a simple solution of the bratu problem a brief survey of the properties and different treatments of the one dimensional 1d and 2d bratu problems is presented different iterative treatments of the resulting nonlinear system of equations are discussed the finite difference treatment of the problem is considered nonstandard finite difference methods with a simple sinusoidal starting function having an appropriate amplitude are recommended bounds on the amplitude for yielding both lower and upper solutions are given

a modified perturbation solution to the one dimensional bratu problem abstract an approximate analytical solution to the one dimensional bratu boundary value problem is introduced in this paper the solution is based on some perturbation expansion methods the first step is taken from the linearized perturbation technique which was developed to solve initial value problem with a nonlinear term and no small parameter an artificial small parameter is embedded and the dependent variable can then be expanded in terms of this parameter however necessary modifications are introduced to implement some techniques to solve a boundary value problem in order to allow for different nonlinearities the current solution showed good convergence at any value of the bratu constant when compared with the exact lower branch of the solution to the one dimensional bratu problem also its recursive nature allows for more iterations and adding more correction terms to the final approximate solution which increases the accuracy

detection and segmentation of colonic polyps on implicit isosurfaces by second principal curvature flow today s computer aided detection systems for computed tomography colonography ctc enable automated detection and segmentation of colorectal polyps we present a paradigm shift by proposing a method that measures the amount of protrudedness of a candidate object in a scale adaptive fashion one of the main results is that the performance of the candidate detection depends only on one parameter the amount of protrusion additionally the method yields correct polyp segmentation without the need of an additional segmentation step the supervised pattern recognition involves a clear distinction between size related features and features related to shape or intensity a mahalanobis transformation of the latter facilitates ranking of the objects using a logistic classifier we evaluate two implementations of the method on 84 patients with a total of 57 polyps larger than or equal to 6 mm we obtained a performance of 95 sensitivity at four false positives per scan for polyps larger than or equal to 6 mm

a novel hybrid linear nonlinear classifier for two class classification theory algorithm and applications classifier design for a given classification task needs to take into consideration both the complexity of the classifier and the size of the dataset that is available for training the classifier with limited training data as often is the situation in computer aided diagnosis of medical images a classifier with simple structure e g a linear classifier is more robust and therefore preferred we propose a novel two class classifier which we call a hybrid linear nonlinear classifier hlnlc that involves two stages the input features are linearly combined to form a scalar variable in the first stage and then the likelihood ratio of the scalar variable is used as the decision variable for classification we first develop the theory of hlnlc by assuming that the feature data follow normal distributions we show that the commonly used fisher s linear discriminant function is generally not the optimal linear function in the first stage of the hlnlc we formulate an optimization problem to solve for the optimal linear function in the first stage of the hlnlc i e the linear function that maximizes the area under the receiver operating characteristic roc curve of the hlnlc for practical applications we propose a robust implementation of the hlnlc by making a loose assumption that the two class feature data arise from a pair of latent rather than explicit multivariate normal distributions the novel hybrid classifier fills a gap between linear discriminant analysis lda and quadratic discriminant analysis qda in the sense that both its theoretical performance and its complexity lie between those of the lda and those of the qda simulation studies show that the hybrid linear nonlinear classifier performs better than lda without increasing the classifier complexity accordingly with a finite number of training samples the hlnlc can perform better than that of the ideal observer due to its simplicity finally we demonstrate the application of the hlnlc in computer aided diagnosis of breast lesions in ultrasound images

equivalence of binormal likelihood ratio and bi chi squared roc curve models a basic assumption for a meaningful diagnostic decision variable is that there is a monotone relationship between it and its likelihood ratio this relationship however generally does not hold for a decision variable that results in a binormal receiver operating characteristic roc curve as a result roc curve estimation based on the assumption of a binormal roc curve model produces improper roc curves which have hooks are not concave over the entire domain and cross the chance line although in practice this improperness is usually not noticeable sometimes it is evident and problematic to avoid this problem metz and pan proposed basing roc curve estimation on the assumption of a binormal likelihood ratio binormal lr model which states that the decision variable is an increasing transformation of the likelihood ratio function of a random variable having normal conditional diseased and nondiseased distributions however their development is not easy to follow i show that the binormal lr model is equivalent to a bi chi squared model in the sense that the families of corresponding roc curves are the same the bi chi squared formulation provides an easier to follow development of the binormal lr roc curve and its properties in terms of well known distributions copyright 2015 john wiley sons ltd

