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A Fast Algorithm for Quaternion Tensor Singular Value Decomposition under the QT-Product with Applications to Color Video Processing
Daochang Zhang, Yue Zhao, Jingqian Li
TL;DR
QT-SVD has a theoretical framework, but its computational efficiency and performance for color-video processing merit further investigation. The paper revisits QT-SVD using complex representations and develops a fast implementation. Across five tested sequences, QT-SVD achieves higher reconstruction quality, while both methods show similar aggregate ciphertext sensitivity and adjacency correlation, and QT-SVD adds less degradation for tested occluded inputs.
Problem
QT-SVD provides a theoretical framework, but its computational efficiency and performance in color-video processing require further investigation, particularly because direct quaternion matrix SVD may be costly for large tensor data.
Source: §1 · p. 2Method
The paper revisits QT-SVD under the QT-product using complex quaternion-matrix representations and applies it to color-video encryption and reconstruction.
Source: §1 · p. 2Results
Across five tested sequences, QT-SVD yields higher PSNR and SSIM after decryption in every video, while both methods have similar aggregate ciphertext sensitivity and low measured R-channel adjacency correlation; QT-SVD adds less average degradation for tested occluded inputs.
Source: §4.7 · p. 23Takeaways & Limitations
Under the tested settings, QT-SVD is a useful representation for the video recovery pipeline, with higher reconstruction quality and less additional degradation for tested occluded inputs.
Source: §4.7 · p. 23The evaluation covers five sequences resized to 256 × 256, one masking/permutation parameter set, and retained auxiliary information, and makes no claim of formal cryptographic security.
Source: §4.7 · p. 23
Abstract
from arXiv · showhide
Based on the complex representation of quaternion matrices, this paper revisits the quaternion tensor singular value decomposition (QT-SVD) under the QT-product and establishes the corresponding $z$-block circulant matrix decomposition. We develop an efficient implementation of the QT-SVD. Numerical experiments show that this implementation is substantially faster than QT-SVD based on the quaternion matrix SVD approach \cite[Theorem~2.3]{AMLzhang2022} while maintaining high numerical accuracy. We further apply the QT-SVD to color video encryption and reconstruction. Experiments on five DAVIS video sequences show that the QT-SVD achieves higher reconstruction quality and shorter decomposition times than the quaternion T-SVD \cite{tsvdarxiv} under the same processing procedure. Both methods exhibit comparable ciphertext statistics and differential sensitivity, while the QT-SVD introduces less additional distortion under the tested input perturbations.
1. Introduction
Quaternion tensors represent multichannel data while retaining correlations across video modes, and QT-SVD provides a decomposition framework under the QT-product. This paper revisits that framework, develops a faster implementation, and evaluates it for color-video encryption and reconstruction.
1. Introduction
- Encoding RGB channels in the imaginary components of quaternion matrices makes QSVD relevant to color-image processing.Quaternion representations have also been used in image processing, signal processing, and computer vision.Source: §1 · p. 1
- For higher-order data such as video, tensor representations naturally retain correlations across multiple modes; quaternion tensor SVDs include octonion-based work, T-product methods, and QT-SVD under the QT-product.Source: §1 · p. 1
- QT-SVD offers a theoretical framework, but its computational efficiency and color-video performance require further investigation.Direct quaternion matrix SVD may be costly for large-scale tensor data.Source: §1 · p. 2
- The paper develops a fast QT-SVD implementation using complex quaternion-matrix representations and compares its speed and accuracy with direct quaternion matrix SVD.It also revisits the QT-SVD theory and establishes the associated z-block circulant matrix decomposition.Source: §1 · p. 2
2. Preliminaries and key lemmas
The preliminaries define quaternion-tensor notation and the QT-product, then connect tensor operations to z-block circulant matrices and Fourier block diagonalization. They also introduce the complex representation used in the paper's decomposition methods.
