closed form solutions for physically based shape modeling and recognition the authors present a closed form physically based solution for recovering a three dimensional 3 d solid model from collections of 3 d surface measurements given a sufficient number of independent measurements the solution is overconstrained and unique except for rotational symmetries the proposed approach is based on the finite element method fem and parametric solid modeling using implicit functions this approach provides both the convenience of parametric modeling and the expressiveness of the physically based mesh formulation and in addition can provide great accuracy at physical simulation a physically based object recognition method that allows simple closed form comparisons of recovered 3 d solid models is presented the performance of these methods is evaluated using both synthetic range data with various signal to noise ratios and using laser rangefinder data

shape analysis of elastic curves in euclidean spaces this paper introduces a square root velocity srv representation for analyzing shapes of curves in euclidean spaces under an elastic metric in this srv representation the elastic metric simplifies to the il2 metric the reparameterization group acts by isometries and the space of unit length curves becomes the unit sphere the shape space of closed curves is the quotient space of a submanifold of the unit sphere modulo rotation and reparameterization groups and we find geodesics in that space using a path straightening approach these geodesics and geodesic distances provide a framework for optimally matching deforming and comparing shapes these ideas are demonstrated using 1 shape analysis of cylindrical helices for studying protein structure 2 shape analysis of facial curves for recognizing faces 3 a wrapped probability distribution for capturing shapes of planar closed curves and 4 parallel transport of deformations for predicting shapes from novel poses

building 3 d statistical shape models by direct optimization statistical shape models are powerful tools for image interpretation and shape analysis a simple yet effective way of building such models is to capture the statistics of sampled point coordinates over a training set of example shapes however a major drawback of this approach is the need to establish a correspondence across the training set in 2 d a correspondence is often defined using a set of manually placed landmarks and linear interpolation to sample the shape in between such annotation is however time consuming and subjective particularly when extended to 3 d in this paper we show that it is possible to establish a dense correspondence across the whole training set automatically by treating correspondence as an optimization problem the objective function we use for the optimization is based on the minimum description length principle which we argue is a criterion that leads to models with good compactness specificity and generalization ability we manipulate correspondence by reparameterizing each training shape we describe an explicit representation of reparameterization for surfaces in 3 d that makes it impossible to generate an illegal i e not one to one correspondence we also describe several large scale optimization strategies for model building and perform a detailed analysis of each approach finally we derive quantitative measures of model quality allowing meaningful comparison between models built using different methods results are given for several different training sets of 3 d shapes which show that the minimum description length models perform significantly better than other approaches

numerical inversion of srnf maps for elastic shape analysis of genus zero surfaces recent developments in elastic shape analysis esa are motivated by the fact that it provides a comprehensive framework for simultaneous registration deformation and comparison of shapes these methods achieve computational efficiency using certain square root representations that transform invariant elastic metrics into euclidean metrics allowing for the application of standard algorithms and statistical tools for analyzing shapes of embeddings of mathbf s 2 in mathbb r 3 jermyn et al 1 introduced square root normal fields srnfs which transform an elastic metric with desirable invariant properties into the mathbb l 2 metric these srnfs are essentially surface normals scaled by square roots of infinitesimal area elements a critical need in shape analysis is a method for inverting solutions deformations averages modes of variations etc computed in srnf space back to the original surface space for visualizations and inferences due to the lack of theory for understanding srnf maps and their inverses we take a numerical approach and derive an efficient multiresolution algorithm based on solving an optimization problem in the surface space that estimates surfaces corresponding to given srnfs this solution is found to be effective even for complex shapes that undergo significant deformations including bending and stretching e g human bodies and animals we use this inversion for computing elastic shape deformations transferring deformations summarizing shapes and for finding modes of variability in a given collection while simultaneously registering the surfaces we demonstrate the proposed algorithms using a statistical analysis of human body shapes classification of generic surfaces and analysis of brain structures

elastic geodesic paths in shape space of parameterized surfaces this paper presents a novel riemannian framework for shape analysis of parameterized surfaces in particular it provides efficient algorithms for computing geodesic paths which in turn are important for comparing matching and deforming surfaces the novelty of this framework is that geodesics are invariant to the parameterizations of surfaces and other shape preserving transformations of surfaces the basic idea is to formulate a space of embedded surfaces surfaces seen as embeddings of a unit sphere in ir3 and impose a riemannian metric on it in such a way that the reparameterization group acts on this space by isometries under this framework we solve two optimization problems one given any two surfaces at arbitrary rotations and parameterizations we use a path straightening approach to find a geodesic path between them under the chosen metric second by modifying a technique presented in 25 we solve for the optimal rotation and parameterization registration between surfaces their combined solution provides an efficient mechanism for computing geodesic paths in shape spaces of parameterized surfaces we illustrate these ideas using examples from shape analysis of anatomical structures and other general surfaces

