Comparative Analysis of Fractional Order Calculus in Image Processing

The malware spreading in Wireless Sensor Network (WSN) has lately attracted the attention of many researchers as a hot problem in nonlinear systems. WSN is a collection of sensor nodes that communicate with each other wirelessly. These nodes are linked in a decentralised and distributed structure, allowing for efficient data collection and communication. Due to their decentralised architecture and limited resources, WSN is vulnerable to security risks, including malware attacks. Malware can attack sensor nodes, causing them to malfunction and consume more energy. These attacks can spread from one infected node to others in the network, making it essential to protect WSN against malware attacks. In this paper, we focus on the analysis of a novel fractional epidemiology model, specifically the fractional order SEIVR epidemic model in the sense of Caputo’s fractional derivative of order <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$0{ &lt; {\alpha }{\leq }}1$ </tex-math></inline-formula> with the goal of examining the efficacy of vaccination strategies and the heterogeneity of a scale-free network on epidemic spreading. First, using the next-generation technique and obtain the basic reproduction number of the proposed epidemic model, which is essential for determining both the locally asymptotically stable equilibrium point of the worm-free system and the unique existence of the endemic equilibrium point. To numerically solve the model, the Adam-Bashforth-Moulton predictor-corrector (ABM) method is applied. The fractional calculus enables us to deal directly with the “memory effect” of numerous phenomena, taking into account the system’s dependence on previous stages. This method provides the results of a complex system. Additionally, research demonstrates that vaccine treatments are quite effective at preventing the spread of malware. The outcome of the study reveals that the applied ABM predictor-corrector method is computationally strong and effective to analyse fractional order dynamical systems in the SEIVR epidemic model for malware propagation in WSN. The results show that the order of the fractional derivative has a significant effect on the dynamic process. Also, from the result, it is obvious that the memory effect is zero for <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${\alpha }$ </tex-math></inline-formula> = 1. When the fractional order <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${\alpha }$ </tex-math></inline-formula> is decreased from 1, the memory effect appears, and its dynamics vary according to the time. This memory effect points out the difference between derivatives of fractional and integer orders. The theorems and their proofs are presented to validate the validity of the proposed model. To validate the proposed model, extensive theoretical study and computational analysis have also been applied.

The calculation of fractional or integer order derivatives and integrals has been demonstrated to be simple and fast in the frequency domain. It is also the most sensible method if one wishes to calculate derivatives or integrals of periodic signals. In this paper, error analysis is carried out for the numerical algorithm for Weyl fractional derivatives. To derive an upper bound for the numerical error, some knowledge of the smoothness of the signal must be known in advance or it must be estimated. The derived error analysis is tested with sampled functions with known regularity and with real vibration measurements from rotating machines. Compared to previous publications which deal with error analysis of integer order numerical derivatives in the frequency domain using L2 errors, the result of this paper is in terms of maximum absolute error and it is based on a novel result on the signal's regularity. The general conclusion using either error estimates is the same: the error of numerical Weyl derivatives is bounded by some constant times the sequence length raised to a negative power. The exponent depends on the smoothness of the signal. This contrasts with using difference quotients in numerical differentiation, in which case the error is bounded by a constant times the sequence length raised to a some fixed negative power and the order of the method defines that exponent.

This chapter describes the basic concepts of partial differential equations (PDEs) based image modelling and their applications to image restoration. The general basic concepts of partial differential equation (PDE)-based image modelling and processing techniques are discussed for image restoration problems. These techniques can also be used in the design and development of efficient tools for various image processing and vision related tasks such as restoration, enhancement, segmentation, registration, inpainting, shape from shading, 3D reconstruction of objects from multiple views, and many more. As a case study, the topic in consideration is oriented towards image restoration using PDEs formalism since image restoration is considered to be an important pre-processing task for 3D surface geometry, reconstruction, and many other applications. An image may be subjected to various types of noises during its acquisition leading to degraded quality of the image, and hence, the noise must be reduced. The noise may be additive or multiplicative in nature. Here, the PDE-based models for removal of both types of noises are discussed. As examples, some PDE-based schemes have been implemented and their comparative study with other existing techniques has also been presented.

Topological derivative: A tool for image processing

Design and Implementation of Fractional Order Integral Filter for Denoising of Echocardiographic Images

Parameter identification of fractional order linear system based on Haar wavelet operational matrix

In this paper, a fractional order LQG benchmark is proposed for the control performance assessment of fractional order control systems. Similar to the conventional LQG benchmark, the fractional order LQG performance benchmark curve is determined by the numerical calculation method, which avoids the calculation of the complex interaction matrix. The fractional order process model is discretized via fractional order calculus. Meanwhile, the fractional order integral is introduced into the conventional LQG cost function. Then solving the linear quadratic Gaussian problem under the fractional order model and fractional control, the optimal input and output variances are determined for different weighting factors and the performance curves can be achieved. The comparison between fractional order LQG and the conventional LQG shows the improvement of the proposed benchmark under the same condition. The proposed benchmark can provide a more direct and superior reference standard to evaluate the performance of fractional order control system. Finally, a case study of fractional order PID(FO-PID) controller in industrial heating furnace temperature control experiment with model matching and model mismatch conditions is used to verify the effectiveness of the proposed benchmark.

