In this paper, we consider the problem of fast and efficient indexing techniques for human activity sequences evolving in Euclidean and non-Euclidean spaces. This problem has several applications in the areas of human activity analysis, where there is a need to perform fast search and recognition in large databases. The problem is made more challenging when features such as landmarks, contours,...

A. Cayley and F. Klein discovered in the nineteenth century that euclidean and non-euclidean geometries can be considered as mathematical structures living inside projective-metric spaces. They outlined this idea with respect to the real projective plane and established (“begründeten”) in this way the hyperbolic and elliptic geometry. The generalization of this approach to projective spaces ove...

A methodology is given to determine the effect of different mass distributions for triatomic reactions using the geometry of shape space. Atomic masses are incorporated into the non-Euclidean shape space metric after the separation of rotations. Using the equations of motion in this non-Euclidean shape space, an averaged field of velocity-dependent fictitious forces is determined. This force fi...

In this paper we study the sub-Finsler geometry as a time-optimal control problem. In particular, we consider non-smooth and non-strictly convex sub-Finsler structures associated with the Heisenberg, Grushin, and Martinet distributions. Motivated by problems in geometric group theory, we characterize extremal curves, discuss their optimality, and calculate the metric spheres, proving their Eucl...

Noisy intermediate-scale quantum computers are useful for various tasks such as state preparation and variational algorithms. However, the non-Euclidean geometry of parameterized circuits is detrimental these applications. Here we introduce natural circuit (NPQC) that can be initialized with a Euclidean geometry. The initial training algorithms substantially sped up gradient equivalent to gradi...

Manifold learning and finding low-dimensional structure in data is an important task. Many algorithms for this purpose embed data in Euclidean space, an approach which is destined to fail on non-flat data. This paper presents a non-iterative algebraic method for embedding the data into hyperbolic and spherical spaces. We argue that these spaces are often better than Euclidean space in capturing...

Kant argued that Euclidean geometry is synthesized on the basis of an a priori intuition of space. This proposal inspired much behavioral research probing whether spatial navigation in humans and animals conforms to the predictions of Euclidean geometry. However, Euclidean geometry also includes concepts that transcend the perceptible, such as objects that are infinitely small or infinitely lar...

—A new multiscale implementation of non-local means filtering for image denoising is proposed. The proposed algorithm also introduces a modification of similarity measure for patch comparison. The standard Euclidean norm is replaced by weighted Euclidean norm for patch based comparison. Assuming the patch as an oriented surface, notion of normal vector patch is being associated with each patch....