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In this article, we determine probabilistic approximations of the entries of random perturbation matrices implementing the Markoff inequality. These approximations are used to derive with prescribed probability asymptotic componentwise perturbation bounds of some orthogonal and unitary matrix decompositions. We show that the probabilistic asymptotic bounds are significantly less conservative than the corresponding deterministic perturbation bounds. As case studies, we consider the determining of probabilistic perturbation bounds of the QR decomposition, the singular value decomposition and the Schur decomposition of a matrix using an unified method for asymptotic componentwise perturbation analysis of these decompositions. It is demonstrated that the probability bounds of the orthogonal transformations, singular values and eigenvalues are much tighter than the corresponding deterministic asymptotic bounds. The probabilistic bounds derived are appropriate for perturbation analysis of large‐order matrix problems. [ABSTRACT FROM AUTHOR]

How are the advantage relations between a set of agents playing a game organized and how do they reflect the structure of the game? In this paper, we illustrate 'Principal Trade-off Analysis' (PTA), a decomposition method that embeds games into a low-dimensional feature space. We argue that the embeddings are more revealing than previously demonstrated by developing an analogy to Principal Component Analysis (PCA). PTA represents an arbitrary two-player zero-sum game as linear combination of simple games via the projection of policy profiles into orthogonal 2D feature planes. We show that the feature planes represent unique strategic trade-offs and truncation of the sequence provides insightful model reduction and visualization. We demonstrate the validity of PTA on a quartet of games (Kuhn poker, RPS + 2, Blotto and Pokemon). In Kuhn poker, PTA clearly identifies the trade-off between bluffing and calling. In Blotto, PTA identifies game symmetries and specifies strategic trade-offs associated with distinct win conditions. These symmetries reveal limitations of PTA unaddressed in previous work. For Pokemon, PTA recovers clusters that naturally correspond to Pokemon types, correctly identifies the designed trade-off between those types, and discovers a rock-paper-scissor (RPS) cycle in the Pokemon generation type – all absent any specific information except game outcomes. [ABSTRACT FROM AUTHOR]

The growing use of the internet results in significant difficulties for the copyright protection of images. To resolve the copyright problem, a scheme is proposed that utilizes the advantages of Zigzag scanning (ZS), Schur decomposition (SD), Discrete cosine transform (DCT) and Redundant discrete wavelet transform (RDWT). In the proposed scheme, Henon map is applied on the watermarked image for its secure transfer and further enhancement of the security. Medical image security assurance is the need of the hour. Therefore, ultrasound images of the liver of six different patients are taken for experimental result analysis. The use of Singular value decomposition (SVD) is avoided here due to its false positive problem. RDWT is used to split the input ultrasound liver image into different sub-bands. Then DCT is applied on the suitable sub-band to improve the robustness of the presented work. The ZS is used to avoid artifacts which may occur at higher value of the gain factor. SD and RDWT are applied to the watermark logo (MNNIT logo) and selective pixel adding is used to embed the watermark logo into the host image. The authentication process is done by using binary robust invariant scalable keypoints (BRISK) features. The presented scheme perform satisfactorily against various attacks such as Histogram equalization, Region of interest (ROI) filtering, Cropping, Scaling, Rotation, Average and Median filtering attacks, Salt and Pepper, Gaussian noise attack, Flipping, etc. The proposed scheme shows better performance as compared to existing schemes in terms of various performance parameters such as Peak signal-to-noise ratio (PSNR), Structural similarity index measurement (SSIM) and Normalized correlation coefficient (NCC). [ABSTRACT FROM AUTHOR]

Schur decompositions and the corresponding Schur forms of a single matrix, a pair of matrices, or a collection of matrices associated with the periodic eigenvalue problem are frequently used and studied. These forms are upper-triangular complex matrices or quasi-upper-triangular real matrices that are equivalent to the original matrices via unitary or, respectively, orthogonal transformations. In general, for theoretical and numerical purposes we often need to reduce, by admissible transformations, a collection of matrices to the Schur form. Unfortunately, such a reduction is not always possible. In this paper we describe all collections of complex (real) matrices that can be reduced to the Schur form by the corresponding unitary (orthogonal) transformations and explain how such a reduction can be done. We prove that this class consists of the collections of matrices associated with pseudoforest graphs. In other words, we describe when the Schur form of a collection of matrices exists and how to find it. [ABSTRACT FROM AUTHOR]

For image information transmission security and copyright disputes, a novel and effective digital image watermarking encryption scheme is designed by combining zero watermarking and digital image encryption. In the aspect of watermark acquisition, the Canny operator is used to extract the edge features of the plaintext image, and the embedded watermark with a small amount of data and high similarity to the plaintext is obtained. This operation enables better post-embed invisibility as well as cost savings on transfer time. In the image encryption part, combined with DCT compression technology, the Fisher shuffling algorithm is improved, which can effectively scramble the position of pixels and reduce the time complexity of its operation. The improved gravity model algorithm is used to complete the diffusion operation so that the plaintext information is further hidden in the pixels. Simulation results and security analysis show that the proposed algorithm performs well in terms of time cost and encryption effect. The results of watermark restoration and reconstruction show that the scheme has an excellent performance in realizing the non-visualization and robustness of the watermark. [ABSTRACT FROM AUTHOR]

