Thomas P.-Y. Yu

Nonlinear pyramid transforms based on median-interpolation

We introduce a nonlinear refinement subdivision scheme based on median-inter- polation. The scheme constructs a polynomial interpolating adjacent block medians of an underlying object. The interpolating polynomial is then used to impute block medians at the next finer triadic scale. Perhaps surprisingly, expressions for the refinement operator can be obtained in closed-form for the scheme interpolating by polynomials of degree D = 2. Despite the nonlinearity of this scheme, convergence and regularity can be established using techniques reminiscent of those developed in analysis of linear refinement schemes. The refinement scheme can be deployed in multiresolution fashion to construct a nonlinear pyra- mid and an associated forward and inverse transform. In this paper we discuss the basic properties of these transforms and their possible use in removing badly non-Gaussian noise. Analytic and computational results are presented to show that in the presence of highly non-Gaussian noise, the coefficients of the nonlinear transform have much better properties than traditional wavelet coeffi- cients.

Multivariate refinable Hermite interpolant

We introduce a general definition of refinable Hermite interpolants and investigate their general properties. We study also a notion of symmetry of these refinable interpolants. Results and ideas from the extensive theory of general refinement equations are applied to obtain results on refinable Hermite interpolants. The theory developed here is constructive and yields an easy-to-use construction method for multivariate refinable Hermite interpolants. Using this method, several new refinable Hermite interpolants with respect to dierent dilation matrices and symmetry groups are constructed and analyzed. Some of the Hermite interpolants constructed here are related to well-known spline interpolation schemes developed in the computer-aided geometric design community (e.g. the Powell-Sabin scheme.) We make

An improved vertex caching scheme for 3D mesh rendering

Modern graphics cards are equipped with a vertex cache to reduce the amount of data needing to be transmitted to the graphics pipeline during rendering. To make effective use of the cache and facilitate rendering, it is key to represent a mesh in a manner that maximizes the cache hit rate. In this paper, we propose a simple yet effective algorithm for generating a sequence for efficient rendering of 3D polygonal meshes based on greedy optimization. The algorithm outperforms the current state-of-the-art algorithms in terms of rendering efficiency of the resultant sequence. We also adapt it for the rendering of progressive meshes. For any simplified version of the original mesh, the rendering sequence is generated by adaptively updating the reordered sequence at full resolution. The resultant rendering sequence is cheap to compute and has reasonably good rendering performance, which is desirable to many complex rendering environments involving continuous rendering of meshes at various level of details. The experimental results on a collection of 3D meshes are provided.

Smoothness Equivalence Properties of Manifold-Valued Data Subdivision Schemes Based on the Projection Approach

Interpolation of manifold-valued data is a fundamental problem which has applications in many fields. The linear subdivision method is an efficient and well-studied method for interpolating or approximating real-valued data in a multiresolution fashion. A natural way to apply a linear subdivision scheme

Smoothness Equivalence Properties of General Manifold-Valued Data Subdivision Schemes

\overline{S}

Noninterpolatory Hermite subdivision schemes

to interpolate manifold-valued data is to first embed the manifold at hand into an Euclidean space and construct a projection operator

Robust nonlinear wavelet transform based on median-interpolation

P

Design of Hermite Subdivision Schemes Aided by Spectral Radius Optimization

that maps points from the ambient space to a closest point on the embedded surface, and then consider the nonlinear subdivision operator

Interpolation of medians

S:= P \circ \overline{S}

Denoising of electron tomographic reconstructions from biological specimens using multidimensional multiscale transforms

. When applied to symmetric spaces such as