Bridging the Geometry Mismatch: Frequency-Aware Anisotropic Serialization for Thin-Structure SSMs¶

Conference: ECCV2026
Paper: https://eccv.ecva.net/virtual/2026/poster/5356
PDF: https://media.eventhosts.cc/Conferences/ECCV2026/pdfs/10140.pdf
Code: https://github.com/kbaijin/FGOS-Net
Area: Segmentation
Keywords: Thin-structure segmentation, state-space models, frequency-aligned scanning, active probes, topology preservation

TL;DR¶

FGOS-Net separates low-frequency topology modeling from high-frequency detail filtering, using deterministic sub-band-matched scanning and topology-conditioned active probes to reduce crack fragmentation and texture false positives; it achieves 91.29% mIoU and 97.12% clDice on DeepCrack, with 80.2 FPS under the paper's specified RTX 3090 evaluation setting.

Background & Motivation¶

Cracks, vessels, and roads are difficult not merely because foreground pixels are scarce, but because small local errors can disrupt an entire structure: a short missing crack segment can sever a connected path, while background texture incorporated into a prediction creates false branches. State-space models (SSMs) propagate long-range information efficiently, but visual models usually flatten two-dimensional features into sequences using a fixed raster order. For thin, curved, strongly directional targets, neighboring points along the structure need not be neighbors in the sequence. The state repeatedly crosses the structure instead of following it, so long-range modeling does not necessarily translate into connectivity.

Adding more scanning directions does not solve the entire problem. Directional details contain both crack boundaries and patterns from water stains, oil stains, and materials. Unconditionally mixing these high-frequency responses into semantic features can produce sharper but incorrect edges. Discarding high frequencies instead erases narrow cracks. The paper therefore separates two questions: in what order should structural information enter the SSM, and which details have sufficient structural support to be restored?

Haar decomposition serves as an interface between frequency and geometry, rather than Haar wavelets or Hilbert curves being claimed as novel individually. Low frequencies carry relatively stable topology cues; high frequencies first help correct positions and are then filtered under topology guidance. Core idea: establish a reliable low-frequency topology carrier, select serialization paths by sub-band orientation, and use probes moving along topology to validate high-frequency details, making both long-range propagation and detail injection support thin-structure connectivity.

Method¶

Overall Architecture¶

After convolutional feature extraction, an input image passes through a four-stage FGOS encoder and ultimately produces a pixel-level segmentation mask. Within each stage, “Frequency Disentanglement and Alignment” produces a calibrated low-frequency carrier and the original high-frequency bands. “Frequency-Aligned Scanning” processes the low-frequency branch, while a gate from “Active Spectral-Geometric Probing” filters the high-frequency branch. Inverse wavelet reconstruction combines the branches before subsequent stages. The four stage outputs then enter “Multi-Scale Fusion and Boundary Refinement” to combine scales in parallel and refine boundaries.

There are two levels of wavelet decomposition, not a single interchangeable operation. The outer decomposition separates the topology carrier from details awaiting filtering; another decomposition inside the low-frequency branch supplies the sub-bands used to select FA-Scan paths. ASGP generates its gate from the calibrated low-frequency carrier, rather than waiting for the final FA-Scan output before processing high frequencies. The diagram summarizes the dual-branch relationship within a stage and the subsequent decoder.

%%{init: {'flowchart': {'rankSpacing': 24, 'nodeSpacing': 28, 'padding': 6, 'wrappingWidth': 400}}}%%
flowchart TD
    A["Current-stage input features"] --> B["Frequency Disentanglement and Alignment"]
    B -->|Calibrated low frequencies| C["Frequency-Aligned Scanning"]
    B -->|Calibrated low and original high frequencies| D["Active Spectral-Geometric Probing"]
    C --> E["Inverse wavelet reconstruction<br/>Continue encoding and retain stage outputs"]
    D --> E
    E --> F["Multi-Scale Fusion and Boundary Refinement"]
    F --> G["Segmentation mask"]

Key Designs¶

1. Frequency Disentanglement and Alignment: use details for localization before mixing in texture

An outer Haar discrete wavelet transform (DWT) separates multi-channel features into low-frequency LL and three directional high-frequency sub-bands: LH, HL, and HH. LL is treated as a relatively stable topology carrier, not as an already correct skeleton label. High frequencies retain local changes but contain both structure and noise. The operation acts on convolutional features rather than raw pixels, so some structural information has already been encoded across channels. Haar's compact support keeps computation inexpensive, while the DWT/IDWT pair is exactly invertible and supports spatial reconstruction within each stage.

