Hermite Transform Research Articles

Bio-Inspired Watermarking Method for Authentication of Fundus Images in Computer-Aided Diagnosis of Retinopathy

Nowadays, medical imaging has become an indispensable tool for the diagnosis of some pathologies and as a health prevention instrument. In addition, medical images are transmitted over all types of computer networks, many of them insecure or susceptible to intervention, making sensitive patient information vulnerable. Thus, image watermarking is a popular approach to embed copyright protection, Electronic Patient Information (EPR), institution information, or other digital image into medical images. However, in the medical field, the watermark must preserve the quality of the image for diagnosis purposes. In addition, the inserted watermark must be robust both to intentional and unintentional attacks, which try to delete or weaken it. This work presents a bio-inspired watermarking algorithm applied to retinal fundus images used in computer-aided retinopathy diagnosis. The proposed system uses the Steered Hermite Transform (SHT), an image model inspired by the Human Vision System (HVS), as a spread spectrum watermarking technique, by leveraging its bio-inspired nature to give imperceptibility to the watermark. In addition, the Singular Value Decomposition (SVD) is used to incorporate the robustness of the watermark against attacks. Moreover, the watermark is embedded into the RGB fundus images through the blood vessel patterns extracted by the SHT and using the luma band of Y’CbCr color model. Also, the watermark was encrypted using the Jigsaw Transform (JST) to incorporate an extra level of security. The proposed approach was tested using the image public dataset MESSIDOR-2, which contains 1748 8-bit color images of different sizes and presenting different Diabetic Retinopathy (DR). Thus, on the one hand, in the experiments we evaluate the proposed bio-inspired watermarking method over the entire MESSIDOR-2 dataset, showing that the embedding process does not affect the quality of the fundus images and the extracted watermark, by obtaining average Peak Signal-to-Noise Ratio (PSNR) values higher to 53 dB for the watermarked images and average PSNR values higher to 32 dB to the extracted watermark for the entire dataset. Also, we tested the method against image processing and geometric attacks successfully extracting the watermarking. A comparison of the proposed method against state-of-the-art was performed, obtaining competitive results. On the other hand, we classified the DR grade of the fundus image dataset using four trained deep learning models (VGG16, ResNet50, InceptionV3, and YOLOv8) to evaluate the inference results using the originals and marked images. Thus, the results show that DR grading remains both in the non-marked and marked images.

A Robust and Secure Watermarking Approach Based on Hermite Transform and SVD-DCT

Currently, algorithms to embed watermarks into digital images are increasing exponentially, for example in image copyright protection. However, when a watermarking algorithm is applied, the preservation of the image’s quality is of utmost importance, for example in medical images, where improper embedding of the watermark could change the patient’s diagnosis. On the other hand, in digital images distributed over the Internet, the owner of the images must also be protected. In this work, an imperceptible, robust, secure, and hybrid watermarking algorithm is presented for copyright protection. It is based on the Hermite Transform (HT) and the Discrete Cosine Transform (DCT) as a spatial–frequency representation of a grayscale image. Besides, it uses a block-based strategy and a perfectibility analysis of the best embedding regions inspired by the Human Vision System (HVS), giving the imperceptibility of the watermark, and a Singular-Value Decomposition (SVD) approach improved robustness against attacks. In addition, the proposed method can embed two watermarks, a digital binary image (LOGO) and information about the owner and the technical data of the original image in text format (MetaData). To secure both watermarks, the proposed method uses the Jigsaw Transform (JST) and the Elementary Cellular Automaton (ECA) to encrypt the image LOGO and a random sequence generator and the XOR operation to encrypt the image MetaData. On the other hand, the proposed method was tested using a public dataset of 49 grayscale images to assess the effectiveness of the watermark embedding and extraction procedures. Furthermore, the proposed watermarking algorithm was evaluated under several processing and geometric algorithms to demonstrate its robustness to the majority, intentional or unintentional, attacks, and a comparison was made with several state-of-the-art techniques. The proposed method obtained average values of PSNR = 40.2051 dB, NCC = 0.9987, SSIM = 0.9999, and MSSIM = 0.9994 for the watermarked image. In the case of the extraction of the LOGO, the proposal gave MSE = 0, PSNR ≫ 60 dB, NCC = 1, SSIM = 1, and MSSIM = 1, whereas, for the image MetaData extracted, it gave BER = 0% and Berror=0. Finally, the proposed encryption method presented a large key space (K=1.2689×1089) for the LOGO image.

