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Digital image authentication is an extremely significant concern for the digital revolution, as it is easy to tamper with any image. In the last few decades, it has been an urgent concern for researchers to ensure the authenticity of digital images. Based on the desired applications, several suitable watermarking techniques have been developed to mitigate this concern. However, it is tough to achieve a watermarking system that is simultaneously robust and secure. This paper gives details of standard watermarking system frameworks and lists some standard requirements that are used in designing watermarking techniques for several distinct applications. The current trends of digital image watermarking techniques are also reviewed in order to find the state-of-the-art methods and their limitations. Some conventional attacks are discussed, and future research directions are given.

In this paper, we address the multi-view subspace clustering problem. Our method utilizes the circulant algebra for tensor, which is constructed by stacking the subspace representation matrices of different views and then rotating, to capture the low rank tensor subspace so that the refinement of the view-specific subspaces can be achieved, as well as the high order correlations underlying multi-view data can be explored. By introducing a recently proposed tensor factorization, namely tensor-Singular Value Decomposition (t-SVD) (Kilmer et al. in SIAM J Matrix Anal Appl 34(1):148–172, 2013), we can impose a new type of low-rank tensor constraint on the rotated tensor to ensure the consensus among multiple views. Different from traditional unfolding based tensor norm, this low-rank tensor constraint has optimality properties similar to that of matrix rank derived from SVD, so the complementary information can be explored and propagated among all the views more thoroughly and effectively. The established model, called t-SVD based Multi-view Subspace Clustering (t-SVD-MSC), falls into the applicable scope of augmented Lagrangian method, and its minimization problem can be efficiently solved with theoretical convergence guarantee and relatively low computational complexity. Extensive experimental testing on eight challenging image datasets shows that the proposed method has achieved highly competent objective performance compared to several state-of-the-art multi-view clustering methods.

A simple nonrecursive form of the tensor decomposition in $d$ dimensions is presented. It does not inherently suffer from the curse of dimensionality, it has asymptotically the same number of parameters as the canonical decomposition, but it is stable and its computation is based on low-rank approximation of auxiliary unfolding matrices. The new form gives a clear and convenient way to implement all basic operations efficiently. A fast rounding procedure is presented, as well as basic linear algebra operations. Examples showing the benefits of the decomposition are given, and the efficiency is demonstrated by the computation of the smallest eigenvalue of a 19-dimensional operator.

In directed graphs, relationships are asymmetric and these asymmetries contain essential structural information about the graph. Directed relationships lead to a new type of clustering that is not feasible in undirected graphs. We propose a spectral co-clustering algorithm called DI-SIM for asymmetry discovery and directional clustering. A Stochastic co-Blockmodel is introduced to show favorable properties of DI-SIM. To account for the sparse and highly heterogeneous nature of directed networks, DI-SIM uses the regularized graph Laplacian and projects the rows of the eigenvector matrix onto the sphere. A nodewise ASYMMETRY SCORE and DI-SIM are used to analyze the clustering asymmetries in the networks of Enron emails, political blogs, and the Caenorhabditis elegans chemical connectome. In each example, a subset of nodes have clustering asymmetries; these nodes send edges to one cluster, but receive edges from another cluster. Such nodes yield insightful information (e.g., communication bottlenecks) about directed networks, but are missed if the analysis ignores edge direction.

Pinwise nuclide number density (ND) data from the lattice physics code CASMO5 is compressed using a preconditioned truncated singular value decomposition (TSVD) to reduce storage requirements. Previously only assembly-average or single-pin NDs were optionally saved in the CMS5 few-group cross section library. However, backend analysis has prompted the desire to have pin-by-pin NDs available in the library for use by the SNF code. Adding this data set significantly increases the size of the library, particularly for lattices modeled in full assembly geometry (that is, not in half or octant symmetry). To reduce the required storage, the SVD is used to approximate the entire data with a reduced basis. In the four test cases, compression ratios of 2.7 to 8.5 were achieved for the PWR cases, with maximum errors less than 0.1%. However, the rodded BWR segment proved more difficult, with an average compression ratio of 1.6. One advantage of this technique is that the compression ratio is higher for full-symmetry cases, where the need for the compression is also highest.