snakes active contour models a snake is an energy minimizing spline guided by external constraint forces and influenced by image forces that pull it toward features such as lines and edges snakes are active contour models they lock onto nearby edges localizing them accurately scale space continuation can be used to enlarge the capture region surrounding a feature snakes provide a unified account of a number of visual problems including detection of edges lines and subjective contours motion tracking and stereo matching we have used snakes successfully for interactive interpretation in which user imposed constraint forces guide the snake near features of interest

incorporating baseline measurements into the analysis of crossover trials with time to event endpoints two period two treatment 2 2 crossover designs are commonly used in clinical trials for continuous endpoints it has been shown that baseline pretreatment measurements collected before the start of each treatment period can be useful in improving the power of the analysis methods to achieve a corresponding gain for censored time to event endpoints have not been adequately studied we propose a method in which censored values are treated as missing data and multiply imputed using prespecified parametric event time models the event times in each imputed data set are then log transformed and analyzed using a linear model suitable for a 2 2 crossover design with continuous endpoints with the difference in period specific baselines included as a covariate results obtained from the imputed data sets are synthesized for point and confidence interval estimation of the treatment ratio of geometric mean event times using model averaging in conjunction with rubin s combination rule we use simulations to illustrate the favorable operating characteristics of our method relative to two other methods for crossover trials with censored time to event data ie a hierarchical rank test that ignores the baselines and a stratified cox model that uses each study subject as a stratum and includes period specific baselines as a covariate application to a real data example is provided

global stability of almost periodic solution of multispecies mutualism system with time delays and impulsive effects abstract this paper discusses an almost periodic multispecies lotka volterra mutualism system with time delays and impulsive effects by using the theory of comparison theorem and constructing a suitable lyapunov functional sufficient conditions which guarantee the existence and uniqueness and global asymptotical stability of almost periodic solution of this system are obtained the results of this paper is completed new an suitable example indicates the feasibility of the main results

the impact of baseline covariates on the efficiency of statistical analyses of crossover designs we investigate the impact of baseline covariates on the efficiency of statistical analyses of crossover designs for practical considerations we contemplate two different baseline methods study baselines and period dependent baselines for each baseline method we establish analytical upper bounds for the relative efficiency of a large class of crossover designs the totally balanced designs under a model with the baseline covariates as compared with the model without the baseline covariates we present numerical details based on these bounds for assorted scenarios and reveal implications of these results copyright 2012 john wiley sons ltd

asymptotic properties of the positive equilibrium of a discrete survival model this paper is concerned with the discrete wazewska and lasota model x n 1 x email protected n pe n x n k where email protected 0 1 n email protected 0 and email protected 0 1 2 we show that every positive solution of is persistent and then we provide sufficient conditions for oscillation of all solutions about x finally we give sufficient conditions for x to be stable or attractive

almost periodic solutions of lasota wazewska type delay differential equation sufficient conditions are derived for the existence of a globally attractive almost periodic solution of a single species model given by the nonautonomous lasota wazewska type delay differential equations x t txt ptfxt in which 0 p t 0 t are continuous almost periodic functions and f is a decreasing positive c1 function conditions of the uniform asymptotical stability for some associated nonautonomous delay differential equations x t txt ptxt are also discussed

periodic solution and almost periodic solution of impulsive lasota wazewska model with multiple time varying delays this paper deals with the impulsive lasota wazewska model with multiple time varying delays our results show that the system is uniformly persistent under some appropriate conditions the sufficient condition for global exponential stability of the system is given applying mawhin s continuation theorem of coincidence degree we prove that the periodic system has at least one strictly positive periodic solution by employing hull theory of impulsive almost periodic system the existence and uniqueness of strictly positive almost periodic solution of the almost periodic system is obtained two examples are provided to illustrate our results

