Bi-frames with 4-fold axial symmetry for quadrilateral surface multiresolution processing

When bivariate filter banks and wavelets are used for surface multiresolution processing, it is required that the decomposition and reconstruction algorithms for regular vertices derived from them have high symmetry. This symmetry requirement makes it possible to design the corresponding multiresolution algorithms for extraordinary vertices. Recently lifting-scheme based biorthogonal bivariate wavelets with high symmetry have been constructed for surface multiresolution processing. If biorthogonal wavelets have certain smoothness, then the analysis or synthesis scaling function or both have big supports in general. In particular, when the synthesis low-pass filter is a commonly used scheme such as Loop’s scheme or Catmull-Clark’s scheme, the corresponding analysis low-pass filter has a big support and the corresponding analysis scaling function and wavelets have poor smoothness. Big supports of scaling functions, or in other words big templates of multiresolution algorithms, are undesirable for surface processing. On the other hand, a frame provides flexibility for the construction of “basis” systems. This paper concerns the construction of wavelet (or affine) bi-frames with high symmetry.In this paper we study the construction of wavelet bi-frames with 4-fold symmetry for quadrilateral surface multiresolution processing, with both the dyadic and refinements considered. The constructed bi-frames have 4 framelets (or frame generators) for the dyadic refinement, and 2 framelets for the refinement. Namely, with either the dyadic or refinement, a frame system constructed in this paper has only one more generator than a wavelet system. The constructed bi-frames have better smoothness and smaller supports than biorthogonal wavelets. Furthermore, all the frame algorithms considered in this paper are given by templates so that one can easily implement them.     

Highly symmetric bi-frames for triangle surface multiresolution processing

In this paper we investigate the construction of dyadic affine (wavelet) bi-frames for triangular-mesh surface multiresolution processing. We introduce 6-fold symmetric bi-frames with 4 framelets (frame generators). 6-fold symmetric bi-frames yield frame decomposition and reconstruction algorithms (for regular vertices) with high symmetry, which is required for the design of the corresponding frame multiresolution algorithms for extraordinary vertices on the triangular mesh. Compared with biorthogonal wavelets, the constructed bi-frames have better smoothness and smaller supports. In addition, we also provide frame multiresolution algorithms for extraordinary vertices. All the frame algorithms considered in this paper are given by templates (stencils) so that they are implementable. Furthermore, we present some preliminary experimental results on surface processing with frame algorithms constructed in this paper.     

The template‐based procedure of biorthogonal ternary wavelets for curve multiresolution processing

Curve multiresolution processing techniques have been widely discussed in the study of subdivision schemes and many applications, such as surface progressive transmission and compression. The ternary subdivision scheme is the more appealing one because it can possess the symmetry, smaller topological support, and certain smoothness, simultaneously. So biorthogonal ternary wavelets are discussed in this paper, in which refinable functions are designed for cure and surface multiresolution processing of ternary subdivision schemes. Moreover, by the help of lifting techniques, the template‐based procedure is established for constructing ternary refinable systems with certain symmetry, and it also gives a clear geometric templates of corresponding multiresolution algorithms by several iterative steps. Some examples with certain smoothness are constructed.     

Compactly supported orthonormal complex wavelets with dilation 4 and symmetry

In this paper, we provide a family of compactly supported orthonormal complex wavelets with dilation 4 such that their generating wavelet functions have symmetry and the shortest possible supports with respect to their increasing orders of vanishing moments.     

A multiresolution analysis for tensor-product splines using weighted spline wavelets

We construct biorthogonal spline wavelets for periodic splines which extend the notion of “lazy” wavelets for linear functions (where the wavelets are simply a subset of the scaling functions) to splines of higher degree. We then use the lifting scheme in order to improve the approximation properties with respect to a norm induced by a weighted inner product with a piecewise constant weight function. Using the lifted wavelets we define a multiresolution analysis of tensor-product spline functions and apply it to image compression of black-and-white images. By performing-as a model problem-image compression with black-and-white images, we demonstrate that the use of a weight function allows to adapt the norm to the specific problem.     

Measures of axial symmetry for ovals

A measure of axial symmetry for ovals is defined, and eleven particular measures are studied. Lower bounds for these measures are determined on the classes of arbitrary ovals, centrally symmetric ovals, and ovals of constant breadth. The proofs of these results make use only of elementary geometry and the properties of convexity.     

On elliptic equations with an axial symmetry

Sunto Si considera l'equazione (*) au_xx+2bu_xy+cu_yy+(d/y)u_y=f dove a, b, c, d L, d > 0, f L². Supponendo che (*) sia uniformemente eliittica e che l'oscillazione del quoziente d/a verifichi una resirizione vicino all'asse della x, si provano stime a priori e un teorema di esistenza per soluzioni W^2,2 di un problema di Dirichlet relativo all'equazione (*).

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Work supported by the Istituto di Analisi Globale e Applicazioni.     