Since multimedia data are now more widely distributed in digital form and easier to copy and change, copyright protection has become an essential need. One of the most common copyright protection techniques is watermarking. Invisibility and robustness are crucial aspects of watermarking; however, many of the currently used techniques simply focus more on invisibility than how robust they are. Moreover, the trade-off between invisibility and robustness is challenging. To this end, this paper proposes a novel watermark technique that efficiently overcomes the idea of a trade-off between robustness and invisibility, thereby increasing both under most attacks. Schur decomposition and a dynamic weighting factors matrix are added to the embedding process to improve the robustness of the proposed technique. Besides that, the embedding function is improved to simultaneously maximize imperceptibility and robustness. Another key contribution of the proposed approach is its use of a trajectory-based optimization algorithm rather than the more prevalent population-based algorithms to determine the optimal scaling factors. Consequently, the proposed technique rapidly identifies the best scaling factors for the embedding function. Statistical analysis is performed using the Friedman test. Experimental results show that the proposed technique outperforms other existing techniques for different sizes, shapes, and types of watermarks. [ABSTRACT FROM AUTHOR]

This paper introduces two robust and optimized watermarking techniques designed to protect the copyright of color digital images. These techniques involve embedding a biometric-based digital watermark, generated from an iris image, within the redistributed invariant discrete wavelet transform (RIDWT) domain. In the proposed algorithms, the Y channel of the color image in the YCbCr color space is initially decomposed using RIDWT. Then, the lowest frequency sub-band is partitioned into non-overlapping blocks. Blocks with lower standard deviations are selected to embed each bit of the watermark derived from the iris image. To conceal these watermark bits, two distinct approaches are used. In the first approach, selected blocks undergo a transformation using QR decomposition, while the second approach utilizes Schur decomposition for the same purpose. Before the embedding step, we optimize the embedding strength of the watermarking using the firefly algorithm, striking a balance between robustness and perceptual transparency. Experimental evaluations demonstrate that the proposed algorithms not only deliver good perceptual quality but also effectively resist various geometric and common image processing attacks. Our results underscore the superior robustness of these techniques compared to existing studies. Additionally, both methods maintain an exceptionally low error rate during ownership verification. [ABSTRACT FROM AUTHOR]

Digital image watermarking provides security, copyright protection, and authenticity to multimedia content by embedding digital content like text, image, and video. In the modern digital age, watermarking of digital images is an essential aspect of providing authenticity to digital content. Digital image watermarking is also used for the authenticity of the medical image in telemedicine applications. In these applications, digital image watermarking provides security to various medical images like Computerized tomography (CT) scans, Magnetic resonance imaging (MRI) images, and X-radiation (X-ray) images. In the past decades, many intellectuals have been working in the field of image watermarking to provide various characteristics features to watermarking like sufficient embedding capacity, strong robustness, and better imperceptibility. This paper covers watermarking techniques based on Schur decomposition and various types of moments like Zernike moment, Fourier harmonic moments, polar harmonic moments, etc. A survey on watermarking techniques suitable for telemedicine applications which include various medical images like CT-scan, MRI image, and X-ray images, is also presented. Various performance measures proposed by different researchers are also discussed, and the suitability of these measures for watermarking schemes is analysed under various attacks. In this paper, novel classification of watermarking schemes, recent essential requirements for different categories of image watermarking applications are covered. A tabular comparison of various watermarking schemes based on different families of moments are presented and also a comparative analysis of outcomes of different categories of moment based watermarking schemes is included. Various challenges and opportunities are also included to make this review more comprehensive. [ABSTRACT FROM AUTHOR]

In order to protect the copyright of color digital images in the current Internet environment, this paper proposes a blind color image watermarking scheme based on nonsubsampled contourlet transform (NSCT) and Schur decomposition. In the stage of watermark embedding, NSCT is applied to acquire low-frequency coefficients of host images, and non-overlapping coefficient blocks of sized 4 × 4 are selected from the layered low-frequency coefficients by using Hash pseudo-random algorithm based on MD5. The watermark information encrypted by Lorenz chaotic mapping is embedded into the maximum eigenvalue of the matrix by quantization index modulation. In the extraction process, watermark information is extracted from the attacked watermarked image without using any original information. The simulation results show that all values of peak signal-to-noise ratio of the proposed watermarking scheme are more than 40 dB under the payload of 1/16 bit/pixel, and the values of Structural Similarity Index Metric are more than 0.97; when subjected to various common attacks, the mean of normalized cross-correlation is above 0.96, and the bit error rate is about 0.05; the key space is more than 2624; the running time is about 1.5 s. Compared with other schemes, on the premise of ensuring invisibility and embedding capacity, the proposed scheme has been significantly improved in terms of security and robustness. [ABSTRACT FROM AUTHOR]

