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M H Loew - One of the best experts on this subject based on the ideXlab platform.
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discrimination of mr images of breast masses with fractal Interpolation Function models
Academic Radiology, 1999Co-Authors: Alan Penn, Lizann Bolinger, Mitchell D Schnall, M H LoewAbstract:Rationale and Objectives. The authors evaluated the feasibility of using statistical fractal-dimension features to improve discrimination between benign and malignant breast masses at magnetic resonance (MR) imaging. Materials and Methods. The study evaluated MR images of 32 malignant and 20 benign breast masses from archived data at the University of Pennsylvania Medical Center. The test set included four cases that were difficult to evaluate on the basis of border characteristics. All diagnoses had been confirmed at excisional biopsy. The fractal-dimension feature was computed as the mean of a sample space of fractal-dimension estimates derived from fractal Interpolation Function models. To evaluate the performance of the fractal-dimension feature, the classification effectiveness of five expert-observer architectural features was compared with that of the fractal dimension combined with four expert-observer features. Feature sets were evaluated with receiver operating characteristic analysis. Discrimination analysis used artificial neural networks and logistic regression. Robustness of the fractal-dimension feature was evaluated by determining changes in discrimination when the algorithm parameters were perturbed. Results. The combination of fractal-dimension and expert-observer features provided a statistically significant improvement in discrimination over that achieved with expert-observer features alone. Perturbing selected parameters in the fractal-dimension algorithm had little effect on discrimination. Conclusion. A statistical fractal-dimension feature appears to be useful in distinguishing MR images of benign and malignant breast masses in cases where expert radiologists may have difficulty. The statistical approach to estimating the fractal dimension appears to be more robust than other fractal measurements on data-limited medical images.
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estimating fractal dimension with fractal Interpolation Function models
IEEE Transactions on Medical Imaging, 1997Co-Authors: A I Penn, M H LoewAbstract:Fractal dimension (FD) is a feature which is widely used to characterize medical images. Previously, researchers have shown that FD separates important classes of images and provides distinctive information about texture. The authors analyze limitations of two principal methods of estimating FD: box-counting (BC) and power spectrum (PS). BC is ineffective when applied to data-limited, low-resolution images; PS is based on a fractional Brownian motion (fBm) model-a model which is not universally applicable. The authors also present background information on the use of fractal Interpolation Function (FIF) models to estimate FD of data which can be represented in the form of a Function. They present a new method of estimating FD in which multiple FIF models are constructed. The mean of the FD's of the FIF models is taken as the estimate of the FD of the original data. The standard deviation of the FD's of the FIF models is used as a confidence measure of the estimate. The authors demonstrate how the new method can be used to characterize fractal texture of medical images. In a pilot study, they generated plots of curvature values around the perimeters of images of red blood cells from normal and sickle cell subjects. The new method showed improved separation of the image classes when compared to BC and PS methods.
Nele Moelans - One of the best experts on this subject based on the ideXlab platform.
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a quantitative and thermodynamically consistent phase field Interpolation Function for multi phase systems
Acta Materialia, 2011Co-Authors: Nele MoelansAbstract:Abstract The aimed properties of the Interpolation Functions used in quantitative phase-field models for two-phase systems do not extend to multi-phase systems. Therefore, a new type of Interpolation Functions is introduced that has a zero slope at the equilibrium values of the non-conserved field variables representing the different phases and allows for a thermodynamically consistent Interpolation of the free energies. The Interpolation Functions are applicable for multi-phase-field and multi-order-parameter representations and can be combined with existing quantitative approaches for alloys. A model for polycrystalline, multi-component and multi-phase systems is formulated using the new Interpolation Functions that accounts in a straightforward way for composition-dependent expressions of the bulk Gibbs energies and diffusion mobilities, and interfacial free energies and mobilities. The numerical accuracy of the approach is analyzed for coarsening and diffusion-controlled parabolic growth in Cu–Sn systems as a Function of R /l, with R grain size and l diffuse interface width.
