Statistical Shape Analysis using Kernel PCA

©2006 SPIE--The International Society for Optical Engineering. One print or electronic copy may be made for personal use only. Systematic reproduction and distribution, duplication of any material in this paper for a fee or for commercial purposes, or modification of the content of the paper are prohibited. The electronic version of this article is the complete one and can be found online at: http://dx.doi.org/10.1117/12.641417DOI:10.1117/12.641417Presented at Image Processing Algorithms and Systems, Neural Networks, and Machine Learning, 16-18 January 2006, San Jose, California, USA.Mercer kernels are used for a wide range of image and signal processing tasks like de-noising, clustering, discriminant analysis etc. These algorithms construct their solutions in terms of the expansions in a high-dimensional feature space F. However, many applications like kernel PCA (principal component analysis) can be used more effectively if a pre-image of the projection in the feature space is available. In this paper, we propose a novel method to reconstruct a unique approximate pre-image of a feature vector and apply it for statistical shape analysis. We provide some experimental results to demonstrate the advantages of kernel PCA over linear PCA for shape learning, which include, but are not limited to, ability to learn and distinguish multiple geometries of shapes and robustness to occlusions

Statistical analysis on high-dimensional spheres and shape spaces

We consider the statistical analysis of data on high-dimensional spheres and shape spaces. The work is of particular relevance to applications where high-dimensional data are available--a commonly encountered situation in many disciplines. First the uniform measure on the infinite-dimensional sphere is reviewed, together with connections with Wiener measure. We then discuss densities of Gaussian measures with respect to Wiener measure. Some nonuniform distributions on infinite-dimensional spheres and shape spaces are introduced, and special cases which have important practical consequences are considered. We focus on the high-dimensional real and complex Bingham, uniform, von Mises-Fisher, Fisher-Bingham and the real and complex Watson distributions. Asymptotic distributions in the cases where dimension and sample size are large are discussed. Approximations for practical maximum likelihood based inference are considered, and in particular we discuss an application to brain shape modeling.Comment: Published at http://dx.doi.org/10.1214/009053605000000264 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Shape Constrained Regularisation by Statistical Multiresolution for Inverse Problems: Asymptotic Analysis

This paper is concerned with a novel regularisation technique for solving linear ill-posed operator equations in Hilbert spaces from data that is corrupted by white noise. We combine convex penalty functionals with extreme-value statistics of projections of the residuals on a given set of sub-spaces in the image-space of the operator. We prove general consistency and convergence rate results in the framework of Bregman-divergences which allows for a vast range of penalty functionals. Various examples that indicate the applicability of our approach will be discussed. We will illustrate in the context of signal and image processing that the presented method constitutes a locally adaptive reconstruction method

3-D Face Analysis and Identification Based on Statistical Shape Modelling

This paper presents an effective method of statistical shape representation for automatic face analysis and identification in 3-D. The method combines statistical shape modelling techniques and the non-rigid deformation matching scheme. This work is distinguished by three key contributions. The first is the introduction of a new 3-D shape registration method using hierarchical landmark detection and multilevel B-spline warping technique, which allows accurate dense correspondence search for statistical model construction. The second is the shape representation approach, based on Laplacian Eigenmap, which provides a nonlinear submanifold that links underlying structure of facial data. The third contribution is a hybrid method for matching the statistical model and test dataset which controls the levels of the model’s deformation at different matching stages and so increases chance of the successful matching. The proposed method is tested on the public database, BU-3DFE. Results indicate that it can achieve extremely high verification rates in a series of tests, thus providing real-world practicality

