Fast 3D Extended Target Tracking using NURBS Surfaces

Benjamin Naujoks; Hans-Joachim Wuensche; Patrick Burger

REVIEW 4 major objections 7 minor 29 references

Fast 3D Extended Target Tracking using NURBS Surfaces

T0 review · 4 major / 7 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read This paper claims that a vehicle's full 3D shape and dynamics can be estimated together in real time from sparse LiDAR points by modeling the target as a NURBS surface whose scaling parameters and optionally weights are part of the filter…

desk verdict Legitimate 3D extended target tracking extension using NURBS surfaces with real-world evaluation, but the measurement model is under-specified and the write-up needs a major revision for reproducibility. read the letter →

arxiv 1909.00767 v2 pith:ACBPGVOS submitted 2019-09-02 eess.SP

Benjamin Naujoks , Patrick Burger , Hans-Joachim Wuensche This is my paper · ORCID

classification eess.SP

keywords extendedtargettrackingNURBSsurfacesLiDARpointcloudsunscentedKalmanfilter3Dshapeestimationrandomhypersurfacemodelvehiclereal-time

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper sets out to show that the full three-dimensional shape of a moving vehicle can be tracked jointly with its position, velocity, and orientation from sparse, noisy LiDAR measurements. The shape is represented as a NURBS surface, a standard CAD surface model, whose axes are scaled, and optionally whose control-point weights are adjusted, by an unscented Kalman filter. Two methods are proposed: one estimates both scaling and weights for better shape fidelity, and a faster one estimates only scaling. On real sensor data of a static and a moving vehicle, both methods match or improve on a state-of-the-art 3D Gaussian-process tracker while running in 3.0-18.8 ms per time step. If the claim holds, autonomous vehicles can use a compact parametric model of an entire 3D obstacle rather than a 2D footprint or bounding box.

What carries the argument

The load-bearing object is the NURBS surface $S_k(u,v,\omega_k,s_k)$ of Eq. (12): a tensor-product rational B-spline whose three Cartesian scaling factors $s_k=(s_x,s_y,s_z)$ and control-point weights $\omega_k$ are part of the tracking state. Around it, the measurement model uses the level-set random hypersurface equation $0 = \alpha_{kl}\, d_{\max}(x_k) - d(x_k, y_{kl}-w_{kl})$ (Eq. 6), where $d$ is a signed Mahalanobis distance to the closest surface point found by minimizing angle difference (Eqs. 20-22). The sign of $d$ is set by membership in the inside set $S_d(x_k)$, so the entire filter update reduces to repeatedly evaluating whether a point lies inside or outside the current NURBS surface. Inference is done with an unscented Kalman filter that processes all measurements in one stacked update to keep the posterior order-independent.

What would settle it

A controlled synthetic test with known ground truth: generate LiDAR points just inside and just outside a known NURBS surface whose true surface and labels are known, run both proposed filters, and check whether a single marginal point whose inside/outside label is deliberately flipped causes the estimated surface to drift or the area error to jump; if the hidden point-in-surface test is inconsistent, the sign of $d$ flips and the pseudo-measurement is biased.

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Extended reading notes

Core claim

The central claim is that a NURBS surface can serve as the shape model in an extended target tracking filter, so that the target's unknown extent and its kinematics are recovered from the same noisy point measurements. The measurements enter through a level-set random hypersurface measurement equation, where each LiDAR point is associated with the surface point closest in angle, and the residual is a signed Mahalanobis distance to that surface point. Method 1 augments the kinematic state with three Cartesian scaling factors plus all NURBS weights, regularized by normalized Gaussian curvature; Method 2 estimates only the three scaling factors. In the reported real-world scenarios, Method 1 reaches an area error of $0.617\,\text{m}^2$ (static) and Method 2 reaches $0.323\,\text{m}^2$ (dynamic), compared with $5.789\,\text{m}^2$ and $3.576\,\text{m}^2$ for the 3D Gaussian-process baseline, with mean runtimes of $18.8\,\text{ms}$ and $3.1\,\text{ms}$ per 50-point update. The authors conclude that both methods are real-time capable and estimate a fully 3D extent rather than a 2D approximation.

