the dirichlet problem for nonlinear second order elliptic equations iii functions of the eigenvalues of the hessian on etudie le probleme de dirichlet dans un domaine borne de r n a frontiere lisse f d 2 u dans u sur

an overdetermined problem for a weighted poisson s equation abstract in this paper we discuss an overdetermined problem for a weighted poisson s equation we prove that if there exists a solution of the weighted poisson s equation u log w u 1 on a smooth bounded domain with both dirichlet and neumann constant boundary condition and the weight function w satisfies some conditions in then is a ball we also study some applications of the overdetermined problems and some overdetermined problems with nonconstant neumann boundary condition

statistical models of partial volume effect statistical models of partial volume effect for systems with various types of noise or pixel value distributions are developed and probability density functions are derived the models assume either gaussian system sampling noise or intrinsic material variances with gaussian or poisson statistics in particular a material can be viewed as having a distinct value that has been corrupted by additive noise either before or after partial volume mixing or the material could have nondistinct values with a poisson distribution as might be the case in nuclear medicine images general forms of the probability density functions are presented for the n material cases and particular forms for two and three material cases are derived these models are incorporated into finite mixture densities in order to more accurately model the distribution of image pixel values examples are presented using simulated histograms to demonstrate the efficacy of the models for quantification modeling of partial volume effect is shown to be useful when one of the materials is present in images mainly as a pixel component

an evaluation of thresholding techniques in fmri analysis abstract this paper reviews and compares individual voxel wise thresholding methods for identifying active voxels in single subject fmri datasets different error rates are described which may be used to calibrate activation thresholds we discuss methods which control each of the error rates at a prespecified level including simple procedures which ignore spatial correlation among the test statistics as well as more elaborate ones which incorporate this correlation information the operating characteristics of the methods are shown through a simulation study indicating that the error rate used has an important impact on the sensitivity of the thresholding method but that accounting for correlation has little impact therefore the simple procedures described work well for thresholding most single subject fmri experiments and are recommended the methods are illustrated with a real bilateral finger tapping experiment

solutions to the overdetermined boundary problem for semilinear equations with position dependent nonlinearities abstract we show that a wide range of overdetermined boundary problems for semilinear equations with position dependent nonlinearities admits nontrivial solutions the result holds true both on r n and on compact riemannian manifolds as a byproduct of the proofs we also obtain some rigidity or partial symmetry results for solutions to overdetermined problems on riemannian manifolds of nonconstant curvature

serrin type overdetermined problems an alternative proof we prove the symmetry of solutions to overdetermined problems for a class of fully nonlinear equations namely the hessian equations in the case of the poisson equation our proof is alternative to the proofs proposed by serrin moving planes and by weinberger moreover our proof makes no direct use of the maximum principle while it sheds light on a relation between the serrin problem and the isoperimetric inequality

solutions of some monge ampere equations with isolated and line singularities in this paper we study existence regularity classification and asymptotic behaviors of solutions of some monge ampere equations with isolated and line singularities we classify all solutions of det 2u 1 in r n with one puncture point this can be applied to characterize ellipsoids in the same spirit of serrin s overdetermined problem for the laplace operator in the case of having k non removable singular points for k 1 modulo affine equivalence the set of all generalized solutions can be identified as an explicit orbifold we also establish existence of global solutions with general singular sets regularity properties and optimal estimates of the second order derivatives of generalized solutions near the singularity consisting of a point or a straight line the geometric motivation comes from singular semi flat calabi yau metrics

