A Laplace-inspired Distribution on SO(3)
for Probabilistic Rotation Estimation
Abstract
Estimating the 3DoF rotation from a single RGB image is an important yet challenging problem. Probabilistic rotation regression has raised more and more attention with the benefit of expressing uncertainty information along with the prediction. Though modeling noise using Gaussian-resembling Bingham distribution and matrix Fisher distribution is natural, they are shown to be sensitive to outliers for the nature of quadratic punishment to deviations.
In this paper, we draw inspiration from multivariate Laplace distribution and propose a novel Rotation Laplace distribution on SO(3). Rotation Laplace distribution is robust to the disturbance of outliers and enforces much gradient to the low-error region, resulting in a better convergence. Our extensive experiments show that our proposed distribution achieves state-of-the-art performance for rotation regression tasks over both probabilistic and non-probabilistic baselines.
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Method
Revisit Matrix Fisher Distribution
Matrix Fisher distribution is a probability distribution over SO(3) for rotation matrices, whose probability density function is in the form of
where parameter A is an arbitrary 3×3 matrix and F(A) is the normalizing constant. The mode and dispersion of the distribution can be computed from computing singular value decomposition of the parameter A.
It is shown that matrix Fisher distribution is highly relevant with zero-mean Gaussian distribution near its mode.
Rotation Laplace Distribution
We propose Rotation Laplace distribution on SO(3), defined aswhere parameter A is an arbitrary 3×3 matrix, S is the diagonal matrix composed of the proper singular values of matrix A and F(A) is the normalizing constant. The mode and dispersion of the distribution can be computed from computing singular value decomposition of the parameter A.
Rotation Laplace distribution is highly relevant with zero-mean multivariate Laplace distribution near its mode.
Discrete Approximation of the Normalization Factor
We approximate the normalization factor of Rotation Laplace distribution through equivolumetric discretization over SO(3) manifold.
In experiments, we discretize SO(3) space into about 37k points.
Quaternion Laplace Distribution
We propose our extension of Laplace-inspired distribution for quaternions, namely, Quaternion Laplace distribution.
where M is a 4x4 orthogonal matrix, and Z is a 4x4 diagonal matrix. Quaternion Laplace distribution is highly relevant with zero-mean multivariate Laplace distribution near its mode.
Our proposed Rotation Laplace distribution is equivalent to Quaternion Laplace distribution, similar to the equivalence of matrix Fisher distribution and Bingham distribution, demonstrating the consistency of our derivations.
Experiment
On common benchmark datasets of rotation estimation from RGB images (ModelNet10-SO(3) and Pascal3D+), we achieve a significant and consistent performance improvement over all baselines.
The predicted distributions can exhibit high uncertainty when the object has rotational symmetry, leading to near 180deg errors (a-c), or the input image is with low resolution (d). Subfigure (e-f) show cases with high certainty and reasonably low errors.