A fractional partial differential equation based multiscale denoising model for texture image

Traditional integer‐order partial differential equation based image denoising approach can easily lead edge and complex texture detail blur, thus its denoising effect for texture image is always not well. To solve the problem, we propose to implement a fractional partial differential equation (FPDE) based denoising model for texture image by applying a novel mathematical method—fractional calculus to image processing from the view of system evolution. Previous studies show that fractional calculus has some unique properties that it can nonlinearly enhance complex texture detail in digital image processing, which is obvious different with integer‐order differential calculus. The goal of the modeling is to overcome the problems of the existed denoising approaches by utilizing the aforementioned properties of fractional differential calculus. Using classic definition and property of fractional differential calculus, we extend integer‐order steepest descent approach to fractional field to implement fractional steepest descent approach. Then, based on the earlier fractional formulas, a FPDE based multiscale denoising model for texture image is proposed and further analyze optimal parameters value for FPDE based denoising model. The experimental results prove that the ability for preserving high‐frequency edge and complex texture information of the proposed fractional denoising model are obviously superior to traditional integral based algorithms, as for texture detail rich images. Copyright © 2013 John Wiley & Sons, Ltd.     

Nonsymmetric and symmetric fractional calculi on arbitrary nonempty closed sets

We introduce a nabla, a delta, and a symmetric fractional calculus on arbitrary nonempty closed subsets of the real numbers. These fractional calculi provide a study of differentiation and integration of non‐integer order on discrete, continuous, and hybrid settings. Main properties of the new fractional operators are investigated and some fundamental results presented, illustrating the interplay between discrete and continuous behaviors. Copyright © 2015 John Wiley & Sons, Ltd.     

Edge detection of noisy images based on cellular neural networks

This paper studies a technique employing both cellular neural networks (CNNs) and linear matrix inequality (LMI) for edge detection of noisy images. Our main work focuses on training templates of noise reduction and edge detection CNNs. Based on the Lyapunov stability theorem, we derive a criterion for global asymptotical stability of a unique equilibrium of the noise reduction CNN. Then we design an approach to train edge detection templates, and this approach can detect the edge precisely and efficiently, i.e., by only one iteration. Finally, we illustrate performance of the proposed methodology from the aspect of peak signal to noise ratio (PSNR) through computer simulations. Moreover, some comparisons are also given to prove that our method outperforms classical operators in gray image edge detection.     

On the Similarity between Volterra Operators and Transformation Operators for Integro-Differential Operators of Fractional Order

We establish the similarity between certain Volterra integral operators and the Riemann--Liouville fractional integration operator as well as the existence of a triangular transformation operator for integro-differential equations of fractional order. The results obtained are consistent with similar results for the case of integer order.     

A conformal mapping based fractional order approach for sub-optimal tuning of PID controllers with guaranteed dominant pole placement

A novel conformal mapping based fractional order (FO) methodology is developed in this paper for tuning existing classical (Integer Order) Proportional Integral Derivative (PID) controllers especially for sluggish and oscillatory second order systems. The conventional pole placement tuning via Linear Quadratic Regulator (LQR) method is extended for open loop oscillatory systems as well. The locations of the open loop zeros of a fractional order PID (FOPID or PI^λD^μ) controller have been approximated in this paper vis-à-vis a LQR tuned conventional integer order PID controller, to achieve equivalent integer order PID control system. This approach eases the implementation of analog/digital realization of a FOPID controller with its integer order counterpart along with the advantages of fractional order controller preserved. It is shown here in the paper that decrease in the integro-differential operators of the FOPID/PI^λD^μ controller pushes the open loop zeros of the equivalent PID controller towards greater damping regions which gives a trajectory of the controller zeros and dominant closed loop poles. This trajectory is termed as “M-curve”. This phenomena is used to design a two-stage tuning algorithm which reduces the existing PID controller’s effort in a significant manner compared to that with a single stage LQR based pole placement method at a desired closed loop damping and frequency.     