Load-bearing premise

The filter assumes that each measurement can be reliably classified as lying inside or outside the currently estimated surface, because that classification sets the sign of the distance residual; the paper never specifies how this classification is done.

Editorial extensions

If this is right

A vehicle's entire 3D extent, not just a 2D footprint, can be estimated in real time from a roof-mounted LiDAR, with each 50-point update taking 18.8 ms for Method 1 and 3.1 ms for Method 2.
The method continues to estimate a useful shape under severe occlusion, since the dynamic scenario's measurements came only from the back of the car.
Estimating NURBS weights in addition to scaling improves static-shape accuracy (area error 0.617 vs 1.433) at the cost of about six times the runtime.
The scaling-only method is claimed to fit approximately cuboid objects such as vehicles, pedestrians, and bicycles.
Because all measurements are incorporated in one stacked UKF update, the resulting estimate is order-independent with respect to the order of the measurements.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The same NURBS machinery could be extended to non-convex targets by replacing the angle-difference closest-point criterion of Eq. (20) with a minimum-distance criterion; the paper notes this possibility in passing but does not implement it.
A practical implementation needs a concrete inside/outside test, such as ray casting to the surface or checking convex hull membership, and the choice of that test will likely dominate tracking quality near the surface.
The fixed process-noise trade-off described in the limitations suggests that an adaptive process-noise schedule for the shape states would be a natural next step, especially for large errors in the initial velocity estimate.

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Signed reviews

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Honest take: this is a legitimate incremental contribution — the first to put NURBS surfaces into level-set RHM for 3D extended target tracking, with scaling-only and scaling-plus-weights variants, and a real-world LiDAR evaluation showing real-time runtimes and lower area error than a 3DGP baseline. Worth reading for anyone doing automotive ETT.

What's good: the NURBS surface representation is natural for vehicles; the two methods give a practical speed/accuracy trade-off (M2 at 3 ms vs 3DGP at 118 ms with 50 points). They honestly report scenarios with occlusions and segmentation errors, and the limitations section is candid about process-noise sensitivity. The comparison against [20] is fair — they correctly note that the 2D B-spline method cannot handle 3D data.

Soft spots, in order of seriousness:

The measurement model is under-specified. Eq. 22 defines the signed distance d through membership in the inside set S_d, and Eq. 5 defines S_d as {z | d ≥ 0}. That is circular on the page. Any implementation has to break the circle with an actual point-in-surface test (ray casting, radial comparison, etc.), but the paper never says which one. This is not just cosmetics: the UKF pseudo-measurement in Eq. 6 depends on the sign of d, and getting it wrong flips the update. A short paragraph specifying the geometric test would fix it.

Second, the weight dynamics in Eq. 18 do not state which surface parameter (u,v) corresponds to weight ω_ij. A control point's weight is associated with a basis function, not a single surface point; the regularization formula needs a defined mapping. Also missing: initial control point coordinates. The figures show the initial surfaces, but the actual coordinates are absent, so the experiments cannot be reproduced from the text.

Third, the results have no error bars or multiple runs, so the area-error gaps (e.g., EA 0.617 vs 5.789 in static) are single-shot numbers. The runtime table is useful but hardware and implementation details are thin.

The stress-test note also flags a conceptual inconsistency between the level-set RHM scaling (α · dmax, which scales the shape radially) and the Mahalanobis distance in Eq. 21, whose level sets are inward offsets rather than scaled copies. For boxy vehicles these two parameterizations can disagree. This is a real issue, but I think it is secondary to the missing inside/outside test; a careful derivation or a clarifying paragraph would address it.

Bottom line: the core idea is sound and the experiments are relevant, but the model description has reproducibility gaps. A serious referee should engage; the paper deserves a major-revision request, not a desk reject. This is a solid 'conditional accept' as is.

Referee Report

4 major / 7 minor

Summary. The paper proposes two NURBS-surface-based methods for 3D extended target tracking from LiDAR point clouds. Method M1 estimates per-axis scaling factors and NURBS weights, while Method M2 estimates scaling factors only. The target state includes kinematics (position, orientation, velocity, curvature) and shape parameters, and a UKF is used for inference. The measurement model is based on a level-set random hypersurface model, with a signed Mahalanobis distance to the closest surface point as the shape function. The methods are evaluated on real-world static and dynamic vehicle scenarios against 3D Gaussian process, 2D B-spline, and point tracking baselines, reporting lower area error and runtimes of 3.0-18.8 ms per step.