Posterior shape models

We present a method to compute the conditional distribution of a statistical shape model given partial data. The result is a "posterior shape model", which is again a statistical shape model of the same form as the original model. This allows its direct use in the variety of algorithms that include prior knowledge about the variability of a class of shapes with a statistical shape model. Posterior shape models then provide a statistically sound yet easy method to integrate partial data into these algorithms. Usually, shape models represent a complete organ, for instance in our experiments the femur bone, modeled by a multivariate normal distribution. But because in many application certain parts of the shape are known a priori, it is of great interest to model the posterior distribution of the whole shape given the known parts. These could be isolated landmark points or larger portions of the shape, like the healthy part of a pathological or damaged organ. However, because for most shape models the dimensionality of the data is much higher than the number of examples, the normal distribution is singular, and the conditional distribution not readily available. In this paper, we present two main contributions: First, we show how the posterior model can be efficiently computed as a statistical shape model in standard form and used in any shape model algorithm. We complement this paper with a freely available implementation of our algorithms. Second, we show that most common approaches put forth in the literature to overcome this are equivalent to probabilistic principal component analysis (PPCA), and Gaussian Process regression. To illustrate the use of posterior shape models, we apply them on two problems from medical image analysis: model-based image segmentation incorporating prior knowledge from landmarks, and the prediction of anatomically correct knee shapes for trochlear dysplasia patients, which constitutes a novel medical application. Our experiments confirm that the use of conditional shape models for image segmentation improves the overall segmentation accuracy and robustness

Evaluating Shape Correspondence for Statistical Shape Analysis: A Benchmark Study

This paper introduces a new benchmark study to evaluate the performance of landmark-based shape correspondence used for statistical shape analysis. Different from previous shape-correspondence evaluation methods, the proposed benchmark first generates a large set of synthetic shape instances by randomly sampling a given statistical shape model that defines a ground-truth shape space. We then run a test shape-correspondence algorithm on these synthetic shape instances to identify a set of corresponded landmarks. According to the identified corresponded landmarks, we construct a new statistical shape model, which defines a new shape space. We finally compare this new shape space against the ground-truth shape space to determine the performance of the test shape-correspondence algorithm. In this paper, we introduce three new performance measures that are landmark independent to quantify the difference between the ground-truth and the newly derived shape spaces. By introducing a ground-truth shape space that is defined by a statistical shape model and three new landmark-independent performance measures, we believe the proposed benchmark allows for a more objective evaluation of shape correspondence than previous methods. In this paper, we focus on developing the proposed benchmark for 2D shape correspondence. However it can be easily extended to 3D cases

Statistical Shape Analysis of Galactic Hii Regions

Hii regions are diffuse nebulae of ionised hydrogen, excited by the extreme ultraviolet emission from massive stars. Due to the embedded nature of massive star formation, there are many observational difficulties involved when investigating such stars. Hii regions, however, are readily observed via their infrared and radio emission. As such, they highlight the location of their massive star sources. Furthermore, Hii region properties are directly resultant of their progenitors and environment. The overall aim of the work presented herein, is to determine whether statistical shape analysis of observational and numerically modelled Hii region data can be used to probe the associated astrophysical properties. Radio continuum and computer simulated synthetic images of Hii regions were analysed using the shape extraction and statistical comparison methods constructed in this work. For the radio data, six morphological groups were identified. Visual inspection and quantitative ordinance techniques confirmed that the shape analysis and grouping procedure were working as intended. It was found that in the first Galactic quadrant, location is mostly independent of group, with a small preference for regions of similar Galactic longitudes to share common morphologies. The shapes are homogeneously distributed across Galactocentric distance and latitude. One group contained regions that are all younger than 0.5 Myr and ionised by relatively low- to intermediate-mass sources. Those in another group are all driven by intermediate- to high-mass sources. One group was distinctly separated from the other five and contained regions at the surface brightness detection limit for the survey. The hierarchical procedure employed was most sensitive to the spatial sampling resolution used, which is determined for each region from its heliocentric distance. The numerical Hii region data was the result of photoionisation and feedback of a 34 M⊙ star, in a 1000 M⊙ cloud. Synthetic observations (SOs) were provided, comprising four evolutionary snapshots (0.1, 0.2, 0.4 and 0.6Myr), and multiple viewing projection angles. The shape analysis results provided conclusive evidence of the efficacy of the numerical simulations. When comparing the shapes of the synthetic regions to their observational counterparts, the SOs were grouped in amongst the Galactic Hii regions by the hierarchical procedure. There was also an association between the evolutionary distribution of regions of the respective samples. This suggested that this method could be further developed for classification of the observational regions by using the synthetic data, with its well defined parameters

