Separable Frequency-Consistent Laplacian Pyramid

• SF-Lap is a neural module that decomposes feature maps into alias-resistant low- and high-frequency streams tailored for MRI reconstruction.
• It leverages depthwise-separable filtering to reduce computational cost while preserving spectral energy and preventing aliasing artifacts.
• Empirical evaluations show notable improvements in PSNR and SSIM, underscoring SF-Lap's effective frequency decoupling in deep neural pipelines.

A separable frequency-consistent Laplacian pyramid (SF-Lap) is a neural module for multiscale decomposition of feature maps into stable, alias-resistant low-frequency and high-frequency streams. In the context of MRI reconstruction, as in HiFi-MambaV2, it enables effective separation and targeted processing of frequency bands within unrolled, data-consistency–regularized neural architectures (Fang et al., 23 Nov 2025). The SF-Lap is distinguished from standard Laplacian pyramids by its efficient depthwise separability, frequency-consistent construction, and explicit integration into hierarchical deep image reconstruction pipelines.

1. Mathematical Formulation and Core Mechanism

The SF-Lap consists of two principal stages: depthwise-separable low-pass filtering and residual high-pass computation. Given an input $\mathbf{x}$, the low-frequency component $\mathbf{L}$ is constructed as: $$\mathbf{L} = \mathrm{Conv}_{v}^{(s=2)}\left(\mathrm{Conv}_{h}^{(s=1)}\bigl(\mathcal{P}_{\mathrm{ref}}(\mathbf{x}),\mathbf{k}\bigr), \mathbf{k}\right)$$ where:

• $\mathrm{Conv}_h$ and $\mathrm{Conv}_v$ are 1D convolutions along horizontal and vertical axes,
• $\mathbf{k} = [1, 4, 6, 4, 1]/16$ is the normalized binomial kernel,
• $\mathcal{P}_{\mathrm{ref}}$ denotes reflect padding,
• stride $s$ implements $2\times$ spatial downsampling.

The resulting $\mathbf{L}$ is upsampled (via bilinear interpolation) to match the input size: $$\tilde{\mathbf{L}} = \mathcal{U}_{\times2}(\mathbf{L})$$ The high-frequency residual is then: $\mathbf{H} = \mathbf{x} - \tilde{\mathbf{L}}$ This split yields two frequency-consistent streams per feature group, with low- and high-frequency content isolated for independent processing.

Depthwise separability, achieved by factorizing $k \times k$ filtering into 1D horizontal and vertical convolutions, reduces computational cost from $\mathcal{O}(k^2HW)$ to $\mathcal{O}(2kHW)$ for spatial support $k$ and input of shape $H \times W$. This design preserves spectral energy and avoids aliasing or checkerboard artifacts that commonly arise in naively implemented pyramids.

2. Integration in Hierarchical Neural Architectures

Within HiFi-MambaV2, SF-Lap operates as a core submodule in each HiFi-Mamba Unit, preceding frequency-specific expert processing. Input features are split into low- and high-frequency representations via SF-Lap; these are then routed (along with global context features) for downstream specialization.

A parallel lightweight squeeze-and-excitation–guided global context path processes the original input as: $$\mathbf{z} = \frac{1}{HW}\sum_{i,j}\mathbf{x}_{i,j}, \quad \mathbf{s} = \sigma(W_2\,\mathrm{ReLU}(W_1\mathbf{z})), \quad \mathbf{x}' = \mathbf{x}\odot\mathbf{s} + \mathbf{x}$$ ensuring each frequency band receives holistic anatomical cues. $\mathbf{x}'$ is injected into the SF-Lap streams, aligning frequency-specific and global information before subsequent mixture-of-experts computation.

3. Motivation and Theoretical Properties

The primary intent behind the SF-Lap design is to address the limitations of prior multistream or Laplacian-based frequency decoupling approaches in deep MRI reconstruction pipelines. Key motivations include:

• Aliasing resistance: The use of binomial kernels with reflective padding yields spectral smoothness and avoids folding artifacts at downsampling boundaries.
• Energy preservation: Each decomposition pass ensures that $\mathbf{x} = \tilde{\mathbf{L}} + \mathbf{H}$, stably partitioning the input's energy.
• Computational efficiency: Depthwise-separable filtering dramatically reduces arithmetic operations and memory, accommodating high-resolution processing in large unrolled architectures.
• Stability across depth: Frequency-consistent separation maintains cross-layer feature balance, preventing drift in spectral statistics as the model propagates through multiple unrolled stages.

4. Ablation, Empirical Analysis, and Quantitative Impact

Ablation studies on benchmarks such as CC359 (brain MR, 8× acceleration) demonstrate that incorporating SF-Lap into a dual-stream HiFi-Mamba backbone increases PSNR from 28.08 dB to 28.31 dB and SSIM from 0.802 to 0.830, compared to a dual-stream $w$-Laplacian baseline (Fang et al., 23 Nov 2025). Further integration of the lightweight global context path and hierarchical shared-routed MoE enhances performance (up to 28.68 dB PSNR, 0.841 SSIM). These results support the claim that SF-Lap delivers measurable gains in alias resistance and high-frequency fidelity.

The following table summarizes ablation results on CC359 (8×, patch size = 2):

Configuration	PSNR (dB)	SSIM	NMSE
Baseline HiFi-Mamba ($w$-Lap)	28.08	0.802	0.027
+ SF-Lap only	28.31	0.830	0.024
+ SF-Lap + LSGP	28.43	0.835	0.023
+ SF-Lap + LSGP + SR-MoE (top-1 routing)	28.68	0.841	0.022
+ Balancing loss ($\mathcal{L}_\mathrm{bal}$)	28.50	0.837	0.023

Incremental gains are directly attributable to the introduction of the SF-Lap and its interaction with cross-band context and adaptive expert routing. A plausible implication is that frequency-consistent decompositions improve signal representation especially for sparse, high-frequency structures typical in medical imaging.

5. Comparison with Prior Architectures and Efficiency

Compared to architectures employing conventional multistream decoupling, standard Laplacian pyramids, or non-separable filtering, SF-Lap exhibits notable improvements in computational efficiency and spectral stability. On fastMRI (8×), HiFi-MambaV2 with SF-Lap achieves 31.73–31.89 dB PSNR and 0.763–0.764 SSIM, outperforming CNN, Transformer, and prior Mamba-based baselines while also reducing computational costs (e.g., $172$ GFLOPs for V2(P2) versus $485$ GFLOPs for comparable prior models) (Fang et al., 23 Nov 2025). These outcomes illustrate the efficacy of SF-Lap in supporting state-of-the-art image quality and practical acceleration.

6. Broader Significance and Limitations

The SF-Lap module’s ability to yield alias-resistant, stable decoupling into low/high-frequency bands renders it a critical building block for high-fidelity image reconstruction, particularly where preservation of fine detail and anatomical coherence is essential. Its design is fundamentally integrable: it operates as a drop-in component that enables frequency-targeted specialization in hierarchical, unrolled, or mixture-of-experts architectures.

This suggests future architectures for image restoration, super-resolution, and compressive sensing could benefit from adopting similar depthwise-separable, frequency-consistent decompositions, especially in domains sensitive to spectral artifacts and with strong requirements for computational efficiency.

No controversy is noted regarding the mathematical validity of the approach; however, the primary evidence for SF-Lap’s advantages is empirical, based on observed improvements in classic MRI reconstruction metrics (PSNR, SSIM, NMSE) and reductions in aliasing. Further exploration in other imaging contexts is a plausible direction.