Topology-Aware Cylindrical Motion Model (TCMM)

• TCMM is a modeling approach that represents data on a cylindrical domain (S¹×ℝ), treating the horizontal coordinate as periodic and the vertical as linear.
• It uses circular differencing and angular lifting within linear Kalman filtering to maintain continuity across panoramic seams and improve state prediction.
• TCMM integrates depth coupling and topological phase-trend decomposition to enhance multi-object tracking performance and unsupervised motion learning.

Searching arXiv for the specified papers to ground the article in the cited literature. Topology-Aware Cylindrical Motion Model (TCMM) denotes a family of formulations that treat motion, geometry, or temporal structure on a cylindrical domain whose topology is $S^1 \times \mathbb{R}$, rather than a purely Euclidean image plane or state space. Across the literature, the defining idea is consistent: the horizontal dimension is periodic and must be modeled as an angular variable on $S^1$, while the vertical, radial, depth, or trend dimension remains Euclidean. In panoramic multi-object tracking, TCMM is instantiated as a seam-consistent motion and association model for equirectangular panoramic video in "CylindTrack: Depth-Aware Cylindrical Motion Modeling for Panoramic Multi-Object Tracking" (Deng et al., 29 Jun 2026). In unsupervised depth and ego-motion learning, closely related cylindrical formulations operate directly on panoramic image domains with horizontal wrap-around and cylindrical projection geometry (Sharma et al., 2020, Sharma et al., 2019). In topological time-series learning, the cylindrical representation appears as a mixed-coordinate model combining one circular and one linear coordinate, yielding $S^1 \times \mathbb{R}$ as an explicit motion manifold (Luo et al., 2023). A distinct physical instantiation appears in heliophysics, where cylindrical symmetry and topology preservation are built into a velocity-modified force-free flux rope model for magnetic clouds at $1\,\mathrm{AU}$ (Wang et al., 2015).

1. Topological basis and problem formulation

The core topological premise of TCMM is that one axis of the data domain is periodic. In equirectangular panoramic video, the left boundary $x \approx 0$ and right boundary $x \approx W$ are physically adjacent on the viewing sphere; the azimuth wraps at $0^\circ/360^\circ$. Accordingly, the horizontal axis is a circle $S^1$, not the real line $\mathbb{R}$ (Deng et al., 29 Jun 2026). Standard planar motion models fail under this topology because a target crossing the seam can exhibit a spurious displacement of approximately $-W$ pixels even when the physical motion is continuous. The reported consequences include corrupted state prediction and filtering, unreliable IoU-based association near the seam, and failure modes for boundary-straddling boxes (Deng et al., 29 Jun 2026).

A closely related topological argument appears in cylindrical panoramic depth-and-pose learning. Cylindrical panoramas are periodic in the horizontal direction and lack spherical pole singularities or severe polar distortion, so standard convolutional layers can be used without distortion-aware modifications; only horizontal circular padding is required to preserve continuity at the seam (Sharma et al., 2020). The formulation is explicitly described as operating on the infinite cylinder $S^1$0, with the horizontal coordinate treated as an angle and the vertical coordinate as linear (Sharma et al., 2019).

In motion-series analysis, the same topology appears in a different guise. "Topological Learning for Motion Data via Mixed Coordinates" introduces “mixed valued coordinates” in $S^1$1; in the simplest case with one linear and one circular coordinate, the representation becomes cylindrical, $S^1$2 (Luo et al., 2023). There, the circular coordinate captures periodic phase and the linear coordinate captures trend.

These strands share a common modeling principle: Euclidean state spaces are insufficient when periodic continuity is intrinsic to the data. This suggests that TCMM is best understood not as a single algorithm, but as a topology-aware modeling pattern for domains with circular-horizontal structure.

2. TCMM in panoramic multi-object tracking

In CylindTrack, TCMM is introduced to address panoramic MOT under equirectangular projection, where horizontal motion is non-Euclidean and seam crossing breaks conventional trackers (Deng et al., 29 Jun 2026). The model “lifts” horizontal motion into a continuous angular state space and performs seam-consistent prediction and association in the periodic panoramic domain.

For motion and box geometry, TCMM uses a constant-velocity state

$S^1$3

where $S^1$4 is horizontal angle, $S^1$5 is the vertical pixel coordinate of the box center, $S^1$6 is aspect ratio, and $S^1$7 is box height in pixels. The angular component is observed and associated on $S^1$8, but internally its evolution is maintained in a lifted, unwrapped angular variable in $S^1$9 for linear Kalman updates; vertical and box-shape components remain Euclidean (Deng et al., 29 Jun 2026).

