Nonlinear Transformed Low-Rank Quaternion Tensor Total Variation for Multidimensional Color Image Completion

Completing multidimensional color images is a fundamental challenge in image processing and computer vision. However, some tensor-based methods often treat RGB channels as independent modes, thereby neglecting their intrinsic correlations. To address this limitation, we represent RGB values as pure quaternions and organize them into a quaternion tensor for holistic modeling that preserves chromatic relationships. To better capture the nonlinear characteristics inherent in visual data and to improve the compactness of low-rank representations, we propose a nonlinear transformation within the quaternion domain. This design enables more expressive modeling compared to conventional linear approaches. In addition, we introduce two novel regularization terms that jointly encode global low-rankness and local smoothness, with the nonlinear transformation further enhancing the exploitation of structural priors. The overall model is optimized via a nonlinear alternating direction method of multipliers (ADMM), with theoretical guarantees of convergence. Extensive experiments on several datasets demonstrate that the proposed method significantly outperforms state-of-the-art low-rank tensor and quaternion tensor recovery techniques in multidimensional color image completion tasks.