optimal crossover designs in a model with self and mixed carryover effects we consider a variant of the usual model for crossover designs with carryover effects instead of assuming that the carryover effect of a treatment is the same regardless of the treatment in the next period the model assumes that the carryover effect of a treatment on itself is different from the carryover effect on other treatments for the traditional model optimal designs tend to have pairs of consecutive identical treatments for the model considered here they tend to avoid such pairs practitioners have long expressed reservations about designs that exhibit such pairs and about the traditional model the new model provides an attractive alternative that leads to appealing optimal designs

a geometric approach to visualization of variability in functional data abstractwe propose a new method for the construction and visualization of boxplot type displays for functional data we use a recent functional data analysis framework based on a representation of functions called square root slope functions to decompose observed variation in functional data into three main components amplitude phase and vertical translation we then construct separate displays for each component using the geometry and metric of each representation space based on a novel definition of the median the two quartiles and extreme observations the outlyingness of functional data is a very complex concept thus we propose to identify outliers based on any of the three main components after decomposition we provide a variety of visualization tools for the proposed boxplot type displays including surface plots we evaluate the proposed method using extensive simulations and then focus our attention on three real data applications including exploratory data analysis of sea surface tempe

computing large deformation metric mappings via geodesic flows of diffeomorphisms this paper examine the euler lagrange equations for the solution of the large deformation diffeomorphic metric mapping problem studied in dupuis et al 1998 and trouve 1995 in which two images i 0 i 1 are given and connected via the diffeomorphic change of coordinates i 0 1 i 1 where 1 is the end point at t 1 of curve t t 0 1 satisfying t v t t t 0 1 with 0 id the variational problem takes the form mathop arg text m in limits upsilon dot phi t upsilon t left dot phi right left int 0 1 left upsilon t right 2 text d t left i 0 circ phi 1 1 i 1 right l 2 2 right where v t v is an appropriate sobolev norm on the velocity field v t and the second term enforces matching of the images with l 2 representing the squared error norm in this paper we derive the euler lagrange equations characterizing the minimizing vector fields v t t 0 1 assuming sufficient smoothness of the norm to guarantee existence of solutions in the space of diffeomorphisms we describe the implementation of the euler equations using semi lagrangian method of computing particle flows and show the solutions for various examples as well we compute the metric distance on several anatomical configurations as measured by 0 1 v t v dt on the geodesic shortest paths

statistical shape analysis clustering learning and testing using a differential geometric treatment of planar shapes we present tools for 1 hierarchical clustering of imaged objects according to the shapes of their boundaries 2 learning of probability models for clusters of shapes and 3 testing of newly observed shapes under competing probability models clustering at any level of hierarchy is performed using a minimum variance type criterion and a markov process statistical means of clusters provide shapes to be clustered at the next higher level thus building a hierarchy of shapes using finite dimensional approximations of spaces tangent to the shape space at sample means we implicitly impose probability models on the shape space and results are illustrated via random sampling and classification hypothesis testing together hierarchical clustering and hypothesis testing provide an efficient framework for shape retrieval examples are presented using shapes and images from eth surrey and amcom databases

symmetry seeking models and 3d object reconstruction we propose models of 3d shape which may be viewed as deformable bodies composed of simulated elastic material in contrast to traditional purely geometric models of shape deformable models are active their shapes change in response to externally applied forces we develop a deformable model for 3d shape which has a preference for axial symmetry symmetry is represented even though the model does not belong to a parametric shape family such as generalized cylinders rather a symmetry seeking property is designed into internal forces that constrain the deformations of the model we develop a framework for 3d object reconstruction based on symmetry seeking models instances of these models are formed from monocular image data through the action of external forces derived from the data the forces proposed in this paper deform the model in space so that the shape of its projection into the image plane is consistent with the 2d silhouette of an object of interest the effectiveness of our approach is demonstrated using natural images

existence and global attractivity of positive periodic solution for an impulsive lasota wazewska model abstract sufficient conditions are obtained for the existence and global attractivity of periodic positive solution of an impulsive lasota wazewska model for the survival of red blood cells

positive almost periodic solution for a class of lasota wazewska model with multiple time varying delays in this paper we study the existence and global exponential convergence of positive almost periodic solutions for the generalized lasota wazewska model with multiple time varying delays under proper conditions we employ a novel proof to establish some criteria to ensure that all solutions of this model converge exponentially to a positive almost periodic solution moreover we give an example to illustrate our main results