Before LL enters the long-range model, the network concatenates the outer high-frequency bands, predicts a two-dimensional offset field with two convolutional layers, and spatially resamples LL through GridSample. High frequencies provide cues about possible boundary displacement rather than being directly added as semantic content. The correction addresses feature misregistration from strided downsampling and local aggregation, not Haar reconstruction error; this distinction also matters when interpreting the alignment ablation. The calibrated LL then supports both scanning and active probing, giving the two branches a shared spatial reference.

2. Frequency-Aligned Scanning: match sub-band orientation instead of learning per-pixel routing

The FA-Block applies another local Haar decomposition only to the calibrated LL. Internal LH uses horizontal scanning, HL uses vertical scanning, and LL and HH use a locality-preserving Hilbert curve. A shared SSM operator processes each serialized sequence. The corresponding inverse indexing restores two-dimensional positions, and an internal IDWT reconstructs the low-frequency branch output. Unlike applying the same four-directional scan to every sub-band, this allocates computation to traversals matched to sub-band orientation rather than adding branches to cover every possibility.

The geometric condition here is the sub-band type. Paths are deterministic, precomputable indices, not dynamic curves tracing the crack centerline in the current image. Horizontal and vertical bands keep responses in their respective directions more contiguous in one dimension, while Hilbert traversal supports two-dimensional locality for components without a single dominant direction. Complementary sub-bands combine during reconstruction to represent oblique structures. This is an inductive bias for thin structures, not exact alignment with every curved trajectory.

After scanning, the LightGate Bottleneck (LGB) provides lightweight channel mixing. It compresses channels to one quarter, applies a depthwise convolution, restores the channel count, and adds the result through a gated residual connection. The two shallow stages use Efficient Channel Attention (ECA), and the two deeper stages use grouped Squeeze-and-Excitation, forming the EEGG policy. Its role is to constrain model size and filter responses by stage, not to introduce another sequence model alongside the scanning mechanism.

3. Active Spectral-Geometric Probing: move probes along topology to reject unsupported high frequencies

Active Spectral-Geometric Probing (ASGP) begins with learnable initial probes querying the calibrated LL. Their attention responses are aggregated into a coarse potential field \(M_0\). High values indicate likely structure, but a static map can miss narrow, weak regions or concentrate responses on the most visible crack segment. The probes therefore move progressively over the coarse field instead of stopping after one query, producing finer structural support.

Each update combines three signals: a semantic offset predicted by an MLP from local low-frequency features, the coarse field's gradient with respect to probe coordinates, and a truncated repulsion force between neighboring probes. The gradient moves probes toward structural responses; repulsion reduces the tendency for all probes to gather on the same thick crack segment. The core update can be typeset consistently from Algorithm 1 as:

\[ \mathbf c_i^{(t+1)}=\operatorname{Clamp}_{[-1,1]}\!\left(\mathbf c_i^{(t)}+\Delta\mathbf c_{\mathrm{sem},i}^{(t)}+\lambda_1\nabla_{\mathbf c_i}M_0(\mathbf c_i^{(t)})+\lambda_2\mathbf F_{\mathrm{div},i}^{(t)}\right). \]

Coordinates are normalized. Repulsion acts only when probes are closer than its radius, pushes them apart, and decays as their separation approaches that radius. The default uses 64 probes and 3 evolution steps, a radius of 0.15, and respective weights of 0.1 and 0.05. This does not separately solve for a ground-truth skeleton during training; it embeds differentiable sampling and geometric constraints into feature processing. If the coarse field misses a structure, subsequent probing can still be limited.

After evolution, probe states are projected back into pixel space to produce a fine map \(M_1\). The coarse and fine maps are combined with equal weights and passed through a sigmoid to form a gate, which multiplies the outer high-frequency bands spatially. Low frequencies thus determine where details deserve preservation, while high frequencies supply the boundary details at those locations. The gate filters responses without providing a strict mathematical connectivity guarantee. Finally, the processed LL and gated high frequencies are recombined through the outer IDWT.

4. Multi-Scale Fusion and Boundary Refinement: avoid repeatedly smoothing narrow structures during fusion

The decoder's Gated Feature Aggregation (GFA) projects all four encoder outputs into a shared channel space and upsamples them to the shallowest stage's spatial size. Each scale independently computes a channel-calibration gate before one-shot summation, rather than passing features progressively through a pyramid with repeated interpolation. Deeper features supply structural semantics, while shallow features retain finer localization. Independent calibration reduces imbalances in feature strength and semantics across scales.

