A multiresolution approach to time warping achieved by a Bayesian prior-posterior transfer fitting strategy.

The procedure known as warping aims at reducing phase variability in a sample of functional curve observations, by applying a smooth bijection to the argument of each of the functions. We propose a natural representation of warping functions in terms of a new type of elementary function named `warping component functions' which are combined into the warping function by composition. A sequential Bayesian estimation strategy is introduced, which fits a series of models and transfers the posterior of the previous fit into the prior of the next fit. Model selection is based on a warping analogue to wavelet thresholding, combined with Bayesian inference.Bayesian inference; Functional data analysis; Markov chain Monte Carlo sampling; Time warping; Warping components; Warping function;

Warping Functional Data in R and C via a Bayesian Multiresolution Approach

Phase variation in functional data obscures the true amplitude variation when a typical cross-sectional analysis of these responses would be performed. Time warping or curve registration aims at eliminating the phase variation, typically by applying transformations, the warping functions τn, to the function arguments. We propose a warping method that jointly estimates a decomposition of the warping function in warping components, and amplitude components. For the estimation routine, adaptive MCMC calculations are performed and implemented in C rather than R to increase computational speed. The R-C interface makes the program user-friendly, in that no knowledge of C is required and all input and output will be handled through R. The R package MRwarping contains all needed files

Statistical shape analysis for bio-structures : local shape modelling, techniques and applications

A Statistical Shape Model (SSM) is a statistical representation of a shape obtained from data to study variation in shapes. Work on shape modelling is constrained by many unsolved problems, for instance, difficulties in modelling local versus global variation. SSM have been successfully applied in medical image applications such as the analysis of brain anatomy. Since brain structure is so complex and varies across subjects, methods to identify morphological variability can be useful for diagnosis and treatment. The main objective of this research is to generate and develop a statistical shape model to analyse local variation in shapes. Within this particular context, this work addresses the question of what are the local elements that need to be identified for effective shape analysis. Here, the proposed method is based on a Point Distribution Model and uses a combination of other well known techniques: Fractal analysis; Markov Chain Monte Carlo methods; and the Curvature Scale Space representation for the problem of contour localisation. Similarly, Diffusion Maps are employed as a spectral shape clustering tool to identify sets of local partitions useful in the shape analysis. Additionally, a novel Hierarchical Shape Analysis method based on the Gaussian and Laplacian pyramids is explained and used to compare the featured Local Shape Model. Experimental results on a number of real contours such as animal, leaf and brain white matter outlines have been shown to demonstrate the effectiveness of the proposed model. These results show that local shape models are efficient in modelling the statistical variation of shape of biological structures. Particularly, the development of this model provides an approach to the analysis of brain images and brain morphometrics. Likewise, the model can be adapted to the problem of content based image retrieval, where global and local shape similarity needs to be measured

Statistical shape analysis for bio-structures : local shape modelling, techniques and applications

A Statistical Shape Model (SSM) is a statistical representation of a shape obtained from data to study variation in shapes. Work on shape modelling is constrained by many unsolved problems, for instance, difficulties in modelling local versus global variation. SSM have been successfully applied in medical image applications such as the analysis of brain anatomy. Since brain structure is so complex and varies across subjects, methods to identify morphological variability can be useful for diagnosis and treatment. The main objective of this research is to generate and develop a statistical shape model to analyse local variation in shapes. Within this particular context, this work addresses the question of what are the local elements that need to be identified for effective shape analysis. Here, the proposed method is based on a Point Distribution Model and uses a combination of other well known techniques: Fractal analysis; Markov Chain Monte Carlo methods; and the Curvature Scale Space representation for the problem of contour localisation. Similarly, Diffusion Maps are employed as a spectral shape clustering tool to identify sets of local partitions useful in the shape analysis. Additionally, a novel Hierarchical Shape Analysis method based on the Gaussian and Laplacian pyramids is explained and used to compare the featured Local Shape Model. Experimental results on a number of real contours such as animal, leaf and brain white matter outlines have been shown to demonstrate the effectiveness of the proposed model. These results show that local shape models are efficient in modelling the statistical variation of shape of biological structures. Particularly, the development of this model provides an approach to the analysis of brain images and brain morphometrics. Likewise, the model can be adapted to the problem of content based image retrieval, where global and local shape similarity needs to be measured.EThOS - Electronic Theses Online ServiceConsejo Nacional de Ciencia y Tecnología (Mexico) (CONACYT)GBUnited Kingdo

