Normalized Quaternion Kernel

• Normalized quaternion kernels are quaternion-valued functions normalized to ensure a unit diagonal while preserving algebraic, geometric, and physical invariants.
• They are constructed within quaternionic reproducing kernel Hilbert spaces and applied in random matrix ensembles, analytic paradigms, and color image processing.
• Applications include the Cauchy–Szegö kernel in Hardy spaces and deblurring in image processing, ensuring stability, invariance, and cross-channel fidelity.

A normalized quaternion kernel is a quaternion-valued kernel function equipped with a normalization that preserves key algebraic, geometric, or physical properties in its domain of application. Normalized quaternion kernels appear across several domains, including quaternionic reproducing kernel Hilbert spaces, quaternionic extensions of classical analytic kernels, random matrix theory, and image processing—particularly for models requiring joint channel and intensity preservation. The normalization typically ensures that the kernel's diagonal value is unity (or a fixed operator), maintains positivity, and satisfies natural invariances or integral constraints analogous to their complex or real counterparts.

1. Formal Construction of Normalized Quaternion Kernels

A general framework for normalized quaternion kernels is given in the theory of quaternionic reproducing kernel Hilbert spaces. Let $X$ be a set, and $\mathcal{H}$ a finite-dimensional right quaternionic Hilbert space, with the inner product $$(\cdot|\cdot)_{\mathcal{H}}:\mathcal{H}\times\mathcal{H}\to\mathbb{H}$$. A function $K:X\times X\to\mathcal{L}(\mathcal{H})$ is a (right) quaternionic reproducing kernel if:

• $K(x,x)$ is strictly positive for each $x\in X$,
• for any $x_1,...,x_m\in X, v_1,...,v_m\in\mathcal{H}$,

$$\sum_{i,j=1}^m (v_i | K(x_i, x_j) v_j)_{\mathcal{H}} \geq 0,$$

• the reproducing property holds:

$$(K(\cdot, x)v | f)_{\mathcal{K}} = (v|f(x))_{\mathcal{H}}, \quad \forall v \in \mathcal{H}, f \in \mathcal{K}.$$

Normalization is achieved via the operation

$$\widetilde K(x,y) = K(x,x)^{-1/2} K(x,y) K(y,y)^{-1/2}$$

which ensures $\widetilde K(x,x) = I_{\mathcal{H}}$, preserves positivity and the reproducing structure, and produces kernels canonically isomorphic to their unnormalized analogs (Thirulogasanthar et al., 2016).

2. Canonical Examples: Hermite and Laguerre Quaternionic Kernels

Concrete normalized kernels can be constructed from quaternionic orthogonal polynomials:

• Hermite case: for $0<\varepsilon<1$, with quaternionic Hermite polynomials $H_n(q)$,

$$K_{\varepsilon}(q,q') = \sum_{n=0}^{\infty}\frac{\varepsilon^n}{2^{n}n!\sqrt{\pi}} H_n(q)H_n(q'),$$

$$\widetilde K_{\varepsilon}(q,q') = \frac{K_{\varepsilon}(q,q')}{\sqrt{K_{\varepsilon}(q,q)K_{\varepsilon}(q',q')}} = \exp\left( \frac{2\varepsilon\Re(q\overline{q'}) - \varepsilon(|q|^2 + |q'|^2)}{1-\varepsilon^2} \right ),$$

with $\widetilde K_{\varepsilon}(q,q) = 1$ (Thirulogasanthar et al., 2016).

• Laguerre case: for $\alpha > -1$, with quaternionic Laguerre polynomials $L_n^{(\alpha)}(q)$, the kernel and ensuing normalization follow similarly, with the diagonal normalization involving Bessel functions.

This methodology preserves positivity, normalization, and the reproducing property, and generalizes directly to more complex settings.

3. Analytic Quaternionic Kernels: The Cauchy–Szegö Paradigm

The normalized quaternionic Cauchy–Szegö kernel on the quaternionic Siegel half-space $\mathcal{U}_n$ provides an analytic archetype. Functions in the Hardy space $H^2(\mathcal{U}_n)$ are left-$\mathbb{H}$-regular, and the kernel is explicitly

$$S(q,w) = s\left( q_{n+1} + \overline{w_{n+1} - 2\overline{w'} \cdot q'} \right ),$$

where the reduced kernel

$$s(v) = \left(\frac{2}{\pi}\right)^{2n} \frac{\partial^{2n}}{\partial x_0^{2n}} \left( \frac{1}{2\pi^2} \frac{\overline{v}}{|v|^4} \right ),$$

and normalization ensures that monomials associated with the kernel are orthonormal in $H^2$. This kernel is unique, Hermitian symmetric ($\overline{S(q,w)} = S(w,q)$), and invariant under dilations, rotations, and Heisenberg translations (Wang et al., 2012). The kernel provides foundational projection operators and defines an orthogonal projection on $L^2(\partial\mathcal{U}_n)$ onto the Hardy space.

