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Tracking the Orientation and Axes Lengths of an Elliptical Extended Object

Shishan Yang, Marcus Baum

arXiv:1805.03276v2eess.SYcs.SY

TL;DR

A single scale-matrix uncertainty cannot distinguish uncertainty in ellipse orientation from uncertainty in semi-axis lengths. MEM-EKF* uses a closed-form Kalman-filter update to estimate ellipse parameters. Simulations report lower error than random-matrix approaches on all three turns, under a nearly static semi-axis assumption.

  • Problem

    A single scalar uncertainty for an ellipse scale matrix cannot distinguish uncertainty in semi-axis lengths from orientation, although practical applications may need that distinction.

    Source: §I · p. 1

  • Method

    MEM-EKF* combines a multiplicative-error measurement equation with analytic moment calculation and linearization for compact closed-form Kalman-filter updates of kinematic state and ellipse parameters.

    Source: A. Contribution · p. 1, §V · p. 8

  • Results

    For all three turns, the proposed tracker has lower error by Gaussian Wasserstein distance than both random-matrix approaches; with turn-rate estimation and a constant-turn model, MEM-EKF* (1) has the least estimation error.

    Source: B. Comparison with Random Matrix… · p. 7, C. Comparison with Random Matrix… · p. 8

  • Takeaways & Limitations

    The full joint covariance allows explicit modeling of temporal evolution and dependencies among ellipse parameters, including fixed unknown semi-axes with varying orientation.

    Source: A. Contribution · p. 1, §V · p. 8

    MEM-EKF* uses more parameters than random-matrix approaches, and its performance is sensitive to their choice.

    Source: C. Comparison with Random Matrix… · p. 8

Abstract

from arXiv · show

Extended object tracking considers the simultaneous estimation of the kinematic state and the shape parameters of a moving object based on a varying number of noisy detections. A main challenge in extended object tracking is the nonlinearity and high-dimensionality of the estimation problem. This work presents compact closed-form expressions for a recursive Kalman filter that explicitly estimates the orientation and axes lengths of an extended object based on detections that are scattered over the object surface (according to a Gaussian distribution). Existing approaches are either based on Monte Carlo approximations or do not allow for explicitly maintaining all ellipse parameters. The performance of the novel approach is demonstrated with respect to the state-of-the-art by means of simulations.

I. INTRODUCTION

Extended object tracking must estimate both motion and shape from spatially distributed detections, but common ellipse models do not separately represent uncertainty in orientation and semi-axis lengths. The paper introduces MEM-EKF*, a compact Kalman-filter approach that explicitly estimates those shape parameters and their joint covariance.

I. INTRODUCTION

  • In high-noise, low-count settings, ellipses are common; random-matrix methods,,, are state-of-the-art for surface scattering, but cannot separate orientation and semi-axis uncertainty.Inverse-Wishart shape uncertainty is encoded in a single scalar.Source: §I · p. 1

A. Contribution

  • The tracker estimates orientation and semi-axis lengths as a 3D vector and retains their 3 × 3 joint covariance, supporting explicit modeling of individual shape dynamics.For example, the semi-axes can be modeled as fixed but unknown while orientation varies.Source: A. Contribution · p. 1
  • MEM-EKF* combines a multiplicative-error measurement model with tailored linearization and moment approximations for a compact Kalman-filter update.The shape model uses multiplicative noise; a problem-tailored second-order approximation avoids complex Hessian calculations.Source: A. Contribution · p. 2

B. Related Work

  • Random-matrix variants, retain inverse-Wishart uncertainty, while approaches, do not use the common spatial-distribution model.The inverse-Wishart representation does not explicitly maintain uncertainty for individual shape parameters.Source: §I · p. 1, B. Related Work · p. 2
  • Random-hypersurface tracking, uses a Cholesky shape-state representation and a source-angle point estimate, which is poor under high measurement noise.The proposed method instead uses two-dimensional scaling rather than the original approach’s one-dimensional scaling factor.Source: B. Related Work · p. 2
  • Other related approaches include particle-based tracking,,, and batch optimization for extended targets or elliptical shapes [31],,.Source: B. Related Work · p. 2

C. Structure

This section introduces the parameterization, measurement model, and process model for tracking one extended object whose shape is approximated by an ellipse. Its shape is represented by orientation and semi-axes, while measurements are modeled as noisy detections related to the object state.