partial volume bayesian classification of material mixtures in mr volume data using voxel histograms the authors present a new algorithm for identifying the distribution of different material types in volumetric datasets such as those produced with magnetic resonance imaging mri or computed tomography ct because the authors allow for mixtures of materials and treat voxels as regions their technique reduces errors that other classification techniques can create along boundaries between materials and is particularly useful for creating accurate geometric models and renderings from volume data it also has the potential to make volume measurements more accurately and classifies noisy low resolution data well there are two unusual aspects to the authors approach first they assume that due to partial volume effects or blurring voxels can contain more than one material e g both muscle and fat the authors compute the relative proportion of each material in the voxels second they incorporate information from neighboring voxels into the classification process by reconstructing a continuous function spl rho x from the samples and then looking at the distribution of values that spl rho x takes on within the region of a voxel this distribution of values is represented by a histogram taken over the region of the voxel the mixture of materials that those values measure is identified within the voxel using a probabilistic bayesian approach that matches the histogram by finding the mixture of materials within each voxel most likely to have created the histogram the size of regions that the authors classify is chosen to match the sparing of the samples because the spacing is intrinsically related to the minimum feature size that the reconstructed continuous function can represent

convexity of solutions and brunn minkowski inequalities for hessian equations in r3 by using minkowski addition of convex functions we prove convexity and rearrangement properties of solutions to some hessian equations in r 3 and brunn minkowski and isoperimetric inequalities for related functionals

variational formula and overdetermined problem for the principle eigenvalue of k hessian operator abstract we first deduce the first variational formula and some overdetermined problems for the principle eigenvalue of the k hessian operator and then prove serrin type symmetry result for our overdetermined problems

a symmetry result for the torsion equation in an unbounded domain in 2d abstract we investigate an overdetermined problem for the torsion equation u 1 with two kinds of free boundary conditions in an unbounded domain and present a symmetry result in the case of 2d the proof is based on the maximum principle in the proof we use the sliding method and some convex arguments

comparison principles for infinity laplace equations in finsler metrics abstract in this paper we study comparison principles for normalized finsler infinity laplace operators with nonhomogeneous terms that depend on solutions and their gradients this is achieved by using a combination of sup convolution methods to approximate viscosity solutions by semiconvex functions and finite difference approximation schemes characterizations of viscosity subsolutions and supersolutions of equations with constant nonhomogeneous terms through comparison with quadratic finsler cone are also discussed

overdetermined elliptic problems in topological disks abstract we introduce a method based on the poincare hopf index theorem to classify solutions to overdetermined problems for fully nonlinear elliptic equations in domains diffeomorphic to a closed disk applications to some well known nonlinear elliptic pdes are provided our result can be seen as the analogue of hopf s uniqueness theorem for constant mean curvature spheres but for the general analytic context of overdetermined elliptic problems

unsupervised learning of finite mixture models this paper proposes an unsupervised algorithm for learning a finite mixture model from multivariate data the adjective unsupervised is justified by two properties of the algorithm 1 it is capable of selecting the number of components and 2 unlike the standard expectation maximization em algorithm it does not require careful initialization the proposed method also avoids another drawback of em for mixture fitting the possibility of convergence toward a singular estimate at the boundary of the parameter space the novelty of our approach is that we do not use a model selection criterion to choose one among a set of preestimated candidate models instead we seamlessly integrate estimation and model selection in a single algorithm our technique can be applied to any type of parametric mixture model for which it is possible to write an em algorithm in this paper we illustrate it with experiments involving gaussian mixtures these experiments testify for the good performance of our approach

a symmetry result for the torsion equation in an unbounded domain in 3d abstract we investigate an overdetermined problem for the torsion equation u 1 in an unbounded domain in r 3 under some additional conditions we show that the solution is a one dimensional quadratic function and that the boundary of the domain is a pair of parallel planes the proof is based on the maximum principle