Significance. If the results hold, the paper offers a real-time capable 3D extended target tracking method that jointly estimates shape and kinematics, demonstrated on real LiDAR data. The evaluation is thorough in terms of real-world data and comparison with state-of-the-art methods. The paper's explicit limitation discussion and reproducible parameter settings are strengths. However, the measurement model and shape function definition contain circular and underspecified steps that currently prevent full verification.

major comments (4)

[Eqs. (5) and (22)] Eqs. (5) and (22) define S_d(x_k) and the sign of d(x_k,z) circularly: S_d is the set with d >= 0, and d is positive exactly for points in S_d. Moreover, dmax in Eq. (6) is the maximum of d over this set, so it inherits the same ambiguity. The manuscript never specifies an independent geometric test for determining whether a measurement lies inside the NURBS surface (e.g., ray casting, winding number, or a signed projection onto the surface normal). Because Eq. (6) is the only observation model feeding the UKF, an unstated or inconsistent test will bias the pseudo-measurement by 2m and can collapse the shape estimate. Please provide the concrete test used and define dmax accordingly.
[Eqs. (6), (21)-(22)] The level-set RHM scaling relation in Eq. (6) assumes that the level sets of the shape function are scaled copies of the boundary. The shape function defined in Eqs. (21)-(22) as the signed closest-surface Mahalanobis distance has level sets that are inward/outward offsets, which are not scaled copies for a general NURBS surface (e.g., a box-like vehicle). The paper does not justify using the offset distance inside the scaled-copy measurement model; this is a fundamental model mismatch that should be either justified or corrected by redefining the shape function (for example, as a radial function in the star-convex framework).
[Eq. (18)] The weight dynamics in Eq. (18) do not specify which surface parameter point (u,v) is associated with the weight omega_{ij}. The paper should state the mapping from control-point indices to parameter values (e.g., Greville abscissae or knot averages) and define the Gaussian curvature argument accordingly. Without this, the evolution of the weights is ambiguous and the regularization process is not reproducible.
[Section V-C] The limitations discussion does not address the circular inside/outside definition or the offset-versus-scaled level-set issue. These are not cosmetic gaps; they directly affect the measurement model and the meaning of the pseudo-measurement. The manuscript should include an explicit statement of the geometric point-in-surface test used and the assumptions under which Eq. (6) is an appropriate measurement generator.

minor comments (7)

[Eq. (9)] In Eq. (9), the stacked vector elements are scalars, so the transpose on each g(...) is unnecessary and notationally confusing. Please stack the scalars directly as a column vector.
[Eq. (20)] The optimization in Eq. (20) is not described; please give the algorithm (e.g., Newton-Raphson or a coarse-to-fine search) and stopping criteria, as this step dominates the runtime and influences the registration accuracy.
[Tables I and III] The area error metric is defined as the area of the encasing rectangle of the estimated shape; this is a 2D measure and does not capture 3D shape accuracy. Please clarify whether this is intentional and discuss its implications for assessing the 3D shape estimation.
[Sections V-A and V-B] Please specify how the random sampling of the 50 (static) and 20 (dynamic) points is performed (e.g., uniform over the point cloud) and how the convex-hull points are chosen, since the number of points affects the reported runtimes and errors.
[Section V-C] The limitations paragraph mentions that M2 is limited to approximately cuboid objects, but it would be useful to also state the constraints of M1 with respect to shape complexity and the chosen NURBS parameterization.
[Figures 5 and 6] In Figure 6 the legend includes GT, but Figure 5 does not; please add the GT curve to the static-scenario plots or explain its absence.
[References] The paper cites reference [10] for level-set RHM; consider also citing the original star-convex RHM work [9] in the derivation of Eq. (6) to clarify the intended level-set scaling interpretation.