Medical data is transferred between hospitals and healthcare providers via telemedicine to improve patient care. This transfer exposes medical data to a number of security risks, and while most existing security solutions, such as cryptographic approaches, protect data from unauthorized access, this protection is only effective when the data is encrypted. In this context, we present a frequency-domain watermarking method for hiding electronic patient records in their associated electrocardiogram (ECG) signals in this paper. The signal is transformed into a 2D picture in this approach, and the frequency content of the image is extracted using the integer wavelet transform. Finally, the acquired coefficients are treated through Schur decomposition, and the watermark bits are integrated by altering the least significant bit of the generated Eigen values. We used the ECG data from the MIT-BIH Arrhythmia Database to test the suggested method. Based on the results of the experiments, we can infer that using the Integer wavelet transform allows for the generation of a watermarked signal that is practically identical to the host signal. The watermark's resistance to several attacks commonly used in watermarking is further confirmed by the robustness results. [ABSTRACT FROM AUTHOR]

One method for securing multimedia data, and images in particular, from malicious attacks is watermarking, which involves adding a signature to the images. Traditional watermarking techniques, however, have drawbacks when used on sensitive images like those used in medicine. A reliable and blind watermarking method is suggested in this work to secure medical image exchanged in telemedicine. Medical image watermarking enables accurate patient identification, prevents scan confusion, and minimizes the risk of diagnostic mistakes that might have detrimental effects. In this method, three distinct transformations are used to get the frequency content of the picture. The low frequency subbands are subsequently subjected to Schur decomposition. In order to incorporate the watermark bits, the resultant upper triangular matrix values are modified. The suggested approaches effectively retain a considerable quality of watermarked images, according to experimental results on imperceptibility and robustness, and are resistant against various common attacks. [ABSTRACT FROM AUTHOR]

Imperceptibility, robustness, and reliability are essential characteristics of image watermarking. In this paper, two image watermarking schemes, first Lifting wavelet transform (LWT)–Discrete cosine transform (DCT) with zigzag scanning-based embedding in Schur domain and second LWT-Schur decomposition-based embedding scheme in Schur domain itself, are proposed. In this paper, two schemes are proposed in which a pre-processed watermark image is generated using LWT and Schur decomposition, where the principle diagonal coefficients of the upper triangular matrix are embedded into the cover image in LWT-Schur domain. The remaining matrix, after removing diagonal components and other unitary matrix of Schur decomposition of LL sub-band of watermark image are used as keys which provide better security to the watermarking scheme. In the LWT-Schur-based scheme principle diagonal coefficients of upper triangular matrix obtained from the Schur decomposition of the LL sub-band of the cover image are modified with the pre-processed watermark in Schur domain. In the LWT–DCT-based scheme, the low and mid-band frequency coefficients are used to embed the watermark using zigzag scanning to avoid any artifacts like diagonal line problem present in the SVD-based watermarking scheme, especially for larger value of the embedding gain factor. The Schur decomposition has lesser computation complexity as compared to SVD and both the schemes are also free from False positive problem. The LWT–DCT with zigzag scanning-based scheme offers higher robustness against filtering and noise attacks as Normalized correlation coefficient (NCC) value is close to 1.0000. Both the proposed schemes provide better imperceptibility and higher robustness as NCC value of more than 0.9800 against various checkmark attacks. Performance analysis shows that these schemes perform better in terms of Peak signal-to-noise ratio, and NCC under most of the attacks and also have lesser computational time than existing schemes. [ABSTRACT FROM AUTHOR]

The field of healthcare is considered to be the most promising application of intelligent sensor networks. However, the security and privacy protection ofmedical images collected by intelligent sensor networks is a hot problem that has attracted more and more attention. Fortunately, digital watermarking provides an effective method to solve this problem. In order to improve the robustness of the medical image watermarking scheme, in this paper, we propose a novel zero-watermarking algorithm with the integer wavelet transform (IWT), Schur decomposition and image block energy. Specifically, we first use IWT to extract low-frequency information and divide them into non-overlapping blocks, then we decompose the sub-blocks by Schur decomposition. After that, the feature matrix is constructed according to the relationship between the image block energy and the whole image energy. At the same time, we encrypt watermarking with the logistic chaotic position scrambling. Finally, the zero-watermarking is obtained by XOR operation with the encrypted watermarking. Three indexes of peak signal-to-noise ratio, normalization coefficient (NC) and the bit error rate (BER) are used to evaluate the robustness of the algorithm. According to the experimental results, most of the NC values are around 0.9 under various attacks, while the BER values are very close to 0. These experimental results show that the proposed algorithm is more robust than the existing zero-watermarking methods, which indicates it is more suitable for medical image privacy and security protection. [ABSTRACT FROM AUTHOR]

