Graph Regularized Non-negative Reduced Biquaternion Matrix Factorization for Color Image Recognition

Tensor completion has emerged as a powerful framework for recovering missing data in multidimensional signals by exploiting low-rank tensor structures. Among existing approaches, linear transform-based tensor nuclear norm (TNN) methods have achieved considerable success by enforcing low-rankness on transformed frontal slices. However, the low-rank structure revealed by linear transforms remains inherently limited. To better capture intrinsic correlations, nonlinear transform-based TNN (NTTNN) models have been proposed, significantly enhancing low-rank representation through composite transforms. Despite their effectiveness, existing NTTNN methods are restricted to real-valued tensors and fail to model quaternion-valued data, which are essential for preserving inter-channel dependencies in color images and videos. Extending nonlinear TNN models to the quaternion domain is challenging due to the non-commutativity of quaternion multiplication and the complexity of quaternion singular value decomposition. To address the limitations encountered in prior works, we propose a quaternion nonlinear transform-induced tensor nuclear norm (QNTTNN) via a real embedding of quaternions, enabling tractable nuclear norm definitions and efficient optimization. Building upon QNTTNN, we formulate a quaternion tensor completion model and develop a proximal alternating minimization algorithm with rigorous convergence guarantees. Extensive experiments on benchmark color video inpainting datasets validate the superior performance of the proposed method over existing approaches.

Division-of-focal-plane (DoFP) color polarization cameras enable snapshot acquisition of color polarization mosaic images, but the inherently sparse sampling pattern makes color polarization demosaicking severely ill-posed. Existing methods often fail to jointly exploit the correlations among polarization channels and the physical constraints inherent in polarization imaging, resulting in noticeable demosaicking artifacts. To address this issue, a quaternion-tensor-based color polarization demosaicking (CPDM) method incorporating Stokes-domain total variation (TV) regularization is proposed. Correlation analysis shows that the correlations among polarization channels are stronger than those among color channels. Accordingly, the color polarization images acquired at $0^\circ$, $45^\circ$, $90^\circ$, and $135^\circ$ are encoded into the four components of a third-order quaternion tensor, with the color channels organized along its third mode. A low-rank prior is then imposed on the quaternion tensor to exploit the global structural redundancy in the color polarization data. Moreover, spatial gradients are mapped to the Stokes domain through an orthogonal transformation to separate intensity, polarization and residual variations, with adaptive quaternion weights enabling component-specific regularization and preserving the energy consistency of the reconstructed Stokes vectors. An efficient optimization algorithm is derived for the resulting model. Extensive experiments demonstrate the superior demosaicking performance of the proposed method.

We investigate the learning of interpretable bases in non-negative matrix factorisation (NMF) by regularising the topology of the learned basis functions. Our approach is motivated by the observation that many data modalities can be viewed as non-negative functions on a structured domain, where the quality of a basis is intrinsically linked to its topology. However, naive methods for incorporating the topology of the support are often hindered by discreteness and threshold dependence, rendering them unsuitable for continuous optimisation. We address these challenges by employing persistent homology as a stable, threshold-free topological quantifier and by designing topological scores that integrate into the NMF objective as regularisers. The resulting framework encompasses spatially coherent image components, periodic time-series structures, and clique-like graph signals within a unified modelling language.