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Quaternion Approximate Networks for Enhanced Image Classification and Oriented Object Detection

Bryce Grant, Peng Wang

arXiv:2509.05512v2cs.CVcs.RO

TL;DR

Robotic perception needs rotation-aware detection, while quaternion networks face implementation overhead and oriented detectors can be computationally demanding. QUAN approximates quaternion operations with real-valued computations and adds quaternion-aware components. It reports state-of-the-art classification performance for quaternion networks with fewer parameters, while noting slightly higher inference time from the approximation.

  • Problem

    Robotic manipulation needs accurate object orientation, but axis-aligned detection omits it and rotation-aware methods can be costly for resource-constrained systems.

    Source: §I · p. 1

  • Method

    QUAN approximates Hamilton-based quaternion convolution with real-valued operations and introduces IQBN and quaternion-aware spatial attention.

    Source: §I · p. 2

  • Results

    95.12% accuracy on CIFAR-10 with 717K parameters; QUAN-WRN outperforms all quaternion baselines.

    Source: C. Image Classification Performance · p. 6

  • Takeaways & Limitations

    The paper presents QUAN as maintaining practical applicability for image classification and oriented object detection through real-valued approximations and independent normalization.

    Source: §V · p. 7

    The approximation increases arithmetic operations and slightly raises inference time, which the authors' CUDA kernels partially mitigate.

    Source: §V · p. 7

Abstract

from arXiv · show

This paper introduces Quaternion Approximate Networks (QUAN), a novel deep learning framework that leverages quaternion algebra for rotation equivariant image classification and object detection. Unlike conventional quaternion neural networks attempting to operate entirely in the quaternion domain, QUAN approximates quaternion convolution through Hamilton product decomposition using real-valued operations. This approach preserves geometric properties while enabling efficient implementation with custom CUDA kernels. We introduce Independent Quaternion Batch Normalization (IQBN) for training stability and extend quaternion operations to spatial attention mechanisms. QUAN is evaluated on image classification (CIFAR-10/100, ImageNet), object detection (COCO, DOTA), and robotic perception tasks. In classification tasks, QUAN achieves higher accuracy with fewer parameters and faster convergence compared to existing convolution and quaternion-based models. For objection detection, QUAN demonstrates improved parameter efficiency and rotation handling over standard Convolutional Neural Networks (CNNs) while establishing the SOTA for quaternion CNNs in this downstream task. These results highlight its potential for deployment in resource-constrained robotic systems requiring rotation-aware perception and application in other domains. Code: https://cwru-aism.github.io/ QUANpaper/

I. INTRODUCTION

Robotic manipulation depends on accurate object orientation, but conventional axis-aligned detection omits it and rotation-aware alternatives can be computationally demanding. QUAN approximates quaternion operations with real-valued computations and introduces quaternion-aware components for classification and oriented detection.

I. INTRODUCTION

  • Robotic manipulation requires accurate object orientation, which axis-aligned bounding boxes discard, affecting grasp planning and assembly.Specialized oriented-detection methods also often require substantial computation, creating a deployment barrier for resource-constrained real-time robots.Source: §I · p. 1
  • Existing pure quaternion networks face implementation overhead from specialized operations that mainstream deep-learning frameworks support poorly.The passage also notes mathematical constraints on bounded non-constant analytic quaternion functions that complicate activation and gradient design.Source: §I · p. 2
  • QUAN approximates Hamilton-based quaternion mixing with real-valued operations to retain rotation equivariance while reducing computational complexity.The paper also describes quaternion-aware components including Independent Quaternion Batch Normalization (IQBN) and Quaternion Partial Spatial Attention (QC2PSA).Source: §I · p. 2
  • QUAN reports state-of-the-art performance for quaternion networks on CIFAR-10, CIFAR-100, and ImageNet with fewer parameters than standard CNNs and existing quaternion approaches.For robotic perception, the paper motivates rotation-aware detection as important for grasp planning and assembly.Source: §I · pp. 1–2
  • QUAN adapts YOLO components QC3k2, QSPPF, and QC2PSA for oriented bounding-box detection on a custom robotic manipulation dataset.Its orientation-aware loss combines quaternion angular metrics with traditional detection objectives for end-to-end location and orientation learning.Source: §I · p. 2
  • The paper presents QUAN as bridging quaternion representations with practical robotic-perception deployment needs.It links parameter efficiency and rotation equivariance to object detection and pose estimation for robotic manipulation.Source: §I · p. 2

II. RELATED WORKS

Prior oriented-detection methods address rotation but often add architectural complexity and computational cost, while quaternion networks offer parameter efficiency but face implementation challenges. Earlier quaternion pose-estimation work had not yet applied quaternion approaches to oriented object detection in practical robotic applications.