2. Preliminaries and key lemmas
- The preliminaries define quaternion tensors, their Frobenius norm, conjugate transpose, identity, unitarity, and f-diagonal structure.A quaternion tensor is represented as A = Ad + jAc, and its frontal slices use MATLAB-style indexing.Source: §2 · pp. 2, 4
- The complex representation of a quaternion matrix preserves its singular values, with each value appearing twice in the complex representation.The representation is defined for A = X + Yj, where X and Y are complex matrices.Source: §2 · p. 2
- The unfolding operator maps a quaternion tensor to the first block column of its block-circulant representation, and folding reverses that mapping.The paper defines the z-block circulant representation and the QT-product using these tensor and matrix constructions.Source: §2 · pp. 3–4
- The normalized mode-3 DFT transforms a quaternion tensor along its third mode, and Fourier block diagonalization relates complex block-circulant matrices to their frequency slices.The associated lemmas give Fourier block diagonalization and algebraic properties of quaternion z-block circulant matrices.Source: §2 · pp. 3–4
- The z-block circulant framework relates tensor multiplication, conjugate transpose, and unitarity to corresponding matrix operations.The listed lemmas cover products, conjugate transpose, and the equivalence between tensor and z-block circulant unitarity.Source: §2 · p. 4
3. Quaternion Tensor Singular Value Decomposition and Its Fast Implementation
The paper establishes QT-SVD under the QT-product, then gives a Fourier-domain implementation that avoids explicitly constructing a large z-block circulant matrix. In tests, that implementation is substantially faster than direct quaternion SVD while retaining high numerical accuracy.
3.1. QT-SVD under the QT-Product
- QT-SVD factors a quaternion tensor into unitary quaternion tensors and an f-diagonal tensor under the QT-product.The associated z-block circulant matrix decomposition has unitary factors and diagonal blocks for the singular-value factor.Source: §3.1 · p. 5
- The proof removes an unnecessary requirement that diag( bS) be diagonal, which would otherwise be too restrictive alongside diagonal frontal slices.The paper notes that diag( bS) need not be diagonal for the stated QT-SVD result.Source: §3.1 · pp. 6, 9
3.2. Algorithms for QT-SVD
- The fast implementation decomposes Fourier-domain frontal slices directly, avoiding explicit construction of the large z-block circulant matrix.It applies qsvdC to each transformed slice and then uses the inverse mode-3 DFT to recover the factors.Source: §3.2 · p. 9
3.3. Comparison of Two QT-SVD Implementations
- The implementation comparison uses the same quaternion FFT, inverse transform, and QT-product, differing only in the matrix SVD computation at each frequency.The direct implementation calls QTFM's quaternion svd; the complex implementation forms a 2n × 2n complex representation for each n × n quaternion matrix.Source: §3.3 · p. 10
- 4.40–15.39 times faster in the tests, the complex implementation retains high numerical accuracy compared with direct quaternion SVD.The largest relative reconstruction errors were 7.735 × 10−15 for direct quaternion and 6.697 × 10−14 for complex implementations; the two implementations compute the same QT-product decomposition.Source: §3.3 · p. 10
4. Applications to Color Video Processing
The paper applies QT-SVD and T-SVD to color-video encryption and recovery using the same processing procedure, retaining the full decomposition and comparing accuracy, ciphertext properties, sensitivity, and time.
4.1. Video Encryption and Decryption Scheme
- The video pipeline decomposes the input, scales and XOR-masks core components, applies an Arnold permutation, then reverses these steps before reconstruction.QT-SVD uses the QT-product for reconstruction; T-SVD uses the T-product and corresponding adjoint.Source: §4.1 · p. 11, §4 · p. 11
4.1.1. Component Normalization and Logistic Keystream Masking
- Each real-valued core slice is normalized and quantized to uint8, then masked with a Logistic keystream generated independently for each frame and component.The keystream is XORed with the component; because masking does not move locations, an invertible Arnold permutation subsequently provides spatial scrambling.Source: §4.1.1 · pp. 11–12
- Quantizing normalized components to uint8 may introduce a small recovery error after inverse normalization.The component extrema are preserved for that inverse normalization.Source: §4.1.1 · p. 11
4.1.2. Arnold Permutation and Core Component Recovery
- The Arnold permutation rearranges core-component locations before recovery; the implementation uses a = 2, b = 1, and 20 iterations.Recovery inverts the permutation, decodes the XOR mask, and inverse-normalizes components before assembling the core tensor for reconstruction.Source: §4.1.2 · p. 12
- Because RGB is encoded in the core tensor’s imaginary components, any confidential information in its real component or auxiliary reconstruction data also requires encryption or secure transmission.The implementation encrypts the three imaginary core components.Source: §4.1.2 · p. 12
4.2. Experimental Setup and Evaluation Metrics
The evaluation uses five DAVIS videos with shared preprocessing and encryption settings for QT-SVD and T-SVD, and defines measures for decomposition accuracy, reconstruction quality, ciphertext properties, and sensitivity to input perturbations.
