Complex elliptical distributions with application to shape analysis

We introduce a general class of complex elliptical distributions on a complex sphere that includes many of the most commonly used distributions, like the complex Watson, Bingham, angular central Gaussian and several others. We study properties of this family of distributions and apply the distribution theory for modeling shapes in two dimensions. We develop maximum likelihood and Bayesian methods of estimation to describe shape and obtain confidence bounds and credible regions for shapes. The methodology is illustrated through an example where estimation of shape of mouse vertebrae is desired.     

Extreme shape analysis

Summary.  We consider the analysis of extreme shapes rather than the more usual mean- and variance-based shape analysis. In particular, we consider extreme shape analysis in two applications: human muscle fibre images, where we compare healthy and diseased muscles, and temporal sequences of DNA shapes from molecular dynamics simulations. One feature of the shape space is that it is bounded, so we consider estimators which use prior knowledge of the upper bound when present. Peaks-over-threshold methods and maximum-likelihood-based inference are used. We introduce fixed end point and constrained maximum likelihood estimators, and we discuss their asymptotic properties for large samples. It is shown that in some cases the constrained estimators have half the mean-square error of the unconstrained maximum likelihood estimators. The new estimators are applied to the muscle and DNA data, and practical conclusions are given.     

Membranes,mitochondria and amoebae: shape models

Most real-world shapes and images are characterized by high variability- they are not rigid, like crystals, for example—but they are strongly structured. Therefore, a fundamental task in the understanding and analysis of such image ensembles is the construction of models that incorporate both variability and structure in a mathematically precise way. The global shape models introduced in Grenander's general pattern theory are intended to do this. In this paper, we describe the representation of two-dimensional mitochondria and membranes in electron microscope photographs, and three-dimensional amoebae in optical sectioning microscopy. There are three kinds of variability to all of these patterns, which these representations accommodate. The first is the variability in shape and viewing orientation. For this, the typical structure is represented via linear, circular and spherical templates, with the variability accomodated via the application of transformations applied to the templates. The transformations form groups: scale, rotation and translation. They are locally applied throughout the continuum and of high dimension. The second is the textural variability; the inside and outside of these basic shapes are subject to random variation, as well as sensor noise. For this, statistical sensor models and Markov random field texture models are used to connect the constituent structures of the shapes to the measured data. The third variability type is associated with the fact that each scene is made up of a variable number of shapes; this number is not assumed to be known a priori. Each scene has a variable number of parameters encoding the transformations of the templates appropriate for that scene. For this, a single posterior distribution is defined over the countable union of spaces representing models of varying numbers of shapes. Bayesian inference is performed via computation of the conditional expectation of the parametrically defined shapes under the posterior. These conditional mean estimates are generated using jump-diffusion processes. Results for membranes, mitochondria and amoebae are shown.     

Membranes, mitochondria and amoebae: shape models

Most real-world shapes and images are characterized by high variability- they are not rigid, like crystals, for example—but they are strongly structured. Therefore, a fundamental task in the understanding and analysis of such image ensembles is the construction of models that incorporate both variability and structure in a mathematically precise way. The global shape models introduced in Grenander's general pattern theory are intended to do this. In this paper, we describe the representation of two-dimensional mitochondria and membranes in electron microscope photographs, and three-dimensional amoebae in optical sectioning microscopy. There are three kinds of variability to all of these patterns, which these representations accommodate. The first is the variability in shape and viewing orientation. For this, the typical structure is represented via linear, circular and spherical templates, with the variability accomodated via the application of transformations applied to the templates. The transformations form groups: scale, rotation and translation. They are locally applied throughout the continuum and of high dimension. The second is the textural variability; the inside and outside of these basic shapes are subject to random variation, as well as sensor noise. For this, statistical sensor models and Markov random field texture models are used to connect the constituent structures of the shapes to the measured data. The third variability type is associated with the fact that each scene is made up of a variable number of shapes; this number is not assumed to be known a priori. Each scene has a variable number of parameters encoding the transformations of the templates appropriate for that scene. For this, a single posterior distribution is defined over the countable union of spaces representing models of varying numbers of shapes. Bayesian inference is performed via computation of the conditional expectation of the parametrically defined shapes under the posterior. These conditional mean estimates are generated using jump-diffusion processes. Results for membranes, mitochondria and amoebae are shown.     

