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 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

Statistical shape analysis of brain structures using spherical wavelets

journal articleWe present a novel method of statistical surface-based morphometry based on the use of non-parametric permutation tests and a spherical wavelet (SWC) shape representation. As an application, we analyze two brain structures, the caudate nucleus and the hippocampus, and compare the results obtained to shape analysis using a sampled point representation. Our results show that the SWC representation indicates new areas of significance preserved under the FDR correction for both the left caudate nucleus and left hippocampus. Additionally, the spherical wavelet representation provides a natural way to interpret the significance results in terms of scale in addition to knowing the spatial location of the regions

Entropy-based particle correspondence for shape populations

Statistical shape analysis of anatomical structures plays an important role in many medical image analysis applications such as understanding the structural changes in anatomy in various stages of growth or disease. Establishing accurate correspondence across object populations is essential for such statistical shape analysis studies

Statistical Shape Analysis of Helices

Consider a sequence of equally spaced points along a helix in three-dimensional space, which are observed subject to statistical noise. In this thesis, maximum likelihood (ML) method is developed to estimate the parameters of the helix. Statistical properties of the estimator are studied and comparisons are made to other estimators found in the literature. Methods are established here for the fitting of unkinked and kinked helices. For an unkinked helix an initial estimate of a helix axis is estimated by a modified eigen-decomposition or a method from the literature. Mardia-Holmes model can be used to estimate the initial helix axis but it is often not very successful one since it requires initial parameters. A better method for initial axis estimation is the Rotfit method. If the the axis is known, we minimize the residual sum of squares (RSS) to estimate the helix parameters and then optimize the axis estimate. For a kinked helix, we specify a test statistic by simulating the null distribution of unkinked helices. If the kink position is known, then the test statistic approximately follows an F-distribution. If the null hypothesis is rejected i.e. the helix has a change point, and then cut the helix into two sub-helices between the change point where the helix has the maximum statistic. Statistics test are studied to test how differ these two sub-helices from each other. Parametric bootstrap procedure is used to study these statistics. The shapes of protein alpha-helices are used to illustrate the procedure