Fourier-Reorganized State Mamba Network

• FRSM is a state-space architecture that reorganizes Mamba-style selective propagation via Fourier-domain operations to incorporate spectral priors.
• It adapts through varied realizations—such as zigzag Fourier scanning, wavelet–Fourier decomposition, and spectral modulation—to address challenges in image restoration, forecasting, and EEG decoding.
• Empirical results across domains show improved performance metrics, underscoring the practical benefits of frequency-aware state reorganization.

Fourier-Reorganized State Mamba Network (FRSM) denotes, in the provided literature, a family of state-space architectures in which Mamba-style selective state propagation is reorganized or conditioned by frequency-domain structure rather than by spatial or temporal order alone. The term appears explicitly in CFSPMNet, where FRSM reorganizes trial token states in the Fourier domain and uses Fourier-derived context to guide Mamba propagation for cross-subject MI-EEG decoding (Wang et al., 11 May 2026). In parallel, several 2024–2025 works use the same design logic under different names or as explicit reinterpretations: FourierMamba performs Mamba scanning in Fourier space for deraining (Li et al., 2024), WalMaFa combines channel-wise Mamba over wavelet low-frequency structure with Fourier adjustment for low-light enhancement (Tan et al., 2024), FLDmamba filters Mamba’s step size $\Delta$ in the Fourier domain and reconstructs transients through an inverse Laplace transform (Zhang et al., 17 Jul 2025), FR-Mamba couples Mamba with Fourier Neural Operators for physical field reconstruction (Long et al., 21 May 2025), and Vim-F injects FFT amplitude spectra into visual Mamba tokens before scanning (Zhang et al., 2024). This makes FRSM less a single canonical block than a recurring architectural principle: spectral reorganization is used to reshape what the state-space model sees, how it propagates, or how its outputs are fused.

1. Terminology and scope

A central point is terminological. FRSM is not presented in the provided sources as a single universally fixed architecture. Instead, the term is explicit in CFSPMNet (Wang et al., 11 May 2026), while other works present FRSM as an interpretive abstraction or technical reframing. FourierMamba is explicitly summarized as a “Fourier-Reorganized State Mamba Network” in the provided overview, with zigzag-based frequency ordering in spatial Fourier space and direct Mamba scanning in channel Fourier space (Li et al., 2024). WalMaFa is similarly connected to a generalized FRSM formulation in which wavelet low-frequency processing and Fourier adjustment are separated into complementary state and frequency routes (Tan et al., 2024).

Two design motifs recur across these systems. The first is a selective state-space backbone, usually Mamba or a closely related SSM, used because its scan is linear in sequence length and can model long-range dependencies. The second is an explicit frequency-domain operation that reorganizes representation before, during, or after the state update. Depending on the task, that reorganization may take the form of zigzag ordering of Fourier coefficients, amplitude–phase refinement, learned spectral filtering of the discretization step size, Fourier-conditioned gating, or direct additive fusion of amplitude spectra into spatial tokens (Li et al., 2024, Tan et al., 2024, Zhang et al., 17 Jul 2025, Zhang et al., 2024).

The scope is correspondingly broad. In the provided material, FRSM-like designs appear in image deraining, low-light image enhancement, remote-sensing super-resolution, image generation, long-horizon forecasting, physical field reconstruction, anomaly detection, and EEG decoding (Zhen et al., 2024, Xiao et al., 2024, Phung et al., 2024, Long et al., 21 May 2025, Wang et al., 19 Nov 2025, Wang et al., 11 May 2026). This suggests that FRSM is best understood as a cross-domain synthesis of spectral priors with state-space sequence modeling rather than as a task-specific module.

2. Core mathematical operators

The state-space component is shared across nearly all variants. A standard discrete SSM is written as

$$h_{t+1} = \bar{A} h_t + \bar{B} u_t, \qquad y_t = C h_t + D u_t,$$

with Mamba-style selective scan making effective parameters input-dependent while preserving linear-time processing (Tan et al., 2024, Li et al., 2024). In FLDmamba, this mechanism is made frequency-aware by filtering the step size $\Delta$ in the Fourier domain before discretization, yielding

$$\bar{A}_F = \exp(\Delta_F A), \qquad \bar{B}_F = \Delta_F A^{-1}\exp(\Delta_F A)\Delta_F B,$$

so periodic structure is injected directly into the state update (Zhang et al., 17 Jul 2025).