The complex Watson distribution and shape analysis

The complex Watson distribution is an important simple distribution on the complex sphere which is used in statistical shape analysis. We describe the density, obtain the integrating constant and provide large sample approximations. Maximum likelihood estimation and hypothesis testing procedures for one and two samples are described. The particular connection with shape analysis is discussed and we consider an application examining shape differences between normal and schizophrenic brains. We make some observations about Bayesian shape inference and finally we describe a more general rotationally symmetric family of distributions.     

Transformation- and label-invariant neural network for the classification of landmark data

One method of expressing coarse information about the shape of an object is to describe the shape by its landmarks, which can be taken as meaningful points on the outline of an object. We consider a situation in which we want to classify shapes into known populations based on their landmarks, invariant to the location, scale and rotation of the shapes. A neural network method for transformation-invariant classification of landmark data is presented. The method is compared with the (non-transformation-invariant) complex Bingham rule; the two techniques are tested on two sets of simulated data, and on data that arise from mice vertebrae. Despite the obvious advantage of the complex Bingham rule because of information about rotation, the neural network method compares favourably.     

Statistical theory of shape under elliptical models via QR decomposition

The statistical shape theory via QR decomposition and based on Gaussian and isotropic models is extended in this paper to the families of non-isotropic elliptical distributions. The new shape distributions are easily computable and then the inference procedure can be studied with the resulting exact densities. An application in Biology is studied under two Kotz models, the best distribution (non-Gaussian) is selected by using a modified Bayesian information criterion (BIC)*.     

Bayesian estimation on semiparametric models with shape restricted B-splines

In this paper, we develop a Bayesian estimation procedure for semiparametric models under shape constrains. The approach uses a hierarchical Bayes framework and characterizations of shape-constrained B-splines. We employ Markov chain Monte Carlo methods for model fitting, using a truncated normal distribution as the prior for the coefficients of basis functions to ensure the desired shape constraints. The small sample properties of the function estimators are provided via simulation and compared with existing methods. A real data analysis is conducted to illustrate the application of the proposed method.     

Testing for brain anomalies: A hippocampus study

A mathematical classification method is presented to show how numerical tests for abnormal anatomical shape change can be used to study geometrical shape changes of the hippocampus in relation to the occurrence of schizophrenia. The method uses the well-known best Bayesian decision rule for two simple hypotheses. Furthermore, the technique is illustrated by applying the hypothesis testing method to some preliminary hippocampal data. The data pool available for the experiment consisted of 10 subjects, five of whom were diagnosed with schizophrenia and five of whom were not schizophrenics. Even though the information used in the experiment is limited and the number of subjects is relatively small, we are confident that the mathematical classification method presented is of significance and can be used successfully, given proper data, as a diagnostic tool.     

Directional statistics and shape analysis

This paper highlights distributional connections between directional statistics and shape analysis. In particular, we provide a test of uniformity for highly dispersed shapes, using the standard techniques of directional statistics. We exploit the isometric transformation from triangular shapes to a sphere in three dimensions, to provide a rich class of shape distributions. A link between the Fisher distribution and the complex Bingham distribution is re-examined. Some extensions to higher-dimensional shapes are outlined.     

Bayesian regression on non-parametric mixed-effect models with shape-restricted Bernstein polynomials

We develop a Bayesian estimation method to non-parametric mixed-effect models under shape-constrains. The approach uses a hierarchical Bayesian framework and characterizations of shape-constrained Bernstein polynomials (BPs). We employ Markov chain Monte Carlo methods for model fitting, using a truncated normal distribution as the prior for the coefficients of BPs to ensure the desired shape constraints. The small sample properties of the Bayesian shape-constrained estimators across a range of functions are provided via simulation studies. Two real data analysis are given to illustrate the application of the proposed method.     

Bayesian Procrustes analysis with applications to hydrology

In this paper, we introduce Procrustes analysis in a Bayesian framework, by treating the classic Procrustes regression equation from a Bayesian perspective, while modeling shapes in two dimensions. The Bayesian approach allows us to compute point estimates and credible sets for the full Procrustes fit parameters. The methods are illustrated through an application to radar data from short-term weather forecasts (nowcasts), a very important problem in hydrology and meteorology.     

