---
title: Topology-Aware Cylindrical Motion Model (TCMM)
url: https://www.emergentmind.com/topics/topology-aware-cylindrical-motion-model-tcmm
type: topic
---

# Topology-Aware Cylindrical Motion Model (TCMM)

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" [2606.30097]. 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 [2010.07704], [1901.00979]. 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 [2310.19960]. 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 $1\,\mathrm{AU}$ [1502.05112].

## 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 $\mathbb{R}$ [2606.30097]. 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 [2606.30097].

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 [2010.07704]. The formulation is explicitly described as operating on the infinite cylinder $S^1 \times \mathbb{R}$, with the horizontal coordinate treated as an angle and the vertical coordinate as linear [1901.00979].

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 $\mathbb{R}^{n_\ell} \times T^{n_c}$; in the simplest case with one linear and one circular coordinate, the representation becomes cylindrical, $S^1 \times \mathbb{R}$ [2310.19960]. 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 [2606.30097]. 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
$$
m_t = (\theta_t, y_t, a_t, h_t, \dot{\theta}_t, \dot{y}_t, \dot{a}_t, \dot{h}_t)^{\top},
$$
where $\theta_t \in S^1$ is horizontal angle, $y_t \in \mathbb{R}$ is the vertical pixel coordinate of the box center, $a_t \in \mathbb{R}$ is aspect ratio, and $h_t \in \mathbb{R}$ is box height in pixels. The angular component is observed and associated on $S^1$, but internally its evolution is maintained in a lifted, unwrapped angular variable in $\mathbb{R}$ for linear Kalman updates; vertical and box-shape components remain Euclidean [2606.30097].

Given an ERP image of width $W$ and detection center $x_c = x + w/2$, horizontal position is mapped to angle by
$$
\theta = 2\pi \left(\frac{x_c}{W} - \frac{1}{2}\right),
$$
and angular box width by
$$
\Delta\theta = \frac{2\pi w}{W}.
$$
Seam-consistent angular differences are computed by
$$
\Delta\theta = \operatorname{wrap}_{2\pi}(\theta_2 - \theta_1)
= ((\theta_2-\theta_1+\pi)\bmod 2\pi)-\pi
= \operatorname{atan2}(\sin(\theta_2-\theta_1),\cos(\theta_2-\theta_1)),
$$
with equivalent circular distance
$$
d_{\mathrm{circ}}(\theta_1,\theta_2)=\min(|\theta_1-\theta_2|,2\pi-|\theta_1-\theta_2|).
$$
The wrap function used throughout is
$$
\operatorname{wrap}_{2\pi}(a)=((a+\pi)\bmod 2\pi)-\pi,
$$
normalizing angular differences to $( -\pi,\pi ]$ [2606.30097].

The motion model is linear in the lifted angular space:
$$
\theta_{t|t-1}^{\mathrm{lift}}=\theta_{t-1}^{\mathrm{lift}}+\dot{\theta}_{t-1}\Delta t+\epsilon_\theta,
$$
with analogous constant-velocity updates for $y,a,h$ 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 [2606.30097].

The measurement for the angular component is lifted relative to the predicted unwrapped angle:
$$
\theta_{\mathrm{obs}}^\star
=
\hat{\theta}_{t|t-1}^{\mathrm{lift}}
+
\operatorname{wrap}_{2\pi}
(\theta_{\mathrm{obs}}-\hat{\theta}_{t|t-1}),
$$
and the angular innovation is
$$
\nu_\theta=\operatorname{wrap}_{2\pi}(\theta_{\mathrm{obs}}-\hat{\theta}_{t|t-1}).
$$
This design specifically targets seam crossing without resorting to nonlinear filtering [2606.30097].

A central implication is that TCMM restores trajectory continuity across the $0^\circ/360^\circ$ seam, stabilizes association for boundary-crossing motion, and avoids identity switches and broken tracks when boxes straddle the boundary [2606.30097].