4.2. Experimental Setup and Evaluation Metrics
- Five DAVIS 2017 TrainVal 480p sequences contribute 355 frames total; frames are converted to double precision, normalized to [0, 1], and resized to 256 × 256.Both branches use the same inputs, preprocessing, and encryption parameters, with no singular-value truncation.Source: §4.2 · pp. 12–13, Table 2 · p. 13
4.2.1. Decomposition Errors
- The absolute Frobenius error between the input tensor and its directly reconstructed decomposition tensor measures numerical decomposition accuracy before RGB conversion and is not normalized by the input norm.The compared QT-SVD and T-SVD branches use complete decompositions without truncation.Source: §4.2.1 · p. 13, §4.2 · p. 13
4.2.2. Reconstruction Quality Metrics
- MSE, PSNR, and SSIM evaluate reconstruction quality; PSNR pools errors from three RGB channels before the logarithm, while SSIM averages channel-wise values; video scores average frames, then equally weight five videos.Source: §4.2.2 · pp. 13–14
4.2.3. Ciphertext Statistics and Differential Sensitivity
- Ciphertext statistics use frame-25 histograms and sampled R-channel adjacent-pixel correlations, while differential sensitivity is measured with channel-averaged NPCR and UACI.A 0.01 change to the first frame’s first-pixel R value triggers rerunning decomposition and encryption; input-dependent Logistic states may also change, so NPCR and UACI characterize the complete procedure.Source: §4.2.3 · pp. 14–15
4.3. Reconstruction Accuracy and Visual Comparison
The QT-SVD and T-SVD both reproduce the input tensors accurately, while QT-SVD yields higher reconstruction quality across the five test sequences. Visual comparisons show scene-preserving reconstructions and noise-like encrypted cores.
4.3.1. Decomposition Accuracy
- QT-SVD errors range from 1.459 × 10^-11 to 2.227 × 10^-11, and T-SVD errors from 3.446 × 10^-12 to 1.054 × 10^-11; both reproduce inputs accurately.The errors are absolute Frobenius errors.Source: §4.3.1 · p. 15, Table 3 · p. 15
4.3.2. Reconstruction Quality
- QT-SVD yields higher PSNR and SSIM than T-SVD across all five sequences, with mean PSNRs of 64.9897 dB and 49.2168 dB, respectively.Mean SSIM values are 0.999809 for QT-SVD and 0.992990 for T-SVD; the five-video mean PSNR difference is 15.7730 dB.Source: §4.3.2 · p. 15, Table 4 · p. 16
4.3.3. Visual Comparison
- At frames 1, 25, and 50, QT-SVD reconstructions preserve scene content, while encrypted cores exhibit noise-like patterns.The visual comparison covers five sequences, with original RGB frames, encrypted-core visualizations, and reconstructed RGB frames arranged in columns and rows.Source: §4.3.3 · p. 16, Figure 1 · p. 16, Figure 2 · p. 17
- At frame 25, QT-SVD and T-SVD both preserve visual content, and their reconstruction differences are small at the displayed scale.The reported quantitative comparisons use double-precision reconstructed RGB frames.Source: §4.3.3 · p. 16, Figure 4 · p. 18
4.4. Statistical Analysis and Differential Sensitivity of the Encrypted Core
The encrypted-core histograms are flatter than the original-frame histograms, and QT-SVD substantially reduces horizontal adjacent-pixel correlation. QT-SVD and T-SVD show comparable measured ciphertext correlations and differential sensitivity.
