Quaternion Tensor DCT (QTDCT)

• QTDCT is a multi-dimensional transform for 3D quaternion tensors that preserves inter-channel correlations and spatial-temporal coherence in color videos.
• It leverages non-commutative quaternion algebra with DCT matrices to achieve effective energy compaction and decorrelation along RGB and temporal modes.
• An ADMM-based optimization framework integrates QTDCT sparsity regularization with low-rank constraints, yielding superior recovery metrics like PSNR and SSIM.

The Quaternion Tensor Discrete Cosine Transform (QTDCT) is a domain-specific linear transform defined for tensors with quaternion-valued entries. It enables structure-preserving and multi-modal sparsity regularization for color video recovery tasks, specifically in the context of low-rank and sparse tensor completion algorithms. QTDCT leverages the non-commutative algebraic properties of quaternions to maintain native inter-channel correlations and spatial-temporal coherence within color video data.

1. Algebraic Definition and Computation of QTDCT

QTDCT operates on 3D quaternion tensors, where each pixel in an RGB video frame is encoded as a pure quaternion (with imaginary components corresponding to R, G, B, and no real part), and frames are stacked along the temporal mode to construct the tensor $$\dot{\mathcal{T}} \in \mathbb{H}^{I_1 \times I_2 \times I_3}$$.

The transform is defined in two algebraic forms due to quaternion non-commutativity, but the left-handed form is adopted for practical use:

• Left-handed QTDCT:

$$\mathcal{C}(\dot{\mathcal{T}})_L \triangleq \dot{u} \cdot (\dot{\mathcal{T}} \times_1 \mathbf{C}_1 \times_2 \mathbf{C}_2 \times_3 \mathbf{C}_3)$$

where $\dot{u}$ is a pure unit quaternion $(\dot{u}^2 = -1)$, $\mathbf{C}_i \in \mathbb{R}^{I_i \times I_i}$ are real DCT matrices (Discrete Cosine Transform along each tensor mode), and $\times_n$ denotes the $n$-mode tensor-matrix product.

• Inverse QTDCT:

$$\mathcal{C}^{-1}(\dot{\mathcal{T}})_L \triangleq \dot{u} \cdot (\dot{\mathcal{T}} \times_1 \mathbf{C}_1^{-1} \times_2 \mathbf{C}_2^{-1} \times_3 \mathbf{C}_3^{-1})$$

Implementation via Cayley-Dickson Representation:

1. Decompose $\dot{\mathcal{T}}$ via Cayley-Dickson into two complex tensors:
   • $$\dot{\mathcal{T}} = \mathcal{T}_p + \mathcal{T}_q j$$, with $$\mathcal{T}_p, \mathcal{T}_q \in \mathbb{C}^{I_1 \times I_2 \times I_3}$$.
2. Apply multidimensional DCT to each:
   • $\text{DCT}_C(\mathcal{T}_p)$ and $\text{DCT}_C(\mathcal{T}_q)$.
3. Recombine: $$\widehat{\mathcal{C}(\dot{\mathcal{T}})}_L = \text{DCT}_C(\mathcal{T}_p) + \text{DCT}_C(\mathcal{T}_q) j$$.
4. Left-multiply by $\dot{u}$ to yield $\mathcal{C}(\dot{\mathcal{T}})_L$.

2. Mathematical Properties and Structural Advantages

QTDCT retains inter-channel and multi-modal dependencies that are intrinsic to color videos:

• Structure-Preserving: It processes the tensor as a whole, rather than decomposing channels, thereby preserving chromatic and spatial relationships.
• Multi-dimensional Decorrelation: DCT applied along each mode achieves effective energy compaction and decorrelation, natively across RGB and time axes.
• Compatibility with Quaternion Algebra: Admitted forms and inverses respect quaternion multiplication's non-commutativity.
• Parseval’s Theorem: Energy in the QTDCT domain is preserved, permitting direct transfer of norm-based constraints and regularization.