segmentation of brain mr images through a hidden markov random field model and the expectation maximization algorithm the finite mixture fm model is the most commonly used model for statistical segmentation of brain magnetic resonance mr images because of its simple mathematical form and the piecewise constant nature of ideal brain mr images however being a histogram based model the fm has an intrinsic limitation no spatial information is taken into account this causes the fm model to work only on well defined images with low levels of noise unfortunately this is often not the the case due to artifacts such as partial volume effect and bias field distortion under these conditions fm model based methods produce unreliable results here the authors propose a novel hidden markov random field hmrf model which is a stochastic process generated by a mrf whose state sequence cannot be observed directly but which can be indirectly estimated through observations mathematically it can be shown that the fm model is a degenerate version of the hmrf model the advantage of the hmrf model derives from the way in which the spatial information is encoded through the mutual influences of neighboring sites although mrf modeling has been employed in mr image segmentation by other researchers most reported methods are limited to using mrf as a general prior in an fm model based approach to fit the hmrf model an em algorithm is used the authors show that by incorporating both the hmrf model and the em algorithm into a hmrf em framework an accurate and robust segmentation can be achieved more importantly the hmrf em framework can easily be combined with other techniques as an example the authors show how the bias field correction algorithm of guillemaud and brady 1997 can be incorporated into this framework to achieve a three dimensional fully automated approach for brain mr image segmentation

bayesian pixel classification using spatially variant finite mixtures and the generalized em algorithm a spatially variant finite mixture model is proposed for pixel labeling and image segmentation for the case of spatially varying mixtures of gaussian density functions with unknown means and variances an expectation maximization em algorithm is derived for maximum likelihood estimation of the pixel labels and the parameters of the mixture densities an a priori density function is formulated for the spatially variant mixture weights a generalized em algorithm for maximum a posteriori estimation of the pixel labels based upon these prior densities is derived this algorithm incorporates a variation of gradient projection in the maximization step and the resulting algorithm takes the form of grouped coordinate ascent gaussian densities have been used for simplicity but the algorithm can easily be modified to incorporate other appropriate models for the mixture model component densities the accuracy of the algorithm is quantitatively evaluated through monte carlo simulation and its performance is qualitatively assessed via experimental images from computerized tomography ct and magnetic resonance imaging mri

non negative matrix factorization with sparseness constraints non negative matrix factorization nmf is a recently developed technique for finding parts based linear representations of non negative data although it has successfully been applied in several applications it does not always result in parts based representations in this paper we show how explicitly incorporating the notion of sparseness improves the found decompositions additionally we provide complete matlab code both for standard nmf and for our extension our hope is that this will further the application of these methods to solving novel data analysis problems

classification of fmri time series in a low dimensional subspace with a spatial prior we propose a new method for detecting activation in functional magnetic resonance imaging fmri data we project the fmri time series on a low dimensional subspace spanned by wavelet packets in order to create projections that are as non gaussian as possible our approach achieves two goals it reduces the dimensionality of the problem by explicitly constructing a sparse approximation to the dataset and it also creates meaningful clusters allowing the separation of the activated regions from the clutter formed by the background time series we use a mixture of gaussian densities to model the distribution of the wavelet packet coefficients we expect activated areas that are connected and impose a spatial prior in the form of a markov random field our approach was validated with in vivo data and realistic synthetic data where it outperformed a linear model equipped with the knowledge of the true hemodynamic response

on the stability of the serrin problem abstract we investigate stability issues concerning the radial symmetry of solutions to serrin s overdetermined problems in particular we show that if u is a solution to u n in a smooth domain r n u 0 on and d u is close to 1 on then is close to the union of a certain number of disjoint unitary balls

skull stripping using graph cuts removal of non brain tissues particularly dura is an important step in enabling accurate measurement of brain structures many popular methods rely on iterative surface deformation to fit the brain boundary and tend to leave residual dura similar to other approaches the method proposed here uses intensity thresholding followed by removal of narrow connections to obtain a brain mask however instead of using morphological operations to remove narrow connections a graph theoretic image segmentation technique was used to position cuts that isolate and remove dura this approach performed well on both the standardized ibsr test data sets and empirically derived data compared to the hybrid watershed algorithm hwa segonne et al 2004 the novel approach achieved an additional 10 30 of dura removal without incurring further brain tissue erosion the proposed method is best used in conjunction with hwa as the errors produced by the two approaches often occur at different locations and cancel out when their masks are combined our experiments indicate that this combination can substantially decrease and often fully avoid cortical surface overestimation in subsequent segmentation