Watermarking approach based on Hermite transform and a sliding window algorithm.

Nowadays, the distribution of huge amounts of medical images through open networks in telemedicine applications has become increasingly faster and easier. Therefore, a number of considerations are introduced related to the risks of the illegal use of these images, as total diagnosis depends on them. Indeed, the patient's data management, storage, and transmission require a technique for boosting security, integrity and privacy measures in telehealthcare services. In fact, in our previous works, we used polynomial decompositions such as Chebychev orthogonal polynomial transform in medical image watermarking. We then customise our tools for finding the best candidate area for embedding the watermark, always seeking to provide the best solution to this issue. In this research, a variation to medical image watermarking approach based on Hermite transform (HT) is suggested for reliable management of medical data. In this approach, the HT is employed as a preprocessing step so as to extract the texture component of the host medical image. Afterward, a sliding window technique is done to select the most suitable regions for watermark embedding. Lastly, the Arnold transform is utilized for encrypting the watermark to strengthen the security of our scheme. Experiments were conducted on various modalities of medical images. Results indicate that the proposed scheme is robust when subjected to various attacks while preserving a high level of security and invisibility factors. Also, our method preserves the quality of medical images with a good embedding capacity. The obtained results support the use of Hermite Polynomials for the implementation of watermarking in the medical imaging context.

New extreme statistics strategy for the extreme value predictions of ship parametric rolling motions through limited model test observations

This work proposes a new extreme statistics strategy to predict the extreme parametric rolling angle through minimal observations from model tests. The core of this strategy is a new synthetic moment method. Based on the Hermite transformation and the Markov chain, the synthetic moment method can provide extreme value predictions for stationary processes. Besides, this work introduces one specific simplification to solve the problem induced by the non-stationarity of the parametric roll. With the help of the new extreme statistics strategy, this work obtains the CDF and PDF of the extreme parametric rolling angle for the C11 container ship. These predictions are derived from only one observed model test time history. The data statistics of other model tests and a vast amount of numerical simulations verify the validities of these predictions. Compared with other methods, the new strategy shows advantages for small sampling predictions. It is possible to replace the widely used ACER method in case of insufficient samples. In addition, further research confirms that the new strategy's intrinsic probability model differs from the widely used GEV model, which is more reasonable for extreme value predictions in a finite time range. The newly presented strategy should effectively predict the extreme value for other complicated stochastic responses.

Hypercomplex Systems and Non-Gaussian Stochastic Solutions with Some Numerical Simulation of χ-Wick-Type (2 + 1)-D C-KdV Equations

In this article, we discuss the (2 + 1)-D coupled Korteweg–De Vries (KdV) equations whose coefficients are variables, and stochastic (2 + 1)-D C-KdV (C-KdV) equations with the χ-Wick-type product. White noise functional solutions (WNFS) are presented with the homogeneous equilibrium principle, Hermite transform (HT), and technicality via the F-expansion procedure. By means of the direct connection between the theory of hypercomplex systems (HCS) and white noise analysis (WNA), we establish non-Gaussian white noise (NGWN) by studying stochastic partial differential equations (PDEs) with NG-parameters. So, by using the F-expansion method we present multiples of exact and stochastic families from variable coefficients of travelling wave and stochastic NG-functional solutions of (2 + 1)-D C-KdV equations. These solutions are Jacobi elliptic functions (JEF), trigonometric, and hyperbolic forms, respectively.

The effectiveness of discrete hermite wavelet filters technique in digital image watermarking

<span>In this work, new discrete wavelets were derived Hermite polynomials for obtained discrete hermite wavelet transformation (DHWT), and their efficiency for use in image processing is demonstrated by proving the realization of important theorems. Moreover, the role of the new and proposed waveforms in their effective effect in placing the watermark with the color image is clarified, and a program was created using MATLAB software by creating a subprogram for constructing the new wavelet and proving its efficiency with an analytic image. The process is repeated using DHWT to analyze the image. The color image has been subjected to various attacks after which the watermark is retrieved from the image after comparing it with the proposed algorithm and it has proven its power faster and better than the previously suggested methods. The final conclusion shows that using new wavelets DHWT better peak signal of noise ratio (PSNR)s can be obtained and that the proposed algorithm fills in better the lack of awareness of the watermark and its strength under different attacks.</span>

Hermite Transform Based Algorithm for Detection and Classification of High Impedance Faults

