Monami Banerjee

A Nonlinear Regression Technique for Manifold Valued Data with Applications to Medical Image Analysis

Regression is an essential tool in Statistical analysis of data with many applications in Computer Vision, Machine Learning, Medical Imaging and various disciplines of Science and Engineering. Linear and nonlinear regression in a vector space setting has been well studied in literature. However, generalizations to manifold-valued data are only recently gaining popularity. With the exception of a few, most existing methods of regression for manifold valued data are limited to geodesic regression which is a generalization of the linear regression in vector-spaces. In this paper, we present a novel nonlinear kernel-based regression method that is applicable to manifold valued data. Our method is applicable to cases when the independent and dependent variables in the regression model are both manifold-valued or one is manifold-valued and the other is vector or scalar valued. Further, unlike most methods, our method does not require any imposed ordering on the manifold-valued data. The performance of our model is tested on a large number of real data sets acquired from Alzhiemers and movement disorder (Parkinsons and Essential Tremor) patients. We present an extensive set of results along with statistical validation and comparisons.

Nonlinear Regression on Riemannian Manifolds and Its Applications to Neuro-Image Analysis

Regression in its most common form where independent and dependent variables are in ℝ n is a ubiquitous tool in Sciences and Engineering. Recent advances in Medical Imaging has lead to a wide spread availability of manifold-valued data leading to problems where the independent variables are manifold-valued and dependent are real-valued or vice-versa. The most common method of regression on a manifold is the geodesic regression, which is the counterpart of linear regression in Euclidean space. Often, the relation between the variables is highly complex, and existing most commonly used geodesic regression can prove to be inaccurate. Thus, it is necessary to resort to a non-linear model for regression. In this work we present a novel Kernel based non-linear regression method when the mapping to be estimated is either from M → ℝ n or ℝ n → M, where M is a Riemannian manifold. A key advantage of this approach is that there is no requirement for the manifold-valued data to necessarily inherit an ordering from the data in ℝ n . We present several synthetic and real data experiments along with comparisons to the state-of-the-art geodesic regression method in literature and thus validating the effectiveness of the proposed algorithm.

Statistics on the space of trajectories for longitudinal data analysis

Statistical analysis of longitudinal data is a significant problem in Biomedical imaging applications. In the recent past, several researchers have developed mathematically rigorous methods based on differential geometry and statistics to tackle the problem of statistical analysis of longitudinal neuroimaging data. In this paper, we present a novel formulation of the longitudinal data analysis problem by identifying the structural changes over time (describing the trajectory of change) to a product Riemannian manifold endowed with a Riemannian metric and a probability measure. We present theoretical results showing that the maximum likelihood estimate of the mean and median of a Gaussian and Laplace distribution respectively on the product manifold yield the Fréchet mean and median respectively. We then present efficient recursive estimators for these intrinsic parameters and use them in conjunction with a nearest neighbor (NN) classifier to classify MR brain scans (acquired from the publicly available OASIS database) of patients with and without dementia.

A Method for Automated Classification of Parkinson's Disease Diagnosis Using an Ensemble Average Propagator Template Brain Map Estimated from Diffusion MRI.

Parkinson’s disease (PD) is a common and debilitating neurodegenerative disorder that affects patients in all countries and of all nationalities. Magnetic resonance imaging (MRI) is currently one of the most widely used diagnostic imaging techniques utilized for detection of neurologic diseases. Changes in structural biomarkers will likely play an important future role in assessing progression of many neurological diseases inclusive of PD. In this paper, we derived structural biomarkers from diffusion MRI (dMRI), a structural modality that allows for non-invasive inference of neuronal fiber connectivity patterns. The structural biomarker we use is the ensemble average propagator (EAP), a probability density function fully characterizing the diffusion locally at a voxel level. To assess changes with respect to a normal anatomy, we construct an unbiased template brain map from the EAP fields of a control population. Use of an EAP captures both orientation and shape information of the diffusion process at each voxel in the dMRI data, and this feature can be a powerful representation to achieve enhanced PD brain mapping. This template brain map construction method is applicable to small animal models as well as to human brains. The differences between the control template brain map and novel patient data can then be assessed via a nonrigid warping algorithm that transforms the novel data into correspondence with the template brain map, thereby capturing the amount of elastic deformation needed to achieve this correspondence. We present the use of a manifold-valued feature called the Cauchy deformation tensor (CDT), which facilitates morphometric analysis and automated classification of a PD versus a control population. Finally, we present preliminary results of automated discrimination between a group of 22 controls and 46 PD patients using CDT. This method may be possibly applied to larger population sizes and other parkinsonian syndromes in the near future.

Robust Fréchet Mean and PGA on Riemannian Manifolds with Applications to Neuroimaging

In this paper, we present novel algorithms to compute robust statistics from manifold-valued data. Specifically, we present algorithms for estimating the robust Frechet Mean (FM) and performing a robust exact-principal geodesic analysis (ePGA) for data lying on known Riemannian manifolds. We formulate the minimization problems involved in both these problems using the minimum distance estimator called the L\(_2\)E. This leads to a nonlinear optimization which is solved efficiently using a Riemannian accelerated gradient descent technique. We present competitive performance results of our algorithms applied to synthetic data with outliers, the corpus callosum shapes extracted from OASIS MRI database, and diffusion MRI scans from movement disorder patients respectively.