and remote sensing image segmentation, deep neural network adaptation, and phase synchronization, all exhibiting superior performance through explicitly frequency-aware, energy-preserving, and alignment-driven designs. The approach is theoretically robust, empirically validated, and extensible across domains.