C. Structure

  • The paper first introduces the single-object elliptical model, then derives recursive measurement and time updates and evaluates the method in later sections.Source: C. Structure · p. 2

A. Parameterization

  • The object state includes its center and potentially velocity or acceleration, while its ellipse is parameterized by orientation and two semi-axes.Source: A. Parameterization · p. 3

B. Measurement Model

  • The measurement model represents detections as arising from the object's extent, with a varying number of independent two-dimensional measurements per time step.Source: B. Measurement Model · p. 3
  • The model approximates the uniform spatial distribution of measurement sources with a Gaussian distribution and assumes the multiplicative error is zero-mean and distributed on a unit circle.Source: B. Measurement Model · p. 3
  • An explicit equation relates a measurement source to the kinematic state and ellipse shape through a multiplicative error term.Source: B. Measurement Model · p. 3
  • With additive Gaussian measurement noise, the stated measurement likelihood is equivalent to the random-matrix likelihood when the extension matrices are matched with Ch=1.Source: B. Measurement Model · p. 3

C. Dynamic Model

The dynamic model allows general temporal models for the kinematic state and shape parameters. For simplicity, this section assumes linear evolution equations with zero-mean Gaussian process noise.

C. Dynamic Model

  • In general, the temporal evolution models for the state and shape parameters are unrestricted.Source: C. Dynamic Model · p. 3
  • For simplicity, the section assumes linear equations for the temporal evolution of the kinematic state and shape parameters.Source: C. Dynamic Model · p. 4
  • The process noises for the state and shape models are assumed to be zero-mean Gaussian with covariance matrices.Source: C. Dynamic Model · p. 4

III. ESTIMATION

The estimation method uses a Kalman-filter measurement and time update while approximating moments needed for nonlinear measurements with multiplicative noise. It decouples kinematics and shape in the density, but retains their interdependencies in the update and prediction formulas.

III. ESTIMATION

  • The method factorizes the joint density of kinematics and extension, avoiding maintenance of their cross-correlation while retaining interdependencies in update and prediction formulas.Source: §III · p. 4
  • Problem-tailored linearization and analytic moment calculations approximate the moments required by the nonlinear Kalman filter with zero-mean multiplicative measurement noise.Source: §III · p. 4

A. Measurement Update

  • Measurements from one scan are incorporated sequentially in arbitrary order, and approximations can make results differ slightly across orderings.Source: A. Measurement Update · p. 4
  • The update uses a pseudo-measurement to update shape parameters because a linear estimator using the actual measurement does not change them after one measurement.Source: A. Measurement Update · p. 4
  • The measurement update algorithm is presented in Table I.Source: Table I · p. 4

1) Kinematic State Update:

The kinematic-state update uses the actual measurement, while nonlinear dependence on shape parameters makes the required measurement and cross-correlation moments difficult to compute. The section approximates these moments through linearization and covariance calculations.

1) Kinematic State Update:

2) Shape Update:

The shape update constructs a pseudo-measurement from each actual measurement and uses it to update the shape parameters with Kalman filter formulas. Its moments are derived from the actual measurement covariance, with the pseudo-measurement uncorrelated with the actual measurement.

2) Shape Update:

  • The shape parameters are updated with a pseudo-measurement constructed from the actual measurement using Kalman filter formulas.The conversion is described as uncorrelated: the pseudo-measurement is uncorrelated with the actual measurement.Source: 2) Shape Update · p. 5
  • The expected pseudo-measurement comprises the centralized second moments of the actual measurements, derived from the actual measurement covariance.The covariance of the pseudo-measurement is also derived, using Isserlis’s theorem [37] for the fourth centralized moments.Source: 2) Shape Update · p. 5
  • The cross-covariance between the pseudo-measurement and shape parameters is approximated by linearizing the measurement-to-shape relation.The resulting measurement-update procedure is given in Table I.Source: 2) Shape Update · p. 5

B. Time Update

Because the kinematic state and shape parameters follow linear temporal evolution models, their time update uses standard Kalman filter formulas.

B. Time Update

  • The linear temporal evolution models for the kinematic state and shape parameters permit standard Kalman filter time-update formulas.Source: B. Time Update · p. 5

IV. EVALUATION

The evaluation examines moment approximations and the proposed shape tracker, using Gaussian Wasserstein distance to assess location and extent together. In a stationary-ellipse simulation, the Monte Carlo moment approximation outperforms the analytic approaches, while the proposed approximation nearly matches exact moments in both noise settings.