With the development of digitalization in healthcare, more and more information is delivered and stored in digital form, facilitating people’s lives significantly. In the meanwhile, privacy leakage and security issues come along with it. Zero watermarking can solve this problem well. To protect the security of medical information and improve the algorithm’s robustness, this paper proposes a robust watermarking algorithm for medical images based on Non-Subsampled Shearlet Transform (NSST) and Schur decomposition. Firstly, the low-frequency subband image of the original medical image is obtained by NSST and chunked. Secondly, the Schur decomposition of lowfrequency blocks to get stable values, extracting the maximum absolute value of the diagonal elements of the upper triangle matrix after the Schur decomposition of each low-frequency block and constructing the transition matrix from it. Then, the mean of the matrix is compared to each element’s value, creating a feature matrix by combining perceptual hashing, and selecting 32 bits as the feature sequence. Finally, the feature vector is exclusive OR (XOR) operated with the encrypted watermark information to get the zero watermark and complete registration with a third-party copyright certification center. Experimental data show that the Normalized Correlation (NC) values of watermarks extracted in random carrier medical images are above 0.5, with higher robustness than traditional algorithms, especially against geometric attacks and achieve watermark information invisibility without altering the carrier medical image. [ABSTRACT FROM AUTHOR]

In this proposed work, a dual image watermarking algorithm is used to protect the data against copyright violations. In this work, the DICOM image is used as a host image. Two watermark images used are the MNNIT logo and the personal data of the patient. This method utilizes the advantages of Schur decomposition, lifting wavelet transform (LWT), discrete cosine transform (DCT) and singular value decomposition (SVD). The scaling factor is a vital parameter of watermarking technique. The firefly optimization technique is used to get the optimized scaling factor. The Speeded-up robust features (SURF) are used for watermarking authentication. To evaluate the performance of the proposed algorithm, peak signal-to-noise ratio (PSNR), normalized correlation coefficient (NCC), and structural similarity index measurement (SSIM) are used. The proposed method is tested against various attacks such as Salt and Pepper noise, Gaussian noise, Gaussian low pass filter, Average filter, Median filter, Histogram equalization, Sharpening, Rotation and Region of interest filtering. The proposed algorithm shows a high level of robustness and imperceptibility. It is found that the features of the input host image and the watermarked image are matching correctly on applying the SURF technique. [ABSTRACT FROM AUTHOR]

Many problems in systems and control theory are related to solvability of Sylvester matrix equations. In many applications, at least some of the parameters of the system should be represented by fuzzy numbers rather than crisp ones. In most of the previous literature, the solutions of fuzzy Sylvester matrix equation are only presented with triangular fuzzy numbers. In this paper, we propose two analytical methods for solving Positive Trapezoidal Fully Fuzzy Sylvester Matrix Equation (PTrFFSME). The PTrFFSME is converted to an equivalent system of crisp Sylvester Matrix Equations (SME) using the existing arithmetic fuzzy multiplication operations. The necessary and sufficient conditions for the existence and uniqueness of the positive fuzzy solutions to the PTrFFSME are investigated. In addition, the equivalency between the solution to the system of SME and the PTrFFSME are discussed. The proposed methods are illustrated by solving one example.

The Schur decomposition of a square matrix A is an important intermediate step of state-of-the-art numerical algorithms for addressing eigenvalue problems, matrix functions, and matrix equations. This work is concerned with the following task: Compute a (more) accurate Schur decomposition of A from a given approximate Schur decomposition. This task arises, for example, in the context of parameter-dependent eigenvalue problems and mixed precision computations. We have developed a Newton-like algorithm that requires the solution of a triangular matrix equation and an approximate orthogonalization step in every iteration. We prove local quadratic convergence for matrices with mutually distinct eigenvalues and observe fast convergence in practice. In a mixed low-high precision environment, our algorithm essentially reduces to only four high-precision matrix-matrix multiplications per iteration. When refining double to quadruple precision, it often needs only 3–4 iterations, which reduces the time of computing a quadruple precision Schur decomposition by up to a factor of 10–20. [ABSTRACT FROM AUTHOR]