## 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 [2606.30097]. A box $b$ is represented by
$$
R(b) = (\theta, y_c, \Delta\theta, h),
$$
with circular horizontal interval
$$
I_\theta(b)=[\theta-\Delta\theta/2,\theta+\Delta\theta/2]_{2\pi}
$$
and vertical interval
$$
I_y(b)=[y_c-h/2,y_c+h/2].
$$

The horizontal periodic overlap length $\ell_\theta(i,j)$ is defined as the length of the intersection of the circular intervals, and the vertical overlap $\ell_y(i,j)$ is the usual pixel overlap:
$$
\ell_y(i,j)
=
\max(0,\min(y_i^+,y_j^+)-\max(y_i^-,y_j^-)).
$$
From these, cylindrical intersection and area are
$$
A_{\cap,\mathrm{hp}}(i,j)=\ell_\theta(i,j)\ell_y(i,j),\qquad
A_{\mathrm{hp}}(i)=\Delta\theta_i h_i.
$$
The resulting Horizontal-periodic, pixel-vertical IoU (HPV-IoU) is
$$
\mathrm{IoU}_{\mathrm{hp}}(i,j)
=
\frac{A_{\cap,\mathrm{hp}}(i,j)}
{A_{\mathrm{hp}}(i)+A_{\mathrm{hp}}(j)-A_{\cap,\mathrm{hp}}(i,j)+\epsilon},
$$
with corresponding cost
$$
C_{\mathrm{hp}}(i,j)=1-\mathrm{IoU}_{\mathrm{hp}}(i,j).
$$
To handle seam-straddling intervals, wrapping intervals are split into at most two non-wrapping arcs and overlap lengths are summed across arcs [2606.30097].

Depth is coupled through Depth-Temporal Trajectory Modeling (DTM), which maintains a trajectory-level depth state
$$
r_t=(d_t,\dot d_t)^\top
$$
with constant-velocity dynamics. The one-dimensional state-space model is
$$
x_t^i = [d_t^i,\dot d_t^i]^\top,\qquad
F=\begin{bmatrix}1&1\\0&1\end{bmatrix},
$$
$$
\hat x_t^i = F x_{t-1}^i,\qquad
\hat P_t^i = F P_{t-1}^i F^\top + Q,
$$
$$
z_t = Hx_t+\eta_t,\qquad
H=[1,0],\qquad
\eta_t\sim N(0,R).
$$
The DTM-predicted depth stabilizes depth cues over time [2606.30097].

The depth consistency cost is
$$
C_z(i,j)=\frac{|\hat d_i-d_j|}{\max_{p,q}|\hat d_p-d_q|+\epsilon},
$$
and the depth-aware cylindrical cost is
$$
C_{d-\mathrm{hp}}(i,j)
=
(1-\lambda_z)C_{\mathrm{hp}}(i,j)+\lambda_z C_z(i,j).
$$
An angular consistency term is further defined as
$$
C_\theta(i,j)=\frac{|\operatorname{wrap}_{2\pi}(\theta_i-\theta_j)|}{\pi},
$$
and the combined association cost becomes
$$
C_{\mathrm{assoc}}(i,j)
=
(1-\lambda_\theta)C_{d-\mathrm{hp}}(i,j)+\lambda_\theta C_\theta(i,j).
$$
Finally, detection confidence $s_j$ adjusts the cost:
$$
\tilde C(i,j)=1-(1-C_{\mathrm{assoc}}(i,j))s_j.
$$
Association uses angular gating $d_{\mathrm{circ}}(\theta_i,\theta_j)\le \gamma_\theta$ and depth gating $|\hat d_i-d_j|\le \tau_d$, followed by Hungarian assignment with threshold $\tau_m$ [2606.30097].

The pipeline performs two-stage association: high-confidence detections with full $C_{\mathrm{assoc}}$, 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 [2606.30097].