using logistic regression procedures for estimating receiver operating characteristic curves summary estimation of a receiver operating characteristic roc curve is usually based either on a fully parametric model such as a normal model or on a fully nonparametric model in this paper we explore a semiparametric approach by assuming a density ratio model for disease and disease free densities this model has a natural connection with the logistic regression model the proposed semiparametric approach is more robust than a fully parametric approach and is more efficient than a fully nonparametric approach two real examples demonstrate that the roc curve estimated by our semiparametric method is much smoother than that estimated by the nonparametric method

a characterization of quadratic forms of type e6 e7 and e8 the goal of this article is to show that quadratic forms of type e6 e7 a nde8 are completely characterized by their even clifford algebras or equivalently by their discriminant and witt invariant 2002 elsevier science usa all rights reserved

iterative methods for large continuation problems abstract the computation of solution paths for continuation problems requires the solution of a sequence of nonlinear systems of equations each nonlinear system can be solved by computing the solution of a succession of linear systems of equations determined by jacobian matrices associated with the nonlinear system of equations points on the solution path where the jacobian matrix is singular are referred to as singular points and require special handling they may be turning points or bifurcation points in order to detect singular points it is essential to monitor the eigenvalues of smallest magnitude of the jacobian matrices generated as the solution path is traversed we describe iterative methods for the computation of solution paths for continuation problems so large that factorization of the jacobian matrices is infeasible or impractical the iterative methods simultaneously solve linear systems of equations determined by the jacobian matrices and compute a few eigenvalue eigenvector pairs associated with the eigenvalues of smallest magnitude of each jacobian a bordering algorithm with a pseudo arclength parametrization is applied in the vicinity of turning points to overcome the singularity of the jacobian a bifurcation perturbation strategy is used to compute solution paths at bifurcation points our iterative methods are based on the block lanczos algorithm and are applicable to problems with large symmetric jacobian matrices

the liouville bratu gelfand problem for radial operators abstract we determine precise existence and multiplicity results for radial solutions of the liouville bratu gelfand problem associated with a class of quasilinear radial operators which includes perturbations of k hessian and p laplace operators

semi parametric maximum likelihood estimates for roc curves of continuous scale tests in this paper we propose a new semi parametric maximum likelihood ml estimate of a receiver operating characteristic roc curve that satisfies the property of invariance of the roc curve and is easy to compute we show that our new estimator is sqrt n consistent and has an asymptotically normal distribution our extensive simulation studies show that the proposed method is efficient and robust finally we illustrate the application of the proposed estimator in a real data set

a new parametric method based on s distributions for computing receiver operating characteristic curves for continuous diagnostic tests receiver operating characteristic roc curves provides a method for evaluating the performance of a diagnostic test these curves represent the true positive ratio that is the true positives among those affected by the disease as a function of the false positive ratio that is the false positives among the healthy corresponding to each possible value of the diagnostic variable when the diagnostic variable is continuous the corresponding roc curve is also continuous however estimation of such curve through the analysis of sample data yields a step line unless some assumption is made on the underlying distribution of the considered variable since the actual distribution of the diagnostic test is seldom known it is difficult to select an appropriate distribution for practical use data transformation may offer a solution but also may introduce a distortion on the evaluation of the diagnostic test in this paper we show that the distribution family known as the s distribution can be used to solve this problem the s distribution is defined as a differential equation in which the dependent variable is the cumulative this special form provides a highly flexible family of distributions that can be used as models for unknown distributions it has been shown that classical statistical distributions can be represented accurately as s distributions and that they occur in a definite subspace of the parameter space corresponding to the whole s distribution family consequently many other distributional forms that do not correspond to known distributions are provided by the s distribution this property can be used to model observed data for unknown distributions and is very useful in constructing parametric roc curves in those cases after fitting an s distribution to the observed samples of diseased and healthy populations roc curve computation is straightforward a roc curve can be considered as the solution of a differential equation in which the dependent variable is the ratio of true positives and the independent variable is the ratio of false positives this equation can be easily obtained from the s distributions fitted to observed data using these results we can compute pointwise confidence bands for the roc curve and the corresponding area under the curve we shall compare this approach with the empirical and the binormal methods for estimating a roc curve to show that the s distribution based method is a useful parametric procedure