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Multimedia manipulation has increased which demand for security across various applications. Majorly image-oriented security issues such as image authentication, proof of ownership, and copyright protection are highly increased. To authenticate and detect the tamped region and to recover the tampered area vision transformer-based hybrid watermarking model is proposed. We proposed a novel model to achieve image authentication, tamper detection, and localization followed by image recovery. In the proposed model, invariant attention-based watermark feature maps are generated using a Vision transformer. We have generated three different watermarks: first using the SVD eigenvalue generated as an authentication watermark, secondly to detect tampered region average 6MSB of each 2*2 block generated by performing the Schur decomposition method on the biometric image, and to locate and recover the image Vision feature maps are generated and average 6MSB of each block is embedded as the tamper detection watermark. Normally the generated watermark is embedded either using an embedding factor or using a suitable embedding location. In the proposed model, watermark embedding is performed by finding the optimal embedding region using high entropy block region. On the original image, curvelet transform is performed followed by invariant integer wavelet transform. The first authentication eigenvalue is embedded on the LH band singular value diagonal matrix obtained by the SVD model. On the LL band, 2*2 blocks Schur decomposition is performed to embed the 6MSB in the 2LSB bit of upper triangular coefficient values. At last, vision feature maps are embedded in the curvelet approximate coefficient high entropy region and inverse curvelet performed, producing a watermarked image. [ABSTRACT FROM AUTHOR]

Medical images became a very important information tool for health professionals. Currently, the medical image acquired in a hospital or an imaging center can be shared among several health professionals to facilitate patient management and improve medical information management. In this work, we proposed a robust and blind watermarking approach to adequately secure medical images exchanged in telemedicine. This approach ensures the traceability and integrity of the medical and essential image for data security in the field of telemedicine. In this approach, the watermark consists of the patient's photography as well as the patient's data and the data related to the medical image acquisition. These combined data will thus guarantee the successful authentication of the image as well as the patient. A hash of this necessary information will be appropriately included in the watermark to ensure the integrity of the hidden data. The proposed watermarking in this approach remains a substitutive process. The frequency content of the image is acquired using transforms. Schur decomposition is then applied to the obtained mid-frequency subbands. Finally, the watermark bits will be substituted to the upper triangular matrix values obtained. Imperceptibility and robustness experimental results show that the proposed methods adequately maintain a significant quality of watermarked images and are remarkably robust against several conventional attacks. Since those schemes offer a reasonable imperceptibility and robustness, they could be useful for copyrights protection of medical images. [ABSTRACT FROM AUTHOR]

The high-speed train bogie is a typical underactuated mechanical system. It has been difficult to design controls to increase the stability against hunting motion with limited sensor measurements and actuation hardware. This paper develops a control design for the stability improvement of the bogie by making use of the balanced truncation reduced order model of the bogie. When the train travels at speed higher than the critical speed, the bogie has unstable modes. The balanced truncation model reduction (BTMR) is usually applied to stable systems. In this paper, we first apply the real Schur decomposition to separate the stable and unstable modes of the bogie. The BTMR is then carried out for the stable subsystem. An augmented X 0 -balanced truncation algorithm is considered to deal with the dependence of the BTMR on initial conditions. The reduced order subsystem is then combined with the unstable subsystem. A linear quadratic regulator control together with a Luenberger observer for the state estimation is designed for the bogie with non-zero initial conditions. Simulation results show that the control designed with the balanced truncation reduced order model performs very well in terms of the stability improvement by significantly increasing the critical train speed. [ABSTRACT FROM AUTHOR]

We propose a simple technique that, if combined with algorithms for computing functions of triangular matrices, can make them more efficient. Basically, such a technique consists in a specific scaling similarity transformation that reduces the departure from normality of a triangular matrix, thus decreasing its norm and in general its function condition number. It can easily be extended to non-triangular matrices, provided that it is combined with algorithms involving a prior Schur decomposition. Situations where the technique should be used or not will be discussed in detail. Special attention is devoted to particular algorithms like the inverse scaling and squaring to the matrix logarithm or inverse cosine and the scaling and squaring to the matrix exponential. The advantages of our proposal are supported by theoretical results and illustrated with numerical experiments, involving matrices of small, medium and large size. [ABSTRACT FROM AUTHOR]

This paper devotes to the generalized eigenvalues for even order tensors. We extend classical spectral theory for matrix pairs to the multilinear case, including the generalized Schur decomposition, the Geršgorin circle theorem, and the Bauer–Fike theorem for regular tensor pairs of even order. We introduce the backward errors and ϵ -pseudospectrums for generalized tensor eigenvalues in normwise and componentwise, respectively, and particularize the application in stability analysis for the generalized multilinear systems. By the normwise pseudospectral theory, we obtain a lower bound for the distance from a regular tensor pair to singularity, and a formulation of the distance from a reachable multilinear time invariant control system to unreachability is given. [ABSTRACT FROM AUTHOR]

Security solutions of stereo images are always a major concern during the transmission and communication. In this manuscript, a simple yet efficient framework for encrypting stereo images is formulated using discrete cosine transform (DCT), generalized logistic map, Schur decomposition and magic square method. The framework initiated with the integration of DCT and generalized logistic map to unify both pair of images. This unified image is then encrypted using the Schur decomposition and magic square method. The various experiments and analysis have been done to explore the validity, proficiency and performance of the purport framework.

