Recent advances in directional statistics

Mainstream statistical methodology is generally applicable to data observed in Euclidean space. There are, however, numerous contexts of considerable scientific interest in which the natural supports for the data under consideration are Riemannian manifolds like the unit circle, torus, sphere and their extensions. Typically, such data can be represented using one or more directions, and directional statistics is the branch of statistics that deals with their analysis. In this paper we provide a review of the many recent developments in the field since the publication of Mardia and Jupp (1999), still the most comprehensive text on directional statistics. Many of those developments have been stimulated by interesting applications in fields as diverse as astronomy, medicine, genetics, neurology, aeronautics, acoustics, image analysis, text mining, environmetrics, and machine learning. We begin by considering developments for the exploratory analysis of directional data before progressing to distributional models, general approaches to inference, hypothesis testing, regression, nonparametric curve estimation, methods for dimension reduction, classification and clustering, and the modelling of time series, spatial and spatio-temporal data. An overview of currently available software for analysing directional data is also provided, and potential future developments discussed.Comment: 61 page

Models of statistical self-similarity for signal and image synthesis

Statistical self-similarity of random processes in continuous-domains is defined through invariance of their statistics to time or spatial scaling. In discrete-time, scaling by an arbitrary factor of signals can be accomplished through frequency warping, and statistical self-similarity is defined by the discrete-time continuous-dilation scaling operation. Unlike other self-similarity models mostly relying on characteristics of continuous self-similarity other than scaling, this model provides a way to express discrete-time statistical self-similarity using scaling of discrete-time signals. This dissertation studies the discrete-time self-similarity model based on the new scaling operation, and develops its properties, which reveals relations with other models. Furthermore, it also presents a new self-similarity definition for discrete-time vector processes, and demonstrates synthesis examples for multi-channel network traffic. In two-dimensional spaces, self-similar random fields are of interest in various areas of image processing, since they fit certain types of natural patterns and textures very well. Current treatments of self-similarity in continuous two-dimensional space use a definition that is a direct extension of the 1-D definition. However, most of current discrete-space two-dimensional approaches do not consider scaling but instead are based on ad hoc formulations, for example, digitizing continuous random fields such as fractional Brownian motion. The dissertation demonstrates that the current statistical self-similarity definition in continuous-space is restrictive, and provides an alternative, more general definition. It also provides a formalism for discrete-space statistical self-similarity that depends on a new scaling operator for discrete images. Within the new framework, it is possible to synthesize a wider class of discrete-space self-similar random fields

Texture Analysis in Magnetic Resonance Imaging: Review and Considerations for Future Applications

Texture analysis is a technique used for the quantification of image texture. It has been successfully used in many fields, and in the past years it has been applied in magnetic resonance imaging (MRI) as a computer-aided diagnostic tool. Quantification of the intrinsic heterogeneity of different tissues and lesions is necessary as they are usually imperceptible to the human eye. In the present chapter, we describe texture analysis as a process consisting of six steps: MRI acquisition, region of interest (ROI) definition, ROI preprocessing, feature extraction, feature selection, and classification. There is a great variety of methods and techniques to be chosen at each step and all of them can somehow affect the outcome of the texture analysis application. We reviewed the literature regarding texture analysis in clinical MRI focusing on the important considerations to be taken at each step of the process in order to obtain maximum benefits and to avoid misleading results

Modeling small objects under uncertainties : novel algorithms and applications.

Active Shape Models (ASM), Active Appearance Models (AAM) and Active Tensor Models (ATM) are common approaches to model elastic (deformable) objects. These models require an ensemble of shapes and textures, annotated by human experts, in order identify the model order and parameters. A candidate object may be represented by a weighted sum of basis generated by an optimization process. These methods have been very effective for modeling deformable objects in biomedical imaging, biometrics, computer vision and graphics. They have been tried mainly on objects with known features that are amenable to manual (expert) annotation. They have not been examined on objects with severe ambiguities to be uniquely characterized by experts. This dissertation presents a unified approach for modeling, detecting, segmenting and categorizing small objects under uncertainty, with focus on lung nodules that may appear in low dose CT (LDCT) scans of the human chest. The AAM, ASM and the ATM approaches are used for the first time on this application. A new formulation to object detection by template matching, as an energy optimization, is introduced. Nine similarity measures of matching have been quantitatively evaluated for detecting nodules less than 1 em in diameter. Statistical methods that combine intensity, shape and spatial interaction are examined for segmentation of small size objects. Extensions of the intensity model using the linear combination of Gaussians (LCG) approach are introduced, in order to estimate the number of modes in the LCG equation. The classical maximum a posteriori (MAP) segmentation approach has been adapted to handle segmentation of small size lung nodules that are randomly located in the lung tissue. A novel empirical approach has been devised to simultaneously detect and segment the lung nodules in LDCT scans. The level sets methods approach was also applied for lung nodule segmentation. A new formulation for the energy function controlling the level set propagation has been introduced taking into account the specific properties of the nodules. Finally, a novel approach for classification of the segmented nodules into categories has been introduced. Geometric object descriptors such as the SIFT, AS 1FT, SURF and LBP have been used for feature extraction and matching of small size lung nodules; the LBP has been found to be the most robust. Categorization implies classification of detected and segmented objects into classes or types. The object descriptors have been deployed in the detection step for false positive reduction, and in the categorization stage to assign a class and type for the nodules. The AAMI ASMI A TM models have been used for the categorization stage. The front-end processes of lung nodule modeling, detection, segmentation and classification/categorization are model-based and data-driven. This dissertation is the first attempt in the literature at creating an entirely model-based approach for lung nodule analysis

