Practical statistical analysis of diffusion tensor images is considered, and we focus primarily on methods that use metrics based on Euclidean distances between powers of diffusion tensors. First we describe a family of anisotropy measures based on a scale invariant power-Euclidean metric, which are useful for visualisation. Some properties of the measures are derived and practical consideratio...

We give a metric characterization of snowflakes of Euclidean spaces. Namely, a metric space is isometric to Rn equipped with a distance (dE) , for some n ∈ N0 and ∈ (0, 1], where dE is the Euclidean distance, if and only if it is locally compact, 2-point isometrically homogeneous, and admits dilations of any factor.

We introduce a notion of the Euclidean and the Minkowski rank for arbitrary metric spaces and we study their behaviour with respect to products. We show that the Minkowski rank is additive with respect to metric products, while additivity of the Euclidean rank does not hold in general.

An embedding of one metric space (X, d) into another (Y, ρ) is an injective map f : X → Y . The central genre of problems in the area of metric embedding is finding such maps in which the distances between points do not change “too much”. Metric Embedding plays an important role in a vast range of application areas such as computer vision, computational biology, machine learning, networking, st...

In solving pattern recognition problem in the Euclidean space, prototypes representing classes are de ned. On the other hand in the metric space, Nearest Neighbor method and K-Nearest Neighbor method are frequently used without de ning any prototypes. In this paper, we propose a new pattern recognition method for the metric space that can use prototypes which are the centroid of any three patte...

In this letter, we introduce the Log-Gaussian Process Implicit Surface (Log-GPIS), a novel continuous and probabilistic mapping representation suitable for surface reconstruction local navigation. Our key contribution is realisation that regularised Eikonal equation can be simply solved by applying logarithmic transformation to GPIS formulation recover accurate Euclidean distance field (EDF) an...

Let (X, d) be a metric space and m ∈ X. Suppose that φ : X×X → R is a nonnegative symmetric function. We define a metric d on X which is equivalent to d. If d is totally bounded, its completion is a compactification of (X, d). As examples, we construct two compactifications of (R, dE), where dE is the Euclidean metric and s ≥ 2. key words. equivalent metric; completion; compactification Mathema...