The lower dimensional Busemann-Petty problem for bodies with the generalized axial symmetry

The lower dimensional Busemann-Petty problem asks, whether n-dimensional centrally symmetric convex bodies with smaller i-dimensional central sections necessarily have smaller volumes. For i = 1, the affirmative answer is obvious. If i > 3, the answer is negative. For i = 2 or i = 3 (n > 4), the problem is still open, however, when the body with smaller sections is a body of revolution, the answer is affirmative. The paper contains a solution to the problem in the more general situation, when the body with smaller sections is invariant under rotations, preserving mutually orthogonal subspaces of dimensions ℓ and n − ℓ, respectively, so that i + ℓ ≤ n. The answer essentially depends on ℓ. The argument relies on the notion of canonical angles between subspaces, spherical Radon transforms, properties of intersection bodies, and the generalized cosine transforms.     

Exact analytic results for the solid angle in systems with axial symmetry

Exact analytical expressions for the solid angle subtended by a circular diaphragm are obtained in terms of complete elliptic integrals. Point, disk and cylindrical sources of radiation are considered.     

High-order numerical solution of the nonlinear Helmholtz equation with axial symmetry

The nonlinear Helmholtz (NLH) equation models the propagation of intense laser beams in a Kerr medium. The NLH takes into account the effects of nonparaxiality and backward scattering that are neglected in the more common nonlinear Schrödinger model. In [G. Fibich, S. Tsynkov, High-order two-way artificial boundary conditions for nonlinear wave propagation with backscattering, J. Comput. Phys., 171 (2001) 632–677] and [G. Fibich, S. Tsynkov, Numerical solution of the nonlinear Helmholtz equation using nonorthogonal expansions, J. Comput. Phys., 210 (2005) 183–224], a novel high-order numerical method for solving the NLH was introduced and implemented in the case of a two-dimensional Cartesian geometry. The NLH was solved iteratively, using the separation of variables and a special nonlocal two-way artificial boundary condition applied to the resulting decoupled linear systems. In the current paper, we propose a major improvement to the previous method. Instead of using LU decomposition after the separation of variables, we employ an efficient summation rule that evaluates convolution with the discrete Green's function. We also extend the method to a three-dimensional setting with cylindrical symmetry, under both Dirichlet and Sommerfeld-type transverse boundary conditions.     

The properties of asymptotic expansions for the parameters of unsteady transonic flows with axial symmetry

A modified equation, which refines the model of unsteady transonic flow with axial symmetry, is proposed, and the following problems are considered: the determination of the wave front when a weak perturbation propagates and the formation of a shock wave from a perturbation with a continuous profile.     

Biorthogonal polynomials for two-matrix models with semiclassical potentials

We consider the biorthogonal polynomials associated to the two-matrix model where the eigenvalue distribution has potentials V₁,V₂ with arbitrary rational derivative and whose supports are constrained on an arbitrary union of intervals (hard-edges). We show that these polynomials satisfy certain recurrence relations with a number of terms d_i depending on the number of hard-edges and on the degree of the rational functions . Using these relations we derive Christoffel–Darboux identities satisfied by the biorthogonal polynomials: this enables us to give explicit formulæ for the differential equation satisfied by d_i+1 consecutive polynomials, We also define certain integral transforms of the polynomials and use them to formulate a Riemann–Hilbert problem for (d_i+1)×(d_i+1) matrices constructed out of the polynomials and these transforms. Moreover, we prove that the Christoffel–Darboux pairing can be interpreted as a pairing between two dual Riemann–Hilbert problems.     

Symmetric orthonormal scaling functions and wavelets with dilation factor 4

It is well known that in the univariate case, up to an integer shift and possible sign change, there is no dyadic compactly supported symmetric orthonormal scaling function except for the Haar function. In this paper we are concerned with the construction of symmetric orthonormal scaling functions with dilation factor d=4. Several examples of such orthonormal scaling functions are provided in this paper. In particular, two examples of C ¹ orthonormal scaling functions, which are symmetric about 0 and 1/6, respectively, are presented. We will then discuss how to construct symmetric wavelets from these scaling functions. We explicitly construct the corresponding orthonormal symmetric wavelets for all the examples given in this paper. This revised version was published online in June 2006 with corrections to the Cover Date.     

On nonnegatively curved 4-manifolds with discrete symmetry

Let M be the closed, simply connected, 4-manifold with nonnegative sectional curvature, called a nonnegatively curved 4-manifold, with an effective and isometric Z _m -action for a positive integer m ≧ 61⁷. Assume that Z _m acts trivially on the homology of M. The goal of this short paper is to prove that if the fixed point set of any nontrivial element of Z _m has at most one two-dimensional component, then M is homeomorphic to S ⁴, # _i ^l =1S ² × S ², l = 1, 2, or # _j ^k = 1 ± CP ², k = 1, 2, 3, 4, 5. The main strategy of this paper is to give an upper bound of the Euler characteristic χ(M) under the homological assumption of a Z _m -action as above by using the Lefschetz fixed point formula.