Describes a study of two fuzzy based segmentation algorithms to model successfully ambiguities and uncertainties inherent in assigning an image pixel to an object of interest and to help choosing the threshold for binarizing images using the index of fuzziness and fuzzy compactness. The methods detect object contours to different degrees of certainty on the basis of the information provided by the user, who define a reference point inside the object around which is construct the contours. First, we obtain a connectedness or a similarity map related to the indicated reference point and then for the pixels surrounding that point, if their membership-values are high they are regarded as being inside the object. We also present the connectedness and similarity maps as a useful tool in image processing, so as to obtain better gradient images.

Directional filtering is commonly known as a technique for selective linear enhancement. In addition to this traditional utility, we present in this paper some examples demonstrating that it can be used as a multi-purpose tool in image processing, by adjusting its parameters. These examples show that: (1) applying directional filtering in conjugate directions simultaneously can lead to the enhancement of specific conjugate linear features on an image, which is useful in interpretation of geophysical and Landsat images, satellite and aerial photographs that contain many conjugate linear features; (2) directional filtering can be used as illuminating sources to produce shaded relief images that show 3D effect; by combining the directional filtered result with its original image, this 3D effect becomes much more profound; (3) the directional filtered result can be weighted by different factors, which allows some subtle features on an image to be identified.

Time fractional order partial derivative equations are the generalization of the traditional integer order partial derivative equations, in which the time fractional order derivative is used to replace the corresponding integer order derivative. In this paper, the variable order time fractional diffusion-wave equation is considered. Firstly, numerical approximation scheme has been proposed by using the method of the piecewise linear interpolation join with a second order approximation of the first order time derivative to discrete the Coimbra variable order time fractional derivative. Then, central difference is used to approximate the second order of the spatial fractional derivatives. Finally, a numerical example has been presented to verify the numerical technique, which shows the efficiency of the numerical method.

Pores are inherent to additively manufactured components and critical especially in technical components. Since they reduce the component's fatigue life, a reliable identification and description of pores is vital to ensure the component's performance. X‐ray computed tomography (CT) is an established and non‐destructive testing method to investigate internal defects. The CT scan process can induce noise and artefacts in the resulting images which afterwards have to be reduced through image processing. To reconstruct the internal defects of a component, the images need to be segmented in defect region and bulk material by applying a threshold. The application of the threshold as well as the previous image processing alter the geometry and size of the identified defects. This contribution aims to quantify the influence of selected commercial image processing and segmentation methods on identified pores in several additively manufactured components made of AlSi10Mg and Ti6Al4V as well as in an artificial CT scan. To that aim, gray value histograms and characteristic parameters thereof are compared for different image processing tools. After the segmentation of the processed images, particle characteristics are compared. The influence of image processing and segmentation on the predicted fatigue life of the material is evaluated through the change of the largest pore in each set of data applying Murakami's empirical ‐parameter model.

The fractional Hamiltonian analysis of 1+1 dimensional field theory is investigated and the fractional Ostrogradski’s formulation is obtained. The fractional path integral of both simple harmonic oscillator with an acceleration-squares part and a damped oscillator are analyzed. The classical results are obtained when fractional derivatives are replaced with the integer order derivatives.

Active contour model (ACM) approaches for image segmentation and feature extraction have emerged as very appealing and powerful tools in image processing. The basis of ACM approach is to evolve a curve, called level set, to extract the desired object (s) under some constraints. In this course, various extensions of earlier Osher's level set model have been suggested in the litareture. More recently, a new ACM model referred to selective binary and Gaussian filtering regularized level set (SBGFRIL) has been put forward as a fruitful combination of geodesic active contour model (GAC) and Chan-Vese (C-V) active contour models. This paper attempts to put forward some appealing performance indices to assess the performances of the suggested SBGFRIL compared with GAC and V-C models. The performance metrics involve the clustering based quality evaluations.

&lt;p&gt;This paper presents a new approach of fractional order sliding mode &lt;br /&gt;controllers (FOSMC) for a class of nonlinear systems which have a single input and two outputs (SITO). Firstly, two fractional order sliding surfaces S1 and S2 were proposed with an intermediate variable z transferred from S2 to S1 in order to hierarchy the two sliding surfaces. Secondly, a control law was determined in order to control the two outputs. A sliding control stability condition was obtained by using the properties of the fractional order calculus. Finally, the effectiveness and robustness of the proposed approach were demonstrated by comparing its performance with the one of the conventional sliding mode controller (SMC), which is based on integer order derivatives. Simulation results were provided for the cases of controlling a ball-beam and inverted pendulum systems.&lt;/p&gt;

The mathematical modeling of damping materials based on fractional calculus has been shown to be very effective in representing the frequency dependence of the properties of these materials. In this model, the integer order derivatives in the constitutive equations of the Kelvin model are replaced by derivatives of fractional order. In this paper, we examine the response of a single degree-of-freedom system in which the damping force is proportional to a derivative of order α &lt; 1 of the displacements. Three methods are proposed to obtain the response: the Laplace and Fourier transform methods, and an operator method that results in a series solution. Some interesting features exhibited by the oscillator’s response due to the fractional representation of the damping are unveiled.