Subdivision schemes with general dilation in the geometric and nonlinear setting

AbstractWe establish results on convergence and smoothness of subdivision rules operating on manifold-valued data which are based on a general dilation matrix. In particular we cover irregular combinatorics. For the regular grid case results are not restricted to isotropic dilation matrices. The nature of the results is that intrinsic subdivision rules which operate on geometric data inherit smoothness properties of their linear counterparts

Point-Normal Subdivision Curves and Surfaces

This paper proposes to generalize linear subdivision schemes to nonlinear subdivision schemes for curve and surface modeling by refining vertex positions together with refinement of unit control normals at the vertices. For each round of subdivision, new control normals are obtained by projections of linearly subdivided normals onto unit circle or sphere while new vertex positions are obtained by updating linearly subdivided vertices along the directions of the newly subdivided normals. Particularly, the new position of each linearly subdivided vertex is computed by weighted averages of end points of circular or helical arcs that interpolate the positions and normals at the old vertices at one ends and the newly subdivided normal at the other ends. The main features of the proposed subdivision schemes are three folds: (1) The point-normal (PN) subdivision schemes can reproduce circles, circular cylinders and spheres using control points and control normals; (2) PN subdivision schemes generalized from convergent linear subdivision schemes converge and can have the same smoothness orders as the linear schemes; (3) PN $C^2$ subdivision schemes generalizing linear subdivision schemes that generate $C^2$ subdivision surfaces with flat extraordinary points can generate visually $C^2$ subdivision surfaces with non-flat extraordinary points. Experimental examples have been given to show the effectiveness of the proposed techniques for curve and surface modeling.Comment: 30 pages, 17 figures, 22.5M

Single basepoint subdivision schemes for manifold-valued data: Timesymmetry without space-symmetry.

Abstract. This paper establishes smoothness results for a class of nonlinear subdivision schemes, known as the single basepoint manifold-valued subdivision schemes, which shows up in the construction of waveletlike transform for manifold-valued data. This class includes the (single basepoint) Log-Exp subdivision scheme as a special case. In these schemes, the exponential map is replaced by a so-called retraction map f from the tangent bundle of a manifold to the manifold. It is known that any choice of retraction map yields a C 2 scheme, provided the underlying linear scheme is C 2 (this is called "C 2 equivalence"). But when the underlying linear scheme is C 3 , Navayazdani and Yu have shown that to guarantee C 3 equivalence, a certain tensor P f associated to f must vanish. They also show that P f vanishes when the underlying manifold is a symmetric space and f is the exponential map. In the present paper, a geometric interpretation of the tensor P f is given. Associated to the retraction map f is a torsion-free affine connection, which in turn defines an exponential map. The condition P f = 0 is shown to be equivalent to the condition that f agrees with the exponential map of the connection up to the 3rd order. In particular, when f is the exponential map of a connection, one recovers the original connection and P f vanishes. It then follows that the condition P f = 0 is satisfied by a wider class of manifolds than was previously known. Under the additional assumption that the subdivision rule satisfies a time-symmetry, it is shown that the vanishing of P f also guarantees C 4 equivalence. Finally, the analysis in the paper strongly indicates that vanishing curvature of the connection associated to f is a necessary condition for C k equivalence for k ≥ 5

Optimal a priori discretization error bounds for geodesic finite elements

We prove optimal bounds for the discretization error of geodesic finite elements for variational partial differential equations for functions that map into a nonlinear space. For this, we first generalize the well-known Céa lemma to nonlinear function spaces. In a second step, we prove optimal interpolation error estimates for pointwise interpolation by geodesic finite elements of arbitrary order. These two results are both of independent interest. Together they yield optimal a priori error estimates for a large class of manifold-valued variational problems. We measure the discretization error both intrinsically using an H1-type Finsler norm and with the H1-norm using embeddings of the codomain in a linear space. To measure the regularity of the solution, we propose a nonstandard smoothness descriptor for manifold-valued functions, which bounds additional terms not captured by Sobolev norms. As an application, we obtain optimal a priori error estimates for discretizations of smooth harmonic maps using geodesic finite elements, yielding the first high-order scheme for this problem

Arithmetic geometry of toric varieties. Metrics, measures and heights

We show that the height of a toric variety with respect to a toric metrized line bundle can be expressed as the integral over a polytope of a certain adelic family of concave functions. To state and prove this result, we study the Arakelov geometry of toric varieties. In particular, we consider models over a discrete valuation ring, metrized line bundles, and their associated measures and heights. We show that these notions can be translated in terms of convex analysis, and are closely related to objects like polyhedral complexes, concave functions, real Monge-Amp\`ere measures, and Legendre-Fenchel duality. We also present a closed formula for the integral over a polytope of a function of one variable composed with a linear form. This allows us to compute the height of toric varieties with respect to some interesting metrics arising from polytopes. We also compute the height of toric projective curves with respect to the Fubini-Study metric, and of some toric bundles.Comment: Revised version, 230 pages, 3 figure