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A Novel Wasserstein Quaternion Generative Adversarial Network for Color Image Generation

Zhigang Jia, Duan Wang, Hengkai Wang, Yajun Xie, Meixiang Zhao, Xiaoyu Zhao

TL;DR

The paper introduces a novel quaternion Wasserstein distance (QWD) and its dual theory via quaternion linear programming, separation theorems, and Farkas lemmas to quantify distribution differences in color images. It then develops the Wasserstein Quaternion GAN (WQGAN) by adopting the dual QWD as the training objective, with discriminators and generators operating on quaternion-valued data. Empirical results on SVHN and CelebA show WQGAN achieves faster convergence and superior image quality (lower FID, competitive IS) compared with GAN, WGAN, QGAN, and DDPM baselines. This work provides both a rigorous quaternion-valued transport framework and a practical algorithm for robust, color-consistent image generation.

Abstract

Color image generation has a wide range of applications, but the existing generation models ignore the correlation among color channels, which may lead to chromatic aberration problems. In addition, the data distribution problem of color images has not been systematically elaborated and explained, so that there is still the lack of the theory about measuring different color images datasets. In this paper, we define a new quaternion Wasserstein distance and develop its dual theory. To deal with the quaternion linear programming problem, we derive the strong duality form with helps of quaternion convex set separation theorem and quaternion Farkas lemma. With using quaternion Wasserstein distance, we propose a novel Wasserstein quaternion generative adversarial network. Experiments demonstrate that this novel model surpasses both the (quaternion) generative adversarial networks and the Wasserstein generative adversarial network in terms of generation efficiency and image quality.

A Novel Wasserstein Quaternion Generative Adversarial Network for Color Image Generation

TL;DR

The paper introduces a novel quaternion Wasserstein distance (QWD) and its dual theory via quaternion linear programming, separation theorems, and Farkas lemmas to quantify distribution differences in color images. It then develops the Wasserstein Quaternion GAN (WQGAN) by adopting the dual QWD as the training objective, with discriminators and generators operating on quaternion-valued data. Empirical results on SVHN and CelebA show WQGAN achieves faster convergence and superior image quality (lower FID, competitive IS) compared with GAN, WGAN, QGAN, and DDPM baselines. This work provides both a rigorous quaternion-valued transport framework and a practical algorithm for robust, color-consistent image generation.

Abstract

Color image generation has a wide range of applications, but the existing generation models ignore the correlation among color channels, which may lead to chromatic aberration problems. In addition, the data distribution problem of color images has not been systematically elaborated and explained, so that there is still the lack of the theory about measuring different color images datasets. In this paper, we define a new quaternion Wasserstein distance and develop its dual theory. To deal with the quaternion linear programming problem, we derive the strong duality form with helps of quaternion convex set separation theorem and quaternion Farkas lemma. With using quaternion Wasserstein distance, we propose a novel Wasserstein quaternion generative adversarial network. Experiments demonstrate that this novel model surpasses both the (quaternion) generative adversarial networks and the Wasserstein generative adversarial network in terms of generation efficiency and image quality.

Paper Structure

This paper contains 13 sections, 6 theorems, 32 equations, 10 figures, 4 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Quaternion and quaternion random variables
  4. Quaternion Wasserstein distance
  5. Quaternion linear programming problem and dual form
  6. Separation theorem of convex quaternion vector sets
  7. Quaternion strong dual theorem
  8. Dual form of quaternion Wasserstein distance with application to Wasserstein quaternion generative model
  9. Dual form of quaternion Wasserstein distance
  10. Wasserstein Quaternion Generation Adversarial Neural Network
  11. Experiments
  12. Experiment
  13. Conclusion

Key Result

Theorem 3.1

Given quaternion vector $\boldsymbol{b} \in \mathbb{Q}^{n}$, nonnegative real vector $C\in \mathbb{R}_+^{m}$ and real matrix $\Upsilon\in \mathbb{R}^{n \times m}$, there is

Figures (10)

  • Figure 1: The architecture of WQGAN.
  • Figure 2: The optimal transportation between $P_r$ and $P_g$.
  • Figure 3: Samples from SVHN database.
  • Figure 4: Samples from CelebA database.
  • Figure 5: Line graph of FID for generated images (SVHN).
  • ...and 5 more figures

Theorems & Definitions (15)

  • Definition 2.1
  • Theorem 3.1
  • proof
  • Definition 3.2
  • Lemma 3.3
  • proof
  • Theorem 3.4
  • proof
  • Definition 3.5
  • Theorem 3.6
  • ...and 5 more