This work proposes a new algorithm to classify high impedance faults (HIFs) in distribution systems. HIFs can be represented by small current magnitudes with non-linear variations, which complicate their detection in distribution grids. The proposed method uses the Hermite transform (HT) as a signal processing technique that offers several advantages, one of the more important being the ability to analyze the signal at multiple resolution levels and at different frequency bands thanks to their filter functions based on Gaussian derivatives. The Hermite coefficients are used to extract signal features that allow efficient identification of the transient behaviour related to HIFs and other typical faults. In this sense, a multichannel approach based on the Hermite transform is proposed to classify different types of faults and HIFs. This analysis is carried out in a distribution network considering photovoltaic (PV) systems considering three different classifiers, which are also used to compare our results with the discrete wavelet transform (DWT) as signal decomposition model; the comparison suggests that our proposal presents better performance discriminating HIFs from typical faults.

WICK-TYPE STOCHASTIC FRACTIONAL SOLITONS SUPPORTED BY QUADRATIC-CUBIC NONLINEARITY

When a random environment with the Gaussian white noise function is considered, the Wick-type stochastic fractional quadratic-cubic nonlinear Schrödinger equation is used to govern the propagation of optical pulse in polarization-preserving fibers. Using a new strategy, namely combining the variable-coefficient fractional Riccati equation method with the fractional derivative, Mittag–Leffler function and Hermite transformation, some special fractional solutions with the Brownian motion function including fractional bright and dark solitons, and fractional combined soliton solutions are given. Under the influence of the stochastic effect from the stochastic Brownian motion function portrayed by using the Lorentz chaotic system, some wave packets randomly appear during the propagation, and thus make fractional bright soliton travel wriggled in the both periodic dispersion system and the exponential dispersion decreasing system. However, the stochastic Brownian motion function has a more significant impact on the propagation of fractional bright soliton in the periodic dispersion system than that in the exponential dispersion decreasing system.

Dark-multi-soliton and soliton molecule solutions of stochastic nonlinear Schrödinger equation in the white noise space

The wick-type nonlinear Schrödinger equation with gain and loss effects is studied. Dark-multi-soliton and dark soliton molecule solutions are obtained via the bilinear method and the Hermite transformation in the white noise space. The effects of gain and loss on the evolution of the dark-two-soliton molecule and the interaction between two dark solitons are analyzed. Moreover, the interaction between a dark-single-soliton and a dark-two-soliton molecule is discussed under the circumstance of the velocity resonance condition. The dark-three-soliton molecule is also considered. The results of this paper are useful for the study of soliton transmission in the white noise environment.

Wick-type stochastic multi-soliton and soliton molecule solutions in the framework of nonlinear Schrödinger equation

The wick-type stochastic nonlinear Schrödinger equation driven from the Brownian motion is considered. Multi-soliton and soliton molecule solutions influenced by Brownian motion function are constructed by using the Hirota method combined with the Hermite transformation. Based on stochastic solutions, the dynamical evolution of solitons produce random wave packets. These results will offer important help for the application of the stochastic physical model in the white noise environment.

Fractional optical solitons with stochastic properties of a wick-type stochastic fractional NLSE driven by the Brownian motion

A new strategy is proposed to analytically construct stochastic fractional solutions of a wick-type SFNLSE driven from the Brownian motion by altogether using the Hermite transformation, modified Riemann–Liouville fractional derivative rule and fractional mapping method with white noise theory. These stochastic fractional solutions include fractional bright, dark solitons and trigonometric function solutions with or without chirped phase. By the dynamical analysis of stochastic solutions, it is found that the evolution of soliton shows randomness and random wave packets appear. Moreover, the randomness of integer order soliton evolution becomes more obvious compared with fractional solitons. They have discontinuous random phenomena with the evolution of time t, and some random independent wave packets appear. These results will play an important role in the related research on the physical application of the random model in the white noise environment.

