Theory Of Radial Basis Functions Research Articles

Multimodality 3D Tumor Segmentation in HCC Patients Treated with TACE

To validate the concordance of a semiautomated multimodality lesion segmentation technique between contrast-enhanced magnetic resonance imaging (CE-MRI), cone-beam computed tomography (CBCT), and multidetector CT (MDCT) in patients with hepatocellular carcinoma (HCC) treated with transarterial chemoembolization (TACE). This retrospective analysis included 45 patients with unresectable HCC who underwent baseline CE-MRI within 1 month before the treatment, intraprocedural CBCT during conventional TACE, and MDCT within 24 hours after TACE. Fourteen patients were excluded because of atypical lesion morphology, portal vein invasion, or small lesion size which precluded sufficient lesion visualization. Thirty-one patients with a total of 40 target lesions were included into the analysis. A tumor segmentation software, based on non-Euclidean geometry and theory of radial basis functions, was used to allow for the segmentation of target lesions in 3D on all three modalities. The algorithm created image-based masks located in a 3D region whose center and size were defined by the user, yielding the nomenclature "semiautomatic". On the basis of that, tumor volumes on all three modalities were calculated and compared using a linear regression model (R(2) values). Residual plots were used to analyze drift and variance of the values. The mean value of tumor volumes was 18.72 ± 19.13 cm(3) (range, 0.41-59.16 cm(3)) on CE-MRI, 21.26 ± 21.99 cm(3) (range, 0.62-86.82 cm(3)) on CBCT, and 19.88 ± 20.88 cm(3) (range, 0.45-75.24 cm(3)) on MDCT. The average volumes of the tumor were not significantly different between CE-MRI and DP-CBCT, DP-CBCT and MDCT, MDCT and CE-MRI (P = .577, .770, and .794, respectively). A strong correlation between volumes on CE-MRI and CBCT, CBCT and MDCT, MDCT and CE-MRI was observed (R(2) = 0.974, 0.992 and 0.983, respectively). When plotting the residuals, no drift was observed for all methods showing deviations of no >10% of absolute volumes (in cm(3)). A semiautomated 3D segmentation of HCC lesions treated with TACE provides high volumetric concordance across all tested imaging modalities.

1:36 PM, Abstract No. 120 - Multimodality 3D tumor segmentation in HCC patients treated with TACE

Rationale and Objectives To validate the concordance of a semiautomated multimodality lesion segmentation technique between contrast-enhanced magnetic resonance imaging (CE-MRI), cone-beam computed tomography (CBCT), and multidetector CT (MDCT) in patients with hepatocellular carcinoma (HCC) treated with transarterial chemoembolization (TACE). Materials and Methods This retrospective analysis included 45 patients with unresectable HCC who underwent baseline CE-MRI within 1 month before the treatment, intraprocedural CBCT during conventional TACE, and MDCT within 24 hours after TACE. Fourteen patients were excluded because of atypical lesion morphology, portal vein invasion, or small lesion size which precluded sufficient lesion visualization. Thirty-one patients with a total of 40 target lesions were included into the analysis. A tumor segmentation software, based on non-Euclidean geometry and theory of radial basis functions, was used to allow for the segmentation of target lesions in 3D on all three modalities. The algorithm created image-based masks located in a 3D region whose center and size were defined by the user, yielding the nomenclature “semiautomatic”. On the basis of that, tumor volumes on all three modalities were calculated and compared using a linear regression model ( R 2 values). Residual plots were used to analyze drift and variance of the values. Results The mean value of tumor volumes was 18.72 ± 19.13 cm 3 (range, 0.41–59.16 cm 3 ) on CE-MRI, 21.26 ± 21.99 cm 3 (range, 0.62–86.82 cm 3 ) on CBCT, and 19.88 ± 20.88 cm 3 (range, 0.45–75.24 cm 3 ) on MDCT. The average volumes of the tumor were not significantly different between CE-MRI and DP-CBCT, DP-CBCT and MDCT, MDCT and CE-MRI ( P = .577, .770, and .794, respectively). A strong correlation between volumes on CE-MRI and CBCT, CBCT and MDCT, MDCT and CE-MRI was observed ( R 2 = 0.974, 0.992 and 0.983, respectively). When plotting the residuals, no drift was observed for all methods showing deviations of no >10% of absolute volumes (in cm 3 ). Conclusions A semiautomated 3D segmentation of HCC lesions treated with TACE provides high volumetric concordance across all tested imaging modalities.