Fractional Nottale’s Scale Relativity and emergence of complexified gravity

Fractional calculus of variations has recently gained significance in studying weak dissipative and nonconservative dynamical systems ranging from classical mechanics to quantum field theories. In this paper, fractional Nottale’s Scale Relativity (NSR) for an arbitrary fractal dimension is introduced within the framework of fractional action-like variational approach recently introduced by the author. The formalism is based on fractional differential operators that generalize the differential operators of conventional NSR but that reduces to the standard formalism in the integer limit. Our main aim is to build the fractional setting for the NSR dynamical equations. Many interesting consequences arise, in particular the emergence of complexified gravity and complex time.     

Image restoration and edge detection by topological asymptotic expansion

A new method for edge detection and image restoration is proposed. This method is based on topological gradient approach. Experimental results obtained on noisy images illustrate the capabilities of this promising method in image processing. To cite this article: L.J. Belaid et al., C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

Zygmund-type estimates for fractional integration and differentiation operators of variable order

We consider non-standard generalized Hölder spaces of functions defined on a segment of the real axis, whose local continuity modulus has a majorant varying from point to point. We establish some properties of fractional integration operators of variable order acting from variable generalized Hölder spaces to those with a “better” majorant, as well as properties of fractional differentiation operators of variable order acting from the same spaces to those with a “worse” majorant.     

PIλDμ controller design for integer and fractional plants using piecewise orthogonal functions

This paper deals with the design of fractional PID controller for integer and fractional plants. A new analytic method is proposed, the developments are based on the expansion of the control loop signals as well as a chosen reference model input and output over a piecewise orthogonal functions, namely, Block pulse, Walsh and Haar wavelets. The generalized operational matrices of differentiation related to these bases which are fitting the Riemann–Liouville definition accurately are used to replace the fractional differential calculus by an algebraic one easier to solve. Thereafter, the controller tuning is elaborated simply with a matrix manipulation manner. At first, a least square is drawn to find only the controller gains, then a nonlinear function defined as a matrix norm is minimized to optimize the whole parameters. A variety of examples covering both integer and fractional systems and reference models are presented to show the validity of the technique.     

Existence of fractional differential chains and factorizations based on transformations

In this work, we deal with the existence of the fractional integrable equations involving two generalized symmetries compatible with nonlinear systems. The method used is based on the Bä cklund transformation or B‐transformation. Furthermore, we shall factorize the fractional heat operator in order to yield the fractional Riccati equation. This is done by utilizing matrix transform Miura type and matrix operators, that is, matrices whose entries are differential operators of fractional order. The fractional differential operator is taken in the sense of Riemann–Liouville calculus. Copyright © 2014 John Wiley & Sons, Ltd.     

Stabilizing periodic orbits of fractional order chaotic systems via linear feedback theory

In this paper stabilizing unstable periodic orbits (UPO) in a chaotic fractional order system is studied. Firstly, a technique for finding unstable periodic orbits in chaotic fractional order systems is stated. Then by applying this technique to the fractional van der Pol and fractional Duffing systems as two demonstrative examples, their unstable periodic orbits are found. After that, a method is presented for stabilization of the discovered UPOs based on the theories of stability of linear integer order and fractional order systems. Finally, based on the proposed idea a linear feedback controller is applied to the systems and simulations are done for demonstration of controller performance.     

How to impose physically coherent initial conditions to a fractional system?

In this paper, it is shown that neither Riemann–Liouville nor Caputo definitions for fractional differentiation can be used to take into account initial conditions in a convenient way from a physical point of view. This demonstration is done on a counter-example. Then the paper proposes a representation for fractional order systems that lead to a physically coherent initialization for the considered systems. This representation involves a classical linear integer system and a system described by a parabolic equation. It is thus also shown that fractional order systems are halfway between these two classes of systems, and are particularly suited for diffusion phenomena modelling.     