The Fourier reorganization itself is not unique. In WalMaFa’s Fast Fourier Adjustment Block, a latent feature map is transformed channel-wise to Fourier space, amplitude and phase are adjusted by learned $1\times1$ convolutional heads, and the feature is reconstructed by inverse FFT:

$$X'_c(u,v) = |X|'_c(u,v)e^{j\angle X'_c(u,v)}, \qquad F'_c = \mathcal{F}^{-1}\{X'_c\}.$$

The same overview also gives an equivalent form

$$X' = (M_A \odot |X|)\cdot \exp\!\big(j(\angle X + \Delta \phi)\big),$$

emphasizing amplitude reweighting and phase correction (Tan et al., 2024). In CFSPMNet, by contrast, the reorganization acts on token states: normalized tokens are transformed by rFFT along the token axis, passed through a learned complex mixer with SoftShrink sparsification,

$$\hat{\Omega}^{(\ell)} = \mathrm{SoftShrink}_{\tau_f}\!\big(\mathcal{M}_f(\Omega^{(\ell)})\big)\odot W_f + \Omega^{(\ell)},$$

and then returned to token space by iFFT, producing an enhanced sequence and a Fourier-derived trial context used to scale and bias Mamba inputs (Wang et al., 11 May 2026).

FourierMamba adds a further variant: it does not merely transform features to frequency space, but explicitly imposes a low-to-high or high-low-high order on spatial Fourier coefficients through zigzag coding, then scans these sequences with Mamba (Li et al., 2024). Vim-F takes the opposite extreme in simplicity: it computes the per-channel 2D FFT amplitude spectrum and adds it back to the feature map,

$$F^{\text{fused}}_c(i,j)=\alpha F_c(i,j)+\beta A_c(i,j),$$

with no iFFT and no phase processing, so the spatial scan operates on tokens already carrying global spectral context (Zhang et al., 2024).

3. Architectural realizations

Architecturally, FRSM-like systems can be divided into several recurring realizations.

A first realization is frequency-space scanning. FourierMamba’s Fourier Spatial Interaction SSM performs FFT, separates amplitude and phase, reorders half-spectrum coefficients with progressive and bilateral zigzag scans, processes the resulting real sequences with a Mamba stack, reconstructs the full spectrum by symmetry, and applies iFFT back to the spatial domain (Li et al., 2024). Its Fourier Channel Evolution SSM applies global average pooling, a 1D FFT along channels, direct Mamba scanning over the ordered channel spectrum, and inverse FFT, using the result as a channel-attention vector. Here, Fourier reorganization is literally the sequence construction step.

A second realization is staged wavelet–Fourier decomposition. WalMaFa uses Haar DWT to split features into $LL$, $$h_{t+1} = \bar{A} h_t + \bar{B} u_t, \qquad y_t = C h_t + D u_t,$$0, $$h_{t+1} = \bar{A} h_t + \bar{B} u_t, \qquad y_t = C h_t + D u_t,$$1, and $$h_{t+1} = \bar{A} h_t + \bar{B} u_t, \qquad y_t = C h_t + D u_t,$$2 bands inside its Wavelet-based Mamba Block. The $$h_{t+1} = \bar{A} h_t + \bar{B} u_t, \qquad y_t = C h_t + D u_t,$$3 branch is processed by channel-wise Mamba to capture global brightness dependencies, while high-frequency subbands receive local $$h_{t+1} = \bar{A} h_t + \bar{B} u_t, \qquad y_t = C h_t + D u_t,$$4 convolutions. In the latent space, the Fast Fourier Adjustment Block refines local texture and ambiguity through learned amplitude and phase correction (Tan et al., 2024). FreqMamba similarly uses a triple interaction structure consisting of spatial Mamba, frequency-band Mamba over wavelet packet sub-bands, and a Fourier global modeling branch that refines amplitude and phase before iFFT (Zhen et al., 2024).

