Optimal biorthogonal wavelet decomposition of wire-frame meshes using box splines, and its application to the hierarchical coding of 3-D surfaces

Application of the convergence of the control net of box splines to scale-space filtering

In this correspondence, we propose a new approach to scale-space filtering using a box spline representation of multidimensional signals. The use of box splines is motivated by their ability to handle complex geometries better than tensor-product B-splines. The box spline we use is defined by a set of vectors invariant under the multiplication by a sampling matrix. We show that such a box spline satisfies a dilation equation which is the basis for the scale-space filtering we propose. Several numerical applications in 2-D conclude the correspondence.

Quasi-interpolating spline models for hexagonally-sampled data

The reconstruction of a continuous-domain representation from sampled data is an essential element of many image processing tasks, in particular, image resampling. Until today, most image data have been available on Cartesian lattices, despite the many theoretical advantages of hexagonal sampling. In this paper, we propose new reconstruction methods for hexagonally sampled data that use the intrinsically 2-D nature of the lattice, and that at the same time remain practical and efficient. To that aim, we deploy box-spline and hex-spline models, which are notably well adapted to hexagonal lattices. We also rely on the quasi-interpolation paradigm to design compelling prefilters; that is, the optimal filter for a prescribed design is found using recent results from approximation theory. The feasibility and efficiency of the proposed methods are illustrated and compared for a hexagonal to Cartesian grid conversion problem.

Hierarchical representation and coding of surfaces using 3-D polygon meshes

This paper presents a novel procedure for the representation and coding of three-dimensional (3-D) surfaces using hierarchical adaptive triangulation. The proposed procedure is based on pyramidal analysis using the quincunx sampling minimum variance interpolation (QMVINT) filters. These are reduced pyramids with quincunx sampling applied to the parametric representation of the surface, chosen so as to minimize the variance of the interpolation error, and thus, when combined with the appropriate encoding of the coefficients, optimize the compression of the mesh information transmitted. At the same time, it produces a hierarchy of meshes based on quincunx sampling where coarse meshes are as similar to their finer versions as possible. This is very much desirable in progressive transmission. Depending on its interpolation error and the available bitrate, each filtered sample is a candidate for becoming a vertex of the mesh. The result is a progressive sequence of meshes consisting of more triangles wherever large variations exist and fewer in uniform regions. Complete correspondence between triangles at each level is identified, resulting in an efficient hierarchical representation of the mesh. The algorithm can be also used for the triangulation of a specific region of interest. Experimental results demonstrate that the proposed scheme provides an improvement in quality (MSE) by a factor of two when compared with other well known adaptive triangulation schemes.