Fractional Order Moments Research Articles

Representing Noisy Image Without Denoising.

A long-standing topic in artificial intelligence is the effective recognition of patterns from noisy images. In this regard, the recent data-driven paradigm considers 1) improving the representation robustness by adding noisy samples in training phase (i.e., data augmentation) or 2) pre-processing the noisy image by learning to solve the inverse problem (i.e., image denoising). However, such methods generally exhibit inefficient process and unstable result, limiting their practical applications. In this paper, we explore a non-learning paradigm that aims to derive robust representation directly from noisy images, without the denoising as pre-processing. Here, the noise-robust representation is designed as Fractional-order Moments in Radon space (FMR), with also beneficial properties of orthogonality and rotation invariance. Unlike earlier integer-order methods, our work is a more generic design taking such classical methods as special cases, and the introduced fractional-order parameter offers time-frequency analysis capability that is not available in classical methods. Formally, both implicit and explicit paths for constructing the FMR are discussed in detail. Extensive simulation experiments and robust visual applications are provided to demonstrate the uniqueness and usefulness of our FMR, especially for noise robustness, rotation invariance, and time-frequency discriminability.

Fractional Negative Order Moment Parameter Estimator of Compound-Gaussian Clutter with Lognormal Texture

Fractional Negative Order Moment Parameter Estimator of Compound-Gaussian Clutter with Lognormal Texture

Recognition of 3D Images by Fusing Fractional-Order Chebyshev Moments and Deep Neural Networks.

In order to achieve efficient recognition of 3D images and reduce the complexity of network parameters, we proposed a novel 3D image recognition method combining deep neural networks with fractional-order Chebyshev moments. Firstly, the fractional-order Chebyshev moment (FrCM) unit, consisting of Chebyshev moments and the three-term recurrence relation method, is calculated separately using successive integrals. Next, moment invariants based on fractional order and Chebyshev moments are utilized to achieve invariants for image scaling, rotation, and translation. This design aims to enhance computational efficiency. Finally, the fused network embedding the FrCM unit (FrCMs-DNNs) extracts depth features to analyze the effectiveness from the aspects of parameter quantity, computing resources, and identification capability. Meanwhile, the Princeton Shape Benchmark dataset and medical images dataset are used for experimental validation. Compared with other deep neural networks, FrCMs-DNNs has the highest accuracy in image recognition and classification. We used two evaluation indices, mean square error (MSE) and peak signal-to-noise ratio (PSNR), to measure the reconstruction quality of FrCMs after 3D image reconstruction. The accuracy of the FrCMs-DNNs model in 3D object recognition was assessed through an ablation experiment, considering the four evaluation indices of accuracy, precision, recall rate, and F1-score.

A NEW GENERALIZATION OF CAUCHY DISTRIBUTION

A new heavy-tailed distribution is considered in this paper: the p-generalized Cauchy distribution. This new 5-parameter distribution family is substantially more flexible than than the classical Cauchy distribution, it contains, in particular, asymmetric distributions. Various numerical characteristics of the new distribution are considered, among them are fractional order moments and “twice incomplete” moments. There were obtained also (using numerical methods) values of quantile-based standardized moments of higher orders: Bowley’s skewness and Moors’ kurtosis. Suitability of the p-generalized Cauchy distribution for modeling real data was confirmed by fitting this distribution to a series of log returns of stock prices. The p-generalized Cauchy distribution had the smaller value of the AIC statistic than the Cauchy distribution, the skew Cauchy distribution, the generalized logistic distribution and the hyperbolic distribution.

Multi-headed deep learning-based estimator for correlated-SIRV Pareto type II distributed clutter

This paper deals with the problem of estimating the parameters of heavy-tailed sea clutter in high-resolution radar, when the clutter is modeled by the correlated Pareto type II distribution. Existing estimators based on the maximum likelihood (ML) approach, integer-order moments (IOM) approach, fractional-order moments (FOM), and log-moments (log-MoM) have shown to be sensitive to changes in data correlation. In this work, we resort to a deep learning (DL) approach based on a multi-headed architecture to overcome this problem. Offline training of the artificial neural networks (ANN) is carried out by using several combinations of the clutter parameters, with different correlation degrees. To assess the performance of the proposed estimator, we resort to Monte Carlo simulation, and we observed that it has superior performance over existing approaches in terms of estimation mean square error (MSE) and robustness to changes of the clutter correlation coefficient.