## 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 $\theta$ and vertical coordinate $h$, with pixel-to-cylinder mapping
$$
\theta = \frac{2\pi}{W}u-\pi,\qquad
h=\frac{v-c_y}{f_y},
$$
and inverse unprojection
$$
\begin{bmatrix}
x_s\\y_s\\z_s
\end{bmatrix}
=
\begin{bmatrix}
d\sin\theta\\dh\\d\cos\theta
\end{bmatrix}
$$
under the paper’s adopted convention [2010.07704]. Equivalent formulations in the earlier version present the cylindrical model directly on $S^1\times\mathbb{R}$ and emphasize that the predicted “depth” is radial distance in the horizontal plane rather than pinhole $Z$-depth [1901.00979].

Ego-motion is represented by a pose $T=[R\mid t]\in SE(3)$, and cylindrical reprojection uses
$$
\theta'=\operatorname{atan2}(x',z'),\qquad
h'=\frac{y'}{\sqrt{x'^2+z'^2}},
$$
followed by
$$
u'=(f_x\theta'+c_x)\bmod W,\qquad
v'=f_y h'+c_y.
$$
The modular arithmetic in $u'$ 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 [2010.07704].

The learning pipeline is topology-aware at multiple levels. Bilinear sampling indexes horizontally modulo $W$; derivatives along the horizontal axis are computed with circular finite differences; and standard CNN layers are retained with circular padding along the horizontal dimension [2010.07704], [1901.00979]. The photometric supervision is based on cylindrical view synthesis with losses of the form
$$
\mathcal{L}_{\mathrm{photo}}
=
\sum_{u,v}
E(u,v)\,
\|I_t(u,v)-I_s(u'(u,v),v'(u,v))\|_1,
$$
combined with smoothness and explainability regularization [2010.07704].

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 $512\times128$, adding horizontal circular padding improved all reported depth metrics: Abs Rel $0.299$ versus $0.301$, Sq Rel $2.514$ versus $2.543$, RMSE $6.037$ versus $6.220$, RMSE log $0.387$ versus $0.395$, $\delta<1.25$ $0.675$ versus $0.668$, $\delta<1.25^2$ $0.835$ versus $0.828$, and $\delta<1.25^3$ $0.906$ versus $0.901$ [2010.07704]. Second, increased field of view improves pose accuracy: for SYNTHIA sequence 02, mean ATE decreased from $0.0324$ at $100^\circ$ to $0.0203$ at $180^\circ$, $0.0150$ at $270^\circ$, and $0.0143/0.0139$ at $360^\circ$ without/with wrap [2010.07704].

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 [2310.19960]. 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 $\mathbb{R}^{n_\ell}\times T^{n_c}$, and in the simplest case a cylindrical parameterization $S^1\times\mathbb{R}$.

For a candidate periodic signal component $s(t)$, delay embedding forms points
$$
x_i=(s(t_i),s(t_{i+\epsilon}),\ldots,s(t_{i+r\epsilon}))\in\mathbb{R}^{r+1}.
$$
A Vietoris–Rips complex is built, a persistent $H^1$ cocycle is selected, and a least-squares smoothing problem is solved:
$$
\min_{\phi\in\mathbb{R}^N}
\sum_{(i,j)\in E}w_{ij}(\phi_j-\phi_i-\alpha_{ij})^2.
$$
The circular coordinate is then
$$
\theta(x_i)=\phi_i \bmod 1 \in S^1.
$$
The linear coordinate is taken from the dominant trend-like PCA component, producing cylindrical coordinates
$$
x(t)\mapsto(\theta(t),z(t))\in S^1\times\mathbb{R},
$$
or the Euclidean embedding
$$
f(t)=[\cos\theta(t),\sin\theta(t),z(t)]^\top.
$$
This model explicitly interprets periodic phase as the circular dimension and drift or progression as the linear dimension [2310.19960].