Statistical Shape Analysis for the Human Back

A thesis submitted to the department of Engineering and Technology in partial fulfilment of the requirements for the degree of Master of Philosophy in Production and Manufacturing Engineering at the University of WolverhamptonIn this research, Procrustes and Euclidean distance matrix analysis (EDMA) have been investigated for analysing the three-dimensional shape and form of the human back. Procrustes analysis is used to distinguish deformed backs from normal backs. EDMA is used to locate the changes occurring on the back surface due to spinal deformity (scoliosis, kyphosis and lordosis) for back deformity patients. A surface topography system, ISIS2 (Integrated Shape Imaging System 2), is available to measure the three-dimensional back surface. The system presents clinical parameters, which are based on distances and angles relative to certain anatomical landmarks on the back surface. Location, rotation and scale definitely influence these parameters. Although the anatomical landmarks are used in the present system to take some account of patient stance, it is still felt that variability in the clinical parameters is increased by the use of length and angle data. Patients also grow and so their back size, shape and form change between appointments with the doctor. Instead of distances and angles, geometric shape that is independent of location, rotation and scale effects could be measured. This research is mainly focusing on the geometric shape and form change in the back surface, thus removing the unwanted effects. Landmarks are used for describing back information and an analysis of the variability in positioning the landmarks has been carried out for repeated measurements. Generalized Procrustes analysis has been applied to all normal backs to calculate a mean Procrustes shape, which is named the standard normal shape (SNS). Each back (normal and deformed) is then translated, rotated and scaled to give a best fit with the SNS using ordinary Procrustes analysis. Riemannian distances are then estimated between the SNS and all individual backs. The highest Riemannian distance value between the normal backs and the SNS is lower than the lowest Riemannian distance value between the deformed backs and the SNS. The results shows that deformed backs can be differentiated from normal backs. EDMA has been used to estimate a mean form, variance-covariance matrix and mean form difference from all the normal backs. This mean form is named the standard normal form (SNF). The influence of individual landmarks for form difference between each deformed back and the SNF is estimated. A high value indicates high deformity on the location of that landmark and a low value close to 1 indicates less deformity. The result is displayed in a graph that provides information regarding the degree and location of the deformity. The novel aspects of this research lie in the development of an effective method for assessing the three-dimensional back shape; extracting automatic landmarks; visualizing back shape and back form differences

Statistical shape analysis of Multi-Object complexes

journal articleAn important goal of statistical shape analysis is the discrimination between populations of objects, exploring group differences in morphology not explained by standard volumetric analysis. Certain applications additionally require analysis of objects in their embedding context by joint statistical analysis of sets of interrelated objects. In this paper, we present a framework for discriminant analysis of populations of 3-D multi-object sets. In view of the driving medical applications, a skeletal object parametrization of shape is chosen since it naturally encodes thickening, bending and twisting. In a multi-object setting, we not only consider a joint analysis of sets of shapes but also must take into account differences in pose. Statistics on features of medial descriptions and pose parameters, which include rotational frames and distances, uses a Riemannian symmetric space instead of the standard Euclidean metric. Our choice of discriminant method is the distance weighted discriminant (DWD) because of its generalization ability in high dimensional, low sample size settings. Joint analysis of 10 sub-cortical brain structures in a pediatric autism study demonstrates that multi-object analysis of shape results in a better group discrimination than pose, and that the combination of pose and shape performs better than shape alone. Finally, given a discriminating axis of shape and pose, we can visualize the differences between the populations