Quaternion Tensor Completion with Sparseness for Color Video Recovery

Abstract: A novel low-rank completion algorithm based on the quaternion tensor is proposed in this paper. This approach uses the TQt-rank of quaternion tensor to maintain the structure of RGB channels throughout the entire process. In more detail, the pixels in each frame are encoded on three imaginary parts of a quaternion as an element in a quaternion matrix. Each quaternion matrix is then stacked into a quaternion tensor. A logarithmic function and truncated nuclear norm are employed to characterize the rank of the quaternion tensor in order to promote the low rankness of the tensor. Moreover, by introducing a newly defined quaternion tensor discrete cosine transform-based (QTDCT) regularization to the low-rank approximation framework, the optimized recovery results can be obtained in the local details of color videos. In particular, the sparsity of the quaternion tensor is reasonably characterized by l1 norm in the QDCT domain. This strategy is optimized via the two-step alternating direction method of multipliers (ADMM) framework. Numerical experimental results for recovering color videos show the obvious advantage of the proposed method over other potential competing approaches.

• The paper presents a novel quaternion tensor completion method that preserves RGB channel structures through TQt-rank analysis for enhanced video recovery.
• It leverages a truncated nuclear norm and logarithmic approximation to maintain low-rank properties while using QTDCT for sparse regularization.
• Experimental results demonstrate superior recovery performance on standard color video datasets even at lower sample rates compared to traditional methods.

Quaternion Tensor Completion with Sparseness for Color Video Recovery

Introduction

The paper "Quaternion Tensor Completion with Sparseness for Color Video Recovery" proposes a novel algorithm for completing color videos leveraging quaternion tensors. It utilizes the TQt-rank of quaternion tensors to maintain the structure of RGB channels across the recovery process. This method involves representing pixels of each frame as elements in a quaternion matrix, subsequently stacked into a quaternion tensor. The algorithm applies a logarithmic function and truncated nuclear norm to approximate the rank of this tensor and adopts a quaternion tensor discrete cosine transform-based (QTDCT) regularization to enhance restoration accuracy.

Figure 1: The illustration of a color video is represented by a three-dimensional quaternion tensor, taking four frames as an example.

Quaternion Tensor Low-Rank Approximation

The quaternion tensor approach extends traditional tensor frameworks by preserving RGB and spatial structures, essential for high-dimensional video data. The TQt-SVD provides a framework for decomposing quaternion tensors; it builds on $N$-th order tensors, allowing a transform domain approach that generalizes t-SVD concepts. The truncated quaternion tensor nuclear norm (QT-RNN) is defined for more accurately depicting rank by not penalizing larger singular values as heavily, enhancing low-rank characteristics pertinent for video recovery processes.

Figure 2: The illustration of the TQt-SVD of an $I_1 \times I_2 \times I_3$ quaternion tensor.

Sparse Regularization Via QTDCT

Sparsity is addressed through the quaternion tensor discrete cosine transform (QTDCT). The sparsity regularization is integrated as a constraint in the recovery model using $l_1$ norm in the QDCT domain, where characteristics of real-world visual data become sparse. This novel sparse depiction facilitates substantial improvements in optimization efficiency and recovery fidelity.

Figure 3: The illustration of the sparsity under QTDCT, taking three color videos (each video has 20 frames) as examples.

Optimization Through ADMM

The model's optimization leverages the Alternating Direction Method of Multipliers (ADMM). The approach involves alternating between optimizing the quaternion tensor for low-rank structure and reinforcing sparsity through QTDCT regularization. This dual approach enables the model to effectively discern and recover missing data, enhancing both global and local video structures. Key operations involve a sequence of optimization equations and iterative updates that align tensor completion with the constraints and objectives encoded by the algorithm.

Experimental Results

Empirical analysis conducted on several standard color video datasets illustrates that the proposed method, particularly QT-RNNS2, achieves superior performance in recovering videos even at lower Sample Rates (SRs) compared to existing completion techniques. These experimental cases highlight the robustness and precision of the quaternion tensor approach in practical video restoration scenarios.

Figure 4: The visual results of recovering bus with SR=0.1 at the first frame.

Discussion

The proposed quaternion tensor model demonstrates a substantial advancement over traditional tensor-based recovery methods due to its ability to preserve intrinsic video data structures without reducing dimensionality, thanks to quaternion representation. The adoption of QT-RNN and sparse regularization enables efficient handling of videos with incomplete data, yielding higher fidelity recovery than previously achieved.

Conclusion

This paper introduces an innovative quaternion tensor recovery model that provides effective color video completion by simultaneously addressing low-rank and sparse characteristics in the quaternion domain. The promising results and robust theoretical foundation propose new directions for exploiting quaternion-based methodologies in video processing and other high-dimensional data applications. Looking forward, expanding upon this quaternion framework with additional tensor decompositions and exploring further local properties can refine and enhance practical implementations.