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.     

Statistics,images, and pattern recognition

Data are increasingly being collected in the form of images, especially in fields using remote sensing and microscopy. Statisticians are becoming interested in developing techniques to handle the highly structured data of images. Statistical work in this area is surveyed, and two problems discussed in more detail. The first is a form of image segmentation, classifying the pixels of a satellite picture by land use. The second is the summarization of electron micrographs.     

Longitudinal Image Analysis of Tumor/Healthy Brain Change in Contrast Uptake Induced by Radiation

This work is motivated by a quantitative Magnetic Resonance Imaging study of the differential tumor/healthy tissue change in contrast uptake induced by radiation. The goal is to determine the time in which there is maximal contrast uptake (a surrogate for permeability) in the tumor relative to healthy tissue. A notable feature of the data is its spatial heterogeneity. Zhang, Johnson, Little, and Cao (2008a and 2008b) discuss two parallel approaches to "denoise" a single image of change in contrast uptake from baseline to one follow-up visit of interest. In this work we extend the image model to explore the longitudinal profile of the tumor/healthy tissue contrast uptake in multiple images over time. We fit a two-stage model. First, we propose a longitudinal image model for each subject. This model simultaneously accounts for the spatial and temporal correlation and denoises the observed images by borrowing strength both across neighboring pixels and over time. We propose to use the Mann-Whitney U statistic to summarize the tumor contrast uptake relative to healthy tissue. In the second stage, we fit a population model to the U statistic and estimate when it achieves its maximum. Our initial findings suggest that the maximal contrast uptake of the tumor core relative to healthy tissue peaks around three weeks after initiation of radiotherapy, though this warrants further investigation.     

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.     

Assessing shape differences in populations of shapes using the complex watson shape distribution

This paper presents a novel Bayesian method based on the complex Watson shape distribution that is used in detecting shape differences between the second thoracic vertebrae for two groups of mice, small and large, categorized according to their body weight. Considering the data provided in Johnson et al. (1988), we provide Bayesian methods of estimation as well as highest posterior density (HPD) estimates for modal vertebrae shapes within each group. Finally, we present a classification procedure that can be used in any shape classification experiment, and apply it for categorizing new vertebrae shapes in small or large groups.     

Towards automated image understanding

In this paper, we discuss an approach to the pattern synthesis and analysis of biological shapes which show a lot of variability, but at the same time also show a characteristic structure. This structure is captured by way of ‘shape classes' which are constructed via a deformable template. Examples of pattern analysis in both two and three dimensions are presented. We define what we call intrinsic and extrinsic understanding of images and apply this to the detection of abnormalities.     

Bayesian texture segmentation of weed and crop images using reversible jump Markov chain Monte Carlo methods

Summary. A Bayesian method for segmenting weed and crop textures is described and implemented. The work forms part of a project to identify weeds and crops in images so that selective crop spraying can be carried out. An image is subdivided into blocks and each block is modelled as a single texture. The number of different textures in the image is assumed unknown. A hierarchical Bayesian procedure is used where the texture labels have a Potts model (colour Ising Markov random field) prior and the pixels within a block are distributed according to a Gaussian Markov random field, with the parameters dependent on the type of texture. We simulate from the posterior distribution by using a reversible jump Metropolis–Hastings algorithm, where the number of different texture components is allowed to vary. The methodology is applied to a simulated image and then we carry out texture segmentation on the weed and crop images that motivated the work.     

Shape spaces for prealigned star-shaped objects—studying the growth of plants by principal components analysis

Summary.  We analyse the shapes of star-shaped objects which are prealigned. This is motivated from two examples studying the growth of leaves, and the temporal evolution of tree rings. In the latter case measurements were taken at fixed angles whereas in the former case the angles were free. Subsequently, this leads to different shape spaces, related to different concepts of size, for the analysis. Whereas several shape spaces already existed in the literature when the angles are fixed, a new shape space for free angles, called spherical shape space , needed to be introduced. We compare these different shape spaces both regarding their mathematical properties and in their adequacy to the data at hand; we then apply suitably defined principal component analysis on these. In both examples we find that the shapes evolve mainly along the first principal component during growth; this is the 'geodesic hypothesis' that was formulated by Le and Kume. Moreover, we could link change-points of this evolution to significant changes in environmental conditions.     

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.     

Functional approaches for predicting land use with the temporal evolution of coarse resolution remote sensing data

The sensor SPOT 4/Végétation gives every day satellite images of Europe with medium spatial resolution, each pixel corresponding to an area of 1 r km 2 1 r km. Such data are useful to characterize the development of the vegetation at a large scale. The pixels, named "mixed' pixels, aggregate information of different crops and thus different themes of interest (wheat, corn, forest, …). We aim at estimating the land use when observing the temporal evolution of reflectances of mixed pixels. The statistical problem is to predict proportions with longitudinal covariates. We compared two functional approaches. The first relies on varying-time regression models and the second is an extension of the multilogit model for functional data. The comparison is achieved on a small area on which the land use is known. Satellite data were collected between March and August 1998. The functional multilogit model gives better predictions and the use of composite vegetation index is more efficient.     

