Compression of Functions Defined on 3D and Higher-Dimensional Manifolds

This talk will introduce a new approach to compression that allows manipulation of scalar functions defined on high-dimensional manifolds of arbitrary topology. This is the first method proposed and developed that allows compression of objects that are not flat (images - 2D) or video (2 1/2D). We show how one can construct a wavelet transform adapted to the geometry of the manifold using Sweldens' second generation wavelets. Compression is achieved with a progressive coding of the resulting wavelet coefficients using a zero-tree based coding scheme. Our approach combines discrete wavelet transforms with zerotree compression, building on ideas from three previous developments: the lifting scheme, spherical wavelets, and embedded zerotree coding methods. Applications lie in the efficient storage and rapid transmission of complex data sets. Typical data sets are earth topography, satellite images, but also parametrizations of complex surfaces. We will outline and discuss few of these applications during the seminar.