A New Four Point Circular-Invariant Corner-Cutting Subdivision for Curve Design

A 4-point nonlinear corner-cutting subdivision scheme is established. It is induced from a special C-shaped biarc circular spline structure. The scheme is circular-invariant and can be effectively applied to 2-dimensional (2D) data sets that are locally convex. The scheme is also extended adaptively to non-convex data. Explicit examples are demonstrated

On a-ary Subdivision for Curve Design: I. 4-Point and 6-Point Interpolatory Schemes

The classical binary 4-point and 6-point interpolatery subdivision schemes are generalized to a-ary setting for any integer a greater than or equal to 3. These new a-ary subdivision schemes for curve design are derived easily from their corresponding two-scale scaling functions, a notion from the context of wavelets

On a-ary Subdivision for Curve Design III. 2m-Point and (2m + 1)-Point Interpolatory Schemes

In this paper, we investigate both the 2m-point a-ary for any a ≥ 2 and (2m + 1)-point a-ary for any odd a ≥ 3 interpolatory subdivision schemes for curve design. These schemes include the extended family of the classical 4- and 6-point interpolatory a-ary schemes and the family of the 3- and 5-point a-ary interpolatory schemes, both having been established in our previous papers (Lian [9]) and (Lian [10])

Bidimensional PR QMF with FIR Filters

Multidimensional perfect reconstruction (PR) quadrature mirror filter (QMF) banks with finite impulse response (FIR) filters induced from systems of biorthogonal multivariate scaling functions and wavelets are investigated. In particular, bivariate scaling functions and wavelets with dilation as an expansive integer matrix whose determinant is two in absolute value are considered. Demonstrative quincunxial examples are explicitly given and new FIR filters are constructed

A New Family of PR Two Channel Filter Banks

A new family of multidimensional dimensional (MD) perfect reconstruction (PR) two channel filter banks with finite impulse response (FIR) filters induced from systems of biorthogonal MD scaling functions and wavelets are introduced. One of the advantages of this construction is that the biorthogonal scaling functions and wavelets are easy to establish due to the interpolatory property of the scaling functions to start with. The other advantage is that all filters can be centrosymmetric or bi-linear phase. Examples of two dimensional (2D) bi-linear phase PR twochannel FIR filter banks will be demonstrated

On a-ary Subdivision for Curve Design II. 3-Point and 5-Point Interpolatory Schemes

The a-ary 3-point and 5-point interpolatery subdivision schemes for curve design are introduced for arbitrary odd integer a greater than or equal to 3. These new schemes further extend the family of the classical 4- and 6-point interpolatory schemes

A Family of Householder Matrices

A Householder transformation, or Householder reflection, or Household matrix, is a reflection about a hyperplane with a unit normal vector. Not only have the Household matrices been used in QR decomposition efficiently but also implicitly and successfully applied in other areas. In the process of investigating a family of unitary filterbanks, a new family of Householder matrices are established. These matrices are produced when a matrix filter is required to preserve certain order of 2d digital polynomial signals. Naturally, they can be applied to image and signal processing among others

Construction of Energy Preserving QMF

Recently, a family of perfect reconstruction (PR) quadrature mirror filterbanks (QMF) with finite impulse response filters (FIR) from systems of biorthogonal refinable functions and wavelets were introduced and also applied to image processing. However, a detailed procedure was absent. The main objective of this paper is to present extensive examples that will provide a thorough process of construction of the new family of PR QMF with FIR filterbanks. These new filters are linearphase due to the symmetry property of their corresponding biorthogonal refinable functions and wavelets. In addition, these filters have odd lengths so that the symmetric extension can be easily applied. Another important feature is that the filters preserve energy (EP) very well. The notion of Condition EP was thus introduced for the purpose of further examining these features

Circular Nonlinear Subdivision Schemes for Curve Design

Two new families of nonlinear 3-point subdivision schemes for curve design are introduced. The first family is ternary interpolatory and the second family is binary approximation. All these new schemes are circular-invariant, meaning that new vertices are generated from local circles formed by three consecutive old vertices. As consequences of the nonlinear schemes, two new families of linear subdivision schemes for curve design are established. The 3-point linear binary schemes, which are corner-cutting depending on the choices of the tension parameter, are natural extensions of the Lane-Riesenfeld schemes. The four families of both nonlinear and linear subdivision schemes are implemented extensively by a variety of examples