Gaussian distributions on Lie groups and their application to statistical shape analysis

An Empirical Bayes Approach to Shrinkage Estimation on the Manifold of Symmetric Positive-Definite Matrices

The James-Stein estimator is an estimator of the multivariate normal mean and dominates the maximum likelihood estimator (MLE) under squared error loss. The original work inspired great interest in developing shrinkage estimators for a variety of problems. Nonetheless, research on shrinkage estimation for manifold-valued data is scarce. In this article, we propose shrinkage estimators for the parameters of the Log-Normal distribution defined on the manifold of N × N symmetric positive-definite matrices. For this manifold, we choose the Log-Euclidean metric as its Riemannian metric since it is easy to compute and has been widely used in a variety of applications. By using the Log-Euclidean distance in the loss function, we derive a shrinkage estimator in an analytic form and show that it is asymptotically optimal within a large class of estimators that includes the MLE, which is the sample Fréchet mean of the data. We demonstrate the performance of the proposed shrinkage estimator via several simulated data experiments. Additionally, we apply the shrinkage estimator to perform statistical inference in both diffusion and functional magnetic resonance imaging problems.

A geometric framework for statistical analysis of trajectories with distinct temporal spans

Analyzing data representing multifarious trajectories is central to the many fields in Science and Engineering; for example, trajectories representing a tennis serve, a gymnast's parallel bar routine, progression/remission of disease and so on. We present a novel geometric algorithm for performing statistical analysis of trajectories with distinct number of samples representing longitudinal (or temporal) data. A key feature of our proposal is that unlike existing schemes, our model is deployable in regimes where each participant provides a different number of acquisitions (trajectories have different number of sample points or temporal span). To achieve this, we develop a novel method involving the parallel transport of the tangent vectors along each given trajectory to the starting point of the respective trajectories and then use the span of the matrix whose columns consist of these vectors, to construct a linear subspace in R ^m . We then map these linear subspaces (possibly of distinct dimensions) of R ^m on to a single high dimensional hypersphere. This enables computing group statistics over trajectories by instead performing statistics on the hypersphere (equipped with a simpler geometry). Given a point on the hypersphere representing a trajectory, we also provide a "reverse mapping" algorithm to uniquely (under certain assumptions) reconstruct the subspace that corresponds to this point. Finally, by using existing algorithms for recursive Fréchet mean and exact principal geodesic analysis on the hypersphere, we present several experiments on synthetic and real (vision and medical) data sets showing how group testing on such diversely sampled longitudinal data is possible by analyzing the reconstructed data in the subspace spanned by the first few principal components.

Computer-assisted hip resurfacing planning using Lie group shape models

INTRODUCTION

Hip resurfacing is a surgical option for osteoarthritis young and active patients. Early failures has been reported due to improper implant placement. Computer-assisted surgery is a promising avenue for more successful procedures.

PURPOSE

This paper presents a novel automatic surgical planning for computer-assisted hip resurfacing procedures. The plan defined the femoral head axis that was used to place the implant. The automatic planning was based on a Lie group statistical shape model.

METHODS

A statistical shape model was constructed using 50 femurs from osteoarthritis patients who underwent computer-assisted hip resurfacing. The model was constructed using product Lie groups representation of shapes and nonlinear analysis on the manifold of shapes. A surgical plan was drawn for the derived base shape. The base shape was transformed to 14 femurs with known manual plans. The transformed base plan was used as the computed plan for each femur. Both actual and computed plans were compared.

RESULTS

The method showed a success by computing plans that differ from the actual plans within the surgical admissible ranges. The minimum crossing distance between the two plans had a mean of 0.75 mm with a standard deviation of 0.54 mm. The angular difference between the two plans had the mean of 5.94° with a standard deviation of 2.145.94°.

CONCLUSION

Product Lie groups shape models were proved to be successful in automatic planning for hip resurfacing computer-assisted surgeries. The method can be extended to other orthopedic and general surgeries.

Visualization of shape motions in shape space

Analysis of dynamic object deformations such as cardiac motion is of great importance, especially when there is a necessity to visualize and compare the deformation behavior across subjects. However, there is a lack of effective techniques for comparative visualization and assessment of a collection of motion data due to its 4-dimensional nature, i.e., timely varying three-dimensional shapes. From the geometric point of view, the motion change can be considered as a function defined on the 2D manifold of the surface. This paper presents a novel classification and visualization method based on a medial surface shape space, in which two novel shape descriptors are defined, for discriminating normal and abnormal human heart deformations as well as localizing the abnormal motion regions. In our medial surface shape space, the geodesic distance connecting two points in the space measures the similarity between their corresponding medial surfaces, which can quantify the similarity and disparity of the 3D heart motions. Furthermore, the novel descriptors can effectively localize the inconsistently deforming myopathic regions on the left ventricle. An easy visualization of heart motion sequences on the projected space allows users to distinguish the deformation differences. Our experimental results on both synthetic and real imaging data show that this method can automatically classify the healthy and myopathic subjects and accurately detect myopathic regions on the left ventricle, which outperforms other conventional cardiac diagnostic methods.