The paper also develops invariance-preserving transformations for circular-coordinate comparison. Mod-$1$ adjustment $T_M$, inversion normalization $T_I$, and translation-invariant comparison $T_L$ are combined into
$$
\Phi(d)(c,\tilde c)=T_L(d)(T_I(T_M(c)),T_I(T_M(\tilde c))).
$$
For mixed coordinates, product metrics combine Euclidean distances on linear coordinates with topology-aware distances on circular coordinates [2310.19960].

Within a multiple-output Gaussian process, these coordinates support topology-induced clustering and cylindrical kernels. One optional refinement is
$$
k_{\mathrm{cyl}}\big((\theta,z),(\theta',z')\big)
=
\sigma^2
\exp\!\left(
-\frac{2\sin^2((\theta-\theta')/2)}{\ell_\theta^2}
\right)
\exp\!\left(
-\frac{(z-z')^2}{2\ell_z^2}
\right),
$$
which respects geodesic distance on $S^1$ and Euclidean distance on $\mathbb{R}$ [2310.19960].

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 [2606.30097]. 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 $+18.91\%$ IDF1 and $+15.20\%$ AssA on QuadTrack, with BCIC $+24.38$ points, and $+10.66\%$ IDF1 and $+18.74\%$ AssA on JRDB, with BCIC $+13.51$ points [2606.30097]. Detector-side integration, termed CylindTrackDet, improves identity-oriented metrics across seven trackers, with average IDF1 gains of $11.83\%$ on QuadTrack and $6.15\%$ on JRDB relative to the DepTR-MOT detector [2606.30097].

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 $30.817$, IDF1 $35.498$, AssA $29.255$. Adding cylindrical motion and topology-aware IoU increases these to HOTA $31.616$, IDF1 $37.026$, AssA $30.794$. Adding angular consistency further increases them to HOTA $33.659$, IDF1 $40.412$, AssA $34.622$. The final HPV-IoU variant reaches HOTA $33.674$, IDF1 $40.446$, AssA $34.665$, MOTA $20.583$ [2606.30097].

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 [2606.30097]. Design choices are correspondingly conservative. Angles are lifted to $\mathbb{R}$ 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 [2606.30097].

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 [2606.30097]. 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 [2010.07704], [1901.00979]. 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 [2310.19960].

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 $S^1$ 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 $1\,\mathrm{AU}$ [1502.05112]. 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
$$
\nabla\times\mathbf{B}=\alpha\mathbf{B},
$$
with
$$
B_r=0,\qquad
\mathbf{B}_\varphi=H B_0 J_1(\alpha r)\,\hat{\boldsymbol\varphi},\qquad
\mathbf{B}_z=B_0 J_0(\alpha r)\,\hat{\mathbf z},
$$
and adds three motion components: linear propagation, cross-sectional expansion, and poloidal motion [1502.05112]. The expansion law
$$
v_r(x)=xv_e,\qquad
R(t)=R(t_0)+v_e(t-t_0),
$$
and poloidal speed
$$
v_\varphi(t,x)=v_p(t_0)\frac{R(t_0)}{R(t)}
$$
for the adopted $f_p(x)=1$ profile define the kinematic part of the model [1502.05112].

The empirical findings are likewise topology-aware in a physical sense: about $62\%$ of magnetic clouds underwent under-expansion at $1\,\mathrm{AU}$, $17\%$ over-expansion, and the expansion rate is about $0.6$ on average; significant non-radial propagation appears in about $15\%$ of events; and the median poloidal speed is about $10\,\mathrm{km\,s^{-1}}$ [1502.05112]. 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 $S^1\times\mathbb{R}$. 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 [2606.30097], extension from $S^1\times\mathbb{R}$ to higher-dimensional tori in motion-series analysis [2310.19960], and relaxation of cylindrical symmetry assumptions in flux-rope modeling [1502.05112]. 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.

Source: https://www.emergentmind.com/topics/topology-aware-cylindrical-motion-model-tcmm