DOA Tracking With Nested Arrays via an Improved Multisource Multi-Bernoulli Filtering in the Presence of Impulsive Noise

Traditional DOA tracking algorithms have poor performance when considering source time-varying and impulsive noise interference. To address these issues, an improved multisource multi-Bernoulli (IMSMB) particle filtering algorithm based on phased fractional lower order moment is proposed. In the IMSMB algorithm, an unscented transform method is used to operate the surviving particles, which increases the diversity of resampling particles. Moreover, the proposed method is extended to nested array to improve tracking accuracy. Simulation results verify the superiority of the proposed method compared to the existing DOA tracking algorithm.

Robust color image watermarking using multiple fractional-order moments and chaotic map

Robust watermarking is an effective method and a promising solution for securing and protecting the copyright of digital images. Moments and moment invariants have become popular tools for robust watermarking due to their geometric invariance and favorable capability of image description. Many moments-based robust watermarking schemes have been proposed. However, there is a challenging problem of these schemes that should be addressed. One of these problems is to improve both imperceptibility and robustness. In contrast, the other problem, most of these schemes used inefficient, traditional computation methods of the moments, resulting in an inaccurate and inefficient performance of the watermarking schemes. To overcome these challenges, in this paper, we propose a novel robust color image-watermarking algorithm based on new multiple fractional multi-channel orthogonal moments, fractional-order exponent moments (MFrEMs), fractional-order polar harmonic transforms (MFrPHTs), and fractional-order radial harmonic Fourier moments (MFrRHFMs). Firstly, highly accurate fractional new multi-channel orthogonal moments are computed for the host color images. Then, more stable and accurate coefficients of fractional new multi-channel orthogonal moments are selected. Finally, a robust color image watermarking approach for multiple watermarks images is proposed based on MFrEMs, MFrPHTs, and MFrRHFMs using a 1D Sine chaotic map. The experimental results demonstrate that the proposed approach provides robustness against various attacks and better imperceptibility than the existing methods.

Anomalous band center localization in the one-dimensional Anderson model with a disordered distribution of infinite variance.

We perform a detailed numerical study of the influence of distributions without a finite second moment on the Lyapunov exponent through the one-dimensional tight-binding Anderson model with diagonal disorder. Using the transfer matrix parametrization method and considering a specific distribution function, we calculate the Lyapunov exponent numerically and demonstrate its relation with the fractional lower order moments of the disorder probability density function. For the lower order of moments of disorder distribution with an infinite variance, we obtain the anomalous behavior near the band center.

Image Analysis by Fractional-Order Gaussian-Hermite Moments.

Moments and moment invariants are effective feature descriptors. They have widespread applications in the field of image processing. The recent researches show that fractional-order moments have notable image representation ability. Hermite polynomials are defined over the interval from negative infinity to positive one. Such unboundedness prevents us from developing fractional-order Gaussian-Hermite moments via the existing ideas or approaches. In this paper, we propose fractional-order Gaussian-Hermite moments by forcing the definition domain of Hermite polynomials to be a bounded interval, meanwhile, resorting to a value-decreasing standard deviation to maintain the orthogonality. Moreover, we successfully develop contrast, translation and rotation invariants from the proposed moments based on the inherent properties of Hermite polynomials. The reconstructions of different types of images demonstrate that the proposed moments have more superior image representation ability to the most existing popular orthogonal moments. Besides, the salient performance in invariant image recognition, noise robustness and region-of-interest feature extraction reflect that these moments and their invariants possess the stronger discrimination power and the better noise robustness in comparison with the existing orthogonal moments. Furthermore, both complexity analysis and time consumption indicate that the proposed moments and their invariants are easy to implement, they are suitable for practical engineering applications.

Some aspects of fractional-order circular moments for image analysis

• The importance of optimal parameters in the fractional-order moments is discussed. • The proposed algorithm searches the optimal fractional parameter. • The optimal parameter found is invariant to changes in scale and rotation. • The experimental analysis demonstrates the superiority of the optimal parameter. • The proposed method performs superbly in rotation-invariant object recognition. In this paper, we briefly review the fractional-order circular moments, such as fractional-order Zernike moments, fractional-order Fourier–Mellin moments, fractional-order Legendre–Fourier moments, and fractional-order Chebyshev–Fourier moments, which can characterize, analyze, and manipulate the information contained in an image with minimal redundancy. Also, they depend on an α parameter for better feature extraction. Therefore, we propose a procedure to find the optimal α in terms of image reconstruction error and classification. We validate the search for the best rotation-invariant features using the MNIST and MNIST-R datasets. Finally, we present the study results and conclusions.