A. General Object Detection and Pose Estimation

  • Oriented-detection methods such as R3Det and gliding vertex address rotated objects, but typically require complex changes and more computation than standard detectors.General object-detection systems commonly use axis-aligned boxes, limiting their utility when orientation matters for robotic manipulation.Source: A. General Object Detection and Pose… · p. 2

B. Quaternion-Based Methods

  • Quaternion neural networks have shown parameter efficiency and comparable or superior performance to standard CNNs, but higher complexity and activation-function constraints hinder adoption.The cited implementation challenges include the non-existence of bounded non-constant analytic quaternion functions.Source: B. Quaternion-Based Methods · pp. 2–3

C. Quaternion Pose Estimation

  • Quaternion-based pose methods include QuatNet for continuous head-pose estimation and PVN3D for 6DoF pose estimation, but had not addressed practical robotic oriented detection.QuatNet uses a multi-regression loss; PVN3D uses point-wise 3D keypoint voting and quaternions for rotation representation.Source: C. Quaternion Pose Estimation · p. 3

D. Alternative Rotation Equivariant Architectures

Alternative rotation-equivariant architectures use specialized designs or incur computational overhead, whereas QUAN uses quaternion algebra within standard deep-learning frameworks. For image inputs, it uses Poincaré mapping and an approximate quaternion convolution that reduces computational complexity.

D. Alternative Rotation Equivariant Architectures

  • QUAN uses quaternion algebra within standard deep-learning frameworks for rotation awareness, contrasting with alternative approaches that require architectural changes or specialized implementations.The cited alternatives include group-equivariant and steerable CNNs, harmonic networks, and oriented response networks.Source: D. Alternative Rotation Equivariant… · p. 3

A. Quaternion Representation and Mapping

  • Poincaré mapping embeds RGB values in the Poincaré ball, where distances increase toward its boundary and the representation better preserves color-space hierarchies than Euclidean embeddings.The paper reports that this mapping performed better than other evaluated strategies.Source: A. Quaternion Representation and Mapping · p. 3
  • Separating quaternion convolution into operations reduces complexity from O(16CHW) to O(4CHW), while IQBN maintains numerical stability without enforcing unit norm during training.The authors report that strict constraints through Riemannian optimization degraded performance.Source: A. Quaternion Representation and Mapping · p. 3

B. Hamilton-Based Quaternion Approximation

QUAN approximates Hamilton-based quaternion convolution with separable real-valued operations, reducing computational complexity while retaining quaternion sign patterns. Custom CUDA kernels fuse these operations, and the implementation is reported to reduce memory-bandwidth requirements with inference time comparable to standard convolutions.

B. Hamilton-Based Quaternion Approximation

  • The full Hamilton product expands quaternion convolution across components using a standard Hamilton matrix and standard convolutions.The section introduces this expansion as the basis for the approximation.Source: B. Hamilton-Based Quaternion… · p. 3
  • QUAN decomposes quaternion convolution into separable operations, relaxing cross-component interactions while preserving quaternion sign patterns.The authors report that the simplification captures rotation-aware features while using optimized tensor operations.Source: B. Hamilton-Based Quaternion… · pp. 3–4
  • The separable QUAN convolution defines component-wise forward outputs for an input tensor and convolutional kernel.The backward pass uses the same separable structure, with a mixing matrix derived from forward-pass coefficients.Source: B. Hamilton-Based Quaternion… · p. 4
  • Fused Hamilton-product operations reduce memory-bandwidth requirements and achieve inference time comparable to standard convolutions.The paper describes custom CUDA kernels for the fusion.Source: B. Hamilton-Based Quaternion… · pp. 4–5

C. Independent Quaternion Batch Normalization

Independent Quaternion Batch Normalization (IQBN) normalizes quaternion components independently while preserving important cross-component relationships. The section also describes maintaining running statistics during training and applying an affine transformation to normalized features.

C. Independent Quaternion Batch Normalization

  • IQBN normalizes each quaternion component independently while preserving important cross-component relationships.The stated input feature has four components, r, i, j, and k.Source: C. Independent Quaternion Batch… · p. 5
  • During training, IQBN maintains running statistics and applies an affine transformation to normalized features.The expression gives the transformation as xnorm multiplied by γ, plus β.Source: C. Independent Quaternion Batch… · p. 5

D. Quaternion Architectural Components

The paper applies QUAN to YOLOv11, replacing standard convolution with approximated quaternion convolution and modifying the architecture to maintain quaternion operations.

