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Are Euler angles a useful rotation parameterisation for pose estimation with Normalizing Flows?

Giorgos Sfikas, Konstantina Nikolaidou, Foteini Papadopoulou, George Retsinas, Anastasios L. Kesidis

TL;DR

This work investigates using Euler angles as a 3-DOF rotation parameterisation for Normalizing Flows in 3D pose estimation on the $ ext{SO}(3)$ manifold. It contrasts an Euler-angle–based NF with rotation-matrix–based baselines across synthetic and real datasets, highlighting an inductive bias that yields competitive log-likelihoods and notably faster training times. The method leverages Möbius transforms applied to each Euler angle in a round-robin coupling flow, with angles encoded via $( ext{cos}, ext{sin})$ to respect periodicity, while acknowledging gimbal-lock-related limitations. Overall, Euler-angle flows offer a simpler and efficient alternative with strong performance, albeit with caveats near singular regions and highly multimodal ground truths.

Abstract

Object pose estimation is a task that is of central importance in 3D Computer Vision. Given a target image and a canonical pose, a single point estimate may very often be sufficient; however, a probabilistic pose output is related to a number of benefits when pose is not unambiguous due to sensor and projection constraints or inherent object symmetries. With this paper, we explore the usefulness of using the well-known Euler angles parameterisation as a basis for a Normalizing Flows model for pose estimation. Isomorphic to spatial rotation, 3D pose has been parameterized in a number of ways, either in or out of the context of parameter estimation. We explore the idea that Euler angles, despite their shortcomings, may lead to useful models in a number of aspects, compared to a model built on a more complex parameterisation.

Are Euler angles a useful rotation parameterisation for pose estimation with Normalizing Flows?

TL;DR

This work investigates using Euler angles as a 3-DOF rotation parameterisation for Normalizing Flows in 3D pose estimation on the manifold. It contrasts an Euler-angle–based NF with rotation-matrix–based baselines across synthetic and real datasets, highlighting an inductive bias that yields competitive log-likelihoods and notably faster training times. The method leverages Möbius transforms applied to each Euler angle in a round-robin coupling flow, with angles encoded via to respect periodicity, while acknowledging gimbal-lock-related limitations. Overall, Euler-angle flows offer a simpler and efficient alternative with strong performance, albeit with caveats near singular regions and highly multimodal ground truths.

Abstract

Object pose estimation is a task that is of central importance in 3D Computer Vision. Given a target image and a canonical pose, a single point estimate may very often be sufficient; however, a probabilistic pose output is related to a number of benefits when pose is not unambiguous due to sensor and projection constraints or inherent object symmetries. With this paper, we explore the usefulness of using the well-known Euler angles parameterisation as a basis for a Normalizing Flows model for pose estimation. Isomorphic to spatial rotation, 3D pose has been parameterized in a number of ways, either in or out of the context of parameter estimation. We explore the idea that Euler angles, despite their shortcomings, may lead to useful models in a number of aspects, compared to a model built on a more complex parameterisation.

Paper Structure

This paper contains 12 sections, 4 equations, 2 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Normalizing Flows on circular and spherical domains
  3. Elements.
  4. Conditional flows.
  5. Handling circular and spherical topologies.
  6. Defining flows on $\mathcal{SO}(3)$.
  7. Proposed Euler Angles-based Flow
  8. Euler angles.
  9. Defining flows with Euler angles.
  10. Experimental Results
  11. Training speed comparison
  12. Conclusion and Future Work

Figures (2)

  • Figure 1: Results for "synthetic" dataset. From top row to bottom row: Ground truth and results for subsets "peak", "cone", "cube". From left column to right column: Ground truth distribution, samples generated by the model proposed in liu2023delving, and samples generated by the proposed Euler angles-based flow model. The rotation matrix of liu2023delving tends to model better disjoint "islands" of mass; however, the proposed Euler angles model is better at estimating variance. (see also comment on \ref{['table:qualitative2']}).
  • Figure 2: Detail from Figure \ref{['table:qualitative']}. From left column to right column: Ground truth distribution, samples generated by the model proposed in liu2023delving, and samples generated by the proposed Euler angles-based flow model. Note the difference in inductive biases between the two models: Liu et alliu2023delving tends to misestimate variance by a considerable margin; the proposed Euler angles flow is closer to the correct value.