3D image recognition using new set of fractional-order Legendre moments and deep neural networks

3D Image representation is an important topic in computer vision and pattern recognition. Recently, 3D image analysis by fractional-order orthogonal moments has provided a new research direction, which has prompted researchers to think about efficient and fast classification. In this paper, the authors derived novel sets of fractional-order Legendre moment invariants (FrLMIs), for 3D object description and recognition. Therefore, an analysis of the performance of reconstruction and classification based on fractional-order moment invariants and Deep Neural Networks (DNNs) by changing the number of descriptors was presented. Accordingly, the performance of these proposed fractional-order moments and moment invariants are evaluated through several appropriate experiments, including 3D image reconstruction, region of interest feature extraction, invariance with respect to the geometric transformations and noisy, and 3D Object classification using different fractional parameters. The superiority of the proposed method is verified by comparing the classification percentages obtained by varying the amount of data used during the training process in comparison with the existing methods. The work presented will help to create new neural network architectures that take advantage of the descriptive capacity of 3D fractional-order moments. Finally, these fractional-order moments are very fast and computationally inexpensive which could be useful in many computer vision applications. Based on these characteristics, the proposed FrOLMs and FrLMIs outperformed all existing orthogonal moments.

An Accuracy estimation of approximation in central limit theorem in the existence of fractional-order moments

An Accuracy estimation of approximation in central limit theorem in the existence of fractional-order moments

Relative Entropy Method Applied for the Fatigue Life Distribution of Carbon Fiber/Epoxy Composites.

This paper verifies the feasibility of the relative entropy method in selecting the most suitable statistical distribution for the experimental data, which do not follow an exponential distribution. The efficiency of the relative entropy method is tested through the fractional order moment and the logarithmic moment in terms of the experimental data of carbon fiber/epoxy composites with different stress amplitudes. For better usage of the relative entropy method, the efficient range of its application is also studied. The application results show that the relative entropy method is not very fit for choosing the proper distribution for non-exponential random data when the heavy tail trait of the experimental data is emphasized. It is not consistent with the Kolmogorov–Smirnov test but is consistent with the residual sum of squares in the least squares method whenever it is calculated by the fractional moment or the logarithmic moment. Under different stress amplitudes, the relative entropy method has different performances.

Fractional‐order generalized Laguerre moments and moment invariants for grey‐scale image analysis

Here, a new set of fractional-order moments, named fractional-order generalized Laguerre moments (FGLM), is introduced. These proposed moments are defined on the Cartesian coordinate system and their basis functions are represented by the fractional-order generalized Laguerre polynomials. Contrary to the classical Chebyshev, Legendre and Gegenbauer moments, which provide only global feature, our proposed FGLM have the ability to extract both global and local features. Moreover, a new set of rotation, scale and translation invariants of the FGLM, is derived and introduced for image classification and invariant pattern recognition. Just as important, we have presented a systematic parameter selection method for finding the optimal fractional parameter values with respect to pattern recognition applications. Finally, several recursive methods for reducing the computation time of our proposed invariants are also provided in this study. Therefore, to demonstrate the performance of the introduced fractional-order moments and moment invariants, a number of experimental analysis are performed in terms of global and local features extraction, robustness to noise, invariance to geometric deformations, object recognition and computational speed. The presented theoretical and experimental results clearly show that the proposed fractional-order moments and their corresponding invariants could be extremely useful in the field of image analysis.

A New Concept of Fractional Order Cumulant and It-Based Signal Processing in α and/or Gaussian Noise

In this article, the concept and definitions of the Fractional Order Moment (FOM) and Fractional Order Cumulant (FOC) are proposed, which is based on the fractional derivative of the fractional order Moment-generating function and the fractional order Cumulant-generating function of stochastic processes. The moment and cumulant are defined on an expanded set from positive integer to the whole positive real. This development not only provides a new technology for signal processing, also complements the existing theory in the field. The properties of the FOC have been derived, and their uniformity and particularity with the High Order Cumulant are compared and commented. In addition, the transformation between the FOM and the FOC are derived and discussed in detail. As one of the applications of the new concept to the α and Gaussian processes, a new method of suppressing α and Gaussian noise is proposed. Furthermore, a FOC-based parameter estimation algorithm is developed for the non-minimum phase ARMA processes in α and/or Gaussian noise. Simulation examples are used to demonstrate the effectiveness of the proposed parameter estimation algorithm.

