scholarly journals Quad/triangle subdivision, nonhomogeneous refinement equation and polynomial reproduction

2012 ◽  
Vol 82 (11) ◽  
pp. 2215-2237 ◽  
Author(s):  
Qingtang Jiang ◽  
Baobin Li
Keyword(s):  
Refinement Equation

Using the refinement equation for the construction of pre-wavelets

1991 ◽  
Vol 1 (1) ◽  
pp. 75-116 ◽  
Author(s):  
Charles A. Micchelli
Keyword(s):  
Refinement Equation

ON SMOOTHNESS FOR A CLASS OF FRACTAL INTERPOLATION SURFACES

Fractals ◽  
2006 ◽  
Vol 14 (03) ◽  
pp. 223-230 ◽  
Author(s):  
HONG-YONG WANG
Keyword(s):  
Wide Class ◽  
Iterated Function Systems ◽  
Fractal Interpolation ◽  
Refinement Equation ◽  
Fractal Interpolation Functions ◽  
Iterated Function ◽  
Interpolation Functions

In this paper, we consider a wide class of iterated function systems in R3, and show that their attractors are a class of fractal interpolation surfaces. Based on a refinement equation, we investigate the properties of smoothness of the fractal interpolation functions, and give the results of the smoothness in several cases.

Smoothness of refinable function vectors on ℝn

2017 ◽  
Vol 15 (05) ◽  
pp. 1750051
Author(s):  
Ramazan Tinaztepe ◽  
Denise Jacobs ◽  
Christopher Heil
Keyword(s):  
Scaling Function ◽  
Joint Spectral Radius ◽  
Refinement Equation ◽  
Dilation Matrix ◽  
Finite Set ◽  
Expansive Matrix ◽  
Refinable Function Vectors ◽  
Multiplicity Formula ◽  
Compactly Supported Solutions ◽  
Order Smoothness

Let [Formula: see text] be a dilation matrix, an [Formula: see text] expansive matrix that maps [Formula: see text] into itself. Let [Formula: see text] be a finite subset of [Formula: see text] and for [Formula: see text] let [Formula: see text] be [Formula: see text] complex matrices. The refinement equation corresponding to [Formula: see text] and [Formula: see text] is [Formula: see text] A solution [Formula: see text] if one exists, is called a refinable vector function or a vector scaling function of multiplicity [Formula: see text] This paper characterizes the higher-order smoothness of compactly supported solutions of the refinement equation, in terms of the [Formula: see text]-norm joint spectral radius of a finite set of finite matrices determined by the coefficients [Formula: see text]

REFINABLE SHIFT INVARIANT SPACES IN ℝd

2005 ◽  
Vol 03 (03) ◽  
pp. 321-345
Author(s):  
CARLOS A. CABRELLI ◽  
SIGRID B. HEINEKEN ◽  
URSULA M. MOLTER
Keyword(s):  
Dimensional Subspace ◽  
Finite Dimensional Subspace ◽  
Invariant Space ◽  
Refinement Equation ◽  
Compactly Supported Function ◽  
Dilation Matrix ◽  
Jordan Basis ◽  
Shift Invariant Spaces ◽  
Invariant Spaces ◽  
Shift Invariant Space

Let φ : ℝd → ℂ be a compactly supported function which satisfies a refinement equation of the form [Formula: see text] where Γ ⊂ ℝd is a lattice, Λ is a finite subset of Γ, and A is a dilation matrix. We prove, under the hypothesis of linear independence of the Γ-translates of φ, that there exists a correspondence between the vectors of the Jordan basis of a finite submatrix of L = [cAi-j]i,j∈Γ and a finite-dimensional subspace [Formula: see text] in the shift-invariant space generated by φ. We provide a basis of [Formula: see text] and show that its elements satisfy a property of homogeneity associated to the eigenvalues of L. If the function φ has accuracy κ, this basis can be chosen to contain a basis for all the multivariate polynomials of degree less than κ. These latter functions are associated to eigenvalues that are powers of the eigenvalues of A-1. Furthermore we show that the dimension of [Formula: see text] coincides with the local dimension of φ, and hence, every function in the shift-invariant space generated by φ can be written locally as a linear combination of translates of the homogeneous functions.

Using the refinement equation for the construction of pre-wavelets III: Elliptic splines

1991 ◽  
Vol 1 (3) ◽  
pp. 331-351 ◽  
Author(s):  
Charles A. Micchelli ◽  
Christophe Rabut ◽  
Florencio I. Utreras
Keyword(s):  
Refinement Equation

Image Denoising in Industry Based on Multiwavelet Riesz Bases

2013 ◽  
Vol 675 ◽  
pp. 59-62
Author(s):  
Qi Chao Song ◽  
Zhi Song Liu ◽  
Chao Ping Wang
Keyword(s):  
Image Denoising ◽  
Riesz Bases ◽  
Refinement Equation ◽  
Compactly Supported

Damage testing of components is a key point in many industry fields. In some cases, endoscope is used to inspect the damage part, while the images are often noised. In this paper, we focus on industrial image denoising based on multiwavelet Riesz bases. Starting from compactly supported vector refinement equation, we provide a characterization to form two Riesz bases and an example is given. Based on example Riesz bases, we research industrial endoscope image denoising and get satisfying result.

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