• An image fusion method was used to reduce the conservation cost of multiple images by fusing them into a single image. • An image encryption method was proposed for encrypting the logo image to enhance the safety of the algorithm. • A combination of FFST-Schur decomposition and image normalization was proposed to improve the resistance against geometric attacks. • Experiments were conducted to show the proposed method outperforms its counterparts. Aiming at issues such as the difficulty of centralized protection of multiple medical images and the high storage cost caused by repetitive operations, we proposed a robust zero-watermarking algorithm for multiple medical images based on fast finite Shearlet transform (FFST)-Schur and Tent mapping. First, multiple medical images were normalized, and the processed images were fused using a gray-weighted average fusion method for image fusion. Second, an FFST is applied to the fused image to divide the low-frequency sub-bands into blocks of the same size and perform Schur decomposition, which generates a feature image using the magnitude between the largest eigenvalue of each block and the overall average value. Finally, an image encryption algorithm based on Tent mapping is proposed to encrypt the logo image, and the encrypted image and feature image are then subjected to an exclusive OR operation to produce a zero-watermarking image. Experimental results demonstrate that the proposed approach effectively safeguards against standard image processing and geometric and combinatorial attacks. The average performance of the anti-attack algorithm was improved by approximately 3.2 % compared to similar algorithms, indicating the superior robustness of the proposed algorithm. [ABSTRACT FROM AUTHOR]

Medical data is transferred between hospitals and healthcare providers via telemedicine to improve patient care. This transfer exposes medical data to a number of security risks. While most existing security solutions, such as cryptographic techniques, protect data from unauthorized access, this protection is only effective when the data is encrypted. In this context, we present a frequency-domain watermarking method for hiding electronic patient records in their associated ECG signals in this paper. The signal is transformed into a 2D image in this method, and the frequency content of the image is extracted using the integer wavelet transform. Finally, the acquired coefficients are subjected to Schur decomposition, and the watermark bits are integrated by altering the least significant bit of the generated Eigen values. We used the ECG data from the MIT-BIH Arrhythmia Database to test the suggested method. Based on the outcomes of the studies, we can conclude that using the Integer wavelet transform allows for the generation of a watermarked signal that is nearly identical to the host signal. The watermark's resistance to various attacks commonly used in watermarking is also confirmed by the robustness results. [ABSTRACT FROM AUTHOR]

The Schur decomposition of the velocity gradient tensor (VGT) is an alternative to the classical Cauchy-Stokes decomposition into rotation rate and strain rate components. Recently, there have been several strands of work that have employed this decomposition to examine the physics of turbulence dynamics, including approaches that combine the Schur and Cauchy-Stokes formalisms. These are briefly reviewed before the latter approach is set out. This partitions the rotation rate and strain rate tensors into normal/local and non-normal/non-local contributions. We then study the relation between the VGT dynamics and ejection-sweep events in a channel flow boundary-layer. We show that the sweeps in particular exhibit novel behaviour compared with either the other quadrants, or the flow in general, with a much-reduced contribution to the dynamics from the non-normal terms above the viscous sub-layer. In particular, the reduction in the production term that is the interaction between the non-normality and the normal straining reduces in the log-layer as a consequence of an absence of alignment between the non-normal vorticity and the strain rate eigenvectors. There have been early forays into using the Schur transform approach for subgrid-scale modelling in large-eddy simulation (LES) and this would appear to be an exciting way forward. [ABSTRACT FROM AUTHOR]

Many problems in systems and control theory are related to solvability of Sylvester matrix equations. In many applications, at least some of the parameters of the system should be represented by fuzzy numbers rather than crisp ones. In most of the previous literature, the solutions of fuzzy Sylvester matrix equation are only presented with triangular fuzzy numbers. In this paper, we propose two analytical methods for solving Positive Trapezoidal Fully Fuzzy Sylvester Matrix Equation (PTrFFSME). The PTrFFSME is converted to an equivalent system of crisp Sylvester Matrix Equations (SME) using the existing arithmetic fuzzy multiplication operations. The necessary and sufficient conditions for the existence and uniqueness of the positive fuzzy solutions to the PTrFFSME are investigated. In addition, the equivalency between the solution to the system of SME and the PTrFFSME are discussed. The proposed methods are illustrated by solving one example. [ABSTRACT FROM AUTHOR]

