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Hierarchical Alternating Least Squares Methods for Quaternion Nonnegative Matrix Factorizations

Junjun Pan

TL;DR

The paper proposes a unified quaternion nonnegative matrix factorization (QNMF) framework to simultaneously handle RGB color and polarization images. It introduces hierarchical nonnegative least squares (HNLS) updates for both the quaternion source matrix $\breve{W}$ and the real nonnegative activation matrix $\mathbf{H}$, with projection operators enforcing polarization or RGB constraints and a convergence guarantee to stationary points. Through extensive experiments on polarization data and RGB facial images, the hierarchical ALS variants demonstrate superior reconstruction quality and favorable efficiency compared to the baseline QALS, with QHALS often providing the best trade-off between accuracy and speed. This work advances practical QNMF for image processing by offering a robust, convergent solver tailored to the physical constraints of Stokes parameters and RGB channels, enabling improved representation and processing of complex quaternion-valued image data.

Abstract

In this report, we discuss a simple model for RGB color and polarization images under a unified framework of quaternion nonnegative matrix factorization (QNMF) and present a hierarchical nonnegative least squares method to solve the factor matrices. The convergence analysis of the algorithm is discussed as well. We test the proposed method in the polarization image and color facial image representation. Compared to the state-of-the-art methods, the experimental results demonstrate the effectiveness of the hierarchical nonnegative least squares method for the QNMF model.

Hierarchical Alternating Least Squares Methods for Quaternion Nonnegative Matrix Factorizations

TL;DR

The paper proposes a unified quaternion nonnegative matrix factorization (QNMF) framework to simultaneously handle RGB color and polarization images. It introduces hierarchical nonnegative least squares (HNLS) updates for both the quaternion source matrix and the real nonnegative activation matrix , with projection operators enforcing polarization or RGB constraints and a convergence guarantee to stationary points. Through extensive experiments on polarization data and RGB facial images, the hierarchical ALS variants demonstrate superior reconstruction quality and favorable efficiency compared to the baseline QALS, with QHALS often providing the best trade-off between accuracy and speed. This work advances practical QNMF for image processing by offering a robust, convergent solver tailored to the physical constraints of Stokes parameters and RGB channels, enabling improved representation and processing of complex quaternion-valued image data.

Abstract

In this report, we discuss a simple model for RGB color and polarization images under a unified framework of quaternion nonnegative matrix factorization (QNMF) and present a hierarchical nonnegative least squares method to solve the factor matrices. The convergence analysis of the algorithm is discussed as well. We test the proposed method in the polarization image and color facial image representation. Compared to the state-of-the-art methods, the experimental results demonstrate the effectiveness of the hierarchical nonnegative least squares method for the QNMF model.
Paper Structure (12 sections, 2 theorems, 30 equations, 5 figures, 2 tables, 3 algorithms)

This paper contains 12 sections, 2 theorems, 30 equations, 5 figures, 2 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Contributions and outline of this paper
  3. Quaternion Matrix Factorization
  4. Problem Statement
  5. Algorithms
  6. Algorithm for Solving $\breve{\mathbf{W}}$
  7. Algorithm for Solving $\mathbf{H}$
  8. Discussion
  9. Numerical Experiments
  10. Polarization Image Representation
  11. RGB Color Image Representation
  12. Conclusion

Key Result

Proposition 1

If $\breve{\mathbf{W}}\in \mathbb{H}^{m\times r}_c$, $\mathbf{H} \in \mathbb{R}^{r\times n}_+$, $\breve{\mathbf{M}}=\breve{\mathbf{W}}\mathbf{H}$, then $\breve{\mathbf{M}}\in \mathbb{H}^{m\times n}_c$.

Figures (5)

  • Figure 1: Top: image "cover"; Bottom: the total intensity $\mathbf{T}^0$ and the normalized polarization $(\mathbf{T}^1/\mathbf{T}^0, \mathbf{T}^2/\mathbf{T}^0)$.
  • Figure 2: Reconstruction results of polarization image "cover" $(r=8)$.
  • Figure 3: The samples of the subset from the AR database
  • Figure 4: The features of AR facial images by the methods when $r=10$.
  • Figure 5: The constructed approximation for AR facial images by the methods when $r=10$.

Theorems & Definitions (7)

  • Proposition 1
  • proof
  • Remark 1
  • Remark 2
  • Theorem 1
  • Remark 3
  • Remark 4