Quaternion Convolution Kernel

• Quaternion convolution kernels are multi-component filters defined as a + bi + cj + dk and utilize Hamilton products to enable joint processing of multidimensional signals.
• They implement both spatial-domain circulant structures and frequency-domain diagonalization via the quaternion Fourier transform to efficiently process and analyze data.
• In neural networks, these kernels reduce redundancy and enhance feature representation by coupling channels, leading to improved computational and learning performance.

A quaternion convolution kernel is a multi-component filter defined over the non-commutative field of quaternions (ℍ), providing a principled mechanism for jointly processing multidimensional signals, most prominently found in applications such as color image analysis and quaternion-valued neural networks. Unlike real or complex convolutional kernels, quaternion kernels intrinsically couple channels through the algebra of Hamilton products, admitting both a spatial-domain and frequency-domain (Fourier or canonical) interpretation. The study of quaternion convolution kernels has yielded rigorous constructions—including circulant matrix formalism, spectral diagonalization via the quaternion Fourier transform (QFT), and advanced kernel normalization—which serve as theoretical and practical foundations for modern hypercomplex signal processing and learning architectures.

1. Algebraic Formulation of Quaternion Convolution Kernels

A quaternion kernel $q$ takes the canonical form $q = a + b\,i + c\,j + d\,k$, $a,b,c,d\in\mathbb{R}$. The Hamilton product operates as: $$q \otimes r = (a e - b f - c g - d h) + (a f + b e + c h - d g)i + (a g - b h + c e + d f)j + (a h + b g - c f + d e)k$$ The quaternion kernel for discrete convolution with length $N$ is a vector $kc\in\mathbb{H}^N$, with circulant matrix representation $K_{i,j}=kc[(i-j)\bmod N]$. For two-dimensional data (e.g., images), spatial-domain product is extended by embedding each color triplet as a pure-imaginary quaternion and the kernel as a sum of four real-valued sub-kernels: $Q(p) = k_0(p) + k_1(p) i + k_2(p) j + k_3(p) k$ This block structure induces cross-channel couplings not present in conventional convolution. Matrix representation of the realized convolution is given by a structured $4 \times 4$ block matrix acting on the 4-dimensional real signal vector, encapsulating rotations and scalings in the feature space (Yang et al., 21 Nov 2025, Altamirano-Gomez et al., 2023).

2. Frequency Domain Diagonalization and the Quaternion Fourier Transform

Quaternion kernels become diagonalizable via the quaternion Fourier transform (QFT). Given a pure unit transform axis $\mu$ ($\mu^2 = -1$), the QFT is the $N \times N$ unitary matrix: $$Q(\mu)_{i,j} = \frac{1}{\sqrt{N}} \exp\left( -2\pi \mu \frac{i j}{N} \right)$$ The key result is that the circulant convolution operator $K$ is simultaneously diagonalized by $Q(\mu)$: $$Q(\mu)^* K Q(\mu) = \mathrm{diag}(\lambda_0, \ldots, \lambda_{N-1})$$ where each $\lambda_u$ is given by the right-QFT of the kernel vector. Consequently, convolution in the QFT domain reduces to entrywise multiplication: $$y = Q(\mu)\left[\,\mathrm{diag}(\lambda_0, \ldots, \lambda_{N-1}) \odot \hat{x}\,\right]$$ This diagonalization not only accelerates computation via quaternion FFT but also provides an analytic means to bound the operator norm (Lipschitz constant) of each layer in quaternion-valued neural networks, since $\|K\|_2 = \max_{u} |\lambda_u|$ (Sfikas et al., 2023).

Non-commutativity necessitates distinguishing left- and right-convolution as well as corresponding left-/right-QFTs. The choice of $\mu$ parameterizes the spectral plane, yielding spectral rotation invariance with respect to the singular-value spectrum.

3. Spatial and Frequency-Domain Kernel Interrelations

Classical quaternion convolution (spatial-domain shifting and weighted summing) and frequency-domain (Mustard) convolution differ fundamentally due to non-commutativity: $(f*g)(x) = \int_{\mathbb{R}^2} f(x-y) g(y) \, dy$ does not, under QFT, transform to pointwise multiplication, motivating Mustard convolution: $$f \star_{p,v} g = 2\pi\, \mathcal{F}_{p,v}^{-1}(\mathcal{F}_{p,v}\{f\}\,\mathcal{F}_{p,v}\{g\})$$ A central result is the decomposition of the classical convolution as a finite linear sum (8 or 64 terms) of Mustard convolutions of reflected and rotated signals, and their QFTs as a corresponding sum of pointwise products. This result underpins the precise transfer of spectral kernel design into spatial implementations (Bie et al., 2015, Bujack et al., 2013).