Hermite polynomial normal transformation for structural reliability analysis

PurposeNormal transformation is often required in structural reliability analysis to convert the non-normal random variables into independent standard normal variables. The existing normal transformation techniques, for example, Rosenblatt transformation and Nataf transformation, usually require the joint probability density function (PDF) and/or marginal PDFs of non-normal random variables. In practical problems, however, the joint PDF and marginal PDFs are often unknown due to the lack of data while the statistical information is much easier to be expressed in terms of statistical moments and correlation coefficients. This study aims to address this issue, by presenting an alternative normal transformation method that does not require PDFs of the input random variables.Design/methodology/approachThe new approach, namely, the Hermite polynomial normal transformation, expresses the normal transformation function in terms of Hermite polynomials and it works with both uncorrelated and correlated random variables. Its application in structural reliability analysis using different methods is thoroughly investigated via a number of carefully designed comparison studies.FindingsComprehensive comparisons are conducted to examine the performance of the proposed Hermite polynomial normal transformation scheme. The results show that the presented approach has comparable accuracy to previous methods and can be obtained in closed-form. Moreover, the new scheme only requires the first four statistical moments and/or the correlation coefficients between random variables, which greatly widen the applicability of normal transformations in practical problems.Originality/valueThis study interprets the classical polynomial normal transformation method in terms of Hermite polynomials, namely, Hermite polynomial normal transformation, to convert uncorrelated/correlated random variables into standard normal random variables. The new scheme only requires the first four statistical moments to operate, making it particularly suitable for problems that are constraint by limited data. Besides, the extension to correlated cases can easily be achieved with the introducing of the Hermite polynomials. Compared to existing methods, the new scheme is cheap to compute and delivers comparable accuracy.

Long-term temperature field of steel-box girder of a long-span bridge: Measurement and simulation

The temperature field of a long-span bridge is variant with an underlying statistical property due to the periodically time-varying solar radiation and the unceasing structural heat exchange with surrounding environment. A comprehensive understanding of the long-term temperature field of a steel-box girder is essential for the design and maintenance of cable-supported bridges. However, it is challenging to obtain the long-term temperature variations because of the complex geometric configuration of the main girder and the complicated heat transfer process. In this context, this paper presents a strategy to overcome this problem by simulating the long-term temperature field of a steel-box girder based on limited monitored data. In the simulation, the long-term temperature field is treated as a stochastic process with prescribed spectral and probabilistic properties. By analyzing the monitored temperature data on Sutong Bridge, the models of power spectral density (PSD), coherence function, and high-order moments are developed. A noteworthy feature captured in measurements is that the temperature is non-Gaussian with considerable kurtosis and skewness. Hence, the temperature field simulation is converted to the simulation of multivariate non-Gaussian stochastic processes. Based on the spectral representation method accompanied with Hermite transformation, the temperature field of the steel-box girder of Sutong Bridge is simulated with a two-year duration considering the non-Gaussianity. Verifications of the simulated field are made through comparisons with the developed spectral and probabilistic models. It is shown that the simulations match well with the measured temperature characteristics, specifically including the time histories and their PSD, PDF, and coherence functions. The good agreement indicates the effectiveness of the simulated long-term temperature field.

Assessment of fatigue damage induced by Non-Gaussian bimodal processes with emphasis on spectral methods

The stress on an engineering structure tends to be random and is usually assumed to be Gaussian or weakly non-Gaussian. In some situations, the power spectrum of stress also exhibits two distinct peaks in a bimodal profile. This study focuses on the fatigue damage estimation of softening non-Gaussian bimodal processes (kurtosis >3) with emphasis on spectral methods. Several classical spectral methods for Gaussian problems and a newly developed method by authors are examined. Correction factors derived from the Hermite transformation are used to handle the non-Gaussianity. Through comprehensive case studies, including ideal bimodal spectra and practical bimodal spectra in ocean engineering, the accuracy of the underlying Gaussian fatigue damage and the corresponding correction factor for each spectral method is investigated separately. The results demonstrate that when the non-Gaussianity is weak (kurtosis between 3 and 6), the correction factor is dominated by the nonlinear mapping between the non-Gaussian stress and the underlying Gaussian stress while the bandwidth effect is relatively limited. Under this situation, the narrow-banded correction factor proposed by Winterstein is conservatively accurate in most cases. Among all spectral methods, the newly developed method coupled with the Winterstein's correction factor renders the best accuracy of non-Gaussian fatigue damage estimation.

