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RIESZ POTENTIAL OPERATORS AND INVERSES VIA FRACTIONAL CENTRED DERIVATIVES Abstract
"... Fractional centred differences and derivatives definitions are proposed generalising to real orders the existing ones valid for even and odd positive integer orders. For each one, suitable integral formulations are obtained. The computations of the involved integrals lead to new generalisations of t ..."
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Fractional centred differences and derivatives definitions are proposed generalising to real orders the existing ones valid for even and odd positive integer orders. For each one, suitable integral formulations are obtained. The computations of the involved integrals lead to new generalisations of the Cauchy integral derivative. To compute this integral, a special two straight line path was used. With this the referred integrals lead to the well known Riesz potential operators and their inverses that emerge as true fractional centred derivatives, but that can be computed through summations similar to the well known GrünwaldLetnikov derivatives Mathematics Subject Classification: 26A33 Keywords: fractional centred difference; fractional centred derivative; GrünwaldLetnikov; generalized Cauchy derivative. 1.
1 On the Fractional Signals and Systems (Invited paper)
"... A look into Fractional Calculus and their applications from the Signal Processing point of view is done in this paper. A coherent approach to the fractional derivative is presented leading to notions that are, not only compatible with the classic, but constitute a true generalization. This means tha ..."
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A look into Fractional Calculus and their applications from the Signal Processing point of view is done in this paper. A coherent approach to the fractional derivative is presented leading to notions that are, not only compatible with the classic, but constitute a true generalization. This means that the classic are recovered when the fractional domain is left. This happens in particular with the impulse response and transfer function. An interesting feature of the systems is in the causality that the fractional derivative imposes. The main properties of the derivatives and their representations are presented. A brief and general study of the fractional linear systems is done, by showing how to compute the impulse, step and frequecy responses, how to test the stability and how to insert the initial conditions. The practical realization problem is focussed and it is shown how to perform the inputouput computations. Some Biomedical applications are described.
An Adaptive Strategy for the Restoration of Textured Images using Fractional Order
"... Abstract. Total variation regularization has good performance in noise removal and edge preservation but lacks in texture restoration. Here we present a texturepreserving strategy to restore images contaminated by blur and noise. According to a texture detection strategy, we apply spatially adaptive ..."
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Abstract. Total variation regularization has good performance in noise removal and edge preservation but lacks in texture restoration. Here we present a texturepreserving strategy to restore images contaminated by blur and noise. According to a texture detection strategy, we apply spatially adaptive fractional order diffusion. A fast algorithm based on the halfquadratic technique is used to minimize the resulting objective function. Numerical results show the effectiveness of our strategy. Key words: illposed problem, deblurring, fractional order derivatives, regularizing iterative method