Given an ERP image of width $S^1 \times \mathbb{R}$0 and detection center $S^1 \times \mathbb{R}$1, horizontal position is mapped to angle by

$S^1 \times \mathbb{R}$2

and angular box width by

$S^1 \times \mathbb{R}$3

Seam-consistent angular differences are computed by

$S^1 \times \mathbb{R}$4

with equivalent circular distance

$S^1 \times \mathbb{R}$5

The wrap function used throughout is

$S^1 \times \mathbb{R}$6

normalizing angular differences to $S^1 \times \mathbb{R}$7 (Deng et al., 29 Jun 2026).

The motion model is linear in the lifted angular space:

$S^1 \times \mathbb{R}$8

with analogous constant-velocity updates for $S^1 \times \mathbb{R}$9 and their derivatives. Gaussian process and measurement noise are used with standard linear Kalman filtering. Only the angular innovation is wrapped before the Kalman gain is applied, preserving linear-Gaussian covariance behavior while enforcing seam consistency (Deng et al., 29 Jun 2026).

The measurement for the angular component is lifted relative to the predicted unwrapped angle:

$1\,\mathrm{AU}$0

and the angular innovation is

$1\,\mathrm{AU}$1

This design specifically targets seam crossing without resorting to nonlinear filtering (Deng et al., 29 Jun 2026).

A central implication is that TCMM restores trajectory continuity across the $1\,\mathrm{AU}$2 seam, stabilizes association for boundary-crossing motion, and avoids identity switches and broken tracks when boxes straddle the boundary (Deng et al., 29 Jun 2026).

3. Cylindrical association, overlap, and depth coupling

Beyond state evolution, TCMM in CylindTrack includes a seam-consistent association model based on cylindrical overlap and angular consistency (Deng et al., 29 Jun 2026). A box $1\,\mathrm{AU}$3 is represented by

$1\,\mathrm{AU}$4

with circular horizontal interval

$1\,\mathrm{AU}$5

and vertical interval

$1\,\mathrm{AU}$6

The horizontal periodic overlap length $1\,\mathrm{AU}$7 is defined as the length of the intersection of the circular intervals, and the vertical overlap $1\,\mathrm{AU}$8 is the usual pixel overlap:

$1\,\mathrm{AU}$9

From these, cylindrical intersection and area are

$x \approx 0$0

The resulting Horizontal-periodic, pixel-vertical IoU (HPV-IoU) is

$x \approx 0$1

with corresponding cost

$x \approx 0$2

To handle seam-straddling intervals, wrapping intervals are split into at most two non-wrapping arcs and overlap lengths are summed across arcs (Deng et al., 29 Jun 2026).

Depth is coupled through Depth-Temporal Trajectory Modeling (DTM), which maintains a trajectory-level depth state

$x \approx 0$3

with constant-velocity dynamics. The one-dimensional state-space model is

$x \approx 0$4

$x \approx 0$5

$x \approx 0$6

The DTM-predicted depth stabilizes depth cues over time (Deng et al., 29 Jun 2026).

The depth consistency cost is

$x \approx 0$7

and the depth-aware cylindrical cost is

$x \approx 0$8

An angular consistency term is further defined as

$x \approx 0$9

and the combined association cost becomes

$x \approx W$0

Finally, detection confidence $x \approx W$1 adjusts the cost:

$x \approx W$2

Association uses angular gating $x \approx W$3 and depth gating $x \approx W$4, followed by Hungarian assignment with threshold $x \approx W$5 (Deng et al., 29 Jun 2026).

The pipeline performs two-stage association: high-confidence detections with full $x \approx W$6, then low-confidence detections with cylindrical IoU alone to recover temporarily weak detections. This architecture integrates TCMM with DTM and with detector-side Spherical Spatio-Temporal Consistency Learning (SSTC), whose Temporal Mixer and Spherical Geometry-aware Attention improve the reliability of per-frame depth observations feeding the trajectory-level model (Deng et al., 29 Jun 2026).

4. Relation to cylindrical projection and panoramic ego-motion learning

Earlier cylindrical panoramic research established the geometric and numerical principles that later TCMM-style tracking models exploit. In unsupervised depth and ego-motion learning from cylindrical panoramic video, the image domain is parameterized by azimuth $x \approx W$7 and vertical coordinate $x \approx W$8, with pixel-to-cylinder mapping

$x \approx W$9

and inverse unprojection

$0^\circ/360^\circ$0

under the paper’s adopted convention (Sharma et al., 2020). Equivalent formulations in the earlier version present the cylindrical model directly on $0^\circ/360^\circ$1 and emphasize that the predicted “depth” is radial distance in the horizontal plane rather than pinhole $0^\circ/360^\circ$2-depth (Sharma et al., 2019).