digital topology and geometry in medical imaging a survey digital topology and geometry refers to the use of topologic and geometric properties and features for images defined in digital grids such methods have been widely used in many medical imaging applications including image segmentation visualization manipulation interpolation registration surface tracking object representation correction quantitative morphometry etc digital topology and geometry play important roles in medical imaging research by enriching the scope of target outcomes and by adding strong theoretical foundations with enhanced stability fidelity and efficiency this paper presents a comprehensive yet compact survey on results principles and insights of methods related to digital topology and geometry with strong emphasis on understanding their roles in various medical imaging applications specifically this paper reviews methods related to distance analysis and path propagation connectivity surface tracking image segmentation boundary and centerline detection topology preservation and local topological properties skeletonization and object representation correction and quantitative morphometry a common thread among the topics reviewed in this paper is that their theory and algorithms use the principle of digital path connectivity path propagation and neighborhood analysis

new smoother to enhance multigrid based methods for bratu problem abstract in this paper we present more investigations of the numerical solution of the 2d bratu equation to obtain the second solution on the upper branch by multigrid classical smoothers such as gauss seidel and weighted jacobi have proven ineffective for obtaining the second solution due to the loss of diagonal dominance and the presence of indefinite jacobian system at some parameter values in this paper we modify the multigrid algorithms by adding and combining some krylov methods as smoothers to enhance the multigrid efficiency though the idea is not new but we could get new enhanced results compared to that presented by hackbusch w hackbusch comparison of different multi grid variants for nonlinear equations zamm z angew math mech 72 1992 148 151 and washio and oosterlee t washio c w oosterlee krylov subspace acceleration for nonlinear multigrid schemes electron trans numer anal 6 1997 271 290

a new construction of moufang quadrangles of type and in the classification of moufang polygons by j tits and r weiss the most intricate case is by far the case of the exceptional moufang quadrangles of type e6 e7 and e8 and in fact the construction that they present is ad hoc and lacking a deeper explanation we will show how tensor products of two composition algebras can be used to construct these moufang quadrangles in characteristic different from 2 as a byproduct we will obtain a method to construct any moufang quadrangle in characteristic different from two from a module for a jordan algebra

bayesian bootstrap estimation of roc curve receiver operating characteristic roc curve is widely applied in measuring discriminatory ability of diagnostic or prognostic tests this makes the roc analysis one of the most active research areas in medical statistics many parametric and semiparametric estimation methods have been proposed for estimating the roc curve and its functionals in this paper we propose the bayesian bootstrap bb a fully nonparametric estimation method for the roc curve and its functionals such as the area under the curve auc the bb method offers a bandwidth free smoothing approach to the empirical estimate and gives credible bounds the accuracy of the estimate of the roc curve in the simulation studies is examined by the integrated absolute error in comparison with other existing curve estimation methods the bb method performs well in terms of accuracy robustness and simplicity we also propose a procedure based on the bb approach to test the binormality assumption

letter to the editor on the integral solution of the one dimensional bratu problem a brief survey of the properties and different treatments of the one dimensional 1d planar bratu problem is presented the treatment using green s function is considered different iterative treatments of the resulting nonlinear system of equations are discussed starting the solution with simple sinusoidal function having appropriate amplitude is recommended then the resulting system is solved using a nonlinear conjugate gradient nl cg method the importance of the proper choice of the starting function as well as the stopping criterion of the iterative solution are stressed

three dimensional computer aided diagnosis scheme for detection of colonic polyps we have developed a three dimensional 3 d computer aided diagnosis scheme for automated detection of colonic polyps in computed tomography ct colonographic data sets and assessed its performance based on colonoscopy as the gold standard in this scheme a thick region encompassing the entire colonic wall is extracted from an isotropic volume reconstructed from the ct images in ct colonography polyp candidates are detected by first computing of 3 d geometric features that characterize polyps folds and colonic walls at each voxel in the extracted colon and then segmenting of connected components corresponding to suspicious regions by hysteresis thresholding based on these geometric features we apply fuzzy clustering to these connected components to obtain the polyp candidates false positive fp detections are then reduced by computation of several 3 d volumetric features characterizing the internal structures of the polyp candidates followed by the application of discriminant analysis to the feature space generated by these volumetric features the locations of the polyps detected by our computerized method were compared to the gold standard of conventional colonoscopy the performance was evaluated based on 43 clinical cases including 12 polyps determined by colonoscopy our computerized scheme was shown to have the potential to detect polyps in ct colonography with a clinically acceptable high sensitivity and a low fp rate