Fatigue life prediction of ship and offshore structures under wide-banded non-Gaussian random loadings: Part II: extension to wide-banded non-Gaussian random processes

Ships and offshore structures are exposed to various cyclic loads such as winds, waves and currents in their lifetime. These loads can result in fatigue failures at hot spots, and to prevent such failures, accurate estimates of the fatigue life of hot spots should be performed in the design stage. The cycle distribution indicative of the probability distribution of cycles is an important factor in estimation of fatigue damage to mechanical systems in frequency-domain methods. It depends on not only the counting method but also the statistical properties of random processes. Many studies deriving both theoretical and approximate models of rainflow-counted cycle distributions in wide-banded non-Gaussian processes have been conducted. However, some of the existing models require multi-dimensional integrations with much computational time or yield inaccurate representations of rainflow-cycle distributions. Therefore, a new model providing accurate cycle distributions is needed for fatigue analysis of offshore structures under wide-banded non-Gaussian random loadings.This paper consists of two parts: Part I is devoted to the development of a new approximation model for the joint probability distribution of mean and amplitude of rainflow-counted cycles in wide-band Gaussian random processes. This probabilistic model derived in Gaussian random processes could be transformed to the corresponding non-Gaussian processes through non-linear transformation technique. Part II discusses the detailed procedure and accuracy of the proposed model in fatigue damage assessment of offshore structures under non-Gaussian random loading. Rainflow cycle distribution in wide-band non-Gaussian processes is derived using the Hermite transformation function. The accuracy of the proposed model is verified through two numerical examples, and compared to other existing models.

Fractional white noise functional soliton solutions of a wick-type stochastic fractional NLSE

The wick-type stochastic fractional nonlinear Schrödinger equation is considered, and some analytical white noise functional solutions are found by the fractional F-expansion method with the Hermite transformation. The chirped and chirp-free fractional bright and dark soliton solutions with the Brownian motion function are constructed.

Applicability of time conformable derivative to Wick-fractional-stochastic PDEs

Fractional-stochastic quadratic-cubic nonlinear Schrödinger equation (QC-NLSE) describing propagation of solitons through optical fibers is analyzed. Hermite transforms, white noise analysis and an improved computational method are used to investigate uncertain solutions for QC-NLSE. Specifically, Hermite transformation is applied to convert fractional-stochastic differential equations by Wick-type into deterministic fractional differential equations with an integral term. Furthermore, inverse Hermite transformation is employed to obtain stochastic solutions in the white noise space. Characteristics of presented equations are shown by using some specific values of physical arguments on obtained solutions.

Predicting loads and dynamic responses of an offshore wind turbine in a nonlinear mixed sea

ABSTRACT In this article the loads and dynamic responses of an offshore wind turbine (OWT) operating in a nonlinear multidirectional mixed sea have been rigorously investigated. It has been demonstrated that nonlinearly simulated realistic waves with bottom effects should be utilised as inputs in the stochastic time domain simulation in order to design an un-conservative support structure for the offshore wind turbine. In order to further improve the simulation efficiency, a novel transformed linear simulation method based on a Hermite transformation model has been first proposed in this article for generating equivalent waves as those obtained from the nonlinear simulation method. The predicted results of the loads and dynamic responses of the offshore wind turbine have been rigorously analysed and systematically compared, and the advantages of using our proposed novel method have been convincingly substantiated.

Image Correlation Using Fractional Hermite Transform

In this paper, we generalize the Hermite transform into a fractional case using the fractional Fourier transform and the fractional convolution. The new methodology was evaluated using phytoplankton images with different illumination patterns and fragmented images. We found that the fractional Hermite transform had a better capability to recognize images. The discrimination coefficient was evaluated for the fractional Hermite transform and the conventional Hermite transform, finding more noise tolerate with the fractional Hermite transform. The Hermite fractional transform, in combination with the extreme phase filter, showed in a study, using fragmented diatom images, a better ability to classify diatoms, even when these had little information.

Estimation of Wave Crest Amplitudes Distribution and Freak Wave Occurrence in A Short Crested Mixed Sea

In this study we have for the first time proposed a novel transformed linear simulation method for the estimation of wave crest amplitudes distribution and freak wave occurrence in a short crested mixed sea with a bimodal 3D spectrum. For implementing the proposed transformed linear simulation method, a Hermite transformation model expressed in a monotonic cubic polynomial has been constructed so that the first four moments of the original true process match the corresponding moments of the transformed model. The proposed novel simulation method has been applied to forecast the freak wave occurrence in two short crested mixed sea states, one with a directional wave spectrum based on the measured surface elevation data at the coast of Yura, and the other one with a typical directional bimodal Torsethaugen wave spectrum. It is shown in the two cases that the proposed novel simulation method can offer more accurate forecasting results than those obtained from the traditional linear simulation method or by using Rayleigh distribution model. It is also demonstrated in this article that the proposed novel simulation method is more efficient than the nonlinear simulation method.