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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)
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)
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)
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)