A third realization is dual-branch spatial–spectral fusion. In FMSR, the Frequency Selection Module performs FFT-based spectral analysis and learned selection, the Vision State Space Module performs 2D selective scan, and the Hybrid Gate Module injects spatially varying local bias, with learnable scaling adaptors recalibrating fusion (Xiao et al., 2024). FR-Mamba applies the same logic at a different scale: a Mamba branch captures long-range temporal dependencies, an FNO branch captures non-local spatial features, and a gating mechanism fuses them before field reconstruction (Long et al., 21 May 2025). DiMSUM, while wavelet-based rather than Fourier-based, is explicitly used in the provided material to motivate an FRSM variant in which Fourier features would replace wavelet subbands and be fused with spatial Mamba through query-swapped cross-attention and a globally shared transformer (Phung et al., 2024).

A fourth realization is spectrally modulated state evolution. FLDmamba is the clearest example. It smooths the input with an RBF kernel, applies FFT to Mamba’s selective step size $$h_{t+1} = \bar{A} h_t + \bar{B} u_t, \qquad y_t = C h_t + D u_t,$$5, multiplies by a learnable spectral kernel $$h_{t+1} = \bar{A} h_t + \bar{B} u_t, \qquad y_t = C h_t + D u_t,$$6, applies iFFT to obtain $$h_{t+1} = \bar{A} h_t + \bar{B} u_t, \qquad y_t = C h_t + D u_t,$$7, and uses $$h_{t+1} = \bar{A} h_t + \bar{B} u_t, \qquad y_t = C h_t + D u_t,$$8 to discretize the SSM. A parallel standard Mamba path is fused with this Fourier-powered FMamba path, and an inverse Laplace transform head reconstructs damped oscillatory components (Zhang et al., 17 Jul 2025). In this design, the reorganization does not reorder tokens; it alters the internal timekeeper of the state transition.

4. Representative systems and application domains

The provided papers instantiate FRSM-like ideas in markedly different regimes, which clarifies the breadth of the concept.

System	Domain	Key FRSM-style mechanism
FourierMamba	Image deraining	Zigzag-reordered Fourier scanning
WalMaFa	Low-light enhancement	Wavelet Mamba + Fourier adjustment
FLDmamba	Time-series forecasting	Fourier-filtered $$h_{t+1} = \bar{A} h_t + \bar{B} u_t, \qquad y_t = C h_t + D u_t,$$9 + ILT
CFSPMNet FRSM	Stroke MI-EEG decoding	Token rFFT reorganization + context-guided Mamba
FR-Mamba	Physical field reconstruction	Mamba temporal branch + FNO spatial branch
Vim-F	Visual backbone	FFT amplitude fused into tokens

In image restoration, the frequency route is tied to degradation structure. FourierMamba argues that rain streak removal benefits from explicitly correlating Fourier coefficients, because prior Fourier-based methods often process frequency bins independently (Li et al., 2024). WalMaFa separates wavelet low-frequency brightness enhancement from Fourier phase-aware texture recovery, reflecting a division between global illumination and local detail (Tan et al., 2024). FreqMamba goes further by combining spatial Mamba, wavelet-packet frequency-band Mamba, and Fourier global modeling in a single deraining backbone (Zhen et al., 2024).

In visual representation learning and generation, the motivation is order awareness. Vim-F adds amplitude spectra to spatial features so that flattened visual tokens carry global context before Mamba scanning (Zhang et al., 2024). DiMSUM shows that a state-space diffusion backbone can benefit from explicit spatial–frequency fusion, and the provided FRSM-style extension replaces wavelet subbands with Fourier-domain bands and reorganizes tokens by radial and angular frequency (Phung et al., 2024).

In temporal modeling, the operative object of reorganization changes. FLDmamba reorganizes the dynamics of the selective scan itself by Fourier-filtering $\Delta$0 and then reconstructs transient dynamics with an inverse Laplace transform (Zhang et al., 17 Jul 2025). CFSPMNet reorganizes token states rather than raw signals, then derives low/high token-frequency context to modulate Mamba propagation in cross-subject EEG decoding (Wang et al., 11 May 2026). FR-Mamba uses Fourier Neural Operators for non-local spatial encoding while Mamba handles temporal evolution, making FRSM-like fusion effective for long-horizon physical field reconstruction (Long et al., 21 May 2025).

5. Empirical evidence

The empirical record in the provided material supports the utility of FRSM-style coupling, although the evidence is task-specific rather than directly comparable across domains.