IV. EVALUATION

  • The evaluation first tests moment approximations, then compares the shape tracker with random-matrix approaches under extension dynamics and turn-rate estimation.It also includes a scenario coupling object extent with its kinematics.Source: §IV · p. 6
  • The evaluation assesses location and extent together with Gaussian Wasserstein distance, combining orientation and axes-length errors into one scalar.Source: §IV · p. 6

A. Evaluation of Moment Approximations

  • The proposed moment approximation nearly matches the exact moments in both low- and high-measurement-noise scenarios, although Monte Carlo approximation performs better.The simulation uses a stationary ellipse and evaluates two measurement-noise covariance matrices.Source: A. Evaluation of Moment Approximations · p. 7

B. Comparison with Random Matrix Approaches using Constant Velocity Motion Models

The comparisons show how explicit shape-parameter modeling performs when an object's orientation or turn rate changes while its size remains nearly fixed. The constant-turn results also show that performance depends on process-noise tuning.

B. Comparison with Random Matrix Approaches using Constant Velocity Motion Models

  • With fixed semi-axes and changing orientation, MEM-EKF* can model low axis-length noise and high orientation noise, unlike the compared random-matrix approaches' single extent-uncertainty parameter.The simulation uses uniformly distributed surface measurements, with a Poisson mean of 20 detections per scan.Source: B. Comparison with Random Matrix… · p. 7
  • Across all three turns, the proposed tracker has lower extent error than both random-matrix approaches, which perform similarly overall.The comparison assumes nearly static semi-axes.Source: B. Comparison with Random Matrix… · p. 7

C. Comparison with Random Matrix Approaches using Constant Turn Motion Models

  • For a manoeuvring target, the prediction can incorporate correlation between orientation and turn rate while assuming the object's size is constant apart from noise.The shape process noise is tuned to allow only slight semi-axis changes.Source: C. Comparison with Random Matrix… · pp. 7–8
  • MEM-EKF* (1) has the least estimation error; MEM-EKF* (2) performs worse when its parameters do not fit, while turn-rate estimation and a constant-turn motion model significantly improve tracking performance.Source: C. Comparison with Random Matrix… · p. 8
  • The variable-turn simulation compares MEM-EKF* with M4 and a random-matrix baseline that does not incorporate turn rate.The turn rate rises from 0 to 20 degrees per second and then returns to 0.Source: C. Comparison with Random Matrix… · p. 8
  • MEM-EKF* uses more parameters than random-matrix approaches, and its performance is sensitive to their choice.The reported results for MEM-EKF* (2) are worse because its parameters do not fit.Source: C. Comparison with Random Matrix… · p. 8

V. CONCLUSION

Surface-scattered measurements and nonlinear estimation make extended object tracking challenging, with closed-form solutions rarely available. The paper introduces MEM-EKF*, a compact closed-form tracker for ellipse orientation and axes lengths, and identifies extensions for future work.

  • Extended object tracking is challenging for surface-scattered measurements because the estimation problem is highly nonlinear and closed-form solutions are rare.Source: §V · p. 8
  • MEM-EKF* uses a multiplicative-error measurement model and tailored moment calculations and linearization to derive a Kalman filter measurement update.It provides an intuitive ellipse parameterization, the full joint covariance of ellipse parameters, and compact closed-form formulas.Source: §V · p. 8
  • Future work will investigate non-elliptical shapes or known extents and embedding the approach in multi-object trackers.Source: §V · p. 8

APPENDIX A DERIVATIONS FOR THE JOINT MOMENTS

The appendix supports the filter derivation by detailing moment calculations and the linearization of the pseudo-measurement equation. It also states the identities and independence assumptions used in deriving Equation (19).

A. Equation (19)

  • For Equation (19), the derivation uses trace cyclic invariance, linearity of expectation, and independence between h and p.Source: A. Equation (19) · p. 9

B. Pseudo-measurement Covariance

  • The pseudo-measurement covariance is calculated from fourth centralized measurement moments, using Isserlis’ theorem, and its entries follow from covariances of pseudo-measurement components.The text defines c_mn as the covariance of centered measurement components.Source: B. Pseudo-measurement Covariance · p. 9

1) Linearization of the Pseudo-measurement Equation:

  • The pseudo-measurement equation is rewritten for linearization, and its cross-covariance with the shape parameters is approximated row by row.Source: 1) Linearization of the… · p. 9
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