This paper is to construct Riesz wavelets with short support. Riesz wavelets with short support are of interests in both theory and application. In theory, it is known that a B-spline of order m has the shortest support among all compactly supported refinable functions with the same regularity. However, it remained open whether a Riesz wavelet with the shortest support and m vanishing moments c...

Hierarchical Powell-Sabin splines are C-continuous piecewise quadratic polynomials defined on a hierarchical triangulation. The mesh is obtained by partitioning an initial conforming triangulation locally with a triadic split, so that it is no longer conforming. We propose a normalized quasi-hierarchical basis for this spline space. The B-spline basis functions have a local support, they form a...

The first type of pseudo-splines were introduced by [Daubechies, Han, Ron and Shen, 2003] (DHRS) to construct tight framelets with desired approximation orders via the unitary extension principle of [Ron and Shen, 1997]. In the spirit of the first type of pseudo-splines, we introduce here a new type (the second type) of pseudo-splines to construct symmetric or antisymmetric tight framelets with...

We present a novel method named truncated hierarchical unstructured splines (THU-splines) that supports both local h-refinement and quadrilateral meshes. In THU-spline construction, an mesh is taken as the input control mesh, where degenerated-patch [20] adopted in irregular regions to define $$C^1$$ -continuous bicubic splines, whereas regular only involve $$C^2$$ B-splines. Irregular are then...

We prove that the Toeplitz binary value matrix inversion associated with mth order E-spline interpolation can be implemented using only 2 (m + 1) additions. We develop pipelined architectures for real time B-spline interpolation based on simple running average filters. We also show that an ideal interpolating function, which is approximated by a truncated sinc function with M half cycles, can b...

We present multiresolution B-spline surfaces of arbitrary order defined over triangular domains. Unlike existing methods, the basic idea of our approach is to construct the triangular basis functions from their tensor product relatives in the spirit of box splines by projecting them onto the barycentric plane. The scheme works for splines of any order where the fundamental building blocks of th...

A novel technique is presented to facilitate the implementation of hierarchical b-splines and their interfacing with conventional finite element implementations. The discrete interpretation of the two-scale relation, as common in subdivision schemes, is used to establish algebraic relations between the basis functions and their coefficients on different levels of the hierarchical b-spline basis...

In this paper we present a methodology to create higher order parametric trivariate representations such as B-splines or T-splines, from closed triangle meshes with higher genus or bifurcations. The input can consist of multiple interior boundaries which represent inner object material attributes. Fundamental to our approach is the use of a midsurface in combination with harmonic functions to d...

The Hybrid Spline method (H-spline) is a method of density estimation which involves regression splines and smoothing splines methods. Using basis functions (B-splines), this method is much faster than Smoothing Spline Density Estimation approach (Gu, 1993). Simulations suggest that with more structured data (e.g., several modes) H-spline method estimates the modes as well as Logspline (Kooperb...

Learning density estimation is important in probabilistic modeling and reasoning with uncertainty. Since B-spline basis functions are piecewise polynomials local support, B-splines shows its advantages when intensive numerical computations involved the subsequent applications. To obtain an optimal B-splines, we need to select bandwidth (i.e., distance of two adjacent knots) for uniform B-spline...