Ego-motion is represented by a pose $0^\circ/360^\circ$3, and cylindrical reprojection uses

$0^\circ/360^\circ$4

followed by

$0^\circ/360^\circ$5

The modular arithmetic in $0^\circ/360^\circ$6 is the direct panoramic analogue of TCMM’s circular handling in tracking: samples that straddle the seam interpolate correctly using circular padding and wrap-around indexing (Sharma et al., 2020).

The learning pipeline is topology-aware at multiple levels. Bilinear sampling indexes horizontally modulo $0^\circ/360^\circ$7; derivatives along the horizontal axis are computed with circular finite differences; and standard CNN layers are retained with circular padding along the horizontal dimension (Sharma et al., 2020, Sharma et al., 2019). The photometric supervision is based on cylindrical view synthesis with losses of the form

$0^\circ/360^\circ$8

combined with smoothness and explainability regularization (Sharma et al., 2020).

This cylindrical formulation established two empirical points relevant to later TCMM developments. First, respecting horizontal wrap-around improves depth estimation. On SYNTHIA panoramic training at $0^\circ/360^\circ$9, adding horizontal circular padding improved all reported depth metrics: Abs Rel $S^1$0 versus $S^1$1, Sq Rel $S^1$2 versus $S^1$3, RMSE $S^1$4 versus $S^1$5, RMSE log $S^1$6 versus $S^1$7, $S^1$8 $S^1$9 versus $\mathbb{R}$0, $\mathbb{R}$1 $\mathbb{R}$2 versus $\mathbb{R}$3, and $\mathbb{R}$4 $\mathbb{R}$5 versus $\mathbb{R}$6 (Sharma et al., 2020). Second, increased field of view improves pose accuracy: for SYNTHIA sequence 02, mean ATE decreased from $\mathbb{R}$7 at $\mathbb{R}$8 to $\mathbb{R}$9 at $-W$0, $-W$1 at $-W$2, and $-W$3 at $-W$4 without/with wrap (Sharma et al., 2020).

A plausible implication is that cylindrical TCMM in tracking inherits not only the topological rationale of seam-consistent angular representation, but also a mature implementation pattern from cylindrical depth-and-pose learning: analytic cylindrical projection, modular horizontal sampling, and circular numerical operators.

5. Mixed coordinates and topological phase–trend decomposition

A conceptually different, but mathematically related, TCMM appears in topological learning for motion data (Luo et al., 2023). There, persistent cohomology is used to recover circular coordinates from delay embeddings of motion or time series, while PCA and Kendall’s tau identify linear trends. The resulting “mixed valued coordinates” yield a representation in $-W$5, and in the simplest case a cylindrical parameterization $-W$6.

For a candidate periodic signal component $-W$7, delay embedding forms points

$-W$8

A Vietoris–Rips complex is built, a persistent $-W$9 cocycle is selected, and a least-squares smoothing problem is solved:

$S^1$00

The circular coordinate is then

$S^1$01

The linear coordinate is taken from the dominant trend-like PCA component, producing cylindrical coordinates

$S^1$02

or the Euclidean embedding

$S^1$03

This model explicitly interprets periodic phase as the circular dimension and drift or progression as the linear dimension (Luo et al., 2023).

The paper also develops invariance-preserving transformations for circular-coordinate comparison. Mod-$S^1$04 adjustment $S^1$05, inversion normalization $S^1$06, and translation-invariant comparison $S^1$07 are combined into

$S^1$08

For mixed coordinates, product metrics combine Euclidean distances on linear coordinates with topology-aware distances on circular coordinates (Luo et al., 2023).

Within a multiple-output Gaussian process, these coordinates support topology-induced clustering and cylindrical kernels. One optional refinement is

$S^1$09

which respects geodesic distance on $S^1$10 and Euclidean distance on $S^1$11 (Luo et al., 2023).

This literature broadens the meaning of TCMM. In tracking and panoramic geometry, the circular coordinate is a physical azimuth. In topological learning, it is a learned phase variable extracted from data. The common structure is nevertheless the same: one periodic coordinate and one linear coordinate, coupled in a cylinder-aware metric or state representation.

6. Empirical performance, design choices, and limitations

The strongest empirical characterization of TCMM as a motion model is given in CylindTrack (Deng et al., 29 Jun 2026). On panoramic MOT benchmarks, full CylindTrack achieves:

Benchmark	HOTA	AssA / IDF1 / FPS
QuadTrack	33.674	AssA 34.665, IDF1 40.446, 28.56 FPS
JRDB	31.117	AssA 31.347, IDF1 34.331, 21.339 FPS

Against the strongest DepTR-MOT+ByteTrack baseline, the reported gains are $S^1$12 IDF1 and $S^1$13 AssA on QuadTrack, with BCIC $S^1$14 points, and $S^1$15 IDF1 and $S^1$16 AssA on JRDB, with BCIC $S^1$17 points (Deng et al., 29 Jun 2026). Detector-side integration, termed CylindTrackDet, improves identity-oriented metrics across seven trackers, with average IDF1 gains of $S^1$18 on QuadTrack and $S^1$19 on JRDB relative to the DepTR-MOT detector (Deng et al., 29 Jun 2026).