NEW LARGE SAMPLE AND BOOTSTRAP METHODS ON SHAPE SPACES IN HIGH LEVEL ANALYSIS OF NATURAL IMAGES

This paper, dedicated to the 80th birthday of Professor C. R. Rao, deals with asymptotic distributions of Fréchet sample means and Fréchet total sample variance that are used in particular for data on projective shape spaces or on 3D shape spaces. One considers the intrinsic means associated with Riemannian metrics that are locally flat in a geodesically convex neighborhood around the support of a probability measure on a shape space or on a projective shape space. Such methods are needed to derive tests concerning variability of planar projective shapes in natural images or large sample and bootstrap confidence intervals for 3D mean shape coordinates of an ordered set of landmarks from laser images.     

A wavelet approach to shape analysis for spinal curves

We present a new method to describe shape change and shape differences in curves, by constructing a deformation function in terms of a wavelet decomposition. Wavelets form an orthonormal basis which allows representations at multiple resolutions. The deformation function is estimated, in a fully Bayesian framework, using a Markov chain Monte Carlo algorithm. This Bayesian formulation incorporates prior information about the wavelets and the deformation function. The flexibility of the MCMC approach allows estimation of complex but clinically important summary statistics, such as curvature in our case, as well as estimates of deformation functions with variance estimates, and allows thorough investigation of the posterior distribution. This work is motivated by multi-disciplinary research involving a large-scale longitudinal study of idiopathic scoliosis in UK children. This paper provides novel statistical tools to study this spinal deformity, from which 5% of UK children suffer. Using the data we consider statistical inference for shape differences between normals, scoliotics and developers of scoliosis, in particular for spinal curvature, and look at longitudinal deformations to describe shape changes with time.     

On clustering shape data

ABSTRACT

Among the statistical methods to model stochastic behaviours of objects, clustering is a preliminary technique to recognize similar patterns within a group of observations in a data set. Various distances to measure differences among objects could be invoked to cluster data through numerous clustering methods. When variables in hand contain geometrical information of objects, such metrics should be adequately adapted. In fact, statistical methods for these typical data are endowed with a geometrical paradigm in a multivariate sense. In this paper, a procedure for clustering shape data is suggested employing appropriate metrics. Then, the best shape distance candidate as well as a suitable agglomerative method for clustering the simulated shape data are provided by considering cluster validation measures. The results are implemented in a real life application.     

Dynamic image analysis using Bayesian shape and texture models

In this paper, we discuss the implementation of fully Bayesian analysis of dynamic image sequences in the context of stochastic deformable templates for shape modelling, Markov/Gibbs random fields for modelling textures, and dynomation.

Throughout, Markov chain Monte Carlo algorithms are used to perform the Bayesian calculations.     

Bayesian estimation of the Cox model under different hazard rate shape assumptions via slice sampling

In this paper, we provide a full Bayesian analysis for Cox's proportional hazards model under different hazard rate shape assumptions. To this end, we select the modified Weibull distribution family to model failure rates. A novel Markov chain Monte Carlo method allows one to tackle both exact and right-censored failure time data. Both simulated and real data are used to illustrate the methods.     

On shape detection in noisy images with particular reference to ultrasonography

We discuss the detection of a connected shape in a noisy image. Two types of image are considered: in the first a degraded outline of the shape is visible, while in the second the data are a corrupted version of the shape itself. In the first type the shape is defined by a thin outline of pixels with records that are different from those at pixels inside and outside the shape, while in the second type the shape is defined by its edge and pixels inside and outside the shape have different records. Our motivation is the identification of cross-sectional head shapes in ultrasound images of human fetuses. We describe and discuss a new approach to detecting shapes in images of the first type that uses a specially designed filter function that iteratively identifies the outline pixels of the head. We then suggest a way based on the cascade algorithm introduced by Jubb and Jennison (1991) of improving and considerably increasing the speed of a method proposed by Storvik (1994) for detecting edges in images of the second type.     

Estimation of reliability of multicomponent stress–strength for a Kumaraswamy distribution

This article deals with the Bayesian and non Bayesian estimation of multicomponent stress–strength reliability by assuming the Kumaraswamy distribution. Both stress and strength are assumed to have a Kumaraswamy distribution with common and known shape parameter. The reliability of such a system is obtained by the methods of maximum likelihood and Bayesian approach and the results are compared using Markov Chain Monte Carlo (MCMC) technique for both small and large samples. Finally, two data sets are analyzed for illustrative purposes.