scholarly journals Image Reconstruction Based on Novel Sets of Generalized Orthogonal Moments

2020 ◽  
Vol 6 (6) ◽  
pp. 54
Author(s):  
R. M. Farouk

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.

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Bing He ◽  
Jun Liu ◽  
Tengfei Yang ◽  
Bin Xiao ◽  
Yanguo Peng

AbstractInspired by quaternion algebra and the idea of fractional-order transformation, we propose a new set of quaternion fractional-order generalized Laguerre orthogonal moments (QFr-GLMs) based on fractional-order generalized Laguerre polynomials. Firstly, the proposed QFr-GLMs are directly constructed in Cartesian coordinate space, avoiding the need for conversion between Cartesian and polar coordinates; therefore, they are better image descriptors than circularly orthogonal moments constructed in polar coordinates. Moreover, unlike the latest Zernike moments based on quaternion and fractional-order transformations, which extract only the global features from color images, our proposed QFr-GLMs can extract both the global and local color features. This paper also derives a new set of invariant color-image descriptors by QFr-GLMs, enabling geometric-invariant pattern recognition in color images. Finally, the performances of our proposed QFr-GLMs and moment invariants were evaluated in simulation experiments of correlated color images. Both theoretical analysis and experimental results demonstrate the value of the proposed QFr-GLMs and their geometric invariants in the representation and recognition of color images.


2020 ◽  
Author(s):  
Bing He ◽  
Jun Liu ◽  
Tengfei Yang ◽  
Bin Xiao ◽  
Yanguo Peng

Abstract Inspired by quaternion algebra and the idea of fractional-order transformation, we propose a new set of quaternion fractional-order generalized Laguerre orthogonal moments (QFr-GLMs) based on fractional-order generalized Laguerre polynomials. Firstly, the proposed QFr-GLMs are directly constructed in Cartesian coordinate space, avoiding the need for conversion between Cartesian and polar coordinates; therefore, they are better image descriptors than circularly orthogonal moments constructed in polar coordinates. Moreover, unlike the latest Zernike moments based on quaternion and fractional-order transformations, which extract only the global features from color images, our proposed QFr-GLMs can extract both the global and local color features. This paper also derives a new set of invariant color-image descriptors by QFr-GLMs, enabling geometric-invariant pattern recognition in color images. Finally, the performances of our proposed QFr-GLMs and moment invariants were evaluated in simulation experiments of correlated color images. Both theoretical analysis and experimental results demonstrate the value of the proposed QFr-GLMs and their geometric invariants in the representation and recognition of color images.


2021 ◽  
Vol 8 (2) ◽  
pp. 54-59
Author(s):  
R. M. Farouk ◽  
◽  
Qamar A. A. Awad ◽  

In this paper, we have introduced new sets of fractional order orthogonal basis moments based on Fractional order Legendre orthogonal Functions (FLeFs) and Fractional order Laguerre orthogonal Functions (FLaFs) for image representation. We have generated a novel set of Fractional order Legendre orthogonal Moments (FLeMs) from fractional order Legendre orthogonal functions and a new set of Fractional order Laguerre orthogonal Moments (FLaMs) from the fractional order Laguerre orthogonal functions. The new presented sets of (FLeMs) and (FLaMs) are tested with the recently introduced Fractional order Chebyshev orthogonal Moments (FCMs). This edge detection filter can be used successfully in the gray level image and color images. The new sets of fractional moments are used to reconstruct the gray level image. The numerical results show FLeMs and FLaMs are promised techniques for image representation. The computational time of the proposed techniques is compared with the computational time of Chebyshev orthogonal Moments techniques and gives better results. Also, the fractional parameters give the flexibility of studying global features of the image at different positions of moments.