3. Role in Low-Rank Quaternion Tensor Completion Framework

QTDCT is integral to the framework for color video recovery under missing data scenarios:

• The completion model solves:

$$\min_{\dot{\mathcal{T}}} \ \|\dot{\mathcal{T}}\|_r + \lambda \|\dot{\mathcal{S}}\|_1 \quad \text{s.t.} \quad P_\Omega(\dot{\mathcal{T}}) = P_\Omega(\dot{\mathcal{O}}), \quad \dot{\mathcal{S}} = \mathcal{C}(\dot{\mathcal{T}})_L$$

where $\|\dot{\mathcal{T}}\|_r$ is the truncated nuclear norm based on TQt-rank, enforcing global low-rank structure; $P_\Omega$ denotes observed entries; $\|\dot{\mathcal{S}}\|_1$ is the $l_1$-norm of QTDCT coefficients promoting sparsity; $\lambda$ is a weighting parameter.

• Sparsity regularization in QTDCT domain exploits the empirical distribution where most QTDCT coefficients are close to zero, concentrating image and video information in a few transform coefficients—facilitating recovery that both avoids over-smoothing and preserves local detail and texture.

4. Optimization via ADMM: Details of QTDCT-Sparse Recovery

ADMM is applied to solve the above model by splitting low-rank and sparsity regularization:

• Auxiliary variable $$\dot{\mathcal{S}} = \mathcal{C}(\dot{\mathcal{T}})_L$$ enables separate handling of rank and sparsity.
• In each iteration, the subproblem for $\dot{\mathcal{S}}$ minimizes:

$$\min_{\dot{\mathcal{S}}} \ \lambda \|\dot{\mathcal{S}}\|_1 + \frac{\beta}{2} \|\dot{\mathcal{S}} - \mathcal{C}(\dot{\mathcal{T}}^{k+1}) + \frac{1}{\beta} \dot{\mathcal{Z}}^k \|_F^2$$

which admits a closed-form solution via soft-thresholding in the QTDCT domain:

$$\dot{\mathcal{S}}^{k+1} = \mathcal{S}_{\frac{4\lambda}{\beta^k}}\left(\mathcal{C}(\dot{\mathcal{T}}^{k+1}) - \frac{1}{\beta^k} \dot{\mathcal{Z}}^k \right)$$

where $\mathcal{S}_\tau(\cdot)$ applies element-wise soft-thresholding.

• Updates alternate between low-rank optimization and QTDCT-sparse steps, with the inverse QTDCT applied as required to revert to spatial/color domain for the next iterate.
• Parseval’s theorem ensures consistent regularization magnitude across domains, maintaining algorithmic stability during decoupling.

5. Effects of QTDCT-Sparsity on Color Video Recovery Performance

QTDCT-based sparsity regularization leads to several documented benefits in color video tensor completion:

• Detail Preservation: Suppresses noise while retaining high-frequency components critical for texture and edge recovery.
• Artifact Reduction: Mitigates common visual artifacts induced by naive low-rank approximations.
• Multi-modal Consistency: Regularizes across spatial, chromatic, and temporal axes, yielding reconstructions with realistic color and motion continuity.
• Experimental results in the referenced work demonstrate strong quantitative improvements (higher PSNR/SSIM) and visually superior reconstructions as compared to methods lacking explicit QTDCT sparsity terms, notably at low observation rates.

6. Pipeline Summary Table: Stages of QTDCT-Based Color Video Recovery

Step	Action	Formula/Remark
1	RGB video $\to$ pure quaternion tensor	RGB encoded as $i, j, k$ parts of quaternion
2	Compute QTDCT	$$\mathcal{C}(\dot{\mathcal{T}})_L = \dot{u} \cdot (\dot{\mathcal{T}} \times_1 \mathbf{C}_1 \times_2 \mathbf{C}_2 \times_3 \mathbf{C}_3)$$
3	Impose QTDCT sparsity	$\lambda \|\mathcal{C}(\dot{\mathcal{T}})_L\|_1$ added to objective
4	ADMM QTDCT step	Soft-thresholding, then inverse QTDCT to update tensor
5	Alternate with low-rank TQt-SVD update	See corresponding ADMM step
6	Iterate to convergence	Jointly enforce low-rank global structure and local sparse texture

7. Concluding Remarks

QTDCT provides a multi-dimensional, quaternion-valued DCT framework engineered for color video tensor recovery applications. By integrating quaternion algebra with transform-domain sparsity regularization, the method advances completion frameworks in preserving both global structure and fine texture, especially under challenging sampling conditions. Its efficient implementation via Cayley-Dickson and DCT matrix computations, and empirical superiority in restoration metrics, underscore its utility in multidimensional visual information recovery scenarios (Yang et al., 2022).