Some families of generating functions for the multiple orthogonal polynomials associated with modified Bessel K-functions

AbstractThe main object of this paper is to derive several substantially more general families of bilinear, bilateral, and mixed multilateral finite-series relationships and generating functions for the multiple orthogonal polynomials associated with the modified Bessel K-functions also known as Macdonald functions. Some special cases of the above statements are also given

Finsler geometry on higher order tensor fields and applications to high angular resolution diffusion imaging.

We study 3D-multidirectional images, using Finsler geometry. The application considered here is in medical image analysis, specifically in High Angular Resolution Diffusion Imaging (HARDI) (Tuch et al. in Magn. Reson. Med. 48(6):1358–1372, 2004) of the brain. The goal is to reveal the architecture of the neural fibers in brain white matter. To the variety of existing techniques, we wish to add novel approaches that exploit differential geometry and tensor calculus. In Diffusion Tensor Imaging (DTI), the diffusion of water is modeled by a symmetric positive definite second order tensor, leading naturally to a Riemannian geometric framework. A limitation is that it is based on the assumption that there exists a single dominant direction of fibers restricting the thermal motion of water molecules. Using HARDI data and higher order tensor models, we can extract multiple relevant directions, and Finsler geometry provides the natural geometric generalization appropriate for multi-fiber analysis. In this paper we provide an exact criterion to determine whether a spherical function satisfies the strong convexity criterion essential for a Finsler norm. We also show a novel fiber tracking method in Finsler setting. Our model incorporates a scale parameter, which can be beneficial in view of the noisy nature of the data. We demonstrate our methods on analytic as well as simulated and real HARDI data

AA-statistical convergence for a class of positive linear operators

In this paper we introduce a sequence of positive linear operators defined on the space $C[0,a]$ $(0<a<1)$ and provide an approximation theorem for these operators via the concept of $A$-statistical convergence. We also compute the rates of convergence of these approximation operators by means of the first and second order modulus of continuity and the elements of the Lipschitz class. Furthermore, by defining the generalization of $r$-th order of these operators we show that the similar approximation properties are preserved on \(C[0,a].\

Complete Set of Invariants of a 4th Order Tensor: The 12 Tasks of HARDI from Ternary Quartics

International audienceInvariants play a crucial role in Diffusion MRI. In DTI (2 nd order tensors), invariant scalars (FA, MD) have been successfully used in clinical applications. But DTI has limitations and HARDI models (e.g. 4 th order tensors) have been proposed instead. These, however, lack invariant features and computing them systematically is challenging. We present a simple and systematic method to compute a functionally complete set of invariants of a non-negative 3D 4 th order tensor with re-spect to SO3. Intuitively, this transforms the tensor's non-unique ternary quartic (TQ) decomposition (from Hilbert's theorem) to a unique canon-ical representation independent of orientation – the invariants. The method consists of two steps. In the first, we reduce the 18 degrees-of-freedom (DOF) of a TQ representation by 3-DOFs via an orthogonal transformation. This transformation is designed to enhance a rotation-invariant property of choice of the 3D 4 th order tensor. In the second, we further reduce 3-DOFs via a 3D rotation transformation of coordinates to arrive at a canonical set of invariants to SO3 of the tensor. The resulting invariants are, by construction, (i) functionally complete, (ii) functionally irreducible (if desired), (iii) computationally efficient and (iv) reversible (mappable to the TQ coefficients or shape); which is the novelty of our contribution in comparison to prior work. Results from synthetic and real data experiments validate the method and indicate its importance

Mixed-Model Noise Removal in 3D MRI via Rotation-and-Scale Invariant Non-Local Means

Mixed noise is a major issue influencing quantitative analysis in different forms of magnetic resonance image (MRI), such as T1 and diffusion image like DWI and DTI. Using different filters sequentially to remove mixed noise will severely deteriorate such medical images. We present a novel algorithm called rotation-and-scale invariant nonlocal means filter (RSNLM) to simultaneously remove mixed noise from different kinds of three-dimensional (3D) MRI images. First, we design a new similarity weights, including rank-ordered absolute difference (ROAD), coming from a trilateral filter (TriF) that is obtained to detect the mixed and high-level noise. Then, we present a shape view to consider the MRI data as a 3D operator, with which the similarity between the patches is calculated with the rigid transformation. The translation, rotation and scale have no influence on the similarity. Finally, the adaptive parameter estimation method of ROAD is illustrated, and the effective proof that validates the proposed algorithm is presented. Experiments using synthetic data with impulse noise, Rician noise, and the real MRI data confirm that the proposed method yields superior performance compared with current state-of-the-art methods