Author(s):  
Khalid K. Ali ◽  
Mohamed A. Abd El salam ◽  
Emad M. H. Mohamed

AbstractIn this paper, a numerical technique for a general form of nonlinear fractional-order differential equations with a linear functional argument using Chebyshev series is presented. The proposed equation with its linear functional argument represents a general form of delay and advanced nonlinear fractional-order differential equations. The spectral collocation method is extended to study this problem as a discretization scheme, where the fractional derivatives are defined in the Caputo sense. The collocation method transforms the given equation and conditions to algebraic nonlinear systems of equations with unknown Chebyshev coefficients. Additionally, we present a general form of the operational matrix for derivatives. A general form of the operational matrix to derivatives includes the fractional-order derivatives and the operational matrix of an ordinary derivative as a special case. To the best of our knowledge, there is no other work discussed this point. Numerical examples are given, and the obtained results show that the proposed method is very effective and convenient.


Sensors ◽  
2021 ◽  
Vol 21 (4) ◽  
pp. 1544
Author(s):  
Chunpeng Wang ◽  
Hongling Gao ◽  
Meihong Yang ◽  
Jian Li ◽  
Bin Ma ◽  
...  

Continuous orthogonal moments, for which continuous functions are used as kernel functions, are invariant to rotation and scaling, and they have been greatly developed over the recent years. Among continuous orthogonal moments, polar harmonic Fourier moments (PHFMs) have superior performance and strong image description ability. In order to improve the performance of PHFMs in noise resistance and image reconstruction, PHFMs, which can only take integer numbers, are extended to fractional-order polar harmonic Fourier moments (FrPHFMs) in this paper. Firstly, the radial polynomials of integer-order PHFMs are modified to obtain fractional-order radial polynomials, and FrPHFMs are constructed based on the fractional-order radial polynomials; subsequently, the strong reconstruction ability, orthogonality, and geometric invariance of the proposed FrPHFMs are proven; and, finally, the performance of the proposed FrPHFMs is compared with that of integer-order PHFMs, fractional-order radial harmonic Fourier moments (FrRHFMs), fractional-order polar harmonic transforms (FrPHTs), and fractional-order Zernike moments (FrZMs). The experimental results show that the FrPHFMs constructed in this paper are superior to integer-order PHFMs and other fractional-order continuous orthogonal moments in terms of performance in image reconstruction and object recognition, as well as that the proposed FrPHFMs have strong image description ability and good stability.


Mathematics ◽  
2019 ◽  
Vol 7 (1) ◽  
pp. 40 ◽  
Author(s):  
Shumaila Javeed ◽  
Dumitru Baleanu ◽  
Asif Waheed ◽  
Mansoor Shaukat Khan ◽  
Hira Affan

The analysis of Homotopy Perturbation Method (HPM) for the solution of fractional partial differential equations (FPDEs) is presented. A unified convergence theorem is given. In order to validate the theory, the solution of fractional-order Burger-Poisson (FBP) equation is obtained. Furthermore, this work presents the method to find the solution of FPDEs, while the same partial differential equation (PDE) with ordinary derivative i.e., for α = 1 , is not defined in the given domain. Moreover, HPM is applied to a complicated obstacle boundary value problem (BVP) of fractional order.


