Multi-Scan 2D: Concepts & Applications

• Multi-Scan 2D (MS2D) is a design idiom that processes 2D data via multiple coordinated scan paths, enhancing fusion, consistency, and robustness.
• It underpins diverse applications, from multi-target tracking and image segmentation in medical imaging to geometric scheduling and industrial inspections.
• By decomposing complex multi-scan assignments into tractable 2D subproblems, MS2D improves computational efficiency and accuracy.

Multi-Scan 2D (MS2D) denotes a family of two-dimensional multi-scan procedures rather than a single universally standardized method. The term is explicit in recent Vision Mamba work, where it names a parameter-free module that varies 2D patch-serialization paths for Mamba-based segmentation (Hardan et al., 15 Jul 2025), and it is also used as a concise exposition term for earlier methods in which two-dimensional data are processed through multiple scan directions, multiple scan windows, or multiple coupled 2D subproblems, including trajectory PMBM/MBM filtering (Xia et al., 2019), geometric scan scheduling in the plane (Fekete et al., 2020), and multi directional scanning for BGA void detection (Ahuja et al., 2019). This suggests that MS2D is best understood as a recurring computational pattern: a 2D object, measurement field, or assignment structure is explored through several coordinated scans, with explicit mechanisms for fusion, consistency, or smoothing.

1. Terminological scope

Across the literature, MS2D appears in several technically distinct but structurally related settings. In some papers it is the author’s explicit terminology; in others it is an interpretive shorthand used to organize a multi-scan 2D procedure. The central commonality is not a single model class, but repeated use of multiple 2D traversals, assignments, or scan coordinates to improve coverage, continuity, or robustness (Ji, 2024).

Domain	Meaning of MS2D	Representative source
Multi-target tracking	Multi-frame association solved as coupled 2D assignments in a sliding window	(Xia et al., 2019)
Labeled RFS smoothing	Multi-scan posterior recursion with per-scan 2D assignment structure	(Vo et al., 2018)
Vision Mamba	Multiple 2D scan routes or patch-order permutations for selective scan	(Ji, 2024, Hardan et al., 15 Jul 2025)
Geometric scheduling	Minimum scan cover in the plane with angular transition costs	(Fekete et al., 2020)
Industrial X-ray inspection	Multi directional scanning, especially polar radial/angular scans	(Ahuja et al., 2019)
Physical acquisition	Two-dimensional beam steering or repeated delay scans for 2D reconstruction	(Patole et al., 2013, Thomas et al., 2023)

A neighboring usage appears in 2D scan matching for Graph-SLAM, where the paper itself does not name the method MS2D, but the exposition explicitly connects it to multi-scan 2D registration through weighted hypotheses in $SE(2)$ and clustering over roto-translations (Iavicoli et al., 2022). That extension is interpretive rather than terminologically canonical.

2. Multi-scan data association and random-finite-set tracking

In trajectory PMBM/MBM filtering, MS2D denotes the multi-frame data association problem solved over a sliding window of length $N$ by decomposing a high-dimensional assignment into a sequence of coupled 2D assignment subproblems, one per scan (Xia et al., 2019). The tracker maintains a track table $T$, each track $i \in T$ has a set of single-trajectory hypotheses $H^i$, and each hypothesis $a^i \in H^i$ carries a weight $w^{i,a^i}$, density $f^{i,a^i}(\cdot)$, and measurement-association history $M(i,a^i) \subseteq M_k$. The best global hypothesis is obtained by minimizing the total log-weight cost,

$$\min \sum_{i\in T}\sum_{a^i\in H^i} c^{i,a^i}\rho^{i,a^i},$$

subject to one-hypothesis-per-track and per-scan measurement-exclusivity constraints, with $N$0 and binary decision variables $N$1 (Xia et al., 2019).