THE ANALYSIS OF TIME SERIES—THE TIME DOMAIN APPROACH1

This paper has three basic aims. The first of these is to give a concise outline of the analysis of time series, from an applied point of view, along the lines proposed by Box and Jenkins (1970). The second aim is to discuss a number of advances in this field since 1970, while the third is to discuss some of the more recent developments in the analysis of time series which are applicable in the physical and social sciences.     

A finite mixture model for image segmentation

In this paper, we propose a model for image segmentation based on a finite mixture of Gaussian distributions. For each pixel of the image, prior probabilities of class memberships are specified through a Gibbs distribution, where association between labels of adjacent pixels is modeled by a class-specific term allowing for different interaction strengths across classes. We show how model parameters can be estimated in a maximum likelihood framework using Mean Field theory. Experimental performance on perturbed phantom and on real benchmark images shows that the proposed method performs well in a wide variety of empirical situations.     

Hidden Markov mesh random field models in image analysis

This paper addresses the image modeling problem under the assumption that images can be represented by third-order, hidden Markov mesh random field models. The range of applications of the techniques described hereafter comprises the restoration of binary images, the modeling and compression of image data, as well as the segmentation of gray-level or multi-spectral images, and image sequences under the short-range motion hypothesis. We outline coherent approaches to both the problems of image modeling (pixel labeling) and estimation of model parameters (learning). We derive a real-time labeling algorithm-based on a maximum, marginal a posteriori probability criterion-for a hidden third-order Markov mesh random field model. Our algorithm achieves minimum time and space complexities simultaneously, and we describe what we believe to be the most appropriate data structures to implement it. Critical aspects of the computer simulation of a real-time implementation are discussed, down to the computer code level. We develop an (unsupervised) learning technique by which the model parameters can be estimated without ground truth information. We lay bare the conditions under which our approach can be made time-adaptive in order to be able to cope with short-range motion in dynamic image sequences. We present extensive experimental results for both static and dynamic images from a wide variety of sources. They comprise standard, infra-red and aerial images, as well as a sequence of ultrasound images of a fetus and a series of frames from a motion picture sequence. These experiments demonstrate that the method is subjectively relevant to the problems of image restoration, segmentation and modeling.     

Detection of Multiple Outbreaks Using Spatio-Temporal EWMA-Ordered Statistics

Spatio-temporal surveillance methods for detecting outbreaks of disease are fairly common in the literature with the scan statistic setting the benchmark. If the shape and size of the outbreaks are known in advance, then the scan approach can be designed to efficiently detect these, however, this is seldom true. Therefore we want to devise plans that are efficient at detecting a number of outbreaks that vary in size and shape. This paper examines plans which use the exponential weighted moving average statistic to build temporal memory into plans and tries to develop robust plans for detecting outbreaks of unknown shapes and sizes.     

Spatially Clustered Outbreak Detection Using the EWMA SCAN Statistics with Multiple Sized Windows

Spatio-temporal surveillance methods for detecting outbreaks of disease are fairly common in the literature with the SCAN statistic setting the benchmark. If the shape and size of the outbreaks are known in advance, then the SCAN statistic can be trained to efficiently detect these, however this is seldom true. Therefore, we want to devise plans that are efficient at detecting a number of outbreaks that vary in size and shape. This article examines plans which use the exponential weighted moving average statistic to build temporal memory into plans and tries to develop robust plans for detecting outbreaks of unknown shapes and sizes.     

A Shape-based Glaucoma Index for Tomographic Images

We examine the use of Confocal Laser Tomographic images for detecting glaucoma. From the clinical aspect, the optic nerve head's (ONH) area contains all the relevant information on glaucoma. The shape of ONH is approximately a skewed cup. We summarize its shape by three biological landmarks on the neural-rim and the fourth landmark as the point of the maximum depth, which is approximately the point where the optic nerve enters this eye cup. These four landmarks are extracted from the images related to some Rhesus monkeys before and after inducing glaucoma. Previous analysis on Bookstein shape coordinates of these four landmarks revealed only marginally significant findings. From clinical experience, it is believed that the ratio depth to diameter of the eye cup provides a useful measure of the shape change. We consider the bootstrap distribution of this normalized 'depth' (G) and give evidence that it provides an appropriate measure of the shape change. This measure G is labelled as the glaucoma index. Further experiments are in progress to validate its use for glaucoma in humans.     

Detecting anomalies in fibre systems using 3-dimensional image data

Statistics and Computing - We consider the problem of detecting anomalies in the directional distribution of fibre materials observed in 3D images. We divide the image into a set of scanning...