[Paper Review] CGA-PoseNet: Camera Pose Regression via a 1D-Up Approach to Conformal Geometric Algebra

Alberto Pepe, Joan Lasenby|arXiv (Cornell University)|Feb 10, 2023

Optical measurement and interference techniques4 citations

TL;DR

CGA-PoseNet proposes a novel camera pose regression method using Conformal Geometric Algebra (CGA) with a 1D-Up approach to unify rotation and translation into a single motor representation, enabling end-to-end training with a simple mean squared error (MSE) loss. The method achieves state-of-the-art accuracy on image-only datasets without requiring 3D point annotations or complex loss weighting.

ABSTRACT

We introduce CGA-PoseNet, which uses the 1D-Up approach to Conformal Geometric Algebra (CGA) to represent rotations and translations with a single mathematical object, the motor, for camera pose regression. We do so starting from PoseNet, which successfully predicts camera poses from small datasets of RGB frames. State-of-the-art methods, however, require expensive tuning to balance the orientational and translational components of the camera pose.This is usually done through complex, ad-hoc loss function to be minimized, and in some cases also requires 3D points as well as images. Our approach has the advantage of unifying the camera position and orientation through the motor. Consequently, the network searches for a single object which lives in a well-behaved 4D space with a Euclidean signature. This means that we can address the case of image-only datasets and work efficiently with a simple loss function, namely the mean squared error (MSE) between the predicted and ground truth motors. We show that it is possible to achieve high accuracy camera pose regression with a significantly simpler problem formulation. This 1D-Up approach to CGA can be employed to overcome the dichotomy between translational and orientational components in camera pose regression in a compact and elegant way.

Motivation & Objective

• To address the challenge of balancing orientational and translational components in camera pose regression, which traditionally requires complex, dataset-specific loss weighting.
• To unify rotation and translation into a single mathematical object— the motor—using Conformal Geometric Algebra (CGA) with a 1D-Up construction.
• To enable high-accuracy camera pose regression using only RGB images and a simple MSE loss, without requiring 3D point annotations or geometric reprojection losses.
• To demonstrate that a well-behaved 4D Euclidean space representation in CGA simplifies training and improves generalization on small datasets.

Proposed method

• Represents camera pose as a motor in 1D-Up Conformal Geometric Algebra, a 4D space with Euclidean signature where all basis vectors square to +1.
• Uses a single 8-parameter motor to encode both rotation and translation, eliminating the need for separate representations of orientation and position.
• Applies a standard mean squared error (MSE) loss between predicted and ground truth motors, avoiding complex, ad-hoc loss weighting schemes.
• Trains a modified PoseNet architecture to regress the motor directly from RGB image inputs, without requiring 3D scene geometry.
• Projects point clouds into spherical space via a conformal transformation to validate pose accuracy using MSE between reconstructed point clouds.
• Employs a spherical space transformation (Eq. 13) to preserve geometric fidelity during pose application and error measurement.

Experimental results

Research questions

• RQ1 Can a unified motor representation in Conformal Geometric Algebra eliminate the need for separate loss weighting between rotation and translation in camera pose regression?
• RQ2 Does using a 1D-Up CGA framework with a Euclidean signature enable simpler, more robust training with standard MSE loss?
• RQ3 Can high-accuracy camera pose regression be achieved without 3D point annotations, relying only on RGB images and motor regression?
• RQ4 How does the performance of a motor-based approach compare to state-of-the-art methods that use geometric reprojection loss with 3D points?

Key findings

• CGA-PoseNet achieves state-of-the-art accuracy on 13 benchmark datasets using only RGB images and a simple MSE loss, without any 3D point annotations.
• The method outperforms or matches approaches that use geometric reprojection loss requiring 2M+ 3D points, such as in the Cambridge Landmarks dataset.
• Positional and rotational errors are highly correlated, confirming the consistency of the motor representation across both components.
• Point cloud reconstruction experiments show that the MSE between predicted and ground truth point clouds is consistent with the positional error reported in the tables, validating the geometric fidelity of the predicted motors.
• The approach yields superior results compared to [20] (which includes model uncertainty) and [48] (which adds an LSTM head), despite its simplicity.
• The median MSE between reconstructed point clouds for landmarks like King’s and Shop Facade is low and consistent with the reported pose errors, confirming robustness across diverse scenes.