Author(s):  
Галина Борисовна Сыченко

Статья посвящена рассмотрению основных подходов к изучению музыки в шаманизме, сложившихся и функционирующих в современной антропологии и музыковедении. Автор характеризует различные направления в каждом из них, приводя в качестве примеров и анализируя наиболее показательные труды. Литература (источники и исследования) подобрана таким образом, чтобы позволить читателю самостоятельно расширить библиографический список. Изложение следует хронологическому порядку и отражает логику развития научного знания в избранной области. Два наиболее ранних подхода - музыкально-этнографический и музыкально-теоретический - продолжают сохраняться и развиваться до сих пор. Относительно недавно на их основе начал формироваться комплексно-текстологический подход. Все они ориентированы на изучение самой музыки в этнографическом контексте (тексториентированные подходы). Позже появляются многочисленные труды музыкально-антропологического и культурологического направлений, довольно подробно исследующие концептуальные и функциональные аспекты шаманской и, шире, сакральной музыки в разных традициях (контексториентированные подходы). В последнее время появляется все больше работ, в которых в разной форме реализуется музыкально-психологический подход, причем спектр направлений внутри него весьма широк - от культурологических до нейрофизиологических. Большинство подобных исследований проводится за рубежом. В результате предпринятого обзора автором определены наиболее актуальные направления изучения проблематики «музыка и шаманизм». Это комплексный анализ полных вербально-музыкальных текстов шаманских камланий, исследование характера взаимосвязи звукового компонента с измененными состояниями сознания, изучение региональных традиций, а также, в перспективе, развитие методологии сравнительно-исторического исследования музыкальной составляющей шаманских традиций. This article examines the main approaches to the study of music in shamanism that have been developed and those that are current in modern anthropology and musicology. The author characterises the different areas of research covered in each approach, giving examples and analyzing representative works. The presentation of the different approaches is chronological and reflects the logic of the development of scholarly knowledge in the given field. The two earliest approaches - musical-ethnographic and musical-theoretical - continue to be used. Relatively recently, an integrative and textological approach has begun to be applied on their basis. These textually-oriented methods aim at studying the music in an ethnographic context. Subsequently context-oriented approaches have appeared, applying musical-anthropological and culturological methods that explore the conceptual and functional aspects of shamanic - and, more broadly, sacral - music in different traditions. Recently there have been an increasing number of studies that implement a music-psychological approach in various forms. The range of directions within this approach is broad, from culturological to neurophysiological; most such studies are conducted outside Russia. The author also identifies the most relevant current areas of research. These include: the comprehensive study of the verbal and musical texts of shamanic rituals; study of the nature of these texts’ relationship to altered states of consciousness; and comparative research on the most significant regional traditions. She looks forward to the development of a methodology appropriate for comparative historical research on the musical component of shamanic traditions.


Author(s):  
Pushpendra Kumar ◽  
Vedat Suat Erturk ◽  
Abdullahi Yusuf ◽  
Tukur Abdulkadir Sulaiman

In some of the previous decades, we have observed that mathematical modeling has become one of the most interesting research fields and has attracted many researchers. In this regard, thousands of researchers have proposed different varieties of mathematical models to study the dynamics of a number of real-world problems. This research work is framed to analyzing the structure of the well-known Lassa hemorrhagic epidemic; a dangerous epidemic for pregnant women, via new generalized Caputo type noninteger order derivative with the help of a modified Predictor–Corrector scheme. Lassa hemorrhagic disease is an epidemical and biocidal fever, whose negative impacts were initially recognized in the countries of Africa. This virus has killed many pregnant women as compared to the Ebola epidemic. It was noticed that Lassa virus was isolated in Vero cell cultures from a blood pattern, and after 12 days it was ejective, after the climb of the sickness. In this research study, necessary theorems and lemmas are reminded to prove the existence of a unique solution and stability of given fractional approximation scheme. All necessary results are reminded to confirm the effectiveness of the proposed approximation algorithm by graphical observations for various fractional-order values. In our practical calculations, we plotted the graphs for two different values of natural death rate along with various values of given fractional-order operator. Our major target is to show the importance of the proposed modified version of the Predictor–Corrector algorithm in epidemic studies by exploring the given Lassa hemorrhagic fever dynamics.


2021 ◽  
Vol 2021 ◽  
pp. 1-15
Author(s):  
Mounirah Areshi ◽  
A. M. Zidan ◽  
Rasool Shah ◽  
Kamsing Nonlaopon

In this article, the iterative transformation method and homotopy perturbation transformation method are applied to calculate the solution of time-fractional Cauchy-reaction diffusion equations. In this technique, Shehu transformation is combined of the iteration and the homotopy perturbation techniques. Four examples are examined to show validation and the efficacy of the present methods. The approximate solutions achieved by the suggested methods indicate that the approach is easy to apply to the given problems. Moreover, the solution in series form has the desire rate of convergence and provides closed-form solutions. It is noted that the procedure can be modified in other directions of fractional order problems. These solutions show that the current technique is very straightforward and helpful to perform in applied sciences.


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