An overview is presented of the medical image processing literature on mutual-information-based registration. The aim of the survey is threefold: an introduction for those new to the field, an overview for those working in the field, and a reference for those searching for literature on a specific application. Methods are classified according to the different aspects of mutual-information-based registration. The main division is in aspects of the methodology and of the application. The part on methodology describes choices made on facets such as preprocessing of images, gray value interpolation, optimization, adaptations to the mutual information measure, and different types of geometrical transformations. The part on applications is a reference of the literature available on different modalities, on interpatient registration and on different anatomical objects. Comparison studies including mutual information are also considered. The paper starts with a description of entropy and mutual information and it closes with a discussion on past achievements and some future challenges.

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Mutual-information-based registration of medical images: a survey

In the paper, we present a new definition of fractional deriva tive with a smooth kernel which takes on two different representations for the temporal and spatial variable. The first works on the time variables; thus it is suitable to use th e Laplace transform. The second definition is related to the spatial va riables, by a non-local fractional derivative, for which it is more convenient to work with the Fourier transform. The interest for this new approach with a regular kernel was born from the prospect that there is a class of non-local systems, which have the ability to descri be the material heterogeneities and the fluctuations of diff erent scales, which cannot be well described by classical local theories or by fractional models with singular kernel.

http://www.naturalspublishing.com/files/published/0gb83k287mo759.pdf

A new Definition of Fractional Derivative without Singular Kernel

Economics and Information Theory

The paper gives some results and improves the derivation of the fractional Taylor's series of nondifferentiable functions obtained recently in the form f (@g + h) = E"@a (h^@aD"@g^@a)f(@g), 0 @a @? 1, where E"@a is the Mittag-Leffier function. Here, one defines fractional derivative as the limit of fractional difference, and by this way one can circumvent the problem which arises with the definition of the fractional derivative of constant using Riemann-Liouville definition. As a result, a modified Riemann-Liouville definition is proposed, which is fully consistent with the fractional difference definition and avoids any reference to the derivative of order greater than the considered one's. In order to support this F-Taylor series, one shows how its first term can be obtained directly in the form of a mean value formula. The fractional derivative of the Dirac delta function is obtained together with the fractional Taylor's series of multivariate functions. The relation with irreversibility of time and symmetry breaking is exhibited, and to some extent, this F-Taylor's series generalizes the fractional mean value formula obtained a few years ago by Kolwantar.

Modified Riemann-Liouville derivative and fractional Taylor series of nondifferentiable functions further results

In this paper, we use the cumulative distribution of a random variable to define its information content and thereby develop an alternative measure of uncertainty that extends Shannon entropy to random variables with continuous distributions. We call this measure cumulative residual entropy (CRE). The salient features of CRE are as follows: 1) it is more general than the Shannon entropy in that its definition is valid in the continuous and discrete domains, 2) it possesses more general mathematical properties than the Shannon entropy, and 3) it can be easily computed from sample data and these computations asymptotically converge to the true values. The properties of CRE and a precise formula relating CRE and Shannon entropy are given in the paper. Finally, we present some applications of CRE to reliability engineering and computer vision.

Cumulative residual entropy: a new measure of information

Table of some basic fractional calculus formulae derived from a modified Riemann–Liouville derivative for non-differentiable functions

By using the new fractional Taylor’s series of fractional order f ( x + h ) = E α ( h α D x α ) f ( x ) where E α ( . ) denotes the Mittag–Leffler function, and D x α is the so-called modified Riemann–Liouville fractional derivative which we introduced recently to remove the effects of the non-zero initial value of the function under consideration, one can meaningfully consider a modeling of fractional stochastic differential equations as a fractional dynamics driven by a (usual) Gaussian white noise. One can then derive two new families of fractional Black–Scholes equations, and one shows how one can obtain their solutions. Merton’s optimal portfolio is once more considered and some new results are contributed, with respect to the modeling on one hand, and to the solution on the other hand. Finally, one makes some proposals to introduce real data and virtual data in the basic equation of stock exchange dynamics.

Guy Jumarie

Papers

Modified Riemann-Liouville derivative and fractional Taylor series of nondifferentiable functions further results

Table of some basic fractional calculus formulae derived from a modified Riemann–Liouville derivative for non-differentiable functions

Derivation and solutions of some fractional Black–Scholes equations in coarse-grained space and time. Application to Merton’s optimal portfolio

On the representation of fractional brownian motion as an integral with respect to (dt) a

Laplace’s transform of fractional order via the Mittag–Leffler function and modified Riemann–Liouville derivative