Quaternion Nuclear Norm Minus Frobenius Norm

• QNMF is a regularization framework that uses quaternions to jointly process RGB channels, effectively preserving inter-channel correlations in color images.
• It employs a hybrid non-convex penalty combining the quaternion nuclear norm and a scaled Frobenius norm, closely approximating true rank minimization.
• The ADMM-based optimization framework in QNMF delivers state-of-the-art performance in denoising, deblurring, inpainting, and impulse noise removal with strong theoretical guarantees.

Quaternion Nuclear Norm Minus Frobenius Norm (QNMF) is a regularization framework designed to improve color image reconstruction by leveraging quaternion algebra to jointly process RGB channels. QNMF addresses inter-channel correlation and enforces low-rank structure through a non-convex penalty. Its formulation enables accurate recovery in tasks such as denoising, deblurring, inpainting, and random impulse noise removal, consistently demonstrating state-of-the-art results in both synthetic and real-world scenarios (Guo et al., 12 Sep 2024).

1. Quaternion Representation of Color Images

Quaternion algebra offers a natural mechanism for encoding color images holistically. A quaternion is written as

$$\dot a = a_0 + a_1\,\mathbf{i} + a_2\,\mathbf{j} + a_3\,\mathbf{k}$$

with the imaginary units satisfying $$\mathbf{i}^2 = \mathbf{j}^2 = \mathbf{k}^2 = \mathbf{i}\mathbf{j}\mathbf{k} = -1$$. The conjugate is $$\dot a^* = a_0 - a_1\,\mathbf{i} - a_2\,\mathbf{j} - a_3\,\mathbf{k}$$, and the modulus is $|\dot a| = \sqrt{a_0^2 + a_1^2 + a_2^2 + a_3^2}$.

A quaternion matrix $\dot X \in \mathbb Q^{m\times n}$ expands as

$$\dot X = X_0 + X_1\,\mathbf{i} + X_2\,\mathbf{j} + X_3\,\mathbf{k}$$

with $X_\ell \in \mathbb R^{m\times n}$. The trace inner product is $$\langle \dot X, \dot Y \rangle = \Re\,\mathrm{tr}(\dot X^T\dot Y)$$, and the Frobenius norm is $$\|\dot X\|_F = \sqrt{\sum_{i,j}|\dot X_{ij}|^2} = \sqrt{\sum_{k=1}^{\min\{m,n\}}\sigma_k(\dot X)^2}$$.

RGB images are encoded as pure quaternion matrices: $$\dot X_{RGB} = X_R\,\mathbf{i} + X_G\,\mathbf{j} + X_B\,\mathbf{k}$$ where $X_R, X_G, X_B$ denote red, green, and blue channel matrices, respectively, and the real part is zero.

2. The QNMF Regularizer: Formulation and Properties

The QNMF regularization term penalizes the difference of the quaternion nuclear norm and a scaled Frobenius norm: $R(\dot X) = \|\dot X\|_* - \alpha \|\dot X\|_F$ with $\alpha > 0$. Here, the quaternion nuclear norm is defined by the sum of singular values, $\|\dot X\|_* = \sum_k \sigma_k(\dot X)$, and the Frobenius norm by $\|\dot X\|_F = \sqrt{\sum_k \sigma_k(\dot X)^2}$.

This penalty is non-convex, constructed as a difference of convex functions. The nuclear norm component encourages singular value sparsity (lower effective rank), while the negative Frobenius norm selectively preserves large singular values, yielding a closer approximation to true rank minimization compared to the nuclear norm alone.

3. Optimization and Algorithmic Framework

Color image reconstruction under QNMF is formulated as a regularized inverse problem. For denoising, the minimization is: $$\min_{\dot X}~\frac{1}{2}\|\dot Y-\dot X\|_F^2 + \lambda(\|\dot X\|_* - \alpha\|\dot X\|_F)$$ and for general linear inverse problems (e.g., deblurring),

$$\min_{\dot X}~\frac{\gamma}{2}\|\dot A \dot X - \dot Y\|_F^2 + \lambda(\|\dot X\|_* - \alpha\|\dot X\|_F)$$

An ADMM splitting is employed, introducing an auxiliary variable $\dot Z$ with $\dot X = \dot Z$. The augmented Lagrangian is

$$\mathcal L(\dot X, \dot Z, \dot\eta) = \frac{\gamma}{2}\|\dot A \dot X - \dot Y\|_F^2 + \lambda(\|\dot Z\|_* - \alpha\|\dot Z\|_F) + \frac{\beta}{2}\|\dot X - \dot Z\|_F^2 + \langle\dot\eta, \dot X - \dot Z\rangle$$

where variables are updated as follows:

• X-subproblem: Closed-form update via quaternion FFT,

$$\dot X^{k+1} = (\gamma \dot A^* \dot A + \beta I)^{-1}(\gamma \dot A^* \dot Y + \beta \dot Z^k - \dot\eta^k)$$

• Z-subproblem (QNMF proximal step):

