Subdivision schemes for curve design and image analysis

Subdivision schemes are able to produce functions, which are smooth up to pixel accuracy, in a few steps through an iterative process. They take as input a coarse control polygon and iteratively generate new points using some algebraic or geometric rules. Therefore, they are a powerful tool for creating and displaying functions, in particular in computer graphics, computer-aided design, and signal analysis. A lot of research on univariate subdivision schemes is concerned with the convergence and the smoothness of the limit curve, especially for schemes where the new points are a linear combination of points from the previous iteration. Much less is known for non-linear schemes: in many cases there are only ad hoc proofs or numerical evidence about the regularity of these schemes. For schemes that use a geometric construction, it could be interesting to study the continuity of geometric entities. Dyn and Hormann propose sufficient conditions such that the subdivision process converges and the limit curve is tangent continuous. These conditions can be satisfied by any interpolatory scheme and they depend only on edge lengths and angles. The goal of my work is to generalize these conditions and to find a sufficient constraint, which guarantees that a generic interpolatory subdivision scheme gives limit curves with continuous curvature. To require the continuity of the curvature it seems natural to come up with a condition that depends on the difference of curvatures of neighbouring circles. The proof of the proposed condition is not completed, but we give a numerical evidence of it. A key feature of subdivision schemes is that they can be used in different fields of approximation theory. Due to their well-known relation with multiresolution analysis they can be exploited also in image analysis. In fact, subdivision schemes allow for an efficient computation of the wavelet transform using the filterbank. One current issue in signal processing is the analysis of anisotropic signals. Shearlet transforms allow to do it using the concept of multiple subdivision schemes. One drawback, however, is the big number of filters needed for analysing the signal given. The number of filters is related to the determinant of the expanding matrix considered. Therefore, a part of my work is devoted to find expanding matrices that give a smaller number of filters compared to the shearlet case. We present a family of anisotropic matrices for any dimension d with smaller determinant than shearlets. At the same time, these matrices allow for the definition of a valid directional transform and associated multiple subdivision schemes

Subdivisions with infinitely supported mask

AbstractIn this paper we investigate the convergence of subdivision schemes associated with masks being polynomially decay sequences. Two-scale vector refinement equations are the formφ(x)=∑α∈Za(α)φ(2x-α),x∈R,where the vector of functions φ=(φ1,…,φr)T is in (L2(R))r and a≕(a(α))α∈Z is polynomially decay sequence of r×r matrices called refinement mask. Associated with the mask a is a linear operator on (L2(R))r given byQaf(x)≔∑α∈Za(α)f(2x-α),x∈R,f=(f1,…,fr)T∈(L2(R))r.By using same methods in [B. Han, R. Q. Jia, Characterization of Riesz bases of wavelets generated from multiresolution analysis, manuscript]; [B. Han, Refinable functions and cascade algorithms in weighted spaces with infinitely supported masks, manuscript]; [R.Q. Jia, Q.T. Jiang, Z.W. Shen, Convergence of cascade algorithms associated with nonhomogeneous refinement equations, Proc. Amer. Math. Soc. 129 (2001) 415–427]; [R.Q. Jia, Convergence of vector subdivision schemes and construction of biorthogonal multiple wavelets, in: Advances in Wavelet, Hong Kong,1997, Springer, Singapore, 1998, pp. 199–227], a characterization of convergence of the sequences (Qanf)n=1,2,… in the L2-norm is given, which extends the main results in [R.Q. Jia, S.D. Riemenschneider, D.X. Zhou, Vector subdivision schemes and multiple wavelets, Math. Comp. 67 (1998) 1533–1563] on convergence of the subdivision schemes associated with a finitely supported mask to the case in which mask a is polynomially decay sequence. As an application, we also obtain a characterization of smoothness of solutions of the refinement equation mentioned above for the case r=1

Ellipse-preserving Hermite interpolation and subdivision

We introduce a family of piecewise-exponential functions that have the Hermite interpolation property. Our design is motivated by the search for an effective scheme for the joint interpolation of points and associated tangents on a curve with the ability to perfectly reproduce ellipses. We prove that the proposed Hermite functions form a Riesz basis and that they reproduce prescribed exponential polynomials. We present a method based on Green's functions to unravel their multi-resolution and approximation-theoretic properties. Finally, we derive the corresponding vector and scalar subdivision schemes, which lend themselves to a fast implementation. The proposed vector scheme is interpolatory and level-dependent, but its asymptotic behaviour is the same as the classical cubic Hermite spline algorithm. The same convergence properties---i.e., fourth order of approximation---are hence ensured

Adaptive mesh refinement techniques for high-order finite-volume WENO schemes

This paper demonstrates the capabilities of Adaptive Mesh Refinement Techniques (AMR) on 2D hybrid unstructured meshes, for high order finite volume WENO methods. The AMR technique developed is a conformal adapting unstructured hybrid quadrilaterals and triangles (quads & tris) technique for resolving sharp flow features in accurate manner for steady-state and time dependent flow problems. In this method, the mesh can be refined or coarsened which depends on an error estimator, making decision at the parent level whilst maintaining a conformal mesh, the unstructured hybrid mesh refinement is done hierarchically.When a numerical method can work on a fixed conformal mesh this can be applied to do dynamic mesh adaptation. Two Refinement strategies have been devised both following a H-P refinement technique, which can be applied for providing better resolution to strong gradient dominated problems. The AMR algorithm has been tested on cylindrical explosion test and forward facing step problems