Image Reconstruction Based on Novel Sets of Generalized Orthogonal Moments.

In this work, we have presented a general framework for reconstruction of intensity images based on new sets of Generalized Fractional order of Chebyshev orthogonal Moments (GFCMs), a novel set of Fractional order orthogonal Laguerre Moments (FLMs) and Generalized Fractional order orthogonal Laguerre Moments (GFLMs). The fractional and generalized recurrence relations of fractional order Chebyshev functions are defined. The fractional and generalized fractional order Laguerre recurrence formulas are given. The new presented generalized fractional order moments are tested with the existing orthogonal moments classical Chebyshev moments, Laguerre moments, and Fractional order Chebyshev Moments (FCMs). The numerical results show that the importance of our general framework which gives a very comprehensive study on intensity image representation based GFCMs, FLMs, and GFLMs. In addition, the fractional parameters give a flexibility of studying global features of images at different positions and scales of the given moments.

First passage time moments of asymmetric Lévy flights

We investigate the first-passage dynamics of symmetric and asymmetric Lévy flights in semi-infinite and bounded intervals. By solving the space-fractional diffusion equation, we analyse the fractional-order moments of the first-passage time probability density function for different values of the index of stability and the skewness parameter. A comparison with results using the Langevin approach to Lévy flights is presented. For the semi-infinite domain, in certain special cases analytic results are derived explicitly, and in bounded intervals a general analytical expression for the mean first-passage time of Lévy flights with arbitrary skewness is presented. These results are complemented with extensive numerical analyses.

New set of fractional-order generalized Laguerre moment invariants for pattern recognition

This article presents a new series of invariant moments, called Fractional-order Generalized Laguerre Moment Invariants (FGLMI), based on Fractional-order Generalized Laguerre polynomials (FGLPs). To begin, we provide the relations and the properties necessary to define the fractional-order generalized Laguerre moments. Then, we present the theoretical framework to derive invariants from fractional-order moments with respect to the change in orientation, size and position based on the algebraic relationships between FGLM and fractional-order geometric moments. In addition, a fast and precise algorithm has been proposed for the calculation of FGLM in order to speed up the calculation time and ensure the numerical stability of the invariant moments. Numerical experiments are carried out to demonstrate the efficiency of FGLM and their proposed invariants compared to existing methods, with regard to the reconstruction of 2D and 3D images, the computation time, the global entity extraction capacity and image localization, invariability property and 2D / 3D image classification performance on different 2D and 3D image databases. The theoretical and experimental results presented clearly show the efficiency of the descriptors proposed for the representation and classification of 2D and 3D images by other types of orthogonal moments.

New fractional-order shifted Gegenbauer moments for image analysis and recognition

Orthogonal moments are used to represent digital images with minimum redundancy. Orthogonal moments with fractional-orders show better capabilities in digital image analysis than integer-order moments. In this work, the authors present new fractional-order shifted Gegenbauer polynomials. These new polynomials are used to define a novel set of orthogonal fractional-order shifted Gegenbauer moments (FrSGMs). The proposed method is applied in gray-scale image analysis and recognition. The invariances to rotation, scaling and translation (RST), are achieved using invariant fractional-order geometric moments. Experiments are conducted to evaluate the proposed FrSGMs and compare with the classical orthogonal integer-order Gegenbauer moments (GMs) and the existing orthogonal fractional-order moments. The new FrSGMs outperformed GMs and the existing orthogonal fractional-order moments in terms of image recognition and reconstruction, RST invariance, and robustness to noise.

New fractional-order Legendre-Fourier moments for pattern recognition applications

Orthogonal moments enable computer-based systems to discriminate between similar objects. Mathematicians proved that the orthogonal polynomials of fractional-orders outperformed their corresponding counterparts in representing the fine details of a given function. In this work, novel orthogonal fractional-order Legendre-Fourier moments are proposed for pattern recognition applications. The basis functions of these moments are defined and the essential mathematical equations for the recurrence relations, orthogonality and the similarity transformations (rotation and scaling) are derived. The proposed new fractional-order moments are tested where their performance is compared with the existing orthogonal quaternion, multi-channel and fractional moments. New descriptors were found to be superior to the existing ones in terms of accuracy, stability, noise resistance, invariance to similarity transformations, recognition rates and computational times.