Directional edge and texture representations for image processing

An efficient representation for natural images is of fundamental importance in image processing and analysis. The commonly used separable transforms such as wavelets axe not best suited for images due to their inability to exploit directional regularities such as edges and oriented textural patterns; while most of the recently proposed directional schemes cannot represent these two types of features in a unified transform. This thesis focuses on the development of directional representations for images which can capture both edges and textures in a multiresolution manner. The thesis first considers the problem of extracting linear features with the multiresolution Fourier transform (MFT). Based on a previous MFT-based linear feature model, the work extends the extraction method into the situation when the image is corrupted by noise. The problem is tackled by the combination of a "Signal+Noise" frequency model, a refinement stage and a robust classification scheme. As a result, the MFT is able to perform linear feature analysis on noisy images on which previous methods failed. A new set of transforms called the multiscale polar cosine transforms (MPCT) are also proposed in order to represent textures. The MPCT can be regarded as real-valued MFT with similar basis functions of oriented sinusoids. It is shown that the transform can represent textural patches more efficiently than the conventional Fourier basis. With a directional best cosine basis, the MPCT packet (MPCPT) is shown to be an efficient representation for edges and textures, despite its high computational burden. The problem of representing edges and textures in a fixed transform with less complexity is then considered. This is achieved by applying a Gaussian frequency filter, which matches the disperson of the magnitude spectrum, on the local MFT coefficients. This is particularly effective in denoising natural images, due to its ability to preserve both types of feature. Further improvements can be made by employing the information given by the linear feature extraction process in the filter's configuration. The denoising results compare favourably against other state-of-the-art directional representations

A global-to-local model for the representation of human faces

In the context of face modeling and face recognition, statistical models are widely used for the representation and modeling of surfaces. Most of these models are obtained by computing Principal Components Analysis (PCA) on a set of representative examples. These models represent novel faces poorly due to their holistic nature (i.e.\ each component has global support), and they suffer from overfitting when used for generalization from partial information. In this work, we present a novel analysis method that breaks the objects up into modes based on spatial frequency. The high-frequency modes are segmented into regions with respect to specific features of the object. After computing PCA on these segments individually, a hierarchy of global and local components gradually decreasing in size of their support is combined into a linear statistical model, hence the name, Global-to-Local model (G2L). We apply our methodology to build a novel G2L model of 3D shapes of human heads. Both the representation and the generalization capabilities of the models are evaluated and compared in a standardized test, and it is demonstrated that the G2L model performs better compared to traditional holistic PCA models. Furthermore, both models are used to reconstruct the 3D shape of faces from a single photograph. A novel adaptive fitting method is presented that estimates the model parameters using a multi-resolution approach. The model is first fitted to contours extracted from the image. In a second stage, the contours are kept fixed and the remaining flexibility of the model is fitted to the input image. This makes the method fast (30 sec on a standard PC), efficient, and accurate

A Contribution to Functional Data Analysis

Functional Principal Component Analysis (FPCA) approximates a sample curve as a linear combination of orthogonal basis functions. It is often possible to describe the essential parts of the variations of functional data by looking only at a usually very small set of principal components and the corresponding principal scores. Two approaches based on FPCA to estimate smooth derivatives of noisy and discretely observed high-dimensional spatial curves are presented. To handle observed data, both approaches rely on local polynomial regressions. The requirements under which the methods are asymptotically equivalent are evaluated. If the curves are contained in a finite-dimensional function space, it is shown that both methods providing better rates of convergence than estimating the curves individually. The methodology is illustrated in a simulation and empirical study, in which state price density (SPD) surfaces from call option prices are estimated. Serious issues deriving to an FPCA decomposition arise in presence of a Registration problem. Registration aims to decompose amplitude and phase variation of samples of curves. Phase variation is captured by warping functions which monotonically transform the domains. Resulting registered curves should then only exhibit amplitude variation. Most existing registration method rely on aligning typical shape features like peaks or valleys to be found in each sample function. It is shown that this is not necessarily an optimal strategy for subsequent statistical data exploration and inference. In this context a major goal is to identify low dimensional linear subspaces of functions that are able to provide accurate approximations of the observed functional data. Problems of identifiability are discussed in detail, and connections to established registration procedures are analyzed. The methodology is applied to simulated and real data for example an analysis of the juggling dataset. Here an elementary landmark registration is used to extract the juggling cycles from the data. The resulting cycles are then registered to functional principal components. After the registration step the focus is then at the functional principal component analysis to explain the amplitude variation of the cycles. More results about the behavior of the juggler's movements of the hand during the juggling trials are obtained by a further investigation of the principal scores