multimodal image coregistration and partitioning a unified framework this paper presents a method for the coregistration and partitioning i e tissue segmentation of brain images that have been acquired in different modalities the basic idea is that instead of matching two images directly one performs intermediate within modality registrations to two template images that are already in register one can use a least squares minimization to determine the affine transformations that map between the templates and the images by incorporating suitable constraints a rigid body transformation which directly maps between the images can be extracted from these more general affine transformations a further refinement capitalizes on the implicit normalization of both images into a standard space this facilitates segmentation or partitioning of both original images into homologous tissue classifications once partitioned the partitions can be jointly matched further increasing the accuracy of the coregistration in short these techniques reduce the between modality problem to a series of simpler within modality problems these methods are relatively robust address a number of problems in image transformations and require no manual intervention

a blind source separation technique using second order statistics separation of sources consists of recovering a set of signals of which only instantaneous linear mixtures are observed in many situations no a priori information on the mixing matrix is available the linear mixture should be blindly processed this typically occurs in narrowband array processing applications when the array manifold is unknown or distorted this paper introduces a new source separation technique exploiting the time coherence of the source signals in contrast with other previously reported techniques the proposed approach relies only on stationary second order statistics that are based on a joint diagonalization of a set of covariance matrices asymptotic performance analysis of this method is carried out some numerical simulations are provided to illustrate the effectiveness of the proposed method

over determined problems for k hessian equations in ring shaped domains abstract in this paper we firstly use a variant of the moving plane method of alexandroff to obtain radial symmetry of solutions for k hessian equations in annulus type domains which can be regarded as a generalization of gidas ni nirenberg result in 1979 then we consider an over determined problem for k hessian equations in ring shaped domains and prove the radial symmetry of the solutions and the domains

quantification of mr brain images by mixture density and partial volume modeling the problem of automatic quantification of brain tissue by utilizing single valued single echo magnetic resonance imaging mri brain scans is addressed it is shown that this problem can be solved without classification or segmentation a method that may be particularly useful in quantifying white matter lesions where the range of values associated with the lesions and the white matter may heavily overlap the general technique utilizes a statistical model of the noise and partial volume effect together with a finite mixture density description of the tissues the quantification is then formulated as a minimization problem of high order with up to six separate densities as part of the mixture this problem is solved by tree annealing with and without partial volume utilized the results compared and the sensitivity of the tree annealing algorithm to various parameters is exhibited the actual quantification is performed by two methods a classification based method called bayes quantification and parameter estimation results from each method are presented for synthetic and actual data

independent factor analysis we introduce the independent factor analysis ifa method for recovering independent hidden sources from their observed mixtures ifa generalizes and unifies ordinary factor analysis fa principal component analysis pca and independent component analysis ica and can handle not only square noiseless mixing but also the general case where the number of mixtures differs from the number of sources and the data are noisy ifa is a two step procedure in the first step the source densities mixing matrix and noise covariance are estimated from the observed data by maximum likelihood for this purpose we present an expectation maximization em algorithm which performs unsupervised learning of an associated probabilistic model of the mixing situation each source in our model is described by a mixture of gaussians thus all the probabilistic calculations can be performed analytically in the second step the sources are reconstructed from the observed data by an optimal nonlinear estimator a variational approximation of this algorithm is derived for cases with a large number of sources where the exact algorithm becomes intractable our ifa algorithm reduces to the one for ordinary fa when the sources become gaussian and to an em algorithm for pca in the zero noise limit we derive an additional em algorithm specifically for noiseless ifa this algorithm is shown to be superior to ica since it can learn arbitrary source densities from the data beyond blind separation ifa can be used for modeling multidimensional data by a highly constrained mixture of gaussians and as a tool for nonlinear signal encoding