Integrated task and motion planning in belief space

We describe an integrated strategy for planning, perception, state estimation and action in complex mobile manipulation domains based on planning in the belief space of probability distributions over states using hierarchical goal regression (pre-image back-chaining). We develop a vocabulary of logical expressions that describe sets of belief states, which are goals and subgoals in the planning process. We show that a relatively small set of symbolic operators can give rise to task-oriented perception in support of the manipulation goals. An implementation of this method is demonstrated in simulation and on a real PR2 robot, showing robust, flexible solution of mobile manipulation problems with multiple objects and substantial uncertainty.

Localized differences in caudate and hippocampal shape are associated with schizophrenia but not antipsychotic type

Caudate and hippocampal volume differences in patients with schizophrenia are associated with disease and antipsychotic treatment, but local shape alterations have not been thoroughly examined. Schizophrenia patients randomly assigned to haloperidol and olanzapine treatment underwent magnetic resonance imaging (MRI) at 3, 6, and 12 months. The caudate and hippocampus were represented as medial representations (M-reps); mesh structures derived from automatic segmentations of high resolution MRIs. Two quantitative shape measures were examined: local width and local deformation. A novel nonparametric statistical method, adjusted exponentially tilted (ET) likelihood, was used to compare the shape measures across the three groups while controlling for covariates. Longitudinal shape change was not observed in the hippocampus or caudate when the treatment groups and controls were examined in a global analysis, nor when the three groups were examined individually. Both baseline and repeated measures analysis showed differences in local caudate and hippocampal size between patients and controls, while no consistent differences were shown between treatment groups. Regionally specific differences in local hippocampal and caudate shape are present in schizophrenic patients. Treatment-related related longitudinal shape change was not observed within the studied timeframe. Our results provide additional evidence for disrupted cortico-basal ganglia-thalamo-cortical circuits in schizophrenia.

CLINICAL TRIAL INFORMATION

This longitudinal study was conducted from March 1, 1997 to July 31, 2001 at 14 academic medical centers (11 in the United States, one in Canada, one in the Netherlands, and one in England). This study was performed prior to the establishment of centralized registries of federally and privately supported clinical trials.

Maximum a posteriori estimation of linear shape variation with application to vertebra and cartilage modeling

The estimation of covariance matrices is a crucial step in several statistical tasks. Especially when using few samples of a high dimensional representation of shapes, the standard maximum likelihood estimation (ML) of the covariance matrix can be far from the truth, is often rank deficient, and may lead to unreliable results. In this paper, we discuss regularization by prior knowledge using maximum a posteriori (MAP) estimates. We compare ML to MAP using a number of priors and to Tikhonov regularization. We evaluate the covariance estimates on both synthetic and real data, and we analyze the estimates' influence on a missing-data reconstruction task, where high resolution vertebra and cartilage models are reconstructed from incomplete and lower dimensional representations. Our results demonstrate that our methods outperform the traditional ML method and Tikhonov regularization.

A statistical analysis of brain morphology using wild bootstrapping

Methods for the analysis of brain morphology, including voxel-based morphology and surface-based morphometries, have been used to detect associations between brain structure and covariates of interest, such as diagnosis, severity of disease, age, IQ, and genotype. The statistical analysis of morphometric measures usually involves two statistical procedures: 1) invoking a statistical model at each voxel (or point) on the surface of the brain or brain subregion, followed by mapping test statistics (e.g., t test) or their associated p values at each of those voxels; 2) correction for the multiple statistical tests conducted across all voxels on the surface of the brain region under investigation. We propose the use of new statistical methods for each of these procedures. We first use a heteroscedastic linear model to test the associations between the morphological measures at each voxel on the surface of the specified subregion (e.g., cortical or subcortical surfaces) and the covariates of interest. Moreover, we develop a robust test procedure that is based on a resampling method, called wild bootstrapping. This procedure assesses the statistical significance of the associations between a measure of given brain structure and the covariates of interest. The value of this robust test procedure lies in its computationally simplicity and in its applicability to a wide range of imaging data, including data from both anatomical and functional magnetic resonance imaging (fMRI). Simulation studies demonstrate that this robust test procedure can accurately control the family-wise error rate. We demonstrate the application of this robust test procedure to the detection of statistically significant differences in the morphology of the hippocampus over time across gender groups in a large sample of healthy subjects.

How to put probabilities on homographies

We present a family of "normal" distributions over a matrix group together with a simple method for estimating its parameters. In particular, the mean of a set of elements can be calculated. The approach is applied to planar projective homographies, showing that using priors defined in this way improves object recognition.

Principal geodesic analysis for the study of nonlinear statistics of shape

A primary goal of statistical shape analysis is to describe the variability of a population of geometric objects. A standard technique for computing such descriptions is principal component analysis. However, principal component analysis is limited in that it only works for data lying in a Euclidean vector space. While this is certainly sufficient for geometric models that are parameterized by a set of landmarks or a dense collection of boundary points, it does not handle more complex representations of shape. We have been developing representations of geometry based on the medial axis description or m-rep. While the medial representation provides a rich language for variability in terms of bending, twisting, and widening, the medial parameters are not elements of a Euclidean vector space. They are in fact elements of a nonlinear Riemannian symmetric space. In this paper, we develop the method of principal geodesic analysis, a generalization of principal component analysis to the manifold setting. We demonstrate its use in describing the variability of medially-defined anatomical objects. Results of applying this framework on a population of hippocampi in a schizophrenia study are presented.