cs.CVOct 24, 2026

Segmentation of Moving Objects During Camera Motion Using the OWL Function

Authors: Daniel Raviv, Juan D. Yepes

Organizations: Electrical Engineering and Computer Science Department, Florida Atlantic University, Boca Raton, Florida, USA

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Abstract

We present a principled, analytical framework for segmenting multiple independently moving rigid objects during single camera motion, built directly upon the OWL perception function. OWL encodes two fundamental visual motion cues, perceived looming LL and perceived rotation ωω, into a single closed-form representation of relative 3D structure. A key consequence of this structure is that the unit direction of relative translation t^\hat{t} is analytically recoverable per image point from LL and ωω alone. All points belonging to a non-rotating rigid body share the same t^\hat{t}, whereas points on different rigid bodies generally yield distinct values. This furnishes a natural segmentation criterion: clustering per-point t^\hat{t} estimates partitions the visible scene into independently moving rigid bodies. Building on the 2D complex form of OWL, we derive the framework in 3D using quaternions, for a single moving camera. The approach is parallel, camera rotation-invariant, and requires no training data, no depth information, and no semantic priors.

Figures & tables

Known per pointConstraintsPoints needed
LL and ω\bm{\omega} (vector)2 (angle + plane)1
LL and ω\omega (scalars)1 (angle only)3
TABLE I: Points Needed to Recover t^\bm{\hat{t}}
Fig. 1: Camera translating through a static environment (with no camera rotation). Top row: a top-down schematic of the camera motion and a representative rendered frame. Middle row, left to right: the ground-truth cues LL , ωθ\omega_{\theta} , and ωϕ\omega_{\phi} . Bottom row: the recovered t^\hat{\mathbf{t}} field visualized as RGB color. Since the scene is a single rigid body undergoing pure camera-relative translation, t^\hat{\mathbf{t}} collapses to a single color (red) everywhere, as predicted in Section III-C .
Fig. 2: Three viewpoints of the RoT point cloud reconstructed for the stationary scene of Fig. 1 , confirming that the recovered scaled 3D geometry is consistent with the underlying structure of the 3D environment.
Fig. 3: Case of translation-only motion. Top row: a representative frame and a top-down schematic showing two independently translating objects. Middle row: the corresponding LL , ωϕ\omega_{\phi} , and ωθ\omega_{\theta} cues, in which each object is visible as a localized deviation from the background. Bottom row: the recovered t^\hat{\mathbf{t}} field, in which each object forms its own uniformly colored cluster distinct from the background, and the corresponding binary object masks. With no self-rotation on either object, each yields a single well-defined t^\hat{\mathbf{t}} , cleanly separable by clustering.
Fig. 4: Two independently moving objects in a degenerate configuration. Top: a top-down schematic of the camera and object motions. Middle: a representative frame, with the robot moving toward the camera boxed. Bottom: the recovered t^\hat{\mathbf{t}} field. The boxed robot moves exactly opposite to the camera, so its t^\hat{\mathbf{t}} coincides with that of the background and it does not appear as a separate cluster (Assumption 3). The second robot forms a distinct cluster.
Fig. 5: Self-rotating object alongside a purely translating object. Top: a top-down schematic marking the self-rotating object and a representative frame. Bottom, from left to right: the recovered t^\hat{\mathbf{t}} field and binary masks. The purely translating object still yields one uniform t^\hat{\mathbf{t}} color, while the self-rotating object’s cluster shows internal color variation, reflecting the local violation of the pure-translation assumption on that body.
Fig. 6: Camera rotating, while translating forward, over a stationary scene. Top: a top-down schematic of the camera’s rotation. Bottom: two frames (Frame 4 and Frame 19) sampled during the rotation, each with its recovered t^\hat{\mathbf{t}} field shown below it. In both frames the stationary background still collapses to a single, uniform t^\hat{\mathbf{t}} color; the color differs between frames because t^\hat{\mathbf{t}} is expressed in the camera’s own frame, which has rotated. Camera rotation does not break the single-cluster structure of the stationary background.
Fig. 7: Camera rotating and translating, with moving objects translating only. Top: a top-down schematic of the camera and object motions. Bottom: two frames (Frame 610 and Frame 640) with their recovered t^\hat{\mathbf{t}} fields. The two moving objects remain visible as distinct, uniformly colored clusters against the background in both frames, indicating that simultaneous camera rotation and translation does not disrupt the recovery of per-object t^\hat{\mathbf{t}} .
Fig. 8: Camera rotating and translating, with two moving objects, the one on the right is translating only and the one on the left rotates only. Top: a top-down schematic of the combined motions at an earlier frame than Frame 322. Bottom: two frames (Frame 322 and Frame 335) with their recovered t^\hat{\mathbf{t}} fields. As in Fig. 5 , the self-rotation of the objects introduces internal color variation within their clusters, now superimposed on the effects of camera rotation and translation.
Fig. 9: A more general case: multiple independently moving objects (two cars, a walking humanoid robot, a delivery robot, and a pedestrian) together with camera motion. Top: a representative frame. Middle: the recovered t^\hat{\mathbf{t}} field, in which each moving object stands out from the uniform background. Bottom: the corresponding binary masks of moving versus static regions. The rigid vehicles produce nearly uniform t^\hat{\mathbf{t}} , while the articulated humanoid and pedestrian, which violate the rigid-body assumption, show internal t^\hat{\mathbf{t}} variation but remain clearly distinct from the background.
Appendix figures & tables1 asset

Supplementary material from the paper’s appendix.

Appendix

Fig. 10: Geometric decomposition of t/∣r∣\mathbf{t}/|\mathbf{r}| in the plane containing t\mathbf{t} and e^r\hat{e}_{r} . The radial component L e^rL\,\hat{e}_{r} (red) is read directly from the looming. The tangential component −e^r×ω-\hat{e}_{r}\times\bm{\omega} (orange) is recovered by a second cross product. Their vector sum (blue) gives t/∣r∣\mathbf{t}/|\mathbf{r}| ; normalizing by L2+ω2\sqrt{L^{2}+\omega^{2}} yields t^\hat{\mathbf{t}} .
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