With the advent of machine learning and its overarching pervasiveness it is imperative to devise ways to represent large datasets efficiently while distilling intrinsic features necessary for subsequent analysis. The primary workhorse used in data dimensionality reduction and feature extraction has been the matrix singular value decomposition (SVD), which presupposes that data have been arranged in matrix format. A primary goal in this study is to show that high-dimensional datasets are more compressible when treated as tensors (i.e., multiway arrays) and compressed via tensor-SVDs under the tensor-tensor product constructs and its generalizations. We begin by proving Eckart–Young optimality results for families of tensor-SVDs under two different truncation strategies. Since such optimality properties can be proven in both matrix and tensor-based algebras, a fundamental question arises: Does the tensor construct subsume the matrix construct in terms of representation efficiency? The answer is positive, as proven by showing that a tensor-tensor representation of an equal dimensional spanning space can be superior to its matrix counterpart. We then use these optimality results to investigate how the compressed representation provided by the truncated tensor SVD is related both theoretically and empirically to its two closest tensor-based analogs, the truncated high-order SVD and the truncated tensor-train SVD.

Recent work by Kilmer and Martin [Linear Algebra Appl., 435 (2011), pp. 641--658] and Braman [Linear Algebra Appl., 433 (2010), pp. 1241--1253] provides a setting in which the familiar tools of linear algebra can be extended to better understand third-order tensors. Continuing along this vein, this paper investigates further implications including (1) a bilinear operator on the matrices which is nearly an inner product and which leads to definitions for length of matrices, angle between two matrices, and orthogonality of matrices, and (2) the use of t-linear combinations to characterize the range and kernel of a mapping defined by a third-order tensor and the t-product and the quantification of the dimensions of those sets. These theoretical results lead to the study of orthogonal projections as well as an effective Gram--Schmidt process for producing an orthogonal basis of matrices. The theoretical framework also leads us to consider the notion of tensor polynomials and their relation to tensor eigentuples defined in the recent article by Braman. Implications for extending basic algorithms such as the power method, QR iteration, and Krylov subspace methods are discussed. We conclude with two examples in image processing: using the orthogonal elements generated via a Golub--Kahan iterative bidiagonalization scheme for object recognition and solving a regularized image deblurring problem. [PUBLICATION ABSTRACT]

Secure image encryption is critical for protecting sensitive data such as satellite imagery, which is pivotal for national security and environmental monitoring. However, existing encryption methods often face challenges such as vulnerability to traffic analysis, limited randomness, and insufficient resistance to attacks. To address these gaps, this article proposes a novel multiple image encryption (MIE) algorithm that integrates hyperchaotic systems, Singular Value Decomposition (SVD), counter mode RC5, a chaos-based Hill cipher, and a custom S-box generated via a modified Blum Blum Shub (BBS) algorithm. The proposed MIE algorithm begins by merging multiple satellite images into an augmented image, enhancing security against traffic analysis. The encryption process splits the colored image into RGB channels, with each channel undergoing four stages: additive confusion using a memristor hyperchaotic key transformed by SVD, RC5 encryption in counter mode with XOR operations, Hill cipher encryption using a 6D hyperchaotic key and invertible matrices mod 256, and substitution with a custom S-box generated by a modified BBS. Experimental results demonstrate the proposed algorithm’s superior encryption efficiency, enhanced randomness, and strong resistance to cryptanalytic, differential, and brute-force attacks. These findings highlight the MIE algorithm’s potential for securing satellite imagery in real-time applications, ensuring confidentiality and robustness against modern security threats.

Singular value decomposition (SVD) can be used both globally and locally to remove random noise in order to improve the signal-to-noise ratio (SNR) of seismic data. However, it can only be applied to seismic data with simple structure such that there is only one dip component in each processing window. We introduce a novel denoising approach that utilizes a structure-oriented SVD, and this approach can enhance seismic reflections with continuous slopes. We create a third dimension for a 2D seismic profile by using the plane-wave prediction operator to predict each trace from its neighbour traces and apply SVD along this dimension. The added dimension is equivalent to flattening the seismic reflections within a neighbouring window. The third dimension is then averaged to decrease the dimension. We use two synthetic examples with different complexities and one field data example to demonstrate the performance of the proposed structure-oriented SVD. Compared with global and local SVDs, and f-x deconvolution, the structure-oriented SVD can obtain much clearer reflections and preserve more useful energy.

This study leverages student performance data and the Funk-singular value decomposition (Funk-SVD) model to identify conceptual weaknesses in first-year calculus learning and generate targeted practice recommendations. Rather than relying on error counts or instructor judgment, the model infers individual learning gaps based on predicted success probabilities. Using data from six exams administered to 850 students, the model achieved strong predictive performance, with an F1-score of 0.794. Simulated intervention analysis revealed that the most substantial learning gains occurred among lower-achieving students. Frequently recommended items indicated persistent difficulties with volume integration, curvature, and Riemann sums. These findings underscore the potential of advanced recommendation models to support scalable, personalized learning–grounded in precise, data-informed diagnosis of conceptual weaknesses–thereby enabling more effective instructional support and promoting long-term academic continuity.