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We prove that a compactly supported spline function of degree k satisses the scaling equation (x) = P N n=0 c(n)(mx?n) for some integer m 2, if and only if (x) = P n p(n)B k (x?n) where p(n) are the coeecients of a polynomial P(z) such that the roots of P(z)(z ? 1) k+1 are mapped into themselves by the mapping z ! z m , and B k is the uniform B-spline of… (More)

- W Lawton, S L Lee, Zuowei Shen
- 1997

This paper characterizes the stability and orthonormality of the shifts of a multidimensional (M; c) reenable function in terms of the eigenvalues and eigenvectors of the transition operator W cau deened by the autocorrelation c au of its reenement mask c; where M is an arbitrary dilation matrix. Another consequence is that if the shifts of form a Riesz… (More)

This paper gives a practical method of extending an n × r matrix P (z), r ≤ n, with Laurent polynomial entries in one complex variable z, to a square matrix also with Laurent polynomial entries. If P (z) has orthonormal columns when z is restricted to the torus T, it can be extended to a paraunitary matrix. If P (z) has rank r for each z ∈ T, it can be… (More)

- W Lawton, S L Lee, Zuowei Shen
- 1998

Necessary and suucient conditions on the spectrum of the restricted transition operators are given for the convergence in L 2 (R d) of the multidimensional cascade algorithm .

Let C denote the complex numbers and L denote the ring of complex-valued Laurent polynomial functions on C n f0g. Furthermore, we denote by L R , L N the subsets of Laurent polynomials whose restriction to the unit circle is real, nonnegative, respectively. We prove that for any two Laurent polynomials P 1 ; P 2 2 L N ; which have no common zeros in C n f0g… (More)

Conjugate quadrature lters with multiple zeros at 1 have classical applications to unitary subband coding of signals using exact reconstruction lter banks. Recent work shows how to construct, given a set of n negative numbers, a CQF whose degree does not exceed 2n-1 and whose zeros contain the speciied negative numbers, and applies such lters to… (More)

Medical images of such branching structures as blood vessels are clear to the human visual system, but easily confuse computer vision programs. We present an intuitive, hand-eye coordinated, reach-in interface that allows the user to sketch central curves for arteries, nerves, etc., detectable in volume data and in 3D space, and have this position/shape… (More)

- WAYNE LAWTON
- 2000

Sufficient conditions for the convergence in distribution of an infinite convolution product µ 1 * µ 2 *. .. of measures on a connected Lie group G with respect to left invariant Haar measure are derived. These conditions are used to construct distributions φ that satisfy T φ = φ where T is a refinement operator constructed from a measure µ and a dilation… (More)