A K B Chand - One of the best experts on this subject based on the ideXlab platform.
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a new class of rational cubic spline fractal Interpolation Function and its constrained aspects
Applied Mathematics and Computation, 2019Co-Authors: S K Katiyar, A K B Chand, Saravana G KumarAbstract:Abstract This paper pertains to the area of shape preservation and sets a theoretical foundation for the applications of preserving constrained nature of a given constraining data in fractal Interpolation Functions (FIFs) techniques. We construct a new class of rational cubic spline FIFs (RCSFIFs) with a preassigned quadratic denominator with two shape parameters, which includes classical rational cubic interpolant [Appl. Math. Comp., 216 (2010), pp. 2036–2049] as special case and improves the sufficient conditions for positivity. Convergence analysis of RCSFIF to the original Function in C 1 is studied. In order to meet the needs of practical design or overcome the drawback of the tension effect in the proposed RCSFIFs, we improve our method by introducing a new tension parameter wi and construct a new class of rational cubic spline FIFs with three shape parameters. The scaling factors and shape parameters have a predictable adjusting role on the shape of curves. The elements of the rational iterated Function system in each subinterval are identified befittingly so that the graph of the resulting C 1 -rational cubic spline FIF constrained (i) within a prescribed rectangle (ii) above a prescribed straight line (iii) between two piecewise straight lines. These parameters include, in particular, conditions on the positivity of the C 1 -rational cubic spline FIF. Several numerical examples are presented to ascertain the correctness and usability of developed scheme and to suggest how these schemes outperform their classical counterparts.
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Bicubic partially blended rational fractal surface for a constrained Interpolation problem
Computational and Applied Mathematics, 2018Co-Authors: A K B Chand, P Viswanathan, N. VijenderAbstract:This paper investigates some univariate and bivariate constrained Interpolation problems using fractal Interpolation Functions. First, we obtain rational cubic fractal Interpolation Functions lying above a prescribed straight line. Using a transfinite Interpolation via blending Functions, we extend the properties of the univariate rational cubic fractal Interpolation Function to generate surfaces that lie above a plane. In particular, the constrained bivariate Interpolation discussed herein includes a method to construct fractal Interpolation surfaces that preserve positivity inherent in a prescribed data set. Uniform convergence of the bivariate fractal interpolant to the original Function which generates the data is proven.
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a mathcal c 1 rational cubic fractal Interpolation Function convergence and associated parameter identification problem
Acta Applicandae Mathematicae, 2015Co-Authors: Pragasam Viswanathan, A K B ChandAbstract:This paper introduces a rational Fractal Interpolation Function (FIF), in the sense that it is obtained using a rational cubic spline transformation involving two shape parameters, and investigates its applicability in some constrained Interpolation problems. We identify suitable values for the parameters of the corresponding Iterated Function System (IFS) so that it generates positive rational FIFs for a given set of positive data. Further, the problem of identifying the rational IFS parameters so as to ensure that its attractor (graph of the corresponding rational FIF) lies in a specified rectangle is also addressed. With the assumption that the data defining Function is continuously differentiable, an upper bound for the Interpolation error (with respect to the uniform norm) for the rational FIF is obtained. As a consequence, the uniform convergence of the rational FIF to the original Function as the norm of the partition tends to zero is proven.
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preserving convexity through rational cubic spline fractal Interpolation Function
Journal of Computational and Applied Mathematics, 2014Co-Authors: P Viswanathan, A K B Chand, Ravi P AgarwalAbstract:We propose a new type of C^1-rational cubic spline Fractal Interpolation Function (FIF) for convexity preserving univariate Interpolation. The associated Iterated Function System (IFS) involves rational Functions of the form P"n(x)Q"n(x), where P"n(x) are cubic polynomials determined through the Hermite Interpolation conditions of the FIF and Q"n(x) are preassigned quadratic polynomials with two shape parameters. The rational cubic spline FIF converges to the original Function @F as rapidly as the rth power of the mesh norm approaches to zero, provided @F^(^r^) is continuous for r=1 or 2 and certain mild conditions on the scaling factors are imposed. Furthermore, suitable values for the rational IFS parameters are identified so that the property of convexity carries from the data set to the rational cubic FIFs. In contrast to the classical non-recursive convexity preserving Interpolation schemes, the present fractal scheme is well suited for the approximation of a convex Function @F whose derivative is continuous but has varying irregularity.
