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Recursive Estimation of Orientation Based on the Bingham Distribution

Gerhard Kurz, Igor Gilitschenski, Simon Julier, Uwe D. Hanebeck

arXiv:1304.8019v1eess.SYcs.ROcs.SY

TL;DR

The paper addresses directional estimation when conventional Gaussian filters ignore periodicity and 180° symmetry. It develops a recursive two-dimensional Bingham filter using approximate hyperspherical composition for prediction and Bingham multiplication for Bayesian updates. The filter outperforms a Kalman filter in most simulated cases, converges faster, and remains fast enough for real-time applications, while extending conceptually to quaternion-based 3D orientation estimation.

  • Problem

    Conventional Kalman-style filters can perform poorly for angular data because Gaussian and linear-space assumptions ignore periodicity, especially under large angular uncertainty.

    Source: §1 · p. 2

  • Method

    The paper develops a recursive two-dimensional Bingham filter using covariance-matched composition for prediction and multiplication closure for Bayesian updates.

    Source: §3 · p. 3

  • Results

    The proposed filter outperforms the Kalman filter in most simulated cases, including every time step in mean angular error, and converges significantly faster.

    Source: §6 · p. 12, §7 · p. 12

  • Takeaways & Limitations

    The Bingham approach supports circular estimation with 180° symmetry, is fast enough for real-time applications, and can be generalized to quaternion-based 3D orientation estimation.

    Source: §1 · p. 2, §6 · p. 12, §7 · p. 12

    The paper evaluates only the two-dimensional case; future 3D use requires efficient Bingham-parameter estimation and confluent hypergeometric-function evaluation.

    Source: §4.5 · p. 7, §7 · p. 12

Abstract

from arXiv · show

Directional estimation is a common problem in many tracking applications. Traditional filters such as the Kalman filter perform poorly because they fail to take the periodic nature of the problem into account. We present a recursive filter for directional data based on the Bingham distribution in two dimensions. The proposed filter can be applied to circular filtering problems with 180 degree symmetry, i.e., rotations by 180 degrees cannot be distinguished. It is easily implemented using standard numerical techniques and suitable for real-time applications. The presented approach is extensible to quaternions, which allow tracking arbitrary three-dimensional orientations. We evaluate our filter in a challenging scenario and compare it to a traditional Kalman filtering approach.

1. Introduction

Angular estimation differs from Euclidean estimation because periodicity and 180° symmetry must be represented explicitly. The paper introduces a two-dimensional recursive Bingham filter for such orientation problems and situates it within prior filtering approaches.

1. Introduction

  • Angular estimation must account for periodicity, non-vector-space geometry, and 180° symmetry when estimating axes or directionless orientations.These issues arise in applications including object tracking, robotics, and augmented reality.Source: §1 · p. 1
  • Kalman, extended Kalman, and unscented Kalman filters can perform poorly for circular data because Gaussian, linear-space assumptions ignore periodicity and may cause divergence under large uncertainty.Existing alternatives reposition or discard measurements, impose constraints, or locally approximate manifolds while often retaining normally distributed noise models.Source: §1 · p. 2
  • The paper presents the first published recursive filter based on the Bingham distribution, focusing on two-dimensional axis estimation.The Bingham distribution is defined on hyperspheres of arbitrary dimension, supporting the paper’s broader extension claim.Source: §1 · p. 2
  • The Bingham-based approach can extend to four-dimensional unit quaternions for representing full three-dimensional orientations with 180° symmetry.Its hyperspherical restriction and antipodal symmetry address quaternion normalization and the equivalence of q and −q.Source: §1 · p. 2
  • The paper reviews directional statistics, derives the recursive Bingham filter, evaluates it in simulation, and compares it with Kalman filtering.The stated organization covers related work, Bingham operations, filtering, simulations, and conclusions.Source: §1 · p. 2

2. Related Work

Related work places the paper within directional statistics and prior orientation filters based on von Mises and wrapped normal distributions. Earlier Bingham research established the distribution and a small set of applications, but not this recursive filter.

