Eurographics Italian Chapter Conference (2010), pp. 1–11 E. Puppo, A. Brogni, and L. De Floriani (Editors) Building and Generalizing Morphological Representations for 2D and 3D Scalar Fields Lidija ˇ Comi´ c 1 , Leila De Floriani 2 and Federico Iuricich 2 1 Faculty of Engineering, Novi Sad, Serbia 2 Department of Computer Science, Genova, Italy Abstract Ascending and descending Morse complexes, defined by the critical points and integral lines of a scalar field f defined on a manifold domain D, induce a subdivision of D into regions of uniform gradient flow, and thus provide a compact description of the morphology of f on D. Here, we propose a dimension independent representation for the ascending and descending Morse complexes, and a data structure which assumes a discrete representation of the field as a simplicial mesh, that we call the incidence-based data structure. We present algorithms for building such data structure for 2D and 3D scalar fields, which make use of a watershed approach to compute the cells of the Morse decompositions. We describe generalization operators for Morse complexes in arbitrary dimensions, we discuss their effect and present results of our implementation of their 2D and 3D instances both on the Morse complexes and on the incidence-based data structure. Categories and Subject Descriptors (according to ACM CCS): I.3.3 [Computer Graphics]: Computational Geometry and Object Modeling—Object Representations Keywords: Morphology, Terrain Models, Scalar Fields, Morse Complexes 1. Introduction Representing morphological information extracted from dis- crete scalar fields is a relevant issue in several application domains, such as terrain modeling, volume data analysis and visualization, and time-varying 3D scalar fields. Morse the- ory offers a natural and intuitive way of analyzing the struc- ture of a scalar field as well as of compactly representing the scalar field through a decomposition of its domain D into meaningful regions associated with the critical points of the field. The ascending and the descending Morse complexes are defined by considering the integral lines emanating from, or converging to the critical points of f , while the Morse- Smale complex describes the subdivision of D into parts characterized by a uniform flow of the gradient between two critical points of f . Computation of an approximation of the Morse and Morse-Smale complexes has been extensively studied in the literature in the 2D case, and recently algo- rithms have been proposed in 3D. The discrete watershed transform is one of the most popular methods used in image segmentation for 2D and 3D images and has been applied to regular Digital Elevation Models (DEMs). Here, we extend the watershed approach by simulated immersion [VS91] to compute the ascending and descending Morse complexes for simplicial meshes, focusing on 2D triangle meshes (forming Triangulated Irregular Networks (TINs) and 3D tetrahedral meshes, discretizing the domain of a 3D scalar field. The ap- proach, however, can be extended to higher dimensions in a straightforward way and our implementation is already di- mension independent. We represent the ascending and descending Morse com- plexes in arbitrary dimensions as a graph, called incidence graph, in which the nodes represent the cells of the Morse complexes in a dual fashion and the arcs their mutual inci- dence relations. We show how, in the discrete case, incidence graph can be effectively combined with a representation of the simplicial decomposition of the underlying domain D. This representation, that we call an incidence-based repre- sentation of the Morse complexes, is based on encoding the incidence relations of the cells of the two complexes, and exploits the duality between the ascending and descending submitted to Eurographics Italian Chapter Conference (2010)