mixture of probabilistic principal component analyzers for shapes from point sets inferring a probability density function pdf for shape from a population of point sets is a challenging problem the lack of point to point correspondences and the non linearity of the shape spaces undermine the linear models methods based on manifolds model the shape variations naturally however statistics are often limited to a single geodesic mean and an arbitrary number of variation modes we relax the manifold assumption and consider a piece wise linear form implementing a mixture of distinctive shape classes the pdf for point sets is defined hierarchically modeling a mixture of probabilistic principal component analyzers ppca in higher dimension a variational bayesian approach is designed for unsupervised learning of the posteriors of point set labels local variation modes and point correspondences by maximizing the model evidence the numbers of clusters modes of variations and points on the mean models are automatically selected using the predictive distribution we project a test shape to the spaces spanned by the local ppca s the method is applied to point sets from i synthetic data ii healthy versus pathological heart morphologies and iii lumbar vertebrae the proposed method selects models with expected numbers of clusters and variation modes achieving lower generalization specificity errors compared to state of the art

covariant image representation with applications to classification problems in medical imaging images are often considered as functions defined on the image domains and as functions their intensity values are usually considered to be invariant under the image domain transforms this functional viewpoint is both influential and prevalent and it provides the justification for comparing images using functional mathbf l p lp norms however with the advent of more advanced sensing technologies and data processing methods the definition and the variety of images has been broadened considerably and the long cherished functional paradigm for images is becoming inadequate and insufficient in this paper we introduce the formal notion of covariant images and study two types of covariant images that are important in medical image analysis symmetric positive definite tensor fields and gaussian mixture fields images whose sample values covary i e jointly vary with image domain transforms rather than being invariant to them we propose a novel similarity measure between a pair of covariant images considered as embedded shapes manifolds in the ambient space a cartesian product of the image and its sample value domains the similarity measure is based on matching the two embedded low dimensional shapes and both the extrinsic geometry of the ambient space and the intrinsic geometry of the shapes are incorporated in computing the similarity measure using this similarity as an affinity measure in a supervised learning framework we demonstrate its effectiveness on two challenging classification problems classification of brain mr images based on patients age and alzheimer s disease status and seizure detection from high angular resolution diffusion magnetic resonance scans of rat brains

model based deformable surface finding for medical images describes a new global shape parameterization for smoothly deformable three dimensional 3 d objects such as those found in biomedical images whose diversity and irregularity make them difficult to represent in terms of fixed features or parts this representation is used for geometric surface matching to 3 d medical image data such as from magnetic resonance imaging mri the parameterization decomposes the surface into sinusoidal basis functions four types of surfaces are modeled tori open surfaces closed surfaces and tubes this parameterization allows a wide variety of smooth surfaces to be described with a small number of parameters extrinsic model based information is incorporated by introducing prior probabilities on the parameters surface finding is formulated as an optimization problem results of the method applied to synthetic images and 3 d medical images of the heart and brain are presented

bayesian multiscale modeling of closed curves in point clouds modeling object boundaries based on image or point cloud data is frequently necessary in medical and scientific applications ranging from detecting tumor contours for targeted radiation therapy to the classification of organisms based on their structural information in low contrast images or sparse and noisy point clouds there is often insufficient data to recover local segments of the boundary in isolation thus it becomes critical to model the entire boundary in the form of a closed curve to achieve this we develop a bayesian hierarchical model that expresses highly diverse 2d objects in the form of closed curves the model is based on a novel multiscale deformation process by relating multiple objects through a hierarchical formulation we can successfully recover missing boundaries by borrowing structural information from similar objects at the appropriate scale furthermore the model s latent parameters help interpret the population indicating dimensions of significant structural variability

the analysis of data from 2 2 cross over trials with baseline measurements an account is given of the analysis of data from 2 x 2 cross over trials which include baseline measurements we show that most of the previously proposed methods can be incorporated into a general framework of least squares estimation with a simple linear model a simple analysis based on ordinary least squares estimators is described which can be used with either two sample t tests and confidence intervals or with the corresponding non parametric procedures it is shown how the use of generalized least squares estimators is equivalent to the use of covariance adjustment these methods require no assumptions about the covariance structure of the measurements from each subject the results of assessing the covariance structure present in examples of data from a number of trials are summarized these results suggest that previously proposed simple covariance structures are unlikely to be appropriate in general