2. Related Work

  • Directional statistics studies random variables on manifolds such as circles and spheres rather than vector spaces.Standard references include Mardia and Jupp and Jammalamadaka and Sengupta.Source: §2 · p. 2
  • Prior two-dimensional orientation work includes recursive von Mises filtering on SO(2) and nonlinear filtering with von Mises and wrapped normal distributions.The literature also includes broader investigations of 2D orientation, such as Krindis et al..Source: §2 · p. 3
  • Bingham research began with Bingham and continued through Kent and Jupp and Mardia, with applications including geology and Monte Carlo pose estimation.The authors note that their implementation is not based on the libbingham library.Source: §2 · p. 3

3. Key Idea of the Bingham Filter

The proposed filter represents all probability densities with the antipodally symmetric Bingham distribution, matching circular estimation with 180° symmetry. Its recursion combines approximate composition for prediction with exact closure under multiplication for Bayesian updating.

3. Key Idea of the Bingham Filter

  • The filter uses the Bingham distribution for all probability densities because its hyperspherical, antipodally symmetric form matches 180°-symmetric angular estimation.Although demonstrated in two dimensions, most presented methods are described as easily generalizable to higher dimensions.Source: §3 · p. 3
  • Prediction is implemented through hyperspherical composition, approximated by matching covariance matrices because Bingham distributions are not closed under composition.Composition provides the analogue of convolving a state with system noise in an additive Euclidean model.Source: §3 · p. 3
  • The Bayesian update multiplies the prior and likelihood, and Bingham distributions remain closed under this multiplication.This closure supplies the posterior density required by the recursive filter.Source: §3 · p. 3

4. Bingham Distribution

The Bingham distribution represents antipodally symmetric directional uncertainty on hyperspheres. This section develops its density, normalization, multiplication, parameter estimation, and approximate composition operations for recursive filtering.

4.1. Probability Density Function ||

  • Bingham densities are defined on the unit hypersphere with orientation matrix M, ordered concentration matrix Z, and a normalization constant.The mode is obtained from the last column of M under the chosen parameterization.Source: §4.1 · p. 4
  • The distribution is invariant to 180° rotations and resembles a Gaussian only for small uncertainty.For z1 < −15, corresponding to about 11°, the Gaussian approximation is nearly indistinguishable; larger uncertainties produce poor approximations.Source: §4.1 · p. 4, Figure 3 · p. 6

4.2. Normalization Constant

  • The normalization constant uses a hypergeometric function of a matrix argument, reducing in two dimensions to a scalar-argument computation.This reduction supports numerical evaluation of the two-dimensional distribution.Source: §4.2 · pp. 4–5

4.3. Multiplication

  • Bingham distributions are closed under multiplication with renormalization, enabling Bayesian products of prior and likelihood densities.The resulting parameters are obtained from the eigenvectors and shifted eigenvalues of the combined concentration matrix.Source: §4.3 · pp. 5–6

4.4. Estimation of Parameters

  • Parameter estimation connects Bingham distributions and covariance matrices through eigendecomposition and numerical solution of the concentration equations.The paper states that each Bingham distribution is uniquely defined by its covariance matrix and vice versa.Source: §4.4 · pp. 6–7

4.5. Composition

  • In the two-dimensional complex representation, composition is complex multiplication and corresponds to adding angles for unit vectors.The operation is required to respect 180° symmetry and to induce an algebraic group structure after identifying antipodal points.Source: §4.5 · pp. 7–8
  • Composition is the directional analogue of adding random variables, but it is approximated because Bingham distributions are not closed under composition.The method composes covariance matrices, then estimates the parameters of the Bingham distribution matching the resulting covariance.Source: §4.5 · pp. 7–8

5. Filter Implementation

The filter implements recursive Bingham estimation through a system prediction followed by a Bayesian measurement update. Both models represent identity transformations disturbed by Bingham-distributed noise, with optional known angular offsets.