For deraining, FourierMamba reports PSNR/SSIM on the Y channel of 31.79 dB / 0.913 on Rain100H, 39.73 dB / 0.986 on Rain100L, 34.23 dB / 0.949 on Test2800, and 34.76 dB / 0.938 on Test1200, with 17.62 M parameters and 22.56 GFLOPs (Li et al., 2024). Its ablations show that removing FSI-SSM or FCE-SSM reduces Rain100L performance to 39.05 dB / 0.9835 and 39.08 dB / 0.9836, respectively, while classic 2D scan in spatial Fourier space gives 38.82 dB / 0.9817 and the combined zigzag strategy reaches 39.73 dB / 0.9856. FreqMamba reports 31.74/0.912 on Rain100H, 39.18/0.981 on Rain100L, 34.25/0.951 on Test2800, and 33.36/0.931 on Test1200, with 14.52M parameters and 36.49 GFlops; its ablations report a $\Delta$1 dB PSNR drop when removing the Fourier branch and a $\Delta$2 dB drop when removing the frequency-band branch (Zhen et al., 2024).

For low-light enhancement, WalMaFa reports 23.27 dB / 0.851 on LOL-v1, 22.49 dB / 0.869 on LOL-v2-real, and 25.56 dB / 0.945 on LOL-v2-syn, with 11.09M parameters, 14.41G FLOPs, and 0.068 s per image (Tan et al., 2024). The reported ablation on LOL-v2-syn gives 23.56/0.912 for only FFAB, 24.12/0.928 for only WMB, 22.47/0.888 for FFAB–WMB–FFAB, and 25.56/0.945 for the full model. This is important because it shows that state-space brightness modeling and Fourier detail adjustment contribute differently and are not interchangeable.

For long-horizon forecasting, FLDmamba is reported to attain state-of-the-art accuracy in the majority of settings, specifically “60/72 or 83.3%,” and in long-lookback ETTh1 testing with $\Delta$3 it reports MSE 0.664 versus 0.715, 0.787, 1.281, and 0.687 for S-Mamba, iTransformer, RLinear, and Autoformer (Zhang et al., 17 Jul 2025). Under injected noise on ETTm1, it is reported to exhibit “about 2.1× higher tolerance than iTransformer.” These claims matter because the FRSM mechanism here acts on $\Delta$4, not on explicit token reordering, yet still yields measurable gains.

For EEG decoding, CFSPMNet reports average accuracies of 68.23% on XW-Stroke and 73.33% on 2019-Stroke, with gains of 5.63 and 8.25 percentage points over the strongest competitors (Wang et al., 11 May 2026). Its FRSM-specific ablations are unusually direct: removing Fourier context guidance yields $\Delta$5 and $\Delta$6 percentage points; removing the High-Frequency branch yields $\Delta$7 and $\Delta$8; removing the Low-Frequency branch yields $\Delta$9 and $$\bar{A}_F = \exp(\Delta_F A), \qquad \bar{B}_F = \Delta_F A^{-1}\exp(\Delta_F A)\Delta_F B,$$0. This is among the clearest explicit demonstrations that Fourier-domain reorganization and context-conditioned Mamba propagation both matter.

For physical field reconstruction, FR-Mamba reports average MAE $$\bar{A}_F = \exp(\Delta_F A), \qquad \bar{B}_F = \Delta_F A^{-1}\exp(\Delta_F A)\Delta_F B,$$1 and average Max-AE $$\bar{A}_F = \exp(\Delta_F A), \qquad \bar{B}_F = \Delta_F A^{-1}\exp(\Delta_F A)\Delta_F B,$$2 on five test intervals for cylinder-wake reconstruction, compared with average MAE 0.91 for PFR-Transformer and 1.69 for Voronoi UNet (Long et al., 21 May 2025). Its ablation without FNO1d and FNO2d gives MAE 0.397 and Max-AE 0.355, with FNO1d only giving 0.315 and 0.125, and the full model giving 0.310 and 0.108.

For visual backbones, Vim-F(H) improves ImageNet-1k Top-1 from 74.7% for reproduced Vim-Ti to 76.0%, and from 79.6% for reproduced Vim-S to 80.4%, while long-sequence fine-tuning yields 76.8% for Vim-Ti-F(H) and 81.2% for Vim-S-F(H) (Zhang et al., 2024). On COCO with Mask R-CNN, Vim-Ti-F(H) reports $$\bar{A}_F = \exp(\Delta_F A), \qquad \bar{B}_F = \Delta_F A^{-1}\exp(\Delta_F A)\Delta_F B,$$3 37.7 and $$\bar{A}_F = \exp(\Delta_F A), \qquad \bar{B}_F = \Delta_F A^{-1}\exp(\Delta_F A)\Delta_F B,$$4 35.7 versus 36.6 and 34.9 for Vim-Ti.

