Input features are conventionally represented as vectors, matrices, or third order tensors in the real field, for color image classification. Inspired by the success of quaternion data modeling for color images in image recovery and denoising tasks, we propose a novel classification method for color image classification, named as the Low-rank Support Quaternion Matrix Machine (LSQMM), in which the RGB channels are treated as pure quaternions to effectively preserve the intrinsic coupling relationships among channels via the quaternion algebra. For the purpose of promoting low-rank structures resulting from strongly correlated color channels, a quaternion nuclear norm regularization term, serving as a natural extension of the conventional matrix nuclear norm to the quaternion domain, is added to the hinge loss in our LSQMM model. An Alternating Direction Method of Multipliers (ADMM)-based iterative algorithm is designed to effectively resolve the proposed quaternion optimization model. Experimental results on multiple color image classification datasets demonstrate that our proposed classification approach exhibits advantages in classification accuracy, robustness and computational efficiency, compared to several state-of-the-art methods using support vector machines, support matrix machines, and support tensor machines.

In 1843, the concept of quaternions was first introduced by Sir William Rowan Hamilton. Nowadays, quaternions have emerged as a crucial mathematical concept in various fields such as Mechanics [1, 36, 41], Optics [4, 16], Modern Computer Graphics [25, 34], Color Image Processing [30, 39, 44], and more. As a new tool for color image representation, quaternions have obtained remarkable results in color image processing. The three imaginary parts of the quaternion correspond to the three channels of the color image, and each color image corresponds to a quaternion matrix, and the quaternion can achieve the fusion of color information by processing the color information of different color channels simultaneously. Therefore, quaternion is applied to convolutional neural networks. The pioneering work of Zhu et al. introduces the Quaternion Convolutional Neural Networks (QCNN) [45], which incorporates network layers such as convolutional layer and fully connected layer. In a similar vein, Parcollet et al. propose the Quaternion Recurrent Duan Wang was with School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou 221116, P. R. China.