5. Filter Implementation

  • The system and measurement models treat state and observations as identity transformations perturbed by Bingham-distributed noise.The process model can include a constant offset or known angular velocity, or apply a rotation before adding unbiased noise.Source: §5 · p. 9

5.1. Prediction

  • The prediction step composes the previous estimate with Bingham-distributed system noise.The composition operation provides the directional analogue of disturbing a state with process noise.Source: §5.1 · p. 9

5.2. Update

  • The measurement update multiplies the prior distribution by the likelihood according to Bayes’ rule and renormalizes the result.The update uses the Bingham multiplication operation to obtain the posterior distribution.Source: §5.2 · p. 10

6. Evaluation

The evaluation compares the proposed Bingham filter with modified Kalman and wrapped-normal filters in a challenging 180°-symmetric axis-estimation scenario. Across 1000 Monte Carlo runs, the Bingham filter converges faster and outperforms the Kalman filter in most cases, while matching the wrapped-normal filter closely.

Experimental setup

  • The Kalman baseline is adapted by restricting estimates to 0 ≤ x_k ≤ π and shifting measurements to handle circularity and 180° symmetry.The simulated example estimates an axis such as a symmetric rotor-blade or robotic-joint axis.Source: §6 · p. 10
  • The simulation runs for 100 time steps with a poor initial estimate and compares confidence intervals against Kalman 2σ bounds.The specified standard deviations are 44° initially, 3° for system noise, and 30° for measurement noise.Source: §6 · p. 11

Results

  • The proposed filter outperforms the Kalman filter in most runs and has a faster convergence rate.This advantage is attributed to modeling the geometry of circular data with the Bingham distribution rather than using a Gaussian model.Source: §6 · p. 12
  • The evaluation is based on 1000 Monte Carlo runs and compares the proposed Bingham filter with a Kalman filter using mean angular error at every time step.Figure 5 also compares the proposed Bingham filter with a wrapped-normal filter, whose angular error is almost indistinguishable from the proposed filter.Source: §6 · p. 12

Computational cost

  • The non-optimized MATLAB implementation requires approximately 60 ms per prediction-update step on a standard laptop.The authors state that faster evaluation of the hypergeometric function could substantially reduce this time.Source: §6 · p. 12

Results

  • The Bingham and wrapped-normal filters have almost indistinguishable angular errors.The wrapped-normal filter was modified for the 180° case, but it cannot be easily generalized to higher dimensions.Source: §6 · p. 12

7. Conclusion

The paper concludes that its recursive Bingham filter handles circular estimation with 180° symmetry and outperforms a modified Kalman approach in simulation. Extending the method to recursive 3D pose estimation remains future work.

Conclusion

  • Simulations show the Bingham filter outperforming a traditional Kalman-filter solution modified for circular estimation.Source: §7 · p. 12
  • The proposed filter applies to circular estimation problems with 180° symmetry.Source: §7 · p. 12

Future work

  • Recursive 3D pose estimation is proposed as future work through the four-dimensional quaternion case.Open challenges include efficient Bingham-parameter estimation and evaluation of the confluent hypergeometric function.Source: §7 · p. 12

Appendix A. Proof of Lemma 2.

The supplied appendix passages do not provide substantive proof content for Lemma 2; they contain a fragmented formula and an unrelated figure caption.

Appendix A

Appendix B. Proof of Lemma 3.

Appendix B derives covariance terms for composed variables by expanding expectations and using independence, linearity, symmetry, and individual matrix-entry calculations.

Appendix B

  • The covariance of the composition is computed by calculating matrix entries individually.Source: Appendix B · p. 15
  • The derivation uses independence of x variables, linearity of expectation, and Bingham symmetry.Source: Appendix B · p. 15
  • The diagonal covariance expression includes c22 = a11b22 + 2a12b12 + a22b11.Source: Appendix B · p. 15
  • The off-diagonal covariance is formed from the expectation of the product minus the product of expectations.Source: Appendix B · p. 15
  • The supplied expressions expand the first covariance entry using second moments and cross-moment terms.Source: Appendix B · p. 15
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