Automated feature extraction from MRI brain scans and diagnosis of Alzheimer's disease are ongoing challenges. With advances in 3D imaging technology, 3D data acquisition is becoming more viable and efficient than its 2D counterpart. Rather than using feature-based vectors, in this paper, for the first time, we suggest a pipeline to extract novel covariance-based descriptors from the cortical surface using the Ricci energy optimization. The covariance descriptors are components of the nonlinear manifold of symmetric positive-definite matrices, thus we focus on using the Gaussian radial basis function to apply manifold-based classification to the 3D shape problem. Applying this novel signature to the analysis of abnormal cortical brain morphometry allows for diagnosing Alzheimer's disease. Experimental studies performed on about two hundred 3D MRI brain models, gathered from Alzheimer's Disease Neuroimaging Initiative (ADNI) dataset demonstrate the effectiveness of our descriptors in achieving remarkable classification accuracy.

Differential privacy has become crucial in the real-world deployment of statistical and machine learning algorithms with rigorous privacy guarantees. The earliest statistical queries, for which differential privacy mechanisms have been developed, were for the release of the sample mean. In Geometric Statistics, the sample Fr\'echet mean represents one of the most fundamental statistical summaries, as it generalizes the sample mean for data belonging to nonlinear manifolds. In that spirit, the only geometric statistical query for which a differential privacy mechanism has been developed, so far, is for the release of the sample Fr\'echet mean: the \emph{Riemannian Laplace mechanism} was recently proposed to privatize the Fr\'echet mean on complete Riemannian manifolds. In many fields, the manifold of Symmetric Positive Definite (SPD) matrices is used to model data spaces, including in medical imaging where privacy requirements are key. We propose a novel, simple and fast mechanism - the \emph{tangent Gaussian mechanism} - to compute a differentially private Fr\'echet mean on the SPD manifold endowed with the log-Euclidean Riemannian metric. We show that our new mechanism has significantly better utility and is computationally efficient -- as confirmed by extensive experiments.

It has been reported repeatedly that discriminative learning of distance metric boosts the pattern recognition performance. A weak point of ITML-based methods is that the distance threshold for similarity/dissimilarity constraints must be determined manually and it is sensitive to generalization performance, although the ITML-based methods enjoy an advantage that the Bregman projection framework can be applied for optimization of distance metric. In this paper, we present a new formulation of metric learning algorithm in which the distance threshold is optimized together. Since the optimization is still in the Bregman projection framework, the Dykstra algorithm can be applied for optimization. A nonlinear equation has to be solved to project the solution onto a half-space in each iteration. Na\"{i}ve method takes $O(LMn^{3})$ computational time to solve the nonlinear equation. In this study, an efficient technique that can solve the nonlinear equation in $O(Mn^{3})$ has been discovered. We have proved that the root exists and is unique. We empirically show that the accuracy of pattern recognition for the proposed metric learning algorithm is comparable to the existing metric learning methods, yet the distance threshold is automatically tuned for the proposed metric learning algorithm.

Covariance and histogram image descriptors provide an effective way to capture information about images. Both excel when used in combination with special purpose distance metrics. For covariance descriptors these metrics measure the distance along the non-Euclidean Riemannian manifold of symmetric positive definite matrices. For histogram descriptors the Earth Mover's distance measures the optimal transport between two histograms. Although more precise, these distance metrics are very expensive to compute, making them impractical in many applications, even for data sets of only a few thousand examples. In this paper we present two methods to compress the size of covariance and histogram datasets with only marginal increases in test error for k-nearest neighbor classification. Specifically, we show that we can reduce data sets to 16% and in some cases as little as 2% of their original size, while approximately matching the test error of kNN classification on the full training set. In fact, because the compressed set is learned in a supervised fashion, it sometimes even outperforms the full data set, while requiring only a fraction of the space and drastically reducing test-time computation.

Human action recognition from videos is a challenging machine vision task with multiple important application domains, such as human-robot/machine interaction, interactive entertainment, multimedia information retrieval, and surveillance. In this paper, we present a novel approach to human action recognition from 3D skeleton sequences extracted from depth data. We use the covariance matrix for skeleton joint locations over time as a discriminative descriptor for a sequence. To encode the relationship between joint movement and time, we deploy multiple covariance matrices over sub-sequences in a hierarchical fashion. The descriptor has a fixed length that is independent from the length of the described sequence. Our experiments show that using the covariance descriptor with an off-the-shelf classification algorithm outperforms the state of the art in action recognition on multiple datasets, captured either via a Kinect-type sensor or a sophisticated motion capture system. We also include an evaluation on a novel large dataset using our own annotation.

Robust visual tracking for long video sequences is a research area that has many important applications. The main challenges include how the target image can be modeled and how this model can be updated. In this paper, we model the target using a covariance descriptor, as this descriptor is robust to problems such as pixel-pixel misalignment, pose and illumination changes, that commonly occur in visual tracking. We model the changes in the template using a generative process. We introduce a new dynamical model for the template update using a random walk on the Riemannian manifold where the covariance descriptors lie in. This is done using log-transformed space of the manifold to free the constraints imposed inherently by positive semidefinite matrices. Modeling template variations and poses kinetics together in the state space enables us to jointly quantify the uncertainties relating to the kinematic states and the template in a principled way. Finally, the sequential inference of the posterior distribution of the kinematic states and the template is done using a particle filter. Our results shows that this principled approach can be robust to changes in illumination, poses and spatial affine transformation. In the experiments, our method outperformed the current state-of-the-art algorithm - the incremental Principal Component Analysis method, particularly when a target underwent fast poses changes and also maintained a comparable performance in stable target tracking cases.

Here novelty detection identifies salient image features to guide autonomous robotic exploration.  There is little advance knowledge of the features in the scene or the proportion that should count as outliers.  A new algorithm addresses this ambiguity by modeling novel data in advance and characterizing regular data at run time.  Detection thresholds adapt dynamically to reduce misclassification risk while accommodating homogeneous and heterogeneous scenes.  Experiments demonstrate the technique on a representative set of navigation images from the Mars Exploration Rover "Opportunity."  An efficient image analysis procedure filters each image using the integral transform.  Pixel-level features are aggregated into covariance descriptors that represent larger regions.  Finally, a distance metric derived from generalized eigenvalues permits novelty detection with kernel density estimation.  Results suggest that exploiting training examples of novel data can improve performance in this domain.