Ablations isolating TCMM show a clear contribution from cylindrical motion filtering, topology-aware IoU, and angular consistency. The image-plane baseline without TCMM reports HOTA $S^1$20, IDF1 $S^1$21, AssA $S^1$22. Adding cylindrical motion and topology-aware IoU increases these to HOTA $S^1$23, IDF1 $S^1$24, AssA $S^1$25. Adding angular consistency further increases them to HOTA $S^1$26, IDF1 $S^1$27, AssA $S^1$28. The final HPV-IoU variant reaches HOTA $S^1$29, IDF1 $S^1$30, AssA $S^1$31, MOTA $S^1$32 (Deng et al., 29 Jun 2026).

The paper emphasizes that the computational overhead is minimal: TCMM adds wrap operations and circular interval logic in association but no heavy learned modules, while maintaining practical online speeds (Deng et al., 29 Jun 2026). Design choices are correspondingly conservative. Angles are lifted to $S^1$33 for Kalman filtering, rather than modeled with circular distributions such as von Mises; standard Kalman filtering is preferred over EKF or UKF because the dynamics are linear and the only nonlinearity is the wrap operation in measurement mapping (Deng et al., 29 Jun 2026).

The limitations are domain-specific. In panoramic MOT, extreme distortions near poles are not explicitly corrected in TCMM’s vertical modeling, and highly nonlinear motion or rapid camera ego-motion may challenge constant-velocity assumptions (Deng et al., 29 Jun 2026). In cylindrical panoramic depth-and-pose learning, the absence of top and bottom polar regions reduces coverage, while dynamic objects, persistent occluders, and pitch/roll can degrade performance (Sharma et al., 2020, Sharma et al., 2019). In mixed-coordinate topological learning, selection thresholds for Kendall’s tau and persistence influence performance, and persistent cohomology on large delay embeddings can be expensive (Luo et al., 2023).

A broader misconception is that cylindrical modeling alone resolves all panoramic geometry issues. The literature does not support that view. Instead, it supports a narrower claim: respecting the $S^1$34 topology removes seam artifacts and discontinuities, while other error sources—depth noise, pole distortion, dynamic occlusion, nonstationarity, or non-self-similar dynamics—still require additional modeling.

7. Broader variants and generalization across domains

The term TCMM is not unique to computer vision. In heliophysics, a topology-aware cylindrical motion model is instantiated as a velocity-modified cylindrical force-free flux rope model for magnetic clouds at $S^1$35 (Wang et al., 2015). There, the cylinder is not an image domain but a physical flux-rope geometry with conserved magnetic topology. The model preserves a Lundquist-type force-free field

$S^1$36

with

$S^1$37

and adds three motion components: linear propagation, cross-sectional expansion, and poloidal motion (Wang et al., 2015). The expansion law

$S^1$38

and poloidal speed

$S^1$39

for the adopted $S^1$40 profile define the kinematic part of the model (Wang et al., 2015).

The empirical findings are likewise topology-aware in a physical sense: about $S^1$41 of magnetic clouds underwent under-expansion at $S^1$42, $S^1$43 over-expansion, and the expansion rate is about $S^1$44 on average; significant non-radial propagation appears in about $S^1$45 of events; and the median poloidal speed is about $S^1$46 (Wang et al., 2015). Although this use of TCMM is distinct from panoramic tracking, it reinforces the broader principle that cylindrical symmetry and periodic coordinates can be coupled with conservation laws or seam-consistent dynamics without flattening the domain to a line.

Across these literatures, TCMM can therefore be characterized by three persistent features. First, it encodes one coordinate as periodic and another as linear, most commonly as $S^1$47. Second, it replaces naïve Euclidean differencing with circular differences, modular indexing, or geometry-respecting kernels. Third, it preserves continuity or topology under motion—whether that motion is target tracking across a panoramic seam, camera ego-motion in cylindrical view synthesis, phase evolution in periodic time series, or plasma motion in a cylindrically symmetric magnetic structure.

Future directions stated in the literature include full spherical motion modeling of both longitude and latitude in panoramic MOT, stronger 3D geometry priors and camera-motion-aware depth filtering, joint learning of detection, depth, motion, and association under panoramic geometry (Deng et al., 29 Jun 2026), extension from $S^1$48 to higher-dimensional tori in motion-series analysis (Luo et al., 2023), and relaxation of cylindrical symmetry assumptions in flux-rope modeling (Wang et al., 2015). This suggests that the enduring value of TCMM lies less in any single implementation than in its explicit insistence that topology is part of the motion model, not a peripheral implementation detail.