The computational step identified with MS2D is the dual-decomposition reduction of this multi-frame assignment into $N$2 coupled subproblems. For each scan $N$3, the subproblem retains only the per-track constraint $N$4 and the single-scan exclusivity constraint $N$5, so it becomes equivalent to a 2D assignment between measurements at scan $N$6 and tracks or dummies in the window. Lagrange multipliers $N$7 enforce agreement across scans, the projected subgradient is

$N$8

and the step size is

$N$9

with stopping based on the relative primal-dual gap (Xia et al., 2019). In implementation, each scan-level subproblem is solved by a standard assignment algorithm such as Hungarian or Auction, while primal recovery repairs the residual cross-scan conflicts.

The tracking formulation is tied to trajectory-valued RFS models. A single-trajectory Bernoulli has

$T$0

the detected-trajectory component is an MBM, and PMBM augments it with a PPP over undetected trajectories. The paper also defines a multi-scan trajectory $T$1 filter, in which the existence probabilities of all Bernoulli components are either $T$2 or $T$3 (Xia et al., 2019). Track-oriented $T$4-scan pruning fixes associations more than $T$5 scans old, shrinks the hypothesis trees entering the next MS2D cycle, and keeps the multi-frame assignment dimension bounded. The simulation study reports that the multi-scan trajectory PMBM showed the best trade-off, with the lowest state/trajectory error and fastest runtime in the challenging proximity scenario, and that the overall MS2D-based trajectory PMBM/MBM framework achieved superior track continuity compared to labelled RFS baselines such as $T$6-GLMB and LMB (Xia et al., 2019).

A closely related construction appears in the multi-scan GLMB posterior recursion, where each scan contributes a 2D track-measurement assignment, but the retained object is the entire association history $T$7 rather than only the current scan’s hypothesis (Vo et al., 2018). The posterior is propagated without marginalizing past states, so smoothing is performed while filtering. In this setting, the paper states explicitly that, unlike the GLMB filter, finding the significant components of a multi-scan GLMB filter is an NP-hard multi-dimensional assignment problem, and proposes Gibbs sampling over association histories to make the multi-scan posterior tractable (Vo et al., 2018). The conceptual link to MS2D is direct: per-scan assignments remain 2D, but the full estimator is multi-scan and history-consistent.

3. Scan-order design in Vision Mamba and medical image segmentation

In Vision Mamba, MS2D refers to constructing several 1D sequences from the same 2D feature map by varying the scan route before applying a linear-time selective scan (Ji, 2024). The MHS-VM formulation projects the embedding into multiple lower-dimensional subspaces,

$T$8

defines $T$9 scan routes $i \in T$0 over the $i \in T$1 grid for each head, and applies a head-specific Mamba block

$i \in T$2

The route-wise outputs are rearranged back to 2D embedding sections and fused. The route-attentive fusion, described as Scan Route Attention, uses the “CV-guided Scaling” rule

$i \in T$3

with

$i \in T$4

and

$i \in T$5

The module instantiates distinct patterns, including zigzag, consecutive horizontal or vertical scan, diagonal scan, and spiral scan, while preserving linear complexity in the sequence length $i \in T$6 (Ji, 2024).

The reported ablations show that, on ISIC18, CV-guided Scaling yields $i \in T$7 mIoU and $i \in T$8 DSC with $i \in T$9M parameters and $H^i$0G FLOPs, and that removing the tail projection reduces the model to $H^i$1M parameters and $H^i$2G FLOPs while maintaining $H^i$3 mIoU and $H^i$4 DSC (Ji, 2024). Relative to VM-UNet’s $H^i$5M parameters and $H^i$6G FLOPs, the proposed MHS-UNet variant is $H^i$7 fewer params and $H^i$8 fewer FLOPs (Ji, 2024). In this strand, MS2D is a multi-route selective-scan design inside a Vision Mamba backbone.