D. Quaternion Architectural Components

  • The YOLOv11 adaptation replaces standard convolution with approximated quaternion convolution and makes additional modifications to maintain quaternion operations throughout the network.The section points to Figure 3 for the architectural modifications.Source: D. Quaternion Architectural Components · p. 5

1) Quaternion Cross Stage Partial with kernel size 2 (QC3k2):

The section describes two quaternion adaptations for YOLO architectures: QC3k2 retains the partial cross-stage design, while QSPPF preserves quaternion structure through pooling and Hamilton-based mixing.

1) Quaternion Cross Stage Partial with kernel size 2 (QC3k2):

2) Quaternion Spatial Pyramid Pooling (QSPPF):

  • QSPPF applies pooling independently to each quaternion component, followed by Hamilton-based mixing, to preserve quaternion structure and capture multi-scale information.The passage states that this approach maintains rotation equivariance.Source: 2) Quaternion Spatial Pyramid Pooling… · p. 5

3) Quaternion Partial Spatial Attention (QC2PSA):

QC2PSA applies quaternion spatial attention to one split feature branch and recombines it; detection training combines localization, orientation, classification, and smoothness objectives.

3) Quaternion Partial Spatial Attention (QC2PSA):

  • QC2PSA splits input features, applies quaternion attention to one branch, then recombines them while preserving rotational information.Its attention mechanism computes spatial weights while respecting quaternion algebra.Source: 3) Quaternion Partial Spatial… · p. 5

E. Loss Functions for Object Detection and Orientation

  • The total detection loss combines classification, complete-IoU localization, quaternion angular, quaternion regularization, and orientation-smoothness terms.The angular term measures geodesic distance between predicted and target quaternion orientations; regularization enforces unit-quaternion constraints.Source: E. Loss Functions for Object Detection… · p. 5

IV. EXPERIMENTS

  • Experiments evaluate QUAN on image classification and object detection using a single NVIDIA RTX 4090 GPU.The stated evaluation aims include parameter efficiency, robotic assembly applicability, and rotation equivariance.Source: §IV · p. 5

1) Image Classification Datasets:

QUAN is evaluated on classification and axis-aligned detection benchmarks, with classification results emphasizing accuracy and parameter efficiency and detection results varying by dataset.

1) Image Classification Datasets:

  • The evaluation uses CIFAR-10/100, a 100-class ImageNet subset, COCO 2017, DOTA-v1.0, and a 1,900-image robotic workspace dataset.The robotic images include oriented and regular annotations for pick-and-place tasks.Source: 1) Image Classification Datasets · p. 5

B. Network Architectures

C. Image Classification Performance

1) Classical Object Detection:

  • On the robotic dataset, QUAN scores 0.9548 vs 0.951 mAP50-95 for YOLO11n using 25.3% of its parameters, but takes 36% more inference time.On COCO, QUAN scores .339 vs .3908 mAP50-95 and takes 12% more inference time; the authors report impaired generalization as dataset scale increases.Source: 1) Classical Object Detection · p. 6, Table II · p. 6
  • The authors lacked capacity to train from scratch with the original models’ batch size and hyperparameters.Source: 1) Classical Object Detection · p. 7

2) Oriented Object Detection:

QUAN improves oriented-detection performance on the robotic dataset relative to YOLO11n-obb, while results on DOTA are slightly lower; ablations favor Poincaré mapping and SiLU.

2) Oriented Object Detection:

  • 76.4 vs. 68.5 on the robotic dataset, QUAN outperforms YOLO11n-obb and reduces inference time by 3.7%.On DOTAv1, QUAN scores 60.8 vs 61.4 map50-95; the authors attribute the impairment likely to the small model size.Source: 2) Oriented Object Detection · p. 7, Table III · p. 7

E. Ablation Studies

  • Poincaré mapping generally outperforms alternatives across activation functions, while SiLU consistently exceeds ReLU and QPReLU.Traditional quaternion batch normalization averaged 0.61% less accuracy than IQBN across CIFAR-10 experiments.Source: E. Ablation Studies · p. 7

V. CONCLUSIONS

  • QUAN approximates quaternion operations with real-valued computations and independent normalization, but adds arithmetic operations that slightly increase inference time.The authors say custom CUDA kernels partially mitigate the added time; Poincaré mapping may not be optimal for every downstream task.Source: §V · p. 7

VI. APPENDIX

The appendix describes five RGB-to-quaternion mappings that differ in how they assign RGB information to quaternion components.

A. Mapping Strategies

  • The evaluated mappings use luminance weights, mean brightness, equally weighted normalized RGB, pure-imaginary RGB components, or projection onto the unit quaternion sphere.The Poincaré mapping is described as numerically stable while preserving relative color relationships.Source: A. Mapping Strategies · p. 8

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