4.4.1. Histograms and Adjacent-Pixel Correlation
- Encrypted-core components have substantially flatter intensity distributions than original RGB frames, but the histograms do not establish exact uniformity.The comparison uses frame 25 across five test videos and 256 intensity bins.Source: §4.4.1 · p. 18, Figure 5 · p. 19
- QT-SVD encrypted-core correlations range from −0.013116 to 0.015532, versus original-frame correlations of 0.920005 to 0.980894, for horizontal adjacent pixels in the R channel.The scatter plots become broadly dispersed after encryption.Source: §4.4.1 · p. 19, Figure 6 · p. 20
- The five-video mean absolute encrypted R-channel correlations are 0.0119 for QT-SVD and 0.0127 for T-SVD, with no consistent per-video advantage for either method.The coefficients are Pearson correlations computed from horizontal adjacent-pixel pairs.Source: §4.4.1 · p. 21, Table 6 · p. 21, Table 5 · p. 21
4.4.2. Differential Sensitivity
- Both methods yield mean NPCR and UACI values close to the theoretical reference values of 99.6094% and 33.4635%, respectively, indicating comparable differential sensitivity.The reported sensitivity concerns changes in the encrypted core after a small plaintext perturbation.Source: §4.4.2 · p. 21
4.5. Stability under Pre-Encryption Input Perturbations
The section tests paired pre-encryption perturbations and finds a clearer QT-SVD advantage under occlusion, while emphasizing that the results do not establish denoising or encrypted-data error tolerance.
4.5. Stability under Pre-Encryption Input Perturbations
- Each perturbation is applied before decomposition, and paired input and reconstructed videos are compared against the same clean reference to isolate processing-related quality change.The tested perturbations are salt-and-pepper noise, Gaussian noise, and a centered 12 × 12 zero occlusion.Source: §4.5 · pp. 21–22
- The slight positive PSNR changes do not by themselves demonstrate a denoising capability.QT-SVD's mean PSNR changes were +0.003292, +0.000558, and −0.002790 dB for salt-and-pepper noise, Gaussian noise, and occlusion, respectively.Source: §4.5 · pp. 21–22
- Under occlusion, mean PSNR decreases by 0.0027 dB for QT-SVD and 0.2187 dB for T-SVD, while mean SSIM decreases are 0.0002 and 0.0067, respectively.The paired perturbation results are averaged equally over five videos.Source: §4.5 · p. 22, Table 7 · p. 22
- The experiments perturb plaintext inputs and recompute factors; they do not establish tolerance to corrupted ciphertext or recovery of missing scene content.The stated scope is stability of processing already disturbed videos.Source: §4.5 · p. 22
4.6. Decomposition Time Comparison
QT-SVD requires less decomposition time than T-SVD for each of the five tested video sequences.
4.6. Decomposition Time Comparison
- QT-SVD takes 16.189 s versus 37.352s for T-SVD on equal-video mean decomposition time, a reduction of approximately 56.7%.Table 8 compares decomposition times for the five video sequences.Source: Table 8 · p. 22, §4.6 · p. 22
4.7. Discussion and Limitations
Across five sequences, QT-SVD performs favorably on reconstruction quality and tested occlusion stability, while the evidence remains bounded by the prototype's evaluation scope.
4.7. Discussion and Limitations
- Across the five-sequence study, both methods accurately reconstruct videos, QT-SVD yields higher post-decryption PSNR and SSIM in every tested video, and both show similar aggregate ciphertext sensitivity and low measured R-channel adjacency correlation.QT-SVD also introduces less additional average degradation for the tested occluded inputs.Source: §4.7 · p. 23
- The evaluation is limited to five sequences resized to 256 × 256, one masking/permutation parameter set, and the prototype's retained auxiliary information.The authors make no claim about lossless floating-point video recovery, rank-compression performance, ciphertext corruption tolerance, or formal cryptographic security.Source: §4.7 · p. 23
Funding
The first author received support from Chinese national, postdoctoral, and Jilin Province research programs.
Funding
- The first author was supported by NSFC grant 11901079, China Postdoctoral Science Foundation grant 2021M700751, and Jilin Province research program grants JJKH20190690KJ, JJKH20220091KJ, and JJKH20250851KJ.The funding statement names these supporting programs and grant numbers.Source: Funding · p. 23