A later MRI study makes MS2D an explicit module name and treats scan order as the primary experimental variable (Hardan et al., 15 Jul 2025). There, MS2D is a parameter-free scan module that replaces VMamba’s default selective-scan-2D unit with a catalogue of patch-order permutations. The unit always runs exactly four parallel Mamba streams; users pick $H^i$9 unique paths, with duplication when $a^i \in H^i$0, so parameters, FLOPs, and memory remain unchanged (Hardan et al., 15 Jul 2025). The benchmark evaluates $a^i \in H^i$1 scan strategies on BraTS 2020, ISLES 2022, and LGG, covering over $a^i \in H^i$2 slices, and reports that scan order is a statistically significant factor by a Friedman test with $a^i \in H^i$3, with performance varying by as much as $a^i \in H^i$4 Dice points (Hardan et al., 15 Jul 2025). The best and worst cases include BraTS 2020 at $a^i \in H^i$5 versus $a^i \in H^i$6, LGG at $a^i \in H^i$7 versus $a^i \in H^i$8, and ISLES 2022 at $a^i \in H^i$9 versus $w^{i,a^i}$0 (Hardan et al., 15 Jul 2025).

The study formalizes the ordering effect through the SSM recurrence

$w^{i,a^i}$1

where the permutation $w^{i,a^i}$2 determines which spatial neighbors become adjacent in sequence space (Hardan et al., 15 Jul 2025). Spatially contiguous paths, especially simple horizontal and vertical rasters, consistently outperform disjointed diagonal scans. The paper recommends the four-direction orthogonal mix $w^{i,a^i}$3 as a robust choice and identifies the single horizontal raster $w^{i,a^i}$4 as a strong single-path baseline (Hardan et al., 15 Jul 2025). In this context, MS2D is neither an attention mechanism nor a higher-order optimizer, but a controlled scan-order bank for 2D-to-1D serialization.

4. Geometric scheduling in the plane

In computational geometry, MS2D denotes the two-dimensional minimum scan cover problem with angular transition costs (Fekete et al., 2020). A graph $w^{i,a^i}$5 is embedded in $w^{i,a^i}$6, and an edge $w^{i,a^i}$7 is scanned only when the two endpoints face each other. Changing the heading of a vertex by a turn angle incurs a time proportional to that angle, and a scan cover is an assignment $w^{i,a^i}$8 satisfying

$w^{i,a^i}$9

for every vertex $f^{i,a^i}(\cdot)$0 and incident edges $f^{i,a^i}(\cdot)$1 and $f^{i,a^i}(\cdot)$2, where $f^{i,a^i}(\cdot)$3 is the smaller angle between the segments $f^{i,a^i}(\cdot)$4 and $f^{i,a^i}(\cdot)$5 at $f^{i,a^i}(\cdot)$6 (Fekete et al., 2020). The makespan is $f^{i,a^i}(\cdot)$7, and the objective is to minimize $f^{i,a^i}(\cdot)$8.

The two-dimensional theory is organized around graph coloring and cut covers. The set of edges scanned at any fixed time induces a bipartite subgraph, the minimum cut-cover number satisfies

$f^{i,a^i}(\cdot)$9

and the paper proves that any schedule of length $M(i,a^i) \subseteq M_k$0 in $M(i,a^i) \subseteq M_k$1 yields a cut cover of size at most $M(i,a^i) \subseteq M_k$2 (Fekete et al., 2020). Equivalently,

$M(i,a^i) \subseteq M_k$3

Combined with constructive upper bounds, this yields the asymptotic statement that, for instances in $M(i,a^i) \subseteq M_k$4D and $M(i,a^i) \subseteq M_k$5D, the minimum scan time lies in $M(i,a^i) \subseteq M_k$6 (Fekete et al., 2020).