linearized perturbation technique and its applications to strongly nonlinear oscillators abstract a new perturbation technique called linearized perturbation method is proposed contrary to the traditional perturbation techniques the unperturbed equations is obtained by linearizing the original nonlinear equation not by setting 0 therefore the obtained results are valid not only for small parameter but also for very large values of the present theory is processed as simple as the straightforward expansion while omits the secular terms completely

bayesian multivariate hierarchical transformation models for roc analysis a bayesian multivariate hierarchical transformation model bmhtm is developed for receiver operating characteristic roc curve analysis based on clustered continuous diagnostic outcome data with covariates two special features of this model are that it incorporates non linear monotone transformations of the outcomes and that multiple correlated outcomes may be analysed the mean variance and transformation components are all modelled parametrically enabling a wide range of inferences the general framework is illustrated by focusing on two problems 1 analysis of the diagnostic accuracy of a covariate dependent univariate test outcome requiring a box cox transformation within each cluster to map the test outcomes to a common family of distributions 2 development of an optimal composite diagnostic test using multivariate clustered outcome data in the second problem the composite test is estimated using discriminant function analysis and compared to the test derived from logistic regression analysis where the gold standard is a binary outcome the proposed methodology is illustrated on prostate cancer biopsy data from a multi centre clinical trial

new perturbation iteration solutions for bratu type equations perturbation iteration theory is systematically generated for both linear and nonlinear second order differential equations and applied to bratu type equations different perturbation iteration algorithms depending upon the number of taylor expansion terms are proposed using the iteration formulas derived using different perturbation iteration algorithms new solutions of bratu type equations are obtained solutions constructed using different perturbation iteration algorithms are contrasted with each other as well as with numerical solutions it is found that algorithms with more taylor series expansion terms yield more accurate results

smooth semiparametric receiver operating characteristic curves for continuous diagnostic tests we propose a semiparametric kernel distribution function estimator based on which a new smooth semiparametric estimator of the receiver operating characteristic roc curve is constructed we derive the asymptotic bias and variance of the newly proposed distribution function estimator and show that it is more efficient than the traditional non parametric kernel distribution estimator we also derive the asymptotic bias and variance of our new roc curve estimator and show that it is more efficient than the smooth non parametric roc curve estimator proposed by zou et al stat med 1997 16 2143 2156 and lloyd j am stat assoc 1998 93 1356 1364 for our proposed estimators we derive data based methods for bandwidth selection in addition we present some results on the analysis of two real data sets finally a simulation study is presented to show that our estimators are better than the non parametric counterparts in terms of bias standard error and mean square error

efficient statistical tests to compare youden index accounting for contingency correlation youden index is widely utilized in studies evaluating accuracy of diagnostic tests and performance of predictive prognostic or risk models however both one and two independent sample tests on youden index have been derived ignoring the dependence association between sensitivity and specificity resulting in potentially misleading findings besides paired sample test on youden index is currently unavailable this article develops efficient statistical inference procedures for one sample independent and paired sample tests on youden index by accounting for contingency correlation namely associations between sensitivity and specificity and paired samples typically represented in contingency tables for one and two independent sample tests the variances are estimated by delta method and the statistical inference is based on the central limit theory which are then verified by bootstrap estimates for paired samples test we show that the estimated covariance of the two sensitivities and specificities can be represented as a function of kappa statistic so the test can be readily carried out we then show the remarkable accuracy of the estimated variance using a constrained optimization approach simulation is performed to evaluate the statistical properties of the derived tests the proposed approaches yield more stable type i errors at the nominal level and substantially higher power efficiency than does the original youden s approach therefore the simple explicit large sample solution performs very well because we can readily implement the asymptotic and exact bootstrap computation with common software like r the method is broadly applicable to the evaluation of diagnostic tests and model performance