The Fourier Transform of the quartic Gaussian [formula omitted]: Hypergeometric functions, power series, steepest descent asymptotics and hyperasymptotics and extensions to [formula omitted

The Fourier Transform of a quartic Gaussian, Φ(k)≡∫-∞∞exp(ikx)exp(-x4)dx, is important in the theory of radial basis functions with ϕ(r)=exp(-r4). We show that the transform is the exact sum of two generalized hypergeometric functions 0F2. This implies that Φ(k) is an entire function and also gives an analytical form for the power series coefficients of arbitrary degree. Using the method of steepest descent for integrals, we derive the first five terms in an asymptotic series in powers of k-4/3. Through a simple hyperasymptotic analysis, we show that the magnitude of the error of the optimally-truncated series [“superasymptotic” error] is Oexp-0.818k4/3. The lowest order approximation, Φ(k)∼27/6π/3k-1/3exp(-(21/33/16)k4/3)cos((33/221/3/16)k4/3-π/6) shows that the transform Φ(k) is an exponentially-decaying oscillation.We show that the steepest descent method [2,27] (Bender and Orszag, 1978; Miller, 2006) [26,33] (Lauwerier, 1966; Ursell, 1965) can be applied to the transform of exp(-Ax2n) for general integer n and give the first two terms.Another theme is that modern computer technology has simplified the steepest descent method. The steepest descent paths can be plotted by merely displaying the contours of the imaginary part of the phase function Ψ(t) where zΨ(t) is the logarithm of the integrand, and z is the large parameter (here proportional to k4/3). The task of inverting w(t) to obtain a series in powers of w for the metric factor dt/dw after the usual change of integration variable can now be done to high order by a few lines of an algebraic manipulation language as illustrated by a short table with the complete Maple code.

The shape parameter in the Gaussian function

This is the fifth of our series of works about the shape parameter. We now explore the parameter β contained in the famous Gaussian function e − β | x | 2 , x ∈ R n . In the theory of radial basis functions (RBFs), the Gaussian is frequently used in virtue of its good error bound and numerical tractability. However, the optimal choice of β has been unknown. People conversant with RBFs know that β is very influential, but do not have a reliable criterion of its choice. The purpose of this paper is to uncover its mystery. In particular, we have greatly improved the result of Madych (1992) in [15], and we present a concrete function of β which shows the influence of β in the error estimate of Gaussian interpolation and with which the optimal β can always be found.

Smoothing splines using compactly supported, positive definite, radial basis functions

In this paper, we develop a fast algorithm for a smoothing spline estimator in multivariate regression. To accomplish this, we employ general concepts associated with roughness penalty methods in conjunction with the theory of radial basis functions and reproducing kernel Hilbert spaces. It is shown that through the use of compactly supported radial basis functions it becomes possible to recover the band structured matrix feature of univariate spline smoothing and thereby obtain a fast computational algorithm. Given n data points in R 2, the new algorithm has complexity O(n 2) compared to O(n 3), the order for the thin plate multivariate smoothing splines.

The crucial constants in the exponential-type error estimates for Gaussian interpolation

vb ≤ (Cd)c/d ∥f∥h where C,c are constants, h is the Gaussian function, s is the interpolating function, and d is called fill distance which, roughly speaking, measures the spacing of the points at which interpolation occurs. This error bound gets small very fast as d → 0. The constants C and c are very sensitive. A slight change of them will result in a huge change of the error bound. The number c can be calculated as shown in [9]. However, C cannot be calculated, or even approximated. This is a famous question in the theory of radial basis functions. The purpose of this paper is to answer this question.

Convergence and error estimates for meshless Galerkin methods

In this paper, the theory of radial basis functions together with Galerkin methods is used to deal with the partial differential equations with Dirichlet boundary conditions. After general description, the convergence and error estimates of this method are given in arbitrary dimensions. Some numerical examples are also given to illustrate the efficacious of the method.

A digital procedure for ground water recharge and discharge pattern recognition and rate estimation.

A digital procedure to estimate recharge/discharge rates that requires relatively short preparation time and uses readily available data was applied to a setting in central Wisconsin. The method requires only measurements of the water table, fluxes such as stream baseflows, bottom of the system, and hydraulic conductivity to delineate approximate recharge/discharge zones and to estimate rates. The method uses interpolation of the water table surface, recharge/discharge mapping, pattern recognition, and a parameter estimation model. The surface interpolator used is based on the theory of radial basis functions with thin-plate splines. The recharge/discharge mapping is based on a mass-balance calculation performed using MODFLOW. The results of the recharge/discharge mapping are critically dependent on the accuracy of the water table interpolation and the accuracy and number of water table measurements. The recharge pattern recognition is performed with the help of a graphical user interface (GUI) program based on several algorithms used in image processing. Pattern recognition is needed to identify the recharge/discharge zonations and zone the results of the mapping method. The parameter estimation program UCODE calculates the parameter values that provide a best fit between simulated heads and flows and calibration head-and-flow targets. A model of the Buena Vista Ground Water Basin in the Central Sand Plains of Wisconsin is used to demonstrate the procedure.