The approximation and hardness landscape is unusually sharp. In $M(i,a^i) \subseteq M_k$7D, it is NP-hard to approximate a minimum scan cover within less than a factor of $M(i,a^i) \subseteq M_k$8, even for bipartite graphs (Fekete et al., 2020). Conversely, the paper gives a $M(i,a^i) \subseteq M_k$9-approximation for MSC on bipartite graphs in the plane, based on sector partitioning and two synchronized sweep phases, and also proves absolute guarantees of $$\min \sum_{i\in T}\sum_{a^i\in H^i} c^{i,a^i}\rho^{i,a^i},$$0 for arbitrary bipartite graphs and $$\min \sum_{i\in T}\sum_{a^i\in H^i} c^{i,a^i}\rho^{i,a^i},$$1 when the bipartition classes are separated by a line (Fekete et al., 2020). More generally, for $$\min \sum_{i\in T}\sum_{a^i\in H^i} c^{i,a^i}\rho^{i,a^i},$$2-colored graphs with $$\min \sum_{i\in T}\sum_{a^i\in H^i} c^{i,a^i}\rho^{i,a^i},$$3, there is an $$\min \sum_{i\in T}\sum_{a^i\in H^i} c^{i,a^i}\rho^{i,a^i},$$4-approximation; for complete graphs on $$\min \sum_{i\in T}\sum_{a^i\in H^i} c^{i,a^i}\rho^{i,a^i},$$5 vertices in $$\min \sum_{i\in T}\sum_{a^i\in H^i} c^{i,a^i}\rho^{i,a^i},$$6D, a refined construction yields an approximation factor

$$\min \sum_{i\in T}\sum_{a^i\in H^i} c^{i,a^i}\rho^{i,a^i},$$7

which tends to $$\min \sum_{i\in T}\sum_{a^i\in H^i} c^{i,a^i}\rho^{i,a^i},$$8 as $$\min \sum_{i\in T}\sum_{a^i\in H^i} c^{i,a^i}\rho^{i,a^i},$$9 (Fekete et al., 2020).

Within this literature, MS2D is literal scheduling in a planar geometric domain. Vertices represent scanning devices or directional antennas, edges represent required communication links or sensing baselines, and scan concurrency is controlled by geometric feasibility and bipartite structure rather than probabilistic data association (Fekete et al., 2020).

5. Physical acquisition: beam steering and rapid-scan spectroscopy

A physically literal form of MS2D appears in beam-steered millimeter-wave imaging, where two-dimensional data are acquired by steering the beam in azimuth $N$00 and elevation $N$01 (Patole et al., 2013). For a planar target $N$02 at $N$03, narrowband operation is sufficient because the two distinct dimensions are supplied by electronic beam steering rather than bandwidth. The measurement model takes the form

$N$04

with the spatial-frequency mapping

$N$05

By choosing acquisition angles from a uniform Cartesian $N$06-grid, the method reconstructs the image by a 2D inverse Fourier transform without interpolation in either wavenumber or spatial domain (Patole et al., 2013). The simulations at $N$07 GHz, with a $N$08 planar array and $N$09 mm, report better performance in the presence of noise compared with the switched array imaging system, and the coherent Tx/Rx beamforming gain for a $N$10 array is described as about $N$11 dB (Patole et al., 2013).

A different acquisition-axis interpretation appears in rapid-scan white light two-dimensional electronic spectroscopy (Thomas et al., 2023). There, MS2D is implemented by rapidly and synchronously scanning the pump optical delay $N$12 while detecting every probe shot at the laser repetition rate. The pump is toggled ON/OFF at $N$13 kHz with an acousto-optical deflector, the CCD line camera reads every shot at $N$14 kHz, and the single-shot 2D signal is formed as

$N$15

The reported data average $N$16 scans to produce one absorptive 2D spectrum, achieve a full 2D spectrum in as much as $N$17 seconds of continuous sample exposure at the fastest scan, and obtain an SNR of $N$18 for optical densities down to $N$19 with $N$20 seconds of averaging at a $N$21 kHz laser repetition rate (Thomas et al., 2023). The method is described as repetition-rate scalable and limited only by the CCD line rate and the maximum stage velocity (Thomas et al., 2023).

These two acquisition systems share only a high-level abstraction. One scans $N$22 to sample a 2D spatial spectrum; the other scans $N$23 repeatedly while recording a spectrally resolved probe axis. This suggests that, in instrumentation, MS2D denotes a scan-acquire-reconstruct strategy rather than a fixed algorithmic primitive.