component analysis approach to estimation of tissue intensity distributions of 3d images many segmentation algorithms in medical imaging rely on accurate modeling and estimation of tissue intensity probability density functions gaussian mixture modeling currently the most common approach has several drawbacks such as reliance on a gaussian model and iterative local optimization used to estimate the model parameters it also does not take advantage of substantially larger amount of data provided by 3d acquisitions which are becoming standard in clinical environment we propose a novel and completely non parametric algorithm to estimate the tissue intensity probabilities in 3d images instead of relying on traditional framework of iterating between classification and estimation we pose the problem as an instance of a blind source separation problem where the unknown distributions are treated as sources and histograms of image subvolumes as mixtures the new approach performed well on synthetic data and real magnetic resonance imaging mri scans of the brain robustly capturing intensity distributions of even small image structures and partial volume voxels

discrete serrin s problem we consider here the discrete analogue of serrin s problem if the equilibrium measure of a network with boundary satisfies that its normal derivative is constant what can be said about the structure of the network and the symmetry of the equilibrium measure in the original serrin s problem the conclusion is that the domain is a ball and the solution is radial to study the discrete serrin s problem we first introduce the notion of radial function and then prove a generalization of the minimum principle which is one of the main tools in the continuous case moreover we obtain similar results to those of the continuous case for some families of networks with a ball like structure which include spider networks with radial conductances distance regular graphs or more generally regular layered networks

a note on an overdetermined system involving mean curvature abstract let be a bounded simply connected domain in r n with connected boundary of class c 2 we show that if there exist n functions satisfying some overdetermined system with boundary conditions involving mean curvature of then is a ball the proof is based on the maximum principle and serrin s theorem

magnetic resonance image tissue classification using a partial volume model abstract we describe a sequence of low level operations to isolate and classify brain tissue within t1 weighted magnetic resonance images mri our method first removes nonbrain tissue using a combination of anisotropic diffusion filtering edge detection and mathematical morphology we compensate for image nonuniformities due to magnetic field inhomogeneities by fitting a tricubic b spline gain field to local estimates of the image nonuniformity spaced throughout the mri volume the local estimates are computed by fitting a partial volume tissue measurement model to histograms of neighborhoods about each estimate point the measurement model uses mean tissue intensity and noise variance values computed from the global image and a multiplicative bias parameter that is estimated for each region during the histogram fit voxels in the intensity normalized image are then classified into six tissue types using a maximum a posteriori classifier this classifier combines the partial volume tissue measurement model with a gibbs prior that models the spatial properties of the brain we validate each stage of our algorithm on real and phantom data using data from the 20 normal mri brain data sets of the internet brain segmentation repository our method achieved average indices of 0 746 0 114 for gray matter gm and 0 798 0 089 for white matter wm compared to expert labeled data our method achieved average indices 0 893 0 041 for gm and 0 928 0 039 for wm compared to the ground truth labeling on 12 volumes from the montreal neurological institute s brainweb phantom

nonlinear blind equalization schemes using complex valued multilayer feedforward neural networks among the useful blind equalization algorithms stochastic gradient iterative equalization schemes are based on minimizing a nonconvex and nonlinear cost function however as they use a linear fir filter with a convex decision region their residual estimation error is high in the paper four nonlinear blind equalization schemes that employ a complex valued multilayer perceptron instead of the linear filter are proposed and their learning algorithms are derived after the important properties that a suitable complex valued activation function must possess are discussed a new complex valued activation function is developed for the proposed schemes to deal with qam signals of any constellation sizes it has been further proven that by the nonlinear transformation of the proposed function the correlation coefficient between the real and imaginary parts of input data decreases when they are jointly gaussian random variables last the effectiveness of the proposed schemes is verified in terms of initial convergence speed and mse in the steady state in particular even without carrier phase tracking procedure the proposed schemes correct an arbitrary phase rotation caused by channel distortion