6. Misconceptions, limitations, and research directions

A common misconception is that FRSM always means “run Mamba after FFT.” The provided literature does not support that simplification. In FourierMamba, the Fourier plane itself is scanned after zigzag reordering (Li et al., 2024). In WalMaFa, Mamba is applied to the wavelet $$\bar{A}_F = \exp(\Delta_F A), \qquad \bar{B}_F = \Delta_F A^{-1}\exp(\Delta_F A)\Delta_F B,$$5 branch, while Fourier operations appear only in the latent texture-refinement stage (Tan et al., 2024). In FLDmamba, the spectral operation modulates the step size $$\bar{A}_F = \exp(\Delta_F A), \qquad \bar{B}_F = \Delta_F A^{-1}\exp(\Delta_F A)\Delta_F B,$$6 rather than the input tokens (Zhang et al., 17 Jul 2025). In Vim-F, there is no inverse FFT and no phase processing; only amplitude is added to spatial features before scanning (Zhang et al., 2024). FRSM therefore refers more broadly to Fourier-informed reorganization of state-space computation.

A second misconception is that Fourier reorganization is uniformly preferable to other frequency decompositions. The provided evidence is mixed. WalMaFa explicitly argues that wavelet low-frequency information is more sensitive to global brightness, while Fourier phase is more sensitive to local details, leading to a staged wavelet–Fourier design rather than a purely Fourier one (Tan et al., 2024). DiMSUM reports that alternative frequency transforms, specifically “DCT with JPEG scanning” and “EinFFT,” underperform wavelet in its ablations, even though the provided overview uses DiMSUM as a basis for an FRSM-style Fourier variant (Phung et al., 2024). This suggests that the optimal spectral parameterization is task-dependent.

The limitations are similarly domain-specific. FourierMamba notes that extremely heavy rain, mist/fog veiling, or non-streak artifacts may require additional priors beyond amplitude manipulations (Li et al., 2024). WalMaFa identifies severe sensor noise, nonuniform illumination, local overexposure, and the possible suboptimality of Haar for certain textures (Tan et al., 2024). FLDmamba notes decomposition sensitivity, the non-ideal separation of periodic and transient components, and the added $$\bar{A}_F = \exp(\Delta_F A), \qquad \bar{B}_F = \Delta_F A^{-1}\exp(\Delta_F A)\Delta_F B,$$7 FFT overhead (Zhang et al., 17 Jul 2025). CFSPMNet highlights artifact susceptibility and sensitivity to the Fourier partition ratio $$\bar{A}_F = \exp(\Delta_F A), \qquad \bar{B}_F = \Delta_F A^{-1}\exp(\Delta_F A)\Delta_F B,$$8 and shrinkage threshold $$\bar{A}_F = \exp(\Delta_F A), \qquad \bar{B}_F = \Delta_F A^{-1}\exp(\Delta_F A)\Delta_F B,$$9 (Wang et al., 11 May 2026). FR-Mamba does not use physics-informed constraints, and domain-shift tests across Reynolds numbers are not reported (Long et al., 21 May 2025). Vim-F itself leaves open whether amplitude-only fusion is sufficient for tasks requiring phase-sensitive detail (Zhang et al., 2024).

The research directions in the provided material are consistent. Several overviews propose more explicit spectral gating, learnable band-pass filters, complex-valued processing, adaptive scan order, or multi-resolution frequency pyramids (Xiao et al., 2024, Zhen et al., 2024, Wang et al., 19 Nov 2025). CFSPMNet suggests that Fourier-domain token-state reorganization can serve as a latent neural-state model for clinical EEG, implying that FRSM may be particularly useful where hidden state organization shifts across subjects or domains (Wang et al., 11 May 2026). A plausible implication is that the most mature future form of FRSM will not be a single monolithic block, but a toolkit of frequency-aware state reorganizations—token reordering, spectral gating, amplitude–phase modulation, and state-discretization control—selected according to the structure of the task rather than according to a fixed recipe.