parameterization invariant shape comparisons of anatomical surfaces we consider 3 d brain structures as continuous parameterized surfaces and present a metric for their comparisons that is invariant to the way they are parameterized past comparisons of such surfaces involve either volume deformations or non rigid matching under fixed parameterizations of surfaces we propose a new mathematical representation of surfaces called q maps such that l2 distances between such maps are invariant to re parameterizations this property allows for removing the parameterization variability by optimizing over the re parameterization group resulting in a proper parameterization invariant distance between shapes of surfaces we demonstrate this method in shape analysis of multiple brain structures for 34 subjects in the detroit fetal alcohol and drug exposure cohort study which results in a 91 classification rate for attention deficit hyperactivity disorder cases and controls this method outperforms some existing techniques such as spherical harmonic point distribution model spharm pdm or iterative closest point icp

a reparameterisation based approach to geodesic constrained solvers for curve matching we present a numerical algorithm for a new matching approach for parameterisation independent diffeomorphic registration of curves in the plane targeted at robust registration between curves that require large deformations this condition is particularly useful for the geodesic constrained approach in which the matching functional is minimised subject to the constraint that the evolving diffeomorphism satisfies the hamiltonian equations of motion this means that each iteration of the nonlinear optimisation algorithm produces a geodesic up to numerical discretisation we ensure that the computed solutions correspond to geodesics in the shape space by enforcing the horizontality condition conjugate momentum is normal to the curve explicitly introducing and solving for a reparameterisation variable allows the use of a point to point matching condition the equations are discretised using the variational particle mesh method we provide comprehensive numerical convergence tests and benchmark the algorithm in the context of large deformations to show that it is a viable efficient and accurate method for obtaining geodesics between curves

existence and global attractivity of unique positive periodic solution for a lasota wazewska model abstract in this paper we consider the lasota wazewska model x t a t x t i 1 m p i t e q i t x t i t by using a fixed point theorem some criteria are established for the existence of the unique positive periodic solution x of the above equation in particular we not only give the conclusion of convergence of x n to x where x n is a successive sequence but also show that x is a global attractor of all other positive solutions

improved power in crossover designs through linear combinations of baselines in a crossover design in the absence of any carryover effect including period specific baselines as covariates in an analysis of covariance is known to increase the precision of the estimated treatment effect the extent of the efficiency gain is a function of the joint covariance structure of the baselines and post treatment responses as well as the metric used to incorporate the baselines into the analysis here we show how the underlying covariance structure can be leveraged to find an optimal linear combination of baselines so as to minimize the theoretical variance of the analysis of covariance based estimated treatment effect our work is relevant to complete designs with up to four periods specifically the 2 2 3 3 and 4 4 given that the optimal linear combination of baselines is a function of the covariance structure which in practice is unknown we propose an adaptive method here the covariance structure is chosen using information criterion to guide the choice of the linear combination of baselines evaluation of the proposed approach suggests that the type i error rate is maintained moreover relative to previously published methods sizeable gains in power are possible with this method results from a 2 2 trial exploring renal function and a 3 3 trial with heart rate as the outcome are used to illustrate the methods copyright 2016 john wiley sons ltd

analysis of planar shapes using geodesic paths on shape spaces for analyzing shapes of planar closed curves we propose differential geometric representations of curves using their direction functions and curvature functions shapes are represented as elements of infinite dimensional spaces and their pairwise differences are quantified using the lengths of geodesics connecting them on these spaces we use a fourier basis to represent tangents to the shape spaces and then use a gradient based shooting method to solve for the tangent that connects any two shapes via a geodesic using the surrey fish database we demonstrate some applications of this approach 1 interpolation and extrapolations of shape changes 2 clustering of objects according to their shapes 3 statistics on shape spaces and 4 bayesian extraction of shapes in low quality images

optimal and efficient crossover designs when subject effects are random most studies on optimal crossover designs are based on models that assume subject effects to be fixed effects in this article we identify and study optimal and efficient designs for a model with random subject effects with the number of periods not exceeding the number of treatments we find that totally balanced designs are universally optimal for treatment effects in a large subclass of competing designs however in the entire class of designs totally balanced designs are in general not optimal and their efficiency depends on the ratio of the subject effects variance and the error variance we develop tools to study the efficiency of totally balanced designs and to identify designs with higher efficiency