Given QSVD $$\dot W = \dot X^{k+1} + \dot\eta^k/\beta = \dot U\,\mathrm{diag}(\sigma)\,\dot V^T$$, the new singular values are updated by

$$\tilde\sigma_i = \begin{cases} 0, & \sigma_i \le \lambda/\beta \ K(\sigma_i - \lambda/2\beta), & \lambda/\beta < \sigma_i \le \frac{K\lambda/\beta}{2(K-1)} \ \sigma_i, & \sigma_i > \frac{K\lambda/\beta}{2(K-1)} \end{cases}$$

with $$K=1+\frac{\alpha\lambda/\beta}{\|\max(\sigma-\lambda/\beta,0)\|_2}$$. The new iterate is $$\dot Z^{k+1} = \dot U~\mathrm{diag}(\tilde\sigma)~\dot V^T$$.

• Multiplier and penalty updates: $$\dot\eta^{k+1}=\dot\eta^k+\beta(\dot X^{k+1}-\dot Z^{k+1})$$, $\beta^{k+1}=\mu\beta^k,\,\mu>1$.

Convergence is guaranteed under monotonic penalty update ($\beta^k\to\infty$), with subproblem Z having a global solution and overall iterates converging such that $\|\dot X^{k+1}-\dot Z^{k+1}\| \to 0$ and $\|\dot X^{k+1}-\dot X^k\| \to 0$ (Guo et al., 12 Sep 2024).

4. Parameterization and Theoretical Guarantees

Parameter selection is data- and task-dependent. For denoising, recommended settings are patch size $m$, number of similar patches $n$ dependent on noise standard deviation $\sigma$, $\lambda=2c$ with $c = \sqrt{5\sqrt{2n}\,\sigma}$, and $\alpha = 4$. For deblurring, penalty and weighting ($\gamma$, $\beta$) are tuned per blur kernel. In inpainting and RPCA, $\alpha$ is kept fixed and $\rho$ is adapted for the $\ell_1$ error term. The framework is theoretically supported by optimality results for the Z-subproblem and general non-convex recovery guarantees.

QNMF is non-convex but structured as a difference of convex functions, enabling tractable optimization. The singular value shrinkage in the Z-step admits closed-form evaluation.

5. Empirical Evaluation and Comparative Results

The efficacy of QNMF is established across multiple benchmarks:

• Synthetic Gaussian Denoising: On CSet12, McMaster, and Kodak datasets, QNMF achieves average PSNR/SSIM improvements over CBM3D, McWNNM, SV-TV, QLRMA, QWNNM, and QWSNM across all noise levels. For CSet12, QNMF yields 31.36/0.8764 vs. QWNNM 31.25/0.8748 and QWSNM 31.30/0.8715.
• Real Image Denoising: On CC, PolyU, and SIDD, QNMF attains leading performance, e.g., 36.53 dB/0.9166 (SIDD).
• Deblurring: QNMF gives best or equivalent PSNR/SSIM for uniform, Gaussian, and motion blur settings, visually minimizing ringing artifacts and producing sharper edges.
• Matrix Completion (MC): At 80% missing, QNMF (patch-based) secures PSNR 31.80/SSIM 0.9397 compared to nearest baseline 25.47/0.6993.
• RPCA: With 10% impulse noise, QNMF-G records 29.34/0.8895 vs. TRPCA 28.80/0.9170 (a trade-off in SSIM).
• Runtime Considerations: For denoising $256 \times 256$ images, QNMF runs in ≈620 s versus QWNNM 580 s and QWSNM 955 s; for deblurring, QNMF (≈2155 s) is notably faster than QWNNM (3831 s). For matrix completion, QNMF (≈10.8 s) is substantially faster than QMC (168 s) (Guo et al., 12 Sep 2024).

6. Strengths, Limitations, and Potential Extensions

QNMF's principal strength lies in its quaternion-based joint processing of RGB channels, preserving color structure and minimizing channel-specific artifacts. The hybrid nuclear–Frobenius penalty approximates rank more effectively than conventional convex relaxations, and the ADMM framework accommodates a range of low-level vision tasks with mathematically guaranteed convergence to stationary points.

However, the approach is constrained by the high computational complexity of QSVD and requires manual parameter tuning (patch size, $\lambda$, etc.). Potential future directions include the development of QSVD-free quaternion factorization techniques, exploration of additional hybrid norms (e.g., other nuclear/Frobenius norm combinations), and integration of quaternion low-rank priors into deep neural network architectures.

7. Context and Research Impact

QNMF advances low-rank color image modeling by embedding the intrinsic correlation of RGB channels via quaternion algebra and leveraging a non-convex low-rank surrogate. The method attains state-of-the-art quantitative and visual outcomes across a comprehensive array of color restoration tasks while upholding rigorous mathematical guarantees (Guo et al., 12 Sep 2024). Its modular optimization scheme and adaptability across different degradation modalities position QNMF as a significant development in color image reconstruction research.