This paper proposes an effective optimal blind colour image watermarking based on Triangular Vertex Transform (TVT), Lifting Wavelet Transform (LWT) and Schur decomposition. Firstly, the RGB channels of the colour host image are processed with TVT to attain U , V , and W coefficients. The image quality is preserved by applying LWT on the W coefficients. As the middle frequency (HL and LH) components of the LWT wavelet domain achieve much robustness, these two components are split into 4 × 4 nonoverlapping blocks further it is processed with Schur decomposition for improving perceptual transparency. The optimal solution attained by an Adaptive Chaotic Grasshopper Optimization Algorithm (ACGOA) is used as the embedding factor to ensure the embedding strength of the watermark. The confidentiality of the watermark is improved by 2D Logistic-Modulated-Sine-Coupling-Logistic Chaotic Map (LSMCL). Subsequently, each bit of encrypted watermark image is inserted in the highest eigenvalue of upper triangular matrix of Schur decomposition using the embedding factor β obtained by ACGOA to accomplish the watermarked image. The watermark recovery is performed using the embedding factor β and the decryption process is done using LSMCL. The efficiency of the ACGOA based scheme is estimated and the highest Peak signal-to-noise ratio (PSNR) value achieved is 54.2987 dB. The Normalized Cross-Correlation (NCC) results show better reliability for different attacks including geometric attacks and the NCC value obtained is close to 1. [ABSTRACT FROM AUTHOR]

This paper introduces a defense approach against end-to-end adversarial attacks developed for cutting-edge speech-to-text systems. The proposed defense algorithm has four steps. First, we use the short-time Fourier transform to represent speech signals with 2D spectrograms. Second, we iteratively find a safe vector using a spectrogram subspace projection operation. This operation minimizes the chordal distance adjustment between spectrograms with an additional regularization term. Third, we synthesize a spectrogram with such a safe vector using a novel GAN architecture trained with Sobolev integral probability metric. We impose an additional constraint on the generator network to improve the model’s performance in terms of stability and the total number of learned modes. Finally, we reconstruct the signal from the synthesized spectrogram and the Griffin-Lim phase approximation technique. We evaluate the proposed defense approach against six strong white and black-box adversarial attacks on DeepSpeech, Kaldi, and Lingvo models. The experimental results show that our algorithm outperforms other state-of-the-art defense algorithms in terms of accuracy and signal quality. [ABSTRACT FROM AUTHOR]

We derive several explicit formulae for finding infinitely many solutions of the equation A X A = X A X , when A is singular. We start by splitting the equation into a couple of linear matrix equations and then show how the projectors commuting with A can be used to get families containing an infinite number of solutions. Some techniques for determining those projectors are proposed, which use, in particular, the properties of the Drazin inverse, spectral projectors, the matrix sign function, and eigenvalues. We also investigate in detail how well-known similarity transformations like Jordan and Schur decompositions can be used to obtain new representations of the solutions. The computation of solutions by the suggested methods using finite precision arithmetic is also a concern. Difficulties arising in their implementation are identified and ideas to overcome them are discussed. Numerical experiments shed some light on the methods that may be promising for solving numerically the matrix equation. [ABSTRACT FROM AUTHOR]

A novel robust tamper detection medical watermarking scheme is proposed in the transform domain for authentication and detecting tamper pixels of medical images. In this scheme, 2-level Discrete Wavelet Transform is applied to a significant image to produce four sub-bands (SB) (CLL,CLH,CHL,CHH). Coefficients of LL sub-bands are considered in embedding the watermark. The SB is partitioned into blocks to overcome image processing attacks. LSB is set to zero for each block and then Schur decomposition is applied in generating Authenticated Block Bits (ABB). In developing confusion to the intruders, the watermark is scrambled using Quantum Hilbert Image Scrambling. Watermarking helps in authentication and tamper detection of the significant image after tampering. The scheme is tested with image processing attacks for robustness. Peak signal to noise ratio (PSNR) and Normalized Cross-Correlation (NCC) metrics are utilized as metrics in evaluating the proposed scheme with PSNR greater than 30dB and NCC values nearer to 1 without attacks and even with attacks, NCC values are greater than 0.95, which shows the robustness of the proposed scheme. [ABSTRACT FROM AUTHOR]

A matrix A is said to possess an UV-displacement structure if rank (A U − V A) is small compared with the rank of A. Estimates for the rank A ◯ # V − U A ◯ # are presented, where A ◯ # is the core inverse of A. The computation of the displacement is also considered. [ABSTRACT FROM AUTHOR]

To improve the robustness of image watermarking algorithms against geometric attacks such as cropping, rotation, scaling, shearing, projective mapping, a robust image watermarking algorithm based on DT CWT and Schur decomposition is proposed. Firstly, we use Dual Tree Complex Wavelet Transform (DT CWT) to decompose the U component of the original image (cover image), the decomposed low frequency component is divided into non-overlapping N × N blocks, and Schur decomposition is applied in each block. Secondly, we modify maximum primary diagonal value of the upper triangular matrix of each block to embed the watermark. Finally, the Scale Invariant Feature Transform (SIFT) features of the watermarked image (stego-image) is saved as the SIFT key for watermark extraction. When extracting the watermark, we first use the SIFT features of stego-image and changed stego-image to obtain the geometric torsion, thereby a spatial synchronization can be guaranteed. And we then use the maximum election statistics to improve the quality of the extracted watermark. Experiment results show that the proposed algorithm can resist geometric attacks, and is more robust than the existing watermarking methods. [ABSTRACT FROM AUTHOR]