The fused representation then passes through the Boundary Refinement Module (BRM). Its context branch uses depthwise-separable convolutions, while its edge branch uses depthwise convolution. Their outputs are concatenated, projected, and added residually to the fused features. GFA handles cross-scale coordination; BRM compensates for boundary softening after upsampling. Both support the encoder's geometric modeling. The paper explicitly pairs VMamba with the same decoder to test whether improvements come only from decoder enhancements.

Loss & Training¶

Training uses the sum of binary cross-entropy and Dice losses, without an additional clDice topology loss; clDice is an evaluation metric here. Models follow official splits and train for 100 epochs using AdamW, an initial learning rate of \(10^{-4}\), and cosine decay. Inputs are resized to \(256\times256\), with random flips, \(\pm15^\circ\) rotations, and photometric distortion.

The intended topology bias therefore resides in feature modeling, scan order, and high-frequency filtering rather than only in output supervision. The main text refers to supplementary material for additional implementation details, runtime breakdowns, and multi-seed checks. Those supplementary details are absent from the current cache, so neither training nor efficiency results can be claimed to have been reproduced.

Key Experimental Results¶

Main Results¶

The table selects mIoU and clDice from the paper's Table 1. Values are percentages, and higher is better. Each baseline is the highest reported value for that dataset and metric, so the two metrics need not share a baseline model. Gains are percentage points, not relative percentages. mIoU measures average region intersection-over-union. clDice is an established skeleton-connectivity metric: its standard interpretation is the harmonic mean of the proportion of predicted skeleton inside the ground-truth region and the proportion of ground-truth skeleton covered by the predicted region. The paper does not introduce a new metric.

Dataset	Metric	FGOS-Net	Best baseline in the table	Gain (percentage points)
DeepCrack	mIoU ↑	91.29	SCSegamba 90.71	+0.58
DeepCrack	clDice ↑	97.12	CT-CrackSeg 96.46	+0.66
Crack500	mIoU ↑	79.15	SimCrack 78.76	+0.39
Crack500	clDice ↑	83.85	CT-CrackSeg 82.98	+0.87
CrackMap	mIoU ↑	80.75	CT-CrackSeg 79.93	+0.82
CrackMap	clDice ↑	92.48	SimCrack 91.80	+0.68
TUT	mIoU ↑	85.73	FFM 84.62	+1.11
TUT	clDice ↑	91.98	CT-CrackSeg 91.31	+0.67

Efficiency comes from Table 2: a single \(256\times256\) input, batch size 1, RTX 3090, and averages over 100 runs, with ASGP iterations included in the full inference path. FGOS-Net has 6.26M parameters, 7.87 GFLOPs, and 80.2 FPS; SCSegamba has 3.05M parameters, 18.16 GFLOPs, and 17.9 FPS. FGOS-Net is therefore not the smallest model by parameter count. GLCP reaches 202.6 FPS, also showing that the lowest FLOPs do not imply the highest throughput.

Table 4 additionally reports mIoU/clDice of 80.45/83.35 on CHASEDB1 and 79.83/80.86 on Massachusetts Roads. The text states that architecture, hyperparameters, and input size remain unchanged, but does not explicitly say in that passage whether crack-trained weights are transferred directly. These results should therefore be treated as evidence of applicability across tasks, not as established zero-shot cross-domain transfer.

Ablation Study¶

The following results come from Tables 3C–D on DeepCrack; both metrics are percentages, and higher is better. The first four rows diagnose scan paths, while the last four diagnose system components. The groups answer different questions and should not be interpreted as one sequence of incrementally added modules.

Diagnostic group	Config	mIoU ↑	ODS ↑	Interpretation
Scan paths	Aligned FA-Scan	90.46	89.73	Hilbert for LL/HH; horizontal/vertical matching for LH/HL
Scan paths	Swap LH/HL directions	88.74	87.96	Same number of paths, mismatched orientations
Scan paths	Hilbert for every sub-band	89.46	88.66	Locality preservation is not sub-band orientation matching
Scan paths	Raster for LL/HH	89.12	88.51	Directional bands still match horizontal/vertical paths
System components	Static gate, coarse map only	88.67	88.08	No active evolution
System components	ASGP without FA-Scan	89.83	89.63	Active evolution compared with static gating
System components	FGOS-Net without alignment	90.75	91.35	Remove high-frequency-guided low-frequency resampling
System components	Full FGOS-Net	91.29	92.46	Full model

ODS is reported as a boundary-related metric. It conventionally refers to the detection score obtained with a single optimal threshold across the dataset; the main text does not specify implementation details such as boundary tolerance. Its original column name and values are retained here, without treating ODS as synonymous with fixed-threshold F1.