from point to local neighborhood polyp detection in ct colonography using geodesic ring neighborhoods existing polyp detection methods rely heavily on curvature based characteristics to differentiate between lesions these assume that the discrete triangulated surface mesh or volume closely approximates a smooth continuous surface however this is often not the case and because curvature is computed as a local feature and a second order differential quantity the presence of noise significantly affects its estimation for this reason a more global feature is required to provide an accurate description of the surface at hand in this paper a novel method incorporating a local neighborhood around the centroid of a surface patch is proposed this is done using geodesic rings which accumulate curvature information in a neighborhood around this centroid this geodesic ring neighborhood approximates a single smooth continuous surface upon which curvature and orientation estimation methods can be applied a new global shape index s is also introduced and computed these curvature and orientation values will be used to classify the surface as either a bulbous polyp ridge like fold or semiplanar structure experimental results show that this method is promising 100 sensitivity 100 specificity for lesions 10 mm for distinguishing between bulbous polyps folds and planar like structures in the colon

an interpretation for the roc curve and inference using glm procedures the accuracy of a medical diagnostic test is often summarized in a receiver operating characteristic roc curve this paper puts forth an interpretation for each point on the roc curve as being a conditional probability of a test result from a random diseased subject exceeding that from a random nondiseased subject this interpretation gives rise to new methods for making inference about roc curves it is shown that inference can be achieved with binary regression techniques applied to indicator variables constructed from pairs of test results one component of the pair being from a diseased subject and the other from a nondiseased subject within the generalized linear model glm binary regression framework roc curves can be estimated and we highlight a new semiparametric estimator covariate effects can also be evaluated with the glm models the methodology is applied to a pancreatic cancer dataset where we use the regression framework to compare two different serum biomarkers asymptotic distribution theory is developed to facilitate inference and to provide insight into factors influencing variability of estimated model parameters

application of task based measures of image quality to optimization and evaluation of three dimensional reconstruction based compensation methods in myocardial perfusion spect in this paper we apply the channelized hotelling observer cho using a defect detection task to the optimization and evaluation of three dimensional iterative reconstruction based compensation methods for myocardial perfusion single photon emission computed tomography spect we used a population of 24 mathematical cardiac torso phantoms that realistically model the activity and attenuation distribution in three classes of patients females and males with flat diaphragms and raised diaphragms projection data were generated and subsequently reconstructed using methods based on the ordered subsets expectation maximization osem algorithm the methods evaluated included compensation for attenuation detector response blurring and scatter in various combinations we applied the cho to optimize the number of iterations for osem and the cutoff frequency and order of a three dimensional postreconstruction butterworth filter using the optimal parameters we then compared the compensation methods the index of comparison in these studies was the area under the receiver operating characteristics curve auc for the cho we found that attenuation compensation with either detector response or scatter compensation gave statistically significant increases in the auc compared to attenuation compensation alone the greatest increase in the auc occurred when all three compensations were applied these results indicate that compensation for detector response and scatter in addition to attenuation compensation will improve defect detectability in myocardial spect images

maximum likelihood estimation of receiver operating characteristic roc curves from continuously distributed data we show that truth state runs in rank ordered data constitute a natural categorization of continuously distributed test results for maximum likelihood ml estimation of roc curves on this basis we develop two new algorithms for fitting binormal roc curves to continuously distributed data a true ml algorithm labroc4 and a quasi ml algorithm labroc5 that requires substantially less computation with large data sets simulation studies indicate that both algorithms produce reliable estimates of the binormal roc curve parameters a and b the roc area index az and the standard errors of those estimates

cosmos a representation scheme for 3d free form objects we address the problem of representing and recognizing 3d free form objects when 1 the object viewpoint is arbitrary 2 the objects may vary in shape and complexity and 3 no restrictive assumptions are made about the types of surfaces on the object we assume that a range image of a scene is available containing a view of a rigid 3d object without occlusion we propose a new and general surface representation scheme for recognizing objects with free form sculpted surfaces in this scheme an object is described concisely in terms of maximal surface patches of constant shape index the maximal patches that represent the object are mapped onto the unit sphere via their orientations and aggregated via shape spectral functions properties such as surface area curvedness and connectivity which are required to capture local and global information are also built into the representation the scheme yields a meaningful and rich description useful for object recognition a novel concept the shape spectrum of an object is also introduced within the framework of cosmos for object view grouping and matching we demonstrate the generality and the effectiveness of our scheme using real range images of complex objects