Circularity Check

1 steps flagged · score 3.0 of 10

The signed NURBS shape function is defined circularly: Eq. 22 sets d's sign by membership in S_d, while Eq. 5 defines S_d by the sign of d, leaving the inside/outside test unstated.

self definitional [Eq. (5) in Sec. III-B and Eq. (22) in Sec. IV-B]
"Sd(xk) = {zkl | d(xk,zkl) ≥ 0}. (5) ... It follows the NURBS shape function as the signed mahalonobis distance: d(xk, ˆzkl) = { m(xk, ˆzkl), if ˆzkl ∈ S d(xk); −m(xk, ˆzkl), else . (22)"

The sign of d in Eq. (22) is defined by whether ˆzkl lies in S_d, while Eq. (5) defines S_d as the set where d ≥ 0. Thus the sign of d is defined through a set whose own definition uses the sign of d. The paper gives no independent point-in-surface test (e.g., ray casting, winding number, or radial comparison), so the two equations do not pin down d uniquely. Since Eq. (6) uses d(xk, ykl − wkl) in every pseudo-measurement, the UKF update inherits this undefined sign; a sign flip would shift the pseudo-measurement by 2m and bias the estimated position, orientation, and surface. This is a genuine definitional circularity, though it does not reduce the shape estimates to fitted constants.

full rationale

The only identified circular step is the mutual definition of the shape function d and the inside set S_d in Eqs. (5) and (22). This is load-bearing because the level-set RHM measurement model, Eq. (6), evaluates d for every measurement and feeds the UKF; without an independent inside/outside rule the sign of d is formally unspecified. However, the paper's central claims rest on external evaluations: M1 and M2 are compared with 3DGP, 2DBS, and SP on real LiDAR data with RMSE tables, and the NURBS scaling/weight states are not fitted constants that make the reported errors true by construction. The self-citations (segmentation, bounding box, prior work) are used as data preprocessing or comparisons, not to justify the measurement model. The circularity is therefore real but local and does not force the numerical conclusions; I assign a moderate score of 3 rather than a higher score that would indicate the whole derivation reduces to its inputs.

Assumptions & free parameters 10 free parameters · 6 assumptions · 0 invented entities

The central claim rests on many hand-tuned parameters, including the control point grid, B-spline degrees, damping factor, process noise covariances, initial surface definition, sample count, and an unspecified measurement noise covariance. It also relies on prior modeling assumptions about level-set RHMs, star-convexity, an inside/outside test, and the availability of dmax. The initial control point positions are not published, and the measurement noise covariance is not given, which limits independent verification. No new physical entities are introduced.

free parameters (10)

Control point grid size (nu, nv) = M1: (7,4); M2: (5,4)
This hand-chosen grid defines the shape space dimension and the initial surface, directly affecting accuracy and runtime.
B-spline degrees (p, q) = M1: (3,3); M2: (2,2)
Hand-chosen to trade off surface smoothness against computational cost.
Damping factor nu = 0.001
Used in the weight dynamics of Eq. (18); hand-chosen with no sensitivity analysis.
Weight process noise Q_wij = 0.1 (static), 0.01 (dynamic)
Hand-set; controls the evolution speed of the NURBS weights and hence shape convergence.
Scaling process noise Q_s = 1e-7
Hand-set; controls the evolution of the scaling factors in the UKF.
Input noise C_v and C_c = 1e-4 static; 0.2 and 0.05 dynamic
Hand-set to reflect expected target dynamics; the limitations section notes sensitivity to these values.
Initial scaling factors and weights = All set to 1
Chosen by hand; the starting surface is already a rough vehicle shape, so the filter begins near the solution.
Initial control point positions P_ij = Not stated in the text; shown only in Figure 4
These coordinates define the initial NURBS surface, yet are not published, so the method cannot be re-implemented from the paper alone.
Number of sampled measurements = 50 (static), 20 (dynamic)
Hand-chosen as a compromise between estimation quality and runtime, as stated in Section V.
Measurement noise covariance R_kl = Not specified
Required in Eqs. (3) and (21); no values are given, yet this covariance enters the Mahalanobis-distance shape function and the UKF update directly.