Fault detection based on fractional order models: Application to diagnosis of thermal systems

The aim of this paper is to propose diagnosis methods based on fractional order models and to validate their efficiency to detect faults occurring in thermal systems. Indeed, it is first shown that fractional operator allows to derive in a straightforward way fractional models for thermal phenomena. In order to apply classical diagnosis methods, such models could be approximated by integer order models, but at the expense of much higher involved parameters and reduced precision. Thus, two diagnosis methods initially developed for integer order models are here extended to handle fractional order models. The first one is the generalized dynamic parity space method and the second one is the Luenberger diagnosis observer. Proposed methods are then applied to a single-input multi-output thermal testing bench and demonstrate the methods efficiency for detecting faults affecting thermal systems.     

Detection of Subsurface Bubbles with Discrete Electromagnetic Geotomography

In electromagnetic geotomography, the inversion is very ill-posed due to the limita- tion of an angular range of projections, and consequently the reconstructed image is different than the true image. Nevertheless, the reconstruction can be much im- proved if we are interested only in a piece of information on the interior volume of the examined object, e.g. detection of some anomalies (bubbles). We assume that the image can be represented by binary values, hence: Discrete Electromagnetic Geotomography (DEG). In this paper, we discuss two discrete image reconstruction algorithms. The first one uses the D.C. programming to find the integer solution to the penalized least-squares problem. The penalty term is based on the Gibbs distribution. In the second one, the mean field annealing is applied to maximize the Gibbs-Boltzmann distribution associated with the penalized least-squares function. The numerical results are presented for noise-free and noisy data.     

A note on phase synchronization in coupled chaotic fractional order systems

The dynamic behaviors of fractional order systems have received increasing attention in recent years. This paper addresses the reliable phase synchronization problem between two coupled chaotic fractional order systems. An active nonlinear feedback control scheme is constructed to achieve phase synchronization between two coupled chaotic fractional order systems. We investigated the necessary conditions for fractional order Lorenz, Lü and Rössler systems to exhibit chaotic attractor similar to their integer order counterpart. Then, based on the stability results of fractional order systems, sufficient conditions for phase synchronization of the fractional models of Lorenz, Lü and Rössler systems are derived. The synchronization scheme that is simple and global enables synchronization of fractional order chaotic systems to be achieved without the computation of the conditional Lyapunov exponents. Numerical simulations are performed to assess the performance of the presented analysis.     

Controlling general projective synchronization of fractional order Rossler systems

This paper proposed a method to achieve general projective synchronization of two fractional order Rossler systems. First, we construct the fractional order Rossler system’s corresponding approximation integer order system. Then, a control method based on a partially linear decomposition and negative feedback of state errors was utilized on the integer order system. Numerical simulations show the effectiveness of the proposed method.     

Synchronization of integer order and fractional order Chua’s systems using robust observer

In this paper we present a fractional order Chua’s circuit that behaves chaotically based on the use of a fractional order low pass filter. Next, an integer order robust observer will be designed to synchronize the fractional order Chua’s circuit as well as integer order Chua’s circuit with unknown nonlinearity. This method consists in designing a Luenberger like observer appended with an estimator of the unknown nonlinear function. The estimator assumes that the nonlinear function is slowly varying and that the observer converges quickly and uses the backward difference formula to approximate the state derivative. The efficiency of the proposed method is confirmed using numerical simulations.     

Partial differential equations and finite-difference methods in image processing,part 1: Image representation

Stochastic representation of discrete images by partial differential equation operators is considered. It is shown that these representations can fit random images, with nonseparable, isotropic covariance functions, better than other common covariance models. Application of these models in image restoration, data compression, edge detection, image synthesis, etc., is possible.Different representations based on classification of partial differential equations are considered. Examples on different images show the advantages of using these representations. The previously introduced notion of fast Karhunen-Loeve transform is extended to images with nonseparable or nearly isotropic covariance functions, or both.     

Interpolation series for fractional derivatives and iterates of functions and operators

For a class of complex valued functions on the real line a fractional derivative is defined which is an entire function of exponential type of the order. It is shown that these derivatives can be found by a Newton interpolation series. For a class of linear operators, a fractional derivative for their resolvents also is defined. These fractional derivatives and the fractional iterates of these operators are related and both can be found by a Newton interpolation series on the nth-order iterates of the operators.