Quaternion Fidelity Term in Color Deblurring

• Quaternion fidelity term is a loss function defined as the squared ℓ2-norm between a quaternion convolution output and an observed blurred image, ensuring cross-channel consistency.
• It leverages the full algebraic structure of quaternions to couple color channels and enforce intensity normalization, facilitating joint estimation of sharp images and blur kernels.
• Empirical results demonstrate improved PSNR, SSIM, and reduced color artifacts over standard deblurring methods, confirming its robust performance in blind deconvolution.

A quaternion fidelity term is a data-constrained loss expression central to recent advances in blind deconvolution for color images, designed to model and preserve both cross-channel dependencies and intensity normalization in the estimation of sharp images and blur kernels. In contrast to conventional approaches that treat RGB channels independently or via channel-wise convolution, the quaternion fidelity term leverages the full algebraic structure of quaternions to encode inter-channel correlations naturally and enforce physically consistent deblurring—a capability that is especially salient for color images subject to complex, spatially varying blur. The quaternion fidelity term is formulated as the squared $\ell_2$-distance between an observed blurred image and the result of convolving a sharp color image with a four-component quaternion kernel. This term, together with normalized kernel constraints and additional regularizers, provides the backbone for robust alternating-minimization algorithms in color image blind deconvolution (Yang et al., 21 Nov 2025).

1. Quaternion Representation in Color Image Deconvolution

Let $$u(x, y) = u_1(x, y)\mathbf{i} + u_2(x, y)\mathbf{j} + u_3(x, y)\mathbf{k}$$ denote a sharp color image encoded in pure-imaginary quaternion form, where $u_1$, $u_2$, and $u_3$ correspond to the red, green, and blue channels, respectively. The observed blurred image $f(x, y)$ is expressed similarly. The convolution kernel is a full quaternion $$Q = Q_0 + Q_1\mathbf{i} + Q_2\mathbf{j} + Q_3\mathbf{k}$$ with four real-valued spatial kernels ($Q_0, Q_1, Q_2, Q_3 \in \mathbb{R}^{m \times n}$). Quaternion convolution, denoted $\odot$, is defined according to quaternion algebra:

$$Q \odot u = -[Q_1 \star u_1 + Q_2 \star u_2 + Q_3 \star u_3] + [Q_0 \star u_1 - Q_3 \star u_2 + Q_2 \star u_3]\mathbf{i} + [Q_3 \star u_1 + Q_0 \star u_2 - Q_1 \star u_3]\mathbf{j} + [-Q_2 \star u_1 + Q_1 \star u_2 + Q_0 \star u_3]\mathbf{k}$$

where $\star$ denotes standard real-valued convolution per channel.

2. Formulation of the Quaternion Fidelity Term

The quaternion fidelity term is defined as the squared $\ell_2$-norm of the difference between the model prediction and the observed image: $F(Q, u, f) = \|Q \odot u - f\|_2^2$ Alternatively, upon vectorizing the RGB channels and employing a block-circulant matrix $T_Q$ built from $Q_0, Q_1, Q_2, Q_3$, this fidelity can be written as: $$F(Q, u, f) = \|T_Q\,\mathrm{vec}(u) - \mathrm{vec}(f)\|_2^2$$ This explicitly couples estimates of the sharp image and the quaternion kernel, facilitating joint optimization (Yang et al., 21 Nov 2025).

3. Normalization of the Quaternion Kernel

Typical real-valued blind deconvolution methods enforce kernel non-negativity and $\ell_1$-sum-to-one normalization to guarantee intensity preservation. For a quaternion kernel, simply enforcing $\|Q\|_1 = \sum_{i=0}^3 \|Q_i\|_1 = 1$ is insufficient for intensity conservation across all channels due to the interaction terms inherent in quaternion convolution. The proposed normalization seeks scalars $t_0, t_1, t_2, t_3 \in \mathbb{R}$ such that the rescaled kernel $$\widetilde{Q} = t_0 Q_0 + t_1 Q_1\mathbf{i} + t_2 Q_2\mathbf{j} + t_3 Q_3\mathbf{k}$$ exactly preserves channelwise $\ell_1$-energy. This is imposed by solving the linear system: $$A \cdot \begin{bmatrix} t_0 \ t_1 \ t_2 \ t_3 \end{bmatrix} = \begin{bmatrix} \|f_1\|_1 \ \|f_2\|_1 \ \|f_3\|_1 \end{bmatrix}$$ where $A$ is a $3 \times 4$ matrix with entries $\|(Q_q \star u_p)\|_1$ (up to sign), computed per quaternion component and channel. This system is typically solved in a least-squares sense via the Moore–Penrose pseudo-inverse, producing the normalized kernel $\widetilde{Q}$.