existence and global attractivity of positive periodic solutions of a survival model of red blood cells in this paper we deal with a model for the survival of red blood cells with periodic coefficients x t email protected t x t p t e r t x t r t t 0 some new sufficient conditions are obtained for the existence and global attractivity of positive periodic solutions of the equation

cortical surface shape analysis based on spherical wavelets in vivo quantification of neuroanatomical shape variations is possible due to recent advances in medical imaging and has proven useful in the study of neuropathology and neurodevelopment in this paper we apply a spherical wavelet transformation to extract shape features of cortical surfaces reconstructed from magnetic resonance images mris of a set of subjects the spherical wavelet transformation can characterize the underlying functions in a local fashion in both space and frequency in contrast to spherical harmonics that have a global basis set we perform principal component analysis pca on these wavelet shape features to study patterns of shape variation within normal population from coarse to fine resolution in addition we study the development of cortical folding in newborns using the gompertz model in the wavelet domain which allows us to characterize the order of development of large scale and finer folding patterns independently given a limited amount of training data we use a regularization framework to estimate the parameters of the gompertz model to improve the prediction performance on new data we develop an efficient method to estimate this regularized gompertz model based on the broyden fletcher goldfarb shannon bfgs approximation promising results are presented using both pca and the folding development model in the wavelet domain the cortical folding development model provides quantitative anatomic information regarding macroscopic cortical folding development and may be of potential use as a biomarker for early diagnosis of neurologic deficits in newborns

elastic model based segmentation of 3 d neuroradiological data sets this paper presents a new technique for the automatic model based segmentation of three dimensional 3 d objects from volumetric image data the development closely follows the seminal work of taylor and cootes on active shape models but is based on a hierarchical parametric object description rather than a point distribution model the segmentation system includes both the building of statistical models and the automatic segmentation of new image data sets via a restricted elastic deformation of shape models geometric models are derived from a sample set of image data which have been segmented by experts the surfaces of these binary objects are converted into parametric surface representations which are normalized to get an invariant object centered coordinate system surface representations are expanded into series of spherical harmonics which provide parametric descriptions of object shapes it is shown that invariant object surface parametrization provides a good approximation to automatically determine object homology in terms of sets of corresponding sets of surface points gray level information near object boundaries is represented by 1 d intensity profiles normal to the surface considering automatic segmentation of brain structures as their driving application the authors choice of coordinates for object alignment was the well accepted stereotactic coordinate system major variation of object shapes around the mean shape also referred to as shape eigenmodes are calculated in shape parameter space rather than the feature space of point coordinates segmentation makes use of the object shape statistics by restricting possible elastic deformations into the range of the training shapes the mean shapes are initialized in a new data set by specifying the landmarks of the stereotactic coordinate system the model elastically deforms driven by the displacement forces across the object s surface which are generated by matching local intensity profiles elastical deformations are limited by setting bounds for the maximum variations in eigenmode space the technique has been applied to automatically segment left and right hippocampus thalamus putamen and globus pallidus from volumetric magnetic resonance scans taken from schizophrenia studies the results have been validated by comparison of automatic segmentation with the results obtained by interactive expert segmentation

periodicity and global dynamics of an impulsive delay lasota wazewska model abstract the periodicity of an impulsive delay lasota wazewska model is discussed sufficient and necessary condition for the existence of a positive periodic solution is established sufficient conditions for its global attractivity are also obtained via the methods of lyapunov function and comparison

impulsive differential equations this paper deals with the periodic solutions problem for impulsive differential equations by using lyapunov s second method and the contraction mapping principle some conditions ensuring the existence and global attractiveness of unique periodic solutions are derived which are given from impulsive control and impulsive perturbation points of view as an application the existence and global attractiveness of unique periodic solutions for hopfield neural networks are discussed finally two numerical examples are provided to demonstrate the effectiveness of the proposed results

boundary finding with parametrically deformable models segmentation using boundary finding is enhanced both by considering the boundary as a whole and by using model based global shape information the authors apply flexible constraints in the form of a probabilistic deformable model to the problem of segmenting natural 2 d objects whose diversity and irregularity of shape make them poorly represented in terms of fixed features or form the parametric model is based on the elliptic fourier decomposition of the boundary probability distributions on the parameters of the representation bias the model to a particular overall shape while allowing for deformations boundary finding is formulated as an optimization problem using a maximum a posteriori objective function results of the method applied to real and synthetic images are presented including an evaluation of the dependence of the method on prior information and image quality