Key Findings¶

Swapping LH/HL scanning directions reduces mIoU from 90.46 to 88.74, a loss of 1.72 percentage points. Degradation with the number of scan paths unchanged supports the importance of orientation matching, rather than merely having more branches.
Comparing static gating with active ASGP raises mIoU from 88.67 to 89.83 and ODS from 88.08 to 89.63, gains of 1.16 and 1.55 percentage points. Removing spatial alignment from the full model costs 0.54 percentage points of mIoU and 1.11 percentage points of ODS, indicating a separate role for geometric calibration.
In Table 3B, holding the probe count at 64 yields 91.29% mIoU with 3 evolution steps, but only 88.30% with 7 steps. More iterations do not consistently help. This agrees with the explanation that excessive probe concentration can impair spatial coverage, although it remains a mechanistic interpretation rather than a proof.

Highlights & Insights¶

Serialization order becomes part of task modeling rather than incidental data preparation. Sharing SSM weights and changing only indexing order introduces a directional prior without requiring a large parameter increase.
High frequencies both support spatial calibration and undergo low-frequency topology validation, but these roles are not collapsed into direct fusion. This separation preserves detail while reducing texture contamination, providing a more specific mechanism than simply adding wavelets.
Ablations test alternatives such as orientation swapping, uniform Hilbert scanning, and static gating rather than only module presence. They help bound the conclusion: frequency–geometry coupling matters, and gains cannot be attributed entirely to branch count or decoder enhancement.

Limitations & Future Work¶

The authors explicitly describe the fixed sub-band-to-path assignment as a bounded inductive bias for thin structures, not a universal routing rule. Their stated future direction is more adaptive geometry-conditioned serialization while retaining the efficiency of deterministic paths.
ASGP depends on the coarse low-frequency potential field supporting target locations. If weak structure has already disappeared in low frequencies, moving probes and gating details do not guarantee recovery. Degradation with too many evolution steps also suggests hyperparameter sensitivity.
Some Precision, Recall, and F1 entries in the main table cannot be directly related through a single confusion matrix, and the main text does not fully specify mIoU classes and aggregation. This note reports original table values without silently correcting them; reproduction still requires checking implementations, thresholds, and aggregation conventions.
The current cache lacks supplementary material, preventing independent inspection of multi-seed stability, ASGP runtime breakdowns, and high-resolution scaling. The code URL is the paper's announced release location; its public availability was not checked online.

vs VMamba / generic Cross-Scan: Generic multidirectional scanning repeats similar traversal patterns across bands, whereas FA-Scan selects deterministic paths by sub-band type. The former has broader applicability; the latter ties computation to a structural prior and consequently has a narrower task scope.
vs GLCP: GLCP preserves connectivity through skeleton and local-discontinuity learning; FGOS-Net primarily changes feature alignment, sequence propagation, and detail filtering. GLCP is faster in the reported efficiency table, while FGOS-Net has higher mIoU and clDice on the listed crack datasets. Neither is uniformly superior on every axis.
vs ordinary wavelet fusion and deformable sampling: Ordinary wavelet fusion can directly inject high-frequency texture; ASGP filters it through a low-frequency potential field. Compared with deformable mechanisms based solely on learned offsets, it additionally uses potential-field gradients and local repulsion. The distinction lies in conditioning and constraints, not in introducing wavelets, attention, or offset sampling for the first time.

Rating¶

Novelty: 4/5. Coupling sub-band-conditioned serialization with active detail filtering targets a clear task-specific problem; the underlying wavelet and scanning tools are established.
Experimental Thoroughness: 4/5. Multiple datasets, efficiency measurements, and mechanistic controls are provided, but metric conventions and cross-domain training details remain insufficiently specified in the available main text.
Writing Quality: 3/5. The central argument is clear and claims about novelty are restrained, but incomplete metric explanations and damaged equation extraction increase verification effort.
Value: 4/5. The work provides concrete architectural guidance for real-time thin-structure segmentation, with deployment conclusions limited to the reported hardware and input size.