4. Complete Loss Function and Regularization

The full energy functional for blind deconvolution with the quaternion fidelity term in its basic form is: $$\min_{u,Q}\;\big\| Q \odot u - f \big\|_2^2 + \lambda\|\nabla u\|_0 + \gamma\|Q\|_2^2$$ subject to normalization: after each $Q$-update, $Q \leftarrow \widetilde{Q}$, where $\widetilde{Q}$ is obtained from the normalization procedure. In more general models, an auxiliary prior $\varphi(u)$ (such as a dark-channel or surface-aware prior) can be incorporated: $$E(u, Q) = \|Q\odot u - f\|_2^2 + \lambda\|\nabla u\|_0 + \gamma\|Q\|_2^2 + \mu\varphi(u)$$ The quaternion fidelity term thus encapsulates the data-consistency requirement, integrating the inter-channel structure enforced by the quaternion convolution.

5. Optimization and Algorithmic Framework

Optimization is performed via alternating minimization. At each iteration, three steps are executed:

1. $u$-step: Fix $Q$, minimize $F(Q, u, f) + \lambda\|\nabla u\|_0$ using half-quadratic splitting; threshold $\nabla u$ to obtain $v$, and solve the resulting linear normal equations (which decouple into $4 \times 4$ per-pixel systems in the Fourier domain).
2. $Q$-step: Fix $u$, solve $\min_Q \|Q \odot u - f\|_2^2 + \gamma\|Q\|_2^2$ via FFT-diagonalization or conjugate-gradient/least-squares techniques.
3. Normalization ($t$-step): Solve $A t = [\|f_1\|_1, \|f_2\|_1, \|f_3\|_1]^T$ for $t \in \mathbb{R}^4$, and update $Q \leftarrow \sum_{i=0}^{3} t_i Q_i\mathbf{e}_i$.

A coarse-to-fine pyramid over image scales is also employed to mitigate local minima (Yang et al., 21 Nov 2025).

6. Theoretical Properties

For fixed $Q$, the map $u \mapsto Q \odot u$ is linear, making the fidelity term $F(Q, u, f)$ convex quadratic in $u$. When optimizing $u$ while splitting off the $\ell_0$-term with auxiliary variables, the normal equations involve block-circulant matrices and are efficiently solved in the frequency domain. For fixed $u$, $F(Q, u, f)$ is likewise convex quadratic in the four real kernels, and the $Q$-subproblem remains a small linear system, usually regularized by $\gamma\|Q\|_2^2$ and then normalized to enforce intensity conservation.

7. Empirical Performance and Significance

Extensive experiments on established benchmarks demonstrate that the quaternion fidelity term delivers superior quantitative and qualitative performance compared to standard channel-coupled kernel (CCK) models. On the Köhler dataset, the quaternion model with normalized kernel produces higher average PSNR (31.39 dB vs. 31.20 dB), lower S-CIELAB error (88364 vs. 96638), and higher SSIM on all tested images. On a 4738-pair real-blur dataset, it yields a significant increase in average PSNR (29.72 dB vs. 28.54 dB), higher SSIM (0.9314 vs. 0.9200), and lower S-CIELAB error (102707 vs. 159427), consistently outperforming baseline methods. Qualitative visual comparisons confirm that capturing cross-channel differences with the quaternion fidelity term results in fewer color artifacts and improved hue preservation relative to traditional $\|k \star u - f\|_2^2$ fidelity measures (Yang et al., 21 Nov 2025).

In summary, the quaternion fidelity term, together with the normalization of the quaternion kernel, enables joint estimation of both sharp color images and structurally informed blur kernels that respect cross-channel interactions, with marked improvements over conventional blind deconvolution approaches for color images.