assumptions (6)

domain assumption The level-set random hypersurface measurement model of [10] (Eq. 6) correctly maps an inside measurement to the boundary via the uniform scaling factor alpha.
Adopted from prior work without independent validation for the NURBS shape function.
domain assumption The target is star-convex, so every direction from the center intersects the surface exactly once, making the angle-difference criterion of Eq. (20) valid.
The paper states 'we deal with dynamic and star-convex targets like cars'; non-convex objects or heavy occlusion violate this.
domain assumption A point can be reliably classified as inside or outside the NURBS surface.
The definition of S_d in Eqs. (5) and (22) is circular; an unstated geometric test is required for implementation.
domain assumption The maximum shape function value dmax in Eq. (6) is available or approximable in real time.
dmax bounds the random scaling factor, but no computation or approximation method is given.
standard math NURBS basis functions and the Gaussian curvature formulas from [25] are standard and correctly evaluated.
Unproved background results from geometric modeling.
domain assumption The constant curvature and velocity (CCV) model is adequate for vehicle motion in the scenarios.
Standard for vehicles, but no model mismatch analysis is provided.

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Cite this review

Pith. "Pith review of Fast 3D Extended Target Tracking using NURBS Surfaces." pith.science (2026). https://pith.science/paper/ACBPGVOS

@misc{pith2026190900767,
  author       = {Pith},
  title        = {Pith review of: Fast 3D Extended Target Tracking using NURBS Surfaces},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ACBPGVOS}},
  note         = {Machine review of arXiv:1909.00767}
}

read the original abstract

This paper proposes fast and novel methods to jointly estimate the target's unknown 3D shape and dynamics. Measurements are noisy and sparsely distributed 3D points from a light detection and ranging (LiDAR) sensor. The methods utilize non-uniform rational B-splines (NURBS) surfaces to approximate the target's shape. One method estimates Cartesian scaling parameters of a NURBS surface, whereas the second method estimates the corresponding NURBS weights, too. Major advantages are the capability of estimating a fully 3D shape as well as the fast processing time. Real-world evaluations with a static and dynamic vehicle show promising results compared to state-of-the-art 3D extended target tracking algorithms.

Figures

Figures reproduced from arXiv: 1909.00767 by the authors.

**Figure 1.** NURBS surface with estimated scaling parameters and weights. Crossed points denote the control points of the NURBS surface, whereas the blue and red points are used as measurements. The used measurements are sampled from the segmented point cloud of the vehicle. dense 3D point cloud of the Velodyne-HDL64, where the 2D approximation does not hold. Therefore, we propose a full three-dimensional estimation of the targe… view at source ↗

**Figure 2.** Exemplary control net of a NURBS surface with 16 control points Pij and corresponding weights ω ij , i = 0, . . . , nu, j = 0, . . . , nv. Number of control points in u-direction nu and v-direction nv equals 4. IV. NURBS SURFACE MODEL In this part the used NURBS surface, the resulting NURBS shape function as well as the state’s shape parts are explained. A. NURBS Surface Function Let Pij ∈ R 3×(nu+1)×(nv+1) be the b… view at source ↗

**Figure 3.** Illustration of different segmentation of a vehicle. The blue color correspond to points originating from the 2D convex hull, whereas red points are randomly sampled over the whole point cloud. (a) M1 with 28 control points. (b) M2 with 20 control points. (c) 3DGP with 60 control points. (d) SP with bounding box [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

Figures from the paper (3 more)

**Figure 4.** Figure 4: The initial shape of the methods is shown in the upper half, where the estimated shape is shown in the lower half. The measurements (red and blue points) only originate from the back of the vehicle. 0 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 320 0.00 0.1…

**Figure 5.** Figure 5: Results of estimating a static vehicle, with velocity vk, area error eA, Cartesian position error epos and orientation error eψ. The area of the estimated shapes is defined as the product of the length and width of the encasing rectangle. Furthermore, one time step k i…

**Figure 6.** Figure 6: Results of estimating a dynamic vehicle, with velocity vk, area error eA, Cartesian position error epos and orientation error eψ. The area of the estimated shapes is defined as the product of the length and width of the encasing rectangle. Furthermore, one time step k …

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Reference graph

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [20]

A Cartesian B-Spline Vehicle Model for Extended Object Tracking,

H. Kaulbersch, J. Honer, and M. Baum, “A Cartesian B-Spline Vehicle Model for Extended Object Tracking,” in Proc. Int. Conf. Information Fusion (FUSION) , 2018

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