4. Normalized Quaternion Kernels in Random Matrix Ensembles

Quaternionic kernels in random matrix theory arise in both determinantal and Pfaffian point processes:

• 3D Ginibre point field: On $\mathbb{R}^3 \cong \mathbb{H}_0$, with orthogonal polynomials $P_n(z)$ built over pure quaternions, the normalized kernel,

$$K_{N}(z, w) = \sum_{k=0}^{N-1} \frac{P_k(z)\overline{P_k(w)}}{h_k},$$

yields normalized expected densities—specifically $\int_{\mathbb{R}^3}K_N(z,z)d\mu(z) = N$—and the normalization ensures determinantal structure and self-duality (Kargin, 2017).

• Quaternion-real Ginibre ensemble: For the 2N×2N real-quaternion Ginibre ensemble, the eigenvalue process is Pfaffian with a $2\times2$ matrix kernel

$$K_N(z, w) = \frac{e^{-\tfrac12(|z|^2+|w|^2)}}{2\sqrt\pi} \begin{pmatrix} (w-z)U & \partial_w\left[(w-z)U\right] \ - \int_z^w (t-z)U dt & (w-z)U(w,z) \end{pmatrix}$$

where $U$ is the Tricomi function, and normalization constants follow from Selberg and Schur polynomial techniques (Forrester, 2013).

5. Normalized Quaternion Kernels in Color Image Processing

In blind deconvolution for color images, normalized quaternion kernels serve as convolutional blur operators on quaternionic representations of RGB images:

• The color image is encoded as $u = u_1 i + u_2 j + u_3 k$.
• The blur kernel $Q = Q_0 + i Q_1 + j Q_2 + k Q_3$, where $Q_0$ is non-negative and models overall blur, while $Q_1,Q_2,Q_3$ encode inter-channel couplings.
• Physical and perceptual fidelity in deblurring requires a normalization that matches total intensity per channel. Rather than enforcing $\|Q\|_1 = 1$, normalization proceeds via a small linear system: after convolution, the intensity per channel is matched by scaling $Q_k$ by factors $t_k$ determined from solving $At = (\|f_1\|_1, \|f_2\|_1, \|f_3\|_1)^T$, where $A$ encodes convolution responses. The normalized kernel is then $$\widetilde Q = t_0 Q_0 + t_1 Q_1 i + t_2 Q_2 j + t_3 Q_3 k$$ (Yang et al., 21 Nov 2025).
• This channel-aware normalization eliminates color shifts and artifacts that arise with unnormalized or naïvely normalized quaternion kernels, improving PSNR, SSIM, and S-CIELAB metrics relative to both real and unnormalized quaternion-based deconvolution.

6. Summary Table: Representative Normalized Quaternion Kernels

Context	Kernel Functional Form	Normalization Principle
Quaternionic RKHS	$$\widetilde K(x,y) = K(x,x)^{-1/2}K(x,y)K(y,y)^{-1/2}$$	$\widetilde K(x,x) = I$, positivity
Hardy space (Cauchy–Szegö)	$$S(q,w) = s\big(q_{n+1} + \overline{w_{n+1} - 2\overline{w'} \cdot q'}\big)$$	Orthonormality of monomials, unit diagonal
3D Ginibre ensemble	$$K_N(z,w) = \sum_{k=0}^{N-1} \frac{P_k(z)\overline{P_k(w)}}{h_k}$$	Trace normalization: $\int K_N(z,z)d\mu(z)=N$
Color image deblurring	$$\widetilde{Q} = t_0 Q_0 + t_1 Q_1 i + t_2 Q_2 j + t_3 Q_3 k$$	Channel-wise intensity matching

7. Theoretical and Practical Implications

Normalized quaternion kernels are critical in imposing well-posedness, stability, and invariance in quaternionic analytic structures, stochastic spatial models, and cross-channel signal/image processing. Quaternionic normalization is nontrivial; direct generalization of real or complex kernel normalization often fails due to the structure of quaternionic modules and the physical interpretation of kernel actions. Context-adapted normalization—operator diagonalization in RKHS, channel intensity constraints in imaging, or trace normalization in spatial ensembles—remains indispensable for successful quaternion kernel design and deployment (Thirulogasanthar et al., 2016, Yang et al., 21 Nov 2025, Wang et al., 2012, Kargin, 2017, Forrester, 2013).

A plausible implication is that future directions in both theoretical quaternionic analysis and signal processing will increasingly emphasize structurally adapted normalization mechanisms to fully exploit the quaternion algebra's expressive power while controlling for cross-component physical and statistical behaviors.