Hierarchical isocontours extraction and compression THOMAS LEWINER 1,2 , H´ ELIO LOPES 1 , LUIZ VELHO 3 AND VIN´ ICIUS MELLO 3 1 Department of Mathematics — Pontif´ ıcia Universidade Cat´ olica — Rio de Janeiro — Brazil 2 G´ eom´ etrica Project — INRIA – Sophia Antipolis — France 3 Visgraf Project — IMPA — Rio de Janeiro — Brazil {tomlew, lopes}@mat.puc--rio.br. {lvelho, vinicius}@visgraf.impa.br. Abstract. In this work, we introduce a new scheme to extract hierarchical isocontours from regular and irregular 2D sampled data and to encode it at single rate or progressively. A dynamic tessellation is used to represent and adapt the 2D data to the isocontour. This adaptation induces a controlled multi–resolution representation of the isocontour. We can then encode this representation and control the geometry and topology of the decoded isocontour. The resulting algorithms form an efficient and flexible isocontour extraction and compression scheme. Keywords: Level sets. Data Compression. Simplicial Methods. Progressive Transmission. Geometry Processing. (a) 21 bytes (b) 31 bytes (c) 46 bytes (d) 156 bytes (e) 690 bytes Figure 1: Progressive compression of an elevation curve of the Sugar Loaf. The tessellation is adapted to the curve to minimize the distortion and to preserve the topology. (a) The shape of the tubular neighborhood of the curve is sent first, with the nodes sign. (b),(c),(d) Then the refinements of the tessellation are encoded with the signs of the new nodes. (e) Finally, the values of the nodes in the tubular neighborhood are refined. 1 Introduction Curves are one of the basic building blocks of geometry processing. They are used to represent shape in 2D images, terrain elevation on maps, and equations in mathematical visualization. In most of those applications, the curves can be interpreted as an isocontour of a 2D dataset, possibly mapped on a more complex space. Those isocontours are flexible objects that can be refined or reduced, that can deform with differential simulations or mathematical morphology, and that can be described for shape classification or automatic diagnostic in medicine or geosciences. Problem statement. Given the sampling ˆ f of scalar func- tion f defined over a domain D embedded in R 2 (such as a 2D image or a discrete surface), the isocontour of an isovalue α is the curve f -1 (α). Such an isocontour corresponds to only a small part of D, but usually covers a large area of the Preprint MAT. 13/04, communicated on May 10 th , 2004 to the Department of Mathematics, Pontif´ ıcia Universidade Cat´ olica — Rio de Janeiro, Brazil. The corresponding work was published in the proceedings of the Sibgrapi 2004, pp. 234–241 IEEE Press, 2004. (a) 1619 bytes (b) 4024 bytes Figure 2: Progressive compression of the cortex of a Computa- tional Tomography image, topology controlled. domain. For example, the cortex corresponds to only specific x–ray scintillation inside the scan of the whole head (see Fig- ure 2), the elevation curve is only a small part inside a topo- graphic map (see Figure 1). Therefore, specific compression techniques for isocontours should provide better compres-