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a constructive approach to cubic hermite fractal Interpolation Function and its constrained aspects
Bit Numerical Mathematics, 2013Co-Authors: A K B Chand, P ViswanathanAbstract:The theory of splines is a well studied topic, but the kinship of splines with fractals is novel. We introduce a simple explicit construction for a Open image in new window-cubic Hermite Fractal Interpolation Function (FIF). Under some suitable hypotheses on the original Function, we establish a priori estimates (with respect to the Lp-norm, 1≤p≤∞) for the Interpolation error of the Open image in new window-cubic Hermite FIF and its first derivative. Treating the first derivatives at the knots as free parameters, we derive suitable values for these parameters so that the resulting cubic FIF enjoys Open image in new window global smoothness. Consequently, our method offers an alternative to the standard moment construction of Open image in new window-cubic spline FIFs. Furthermore, we identify appropriate values for the scaling factors in each subinterval and the derivatives at the knots so that the graph of the resulting Open image in new window-cubic FIF lies within a prescribed rectangle. These parameters include, in particular, conditions for the positivity of the cubic FIF. Thus, in the current article, we initiate the study of the shape preserving aspects of fractal Interpolation polynomials. We also provide numerical examples to corroborate our results.
O Hyongchol - One of the best experts on this subject based on the ideXlab platform.
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construction of recurrent fractal Interpolation surfaces with Function scaling factors and estimation of box counting dimension on rectangular grids
Fractals, 2015Co-Authors: Huichol Choi, O HyongcholAbstract:We consider a construction of recurrent fractal Interpolation surfaces (RFISs) with Function vertical scaling factors and estimation of their box-counting dimension. A RFIS is an attractor of a recurrent iterated Function system (RIFS) which is a graph of bivariate Interpolation Function. For any given dataset on rectangular grids, we construct general RIFSs with Function vertical scaling factors and prove the existence of bivariate Functions whose graph are attractors of the above-constructed RIFSs. Finally, we estimate lower and upper bounds for the box-counting dimension of the constructed RFISs.
Huichol Choi - One of the best experts on this subject based on the ideXlab platform.
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construction of recurrent fractal Interpolation surfaces with Function scaling factors and estimation of box counting dimension on rectangular grids
Fractals, 2015Co-Authors: Huichol Choi, O HyongcholAbstract:We consider a construction of recurrent fractal Interpolation surfaces (RFISs) with Function vertical scaling factors and estimation of their box-counting dimension. A RFIS is an attractor of a recurrent iterated Function system (RIFS) which is a graph of bivariate Interpolation Function. For any given dataset on rectangular grids, we construct general RIFSs with Function vertical scaling factors and prove the existence of bivariate Functions whose graph are attractors of the above-constructed RIFSs. Finally, we estimate lower and upper bounds for the box-counting dimension of the constructed RFISs.
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construction of recurrent fractal Interpolation surfaces with Function scaling factors and estimation of box counting dimension on rectangular grids
arXiv: Dynamical Systems, 2013Co-Authors: Huichol Choi, Hyongchol OAbstract:We consider a construction of recurrent fractal Interpolation surfaces with Function vertical scaling factors and estimation of their box-counting dimension. A recurrent fractal Interpolation surface (RFIS) is an attractor of a recurrent iterated Function system (RIFS) which is a graph of bivariate Interpolation Function. For any given data set on rectangular grids, we construct general recurrent iterated Function systems with Function vertical scaling factors and prove the existence of bivariate Functions whose graph are attractors of the above constructed RIFSs. Finally, we estimate lower and upper bounds for the box-counting dimension of the constructed RFISs.