6. Industrial inspection and adjacent registration uses

In electronic manufacturing inspection, MS2D corresponds exactly to the “multi directional scanning” approach for robust BGA void detection in 2D X-ray images (Ahuja et al., 2019). The method operates within each solder ball ROI and prioritizes polar scans aligned with the ball geometry: radial $N$24 and angular $N$25. With polar mapping

$N$26

angular sampling at fixed radius is $N$27 (Ahuja et al., 2019). A positive edge at $N$28 is declared when $N$29 is an LoG edge and

$N$30

The algorithm then scans angularly to the corresponding negative edge and accepts the arc $N$31 as a 1D void if

$N$32

with $N$33 (Ahuja et al., 2019). Accepted 1D arcs are assembled into 2D voids through $N$34-connectivity and intensity consistency, and components with area below $N$35 pixels are removed (Ahuja et al., 2019). The paper emphasizes robustness under low resolution, with solder balls as small as $N$36 pixels, and low contrast, with void-to-background contrast on the order of $N$37 out of $N$38 levels (Ahuja et al., 2019). The qualitative result reported is that the method detects more voids and no void will be missed compared to the reference examples (Ahuja et al., 2019).

A looser, adjacent usage appears in multi-hypothesis 2D scan matching for Graph-SLAM (Iavicoli et al., 2022). The paper proposes a Monte-Carlo generation of weighted displacement hypotheses $N$39 in $N$40, followed by clustering through a factorized KDE with Gaussian translation kernels and a von Mises orientation kernel,

$N$41

and mean-shift or EM-like updates for the dominant mode (Iavicoli et al., 2022). The method is not explicitly named MS2D in the source paper, but the exposition places it in a multi-scan 2D registration context. On the Intel Research Lab dataset, the reported average RMS translation error is $N$42 m versus $N$43 m for IDC, and the average runtime per pair match is $N$44 ms versus $N$45 ms (Iavicoli et al., 2022). A plausible implication is that MS2D can also function as a broad label for multi-scan 2D registration pipelines when the underlying object is scan-to-scan displacement rather than image segmentation or planar scheduling.

7. Cross-domain principles and major distinctions

Taken together, the literature suggests three recurring constructions behind the label MS2D. The first is coupled 2D assignment, where a difficult multi-frame problem is decomposed into several scan-indexed 2D subproblems and then re-coupled by dual variables or association histories, as in trajectory PMBM/MBM and multi-scan GLMB (Xia et al., 2019). The second is multi-route serialization, where a 2D field is flattened along several scan paths and processed by selective scan kernels, as in Vision Mamba and MRI segmentation (Hardan et al., 15 Jul 2025). The third is literal scan acquisition, where physical devices sweep a 2D angular, spatial, or delay coordinate system and reconstruct a 2D object or spectrum from the measured samples (Patole et al., 2013).

Several misconceptions follow from treating the acronym as if it named a single algorithm. “Multi-scan” does not always mean multiple sensors, multiple passes of the same instrument, or multi-frame smoothing. In different papers it denotes an $N$46-scan data-association window, four parallel Mamba streams with duplicated or distinct scan paths, repeated $N$47 sweeps averaged into one 2D spectrum, or polar radial and angular traversals inside a solder-ball ROI (Xia et al., 2019). Likewise, “2D” does not have a unique referent: it may mean image grids, planar graphs, range-azimuth data, $N$48 scan-matching variables, or the two steering coordinates $N$49 (Fekete et al., 2020).

Despite that heterogeneity, the same engineering tensions recur. The literature repeatedly contrasts locality-preserving and locality-disrupting traversals, as in the superiority of horizontal and vertical rasters over diagonal scans in MRI (Hardan et al., 15 Jul 2025). It also repeatedly introduces explicit complexity-control devices: track-oriented $N$50-scan pruning in multi-target tracking, a fixed four-stream budget and parameter-free permutations in Vision Mamba, $N$51 approximation frameworks in geometric scheduling, and shot binning with hardware synchronization in spectroscopy (Xia et al., 2019). This suggests that MS2D is best regarded as a context-dependent design idiom for making 2D scanning, assignment, or traversal both computationally tractable and structurally informative.