Image watermarking is a potentially effective and powerful solution for multimedia security. It is a technique that plays a fundamental role in protecting copyright, proofing ownership, and helps to authenticate sensitive data. Watermarking can cause a decrease in image quality, especially when it comes to medical images. This could provoke false diagnoses, which can lead to serious consequences on the patient's health. In fact, it becomes necessary to think for new more imperceptible, secure watermarking techniques. To provide such techniques, we proposed a novel blind watermarking scheme for medical images, based on Schur decomposition and chaotic sequence (CS). An efficient chaotic method is applied on the watermark and the cover image to obtain encrypted images divided into subblocks. A decomposition based on Schur is utilized to embed the encrypted watermark bits in the cyphered cover image blocks. The same CS is applied to extract the original watermark. We compared the proposed technique with other watermarking schemes. The experimentation results demonstrate that the suggested method achieving good image fidelity and acceptable robustness. [ABSTRACT FROM AUTHOR]

Security solutions of stereo images are always a major concern during the transmission and communication. In this manuscript, a simple yet efficient framework for encrypting stereo images is formulated using discrete cosine transform (DCT), generalized logistic map, Schur decomposition and magic square method. The framework initiated with the integration of DCT and generalized logistic map to unify both pair of images. This unified image is then encrypted using the Schur decomposition and magic square method. The various experiments and analysis have been done to explore the validity, proficiency and performance of the purport framework. [ABSTRACT FROM AUTHOR]

We discuss the use of a matrix‐oriented approach for numerically solving the dense matrix equation AX + XAT + M1XN1 + ... + MℓXNℓ = F, with ℓ ≥ 1, and Mi, Ni, i = 1, ... , ℓ of low rank. The approach relies on the Sherman–Morrison–Woodbury formula formally defined in the vectorized form of the problem, but applied in the matrix setting. This allows one to solve medium size dense problems with computational costs and memory requirements dramatically lower than with a Kronecker formulation. Application problems leading to medium size equations of this form are illustrated and the performance of the matrix‐oriented method is reported. The application of the procedure as the core step in the solution of the large‐scale problem is also shown. In addition, a new explicit method for linear tensor equations is proposed, that uses the discussed matrix equation procedure as a key building block. [ABSTRACT FROM AUTHOR]

The objective of this paper is to contribute to the sharing, transfer, and secured indexing of large color images. Watermarking and digital encryption have emerged as alternative and complementary solutions to ensure authorized access, facilitate authentication of content, or prevent illegal reproduction. For this reason, we suggest a new system based on four techniques derived from various transformation and decomposition methods, such as the discrete orthogonal Racah moment transformations (DORMT), which allows the information capacity to be reduced to a number of coefficients, as well as the discrete wavelet transform (DWT), which makes it possible to decompose into a certain number of multilevel sub-bands, to have the sub-band that represents most of the image information. In addition, Schur decomposition (SD) and singular value decomposition (SVD) have been used to increase security. The proposed system consists of two main phases. The first phase is the integration of a color watermark, first applying DORMT-SD-SVD to a color host image and then applying DWT-SD-SVD to achieve a compromise between capacity, invisibility and robustness in terms of quality and privacy. The second phase is the encryption of the watermarked image with an eπ map to increase security for sharing, transfer and indexing. This system uses an Aquila Optimization Algorithm (AOA) as an optimization procedure to find the best set of DORMT parameters and provide a dynamic and adaptive scale factor. The experimental results, in which the proposed system's properties are tested and compared to those of other systems, make it clear that the proposed system is useful in terms of its level of security, capacity, invisibility, and resistance to attacks that use signal processing. [ABSTRACT FROM AUTHOR]

Perturbation analysis has been primarily considered to be one of the main issues in many fields. The Schur decomposition can factor a square matrix as the product of a unitary matrix and an upper triangular matrix, which contains eigenvalues of the square matrix. In view of the importance of T-eigenvalue problems, this paper discusses a tensor Schur decomposition (T-Schur), the T-Schur decomposition = U ∗ T ∗ U ∗ , which is based on T-product multiplication of third-order tensors. We present the normwise and componentwise perturbation analysis for the unitary tensor U , the upper triangular tensor T and T-eigenvalues of the tensor A. We explore some applications of the T-Schur decomposition and perform the T-Schur form to solve the tensor pole assignment. We give an algorithm to solve the tensor based T-Sylvester equation and present its perturbation bound and the backward error. The T-Sylvester equation can also be used to estimate the condition of the T-sign function. We apply the T-Schur decomposition to facial recognation and compare it with other types of tensor decompositions. [ABSTRACT FROM AUTHOR]