The Embedding Theory of Native Spaces

In the theory of radial basis functions, mathematicians use linear combinations of the translates of the radial basis functions as interpolants. The set of these linear combinations is a normed vector space. This space can be completed and become a Hilbert space, called native space, which is of great importance in the last decade. The native space then contains some abstract elements which are not linear combinations of radial basis functions. The meaning of these abstract elements is not fully known. This paper presents some interpretations for the these elements. The native spaces are embedded into some well-known spaces. For example, the Sobolev-space is shown to be a native space. Since many differential equations have solutions in the Sobolev-space, we can therefore approximate the solutions by linear combinations of radial basis functions. Moreover, the famous question of the embedding of the native space into L2(Ω) is also solved by the author.

The Equivalence Theory of Native Spaces

In this paper some equivalence definitions are given for native spaces which were introduced by Madych and Nelson and have become influential in the theory of radial basis functions. The abstract elements in native spaces are interpreted. Moreover, Weinrich and Iske's theories are unified.

A unified theory of radial basis functions: Native Hilbert spaces for radial basis functions II

This contribution provides a new formulation of the theory of radial basis functions in the context of integral operators. Instead of Fourier transforms, the most important tools now are expansions into eigenfunctions. This unifies the theory of radial basis functions in R d with the theory of zonal functions on the sphere S d−1 and the theory of kernel functions on Riemannian manifolds. New characterizations of native spaces and positive definite functions are included. The paper is a self-contained continuation of an earlier survey (R. Schaback, International Series of Numerical Mathematics, Vol. 132, Birkhäuser, Basel, 1999, pp. 255–282) over the native spaces associated to (not necessarily radial) basis functions.

Meshless Galerkin methods using radial basis functions

We combine the theory of radial basis functions with the field of Galerkin methods to solve partial differential equations. After a general description of the method we show convergence and derive error estimates for smooth problems in arbitrary dimensions.

Axisymmetric augmented thin plate splines

This paper reviews the previous axisymmetric global interpolation functions used in the context of the dual reciprocity boundary element method and the axisymmetric Laplace operator. It upgrades the previous heuristic attempts with the axisymmetric form of the augmented thin plate splines. This new approach, based on the theory of radial basis functions, gives more formal mathematical support to this class of problems. The basic equations are accompanied by a set of related expressions that permit straightforward use of the developed global interpolation functions in a broad spectrum of dual reciprocity boundary element methods like discrete approximative procedures.

Superfast autoconfiguring artificial neural networks and their application to power systems

Abstract This paper presents the new generation of artificial neural netoworks (ANNS) for solving the task of power system operation planning. Today the error back-propagation ANNs are used most because of their simplicity and the possibility of parallel implementation on neuro-computers for high-speed execution. In spite of their popularity they have two major drawbacks: the learning process is time consuming and there is no exact rule for setting the number of neurons to avoid overfitting or underfitting and to achieve, hopefully, a converging learning phase. To avoid these difficulties, a new generation of ANNs has been developed based on the theory of radial basis functions for approximations. A comparison test on an actual problem in power system operation was performed. The results show that this new algorithm is superior to back-propagation ANNs and optimal configured back-propagation ANNs achieved with genetic algorithms.

Automatic person recognition by acoustic and geometric features

This paper describes a multisensorial person-identification system in which visual and acoustic cues are used jointly for person identification. A simple approach, based on the fusion of the lists of scores produced independently by a speaker-recognition system and a face-recognition system, is presented. Experiments are reported that show that the integration of visual and acoustic information enhances both the performance and the reliability of the separate systems. Finally, two network architectures, based on radial basis-function theory, are proposed to describe integration at various levels of abstraction.

Initializing back propagation networks with prototypes

This paper addresses the problem of initializing the weights in back propagation networks with one hidden layer. The proposed method relies on the use of reference patterns, or prototypes, and on a transformation which maps each vector in the original feature space onto a unit-length vector in a space with one additional dimension. This scheme applies to pattern recognition tasks, as well as to the approximation of continuous functions. Issues related to the preprocessing of input patterns and to the generation of prototypes are discussed, and an algorithm for building appropriate prototypes in the continuous case is described. Also examined is the relationship between this approach and the theory of radial basis functions. Finally, simulation results are presented, showing that initializing back propagation networks with prototypes generally results in (a) drastic reductions in training time, (b) improved robustness against local minima, and (c) better generalization.

Weighted polynomials, radial basis functions and potentials on locally compact spaces

We discuss the classical notions of capacity, transfinite diameter, Chebyshev constant and equilibrium distributions in the context of locally compact spaces. We also demonstrate how the analogues of certain theorems in classical potential theory can be extended to this framework and applied to various problems in approximation theory. In particular, we explore the similarities between the theory of weighted polynomials and the theory of radial basis functions from a potential theoretic perspective.