Champs à phase aléatoire et champs gaussiens pour la mesure de netteté d’images et la synthèse rapide de textures

This thesis deals with the Fourier phase structure of natural images, and addresses no-reference sharpness assessment and fast texture synthesis by example. In Chapter 2, we present several models of random fields in a unified framework, like the spot noise model and the Gaussian model, with particular attention to the spectral representation of these random fields. In Chapter 3, random phase models are used to perform by-example synthesis of microtextures (textures with no salient features). We show that a microtexture can be summarized by a small image that can be used for fast and flexible synthesis based on the spot noise model. Besides, we address microtexture inpainting through the use of Gaussian conditional simulation. In Chapter 4, we present three measures of the global Fourier phase coherence. Their link with the image sharpness is established based on a theoretical and practical study. We then derive a stochastic optimization scheme for these indices, which leads to a blind deblurring algorithm. Finally, in Chapter 5, after discussing the possibility of direct phase analysis or synthesis, we propose two non random phase texture models which allow for synthesis of more structured textures and still have simple mathematical guarantees.Dans cette thèse, on étudie la structuration des phases de la transformée de Fourier d'images naturelles, ce qui, du point de vue applicatif, débouche sur plusieurs mesures de netteté ainsi que sur des algorithmes rapides pour la synthèse de texture par l'exemple. Le Chapitre 2 présente dans un cadre unifié plusieurs modèles de champs aléatoires, notamment les champs spot noise et champs gaussiens, en prêtant une attention particulière aux représentations fréquentielles de ces champs aléatoires. Le Chapitre 3 détaille l'utilisation des champs à phase aléatoire à la synthèse de textures peu structurées (microtextures). On montre qu'une microtexture peut être résumée en une image de petite taille s'intégrant à un algorithme de synthèse très rapide et flexible via le modèle spot noise. Aussi on propose un algorithme de désocclusion de zones texturales uniformes basé sur la simulation gaussienne conditionnelle. Le Chapitre 4 présente trois mesures de cohérence globale des phases de la transformée de Fourier. Après une étude théorique et pratique établissant leur lien avec la netteté d'image, on propose un algorithme de déflouage aveugle basé sur l'optimisation stochastique de ces indices. Enfin, dans le Chapitre 5, après une discussion sur l'analyse et la synthèse directe de l'information de phase, on propose deux modèles de textures à phases cohérentes qui permettent la synthèse de textures plus structurées tout en conservant quelques garanties mathématiques simples

The transformation of one-dimensional and two-dimensional autoregressive random fields under coordinate scaling and rotation

A practical problem in computer graphics is that of representing a textured surface at arbitrary scales. I consider the underlying mathematical problem to be that of interpolating autoregressive random fields under arbitrary coordinate transformations. I examine the theoretical basis for the transformations that autoregressive parameters exhibit when the associated stationary random fields are scaled or rotated. The basic result is that the transform takes place in the continuous autocovariance domain, and that the spectral density and associated autoregressive parameters proceed directly from sampling the continuous autocovariance on a transformed grid. I show some real-world applications of these ideas, and explore how they allow us to interpolate into a random field. Along the way, I develop interesting ways to estimate simultaneous autoregressive parameters, to calculate the distorting effects of linear interpolation algorithms, and to interpolate random fields without altering their statistics

Local, Semi-Local and Global Models for Texture, Object and Scene Recognition

This dissertation addresses the problems of recognizing textures, objects, and scenes in photographs. We present approaches to these recognition tasks that combine salient local image features with spatial relations and effective discriminative learning techniques. First, we introduce a bag of features image model for recognizing textured surfaces under a wide range of transformations, including viewpoint changes and non-rigid deformations. We present results of a large-scale comparative evaluation indicating that bags of features can be effective not only for texture, but also for object categization, even in the presence of substantial clutter and intra-class variation. We also show how to augment the purely local image representation with statistical co-occurrence relations between pairs of nearby features, and develop a learning and classification framework for the task of classifying individual features in a multi-texture image. Next, we present a more structured alternative to bags of features for object recognition, namely, an image representation based on semi-local parts, or groups of features characterized by stable appearance and geometric layout. Semi-local parts are automatically learned from small sets of unsegmented, cluttered images. Finally, we present a global method for recognizing scene categories that works by partitioning the image into increasingly fine sub-regions and computing histograms of local features found inside each sub-region. The resulting spatial pyramid representation demonstrates significantly improved performance on challenging scene categorization tasks