4. Quaternion Kernels in Neural Network Architectures

Quaternion convolution kernels underpin layers of quaternion convolutional neural networks (QCNNs), where feature maps and kernels are quaternion-valued. For a spatial location $(i,j)$ and output channel $k$: $$y_{k,i,j} = \sum_{c=1}^{C_{\rm in}} \sum_{u=1}^{K} \sum_{v=1}^{K} W_{k,c}[u,v]\,\otimes\,X_{c}[i+u-1,j+v-1] + b_k$$ The convolution is expanded such that every kernel $W_{k,c}[u,v]$ is a quaternion, yielding four real parameter maps per filter. Gradients in backpropagation utilize quaternion conjugation to appropriately rotate derivatives (Zhu et al., 2019, Altamirano-Gomez et al., 2023).

Compared to real-valued convolution, the Hamiltonian structure inherently couples channels, encoding geometric transformations: rotations and scalings in color- or feature-space. Empirically, this often reduces redundancy, improves representational capacity, and can significantly reduce parameter count (for equal number of channels, a 75% reduction in the simplest cases) (Altamirano-Gomez et al., 2023, Zhu et al., 2019).

5. Kernel Normalization, Design, and Applications

In image deconvolution or restoration, quaternion kernels require normalization principles that extend beyond standard scalar constraints. For color image blind deconvolution, the quaternion kernel is structured as $k_0(p)\geq 0$ (grayscale analog) plus three unconstrained kernels $k_1(p),k_2(p),k_3(p)$ responsible for inter-channel correction. Proper normalization employs solving a constrained linear system to match per-channel intensities based on the action of the kernel on the input, mitigating hue artifacts and channel-wise bias that arise from naive normalization schemes (Yang et al., 21 Nov 2025).

The structure of quaternion kernels enables advanced regularization and kernel constraint strategies in both spectral and spatial domains. In denoising, edge detection, and in complex color manipulation tasks, the algebraic expressivity of the quaternion kernel is exploited for joint feature transformations and adaptive filter design (Yang et al., 21 Nov 2025, Zhu et al., 2019).

6. Generalizations: Canonical Transforms and Correlation Kernels

Beyond QFT, quaternion convolution kernels have been systematically generalized to the quaternion linear canonical transform (QLCT) and offset linear canonical transform (QOLCT) frameworks. In these settings, kernels are modulated by quaternion-valued chirp functions, and spatial convolutions become weighted integrals: $$(f\circledast_{A_1,A_2}^{\mu,\nu} g)(x,y) = \int_{\mathbb{R}^2} W_{A_1}^\mu(x,\tau_1) f(\tau_1,\tau_2) g(x-\tau_1, y-\tau_2) W_{A_2}^\nu(y,\tau_2) d\tau_1 d\tau_2$$ Spectral-domain (“Type II”) convolutions are constructed to restore pointwise product theorems in QLCT space, with explicit transforms specified in the literature. These frameworks maintain associativity and support efficient filtering, Fredholm-equation solution, and system identification for quaternionic PDEs, facilitating both theoretical analysis and engineering applications (Hu et al., 2022, Bhat et al., 2021, Bhat et al., 2021).

7. Implementation and Computational Aspects

Efficient realization of quaternion convolution kernels leverages block decomposition: each quaternion kernel and signal is unpacked to four real-valued channels, and the Hamilton product or corresponding kernel matrix is used to recombine outputs. In spectral implementations, quaternion FFTs (often built atop four real FFTs and appropriate channel recombination) achieve computational costs matching those of standard FFT-based convolutions, i.e., $O(N^2 \log N)$ for 2D signals (Altamirano-Gomez et al., 2023, Bujack et al., 2013). Current deep learning frameworks can encapsulate quaternion layers via low-level extensions, enabling practical usage in GPU-accelerated environments.

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References:

• (Sfikas et al., 2023) On the Matrix Form of the Quaternion Fourier Transform and Quaternion Convolution
• (Zhu et al., 2019) Quaternion Convolutional Neural Networks
• (Bie et al., 2015) Connecting spatial and frequency domains for the quaternion Fourier transform
• (Yang et al., 21 Nov 2025) Blind Deconvolution for Color Images Using Normalized Quaternion Kernels
• (Bujack et al., 2013) Convolution products for hypercomplex Fourier transforms
• (Altamirano-Gomez et al., 2023) Quaternion Convolutional Neural Networks: Current Advances and Future Directions
• (Hu et al., 2022) Convolution theorems associated with quaternion linear canonical transform and applications
• (Bhat et al., 2021) Convolution and Correlation Theorems for Wigner-Ville Distribution Associated with the Quaternion Offset Linear Canonical Transform
• (Bhat et al., 2021) Quaternion Offset Linear Canonical Transform in One dimensional Setting