Univariate data visualization

In the univariate case, that is when a single scalar field is taken into account, topology has already revealed its advantages. In particular, it provides simplified and robust tools for feature-tracking, isosurface traversal and segmentation of the domain based on Morse Theory. We classify the methods in univariate topology-based data visualization into two main paradigms:

• value-driven are those methods referring to the notion of Reeb graph [159] (Definition7.1.1). The retrieval of the Reeb graph provides a compact representation of the scalar field based on its level sets. For this reason, we call the topological methods related directly or indirectly to the Reeb graph as value-driven;

• monotony-driven are those method referring to the notion of an ascending/descending Morse complex(Definition7.1.2). The paradigm consists in simplifying the representation of the scalar field under study by considering as equal all points in the domain with the same behavior with respect to the gradient of the scalar field of interest. For this reason, we call the topological methods related directly or indirectly to the Morse complex as monotony-driven.

7.2.1.1 Value-driven Paradigm

Most of value-driven applications refer to the notion of contour tree, which is a Reeb graph in the special situation of a simply connected domain D. Contour trees [129] are easier to compute and their retrieval can be performed in any dimension of the domain [38]. Moreover, a method

that makes a Reeb graph into a contour tree for volumetric meshes only is proposed in [189]. Among the possible applications to visualization, we report those for isosurface traversal [129], volumetric rendering simplification [39], transfer function design for volume rendering [192], and similarity estimation among different isosurfaces [186,110]. Comprehensive surveys of the Reeb graph algorithms applied to visualization can be found in [19,157]. We subdivide value-driven techniques into:

• vertex-based [172];

• approximate [18,110,196]; • critical-based [158,72]

Vertex-based. The first combinatorial algorithm for 2D meshes and PL scalar functions for the Reeb graph retrieval is proposed in [172]. It considers a level set for each vertex in the domain and then use adjacency relation on the domain to define the arcs in the Reeb graph.

Approximate. Approximate algorithms for the Reeb graph retrieval are proposed in [18,110, 196] for PL domains. These methods speed up the computations. Their result is not precise but depends on a tolerance parameter set by the user which defines how far apart to sample values on the target space. Then, connected components of the respective level sets are computed and then connected according to adjacency relation in the domain D.

Critical-based. The topology of one level set can only change when changing the scalar value by moving across some critical value. Algorithms exploiting the role of critical points has shown to be more accurate and fast [158, 72]. They compute, for each critical point, the connected component of the level set passing through it. Next, it suffices to consider the adjacencies in the segmentation induced by the critical level sets cuttings in the domain to retrieve adjacency among nodes.

7.2.1.2 Monotony-driven Paradigm

The second abstract structure exploited in topology-based approaches is the Morse complex in either the ascending or descending forms. We subdivide monotony-driven techniques into:

• ascending or descending PL methods [60,59,137,105,97,68,140,191,163] where scalar fields over vertexes are extended to higher dimensional cells linearly with respect to the barycentric coordinates of the cell. These techniques refer to the notion of PL critical cell [6, 7,79];

• Morse-Smale complex PL methods [182,79,77,24,156,14,165,166, 5] where scalar field is PL-extended from vertexes and the purpose is that obtaining a Morse-Smale domain decomposition which combines ascending and descending decompositions [79,77]; • discrete methods [43, 123, 101, 162, 102, 99, 173, 175, 193] where scalar fields over

vertexes are extended to higher dimensional cells by the maximum-rule in Definition3.1.2. These techniques refer to discrete Morse Theory [87].

Ascending or descending PL methods. Methods for PL functions refer to the definition of critical point according to Banchoff [6, 7], then, extended to higher dimensional spaces by Edelsbrunner et al.[79]. A comprehensive survey for Morse-based segmentation techniques can be found in [61]. Algorithms based on piecewise linear Morse Theory, and compute the ascending (or descending) Morse complex by growing the top cells, here called regions, from seeds located at the minima (or maxima). [60,59,137,105]. The construction of another approach, called stable flow complex [97], can be obtained through a region-growing approach [68]. The watershed transform is an alternative framework to Morse Theory analogue to the ascending Morse complex which segment the domain according to the so-called topographic distance. It has been first defined for grey-scale images. [140,191]. The watershed transform has also been defined for C2-differentiable functions over a connected domain, having the property that the gradient is non-null everywhere except possibly at some isolated points. This includes Morse functions.[140, 163].

Morse-Smale complex PL methods If each non-empty intersection of a descending and an ascending cell is transversal, their connected components define the Morse-Smale (MS) complex, which encodes both dual constructions, ascending or descending. In the MS complex, two points are equal if their respective integral lines share the same critical extrema points in their support. The notion of Morse-Smale complex has found applications to different fields [130,103,100, 104] such as fluid dynamics and distance field definition. However, the Morse-Smale complex construction is feasible only up to dimension 3 in the domain with some limitations in feasibility over real-size data. The notion of Quasi Morse-Smale complex as the PL counterpart to the MS complex is introduced in [79, 77] for PL domains. Algorithms for the retrieval of the Quasi Morse-Smale complex can be subdivided according to the kind of domain. For triangular and tetrahedral meshes, we have [182,79,77,24, 156,14]. For regular grids, we have algorithm computing derivatives numerically [5] or analitically [165,166].

Discrete methods. Methods for discrete functions refer to the definition of critical point accord- ing to Forman’s discrete Morse Theory [87]. Methods from discrete Morse Theory allows to retrieve all the Morse and Morse-Smale cells in a derivative-free way. The subdivision is obtained by traversing the V-paths, that is discrete analogues for integral lines available when the discrete

gradient is computed. The algorithms of this kind [43,123,101,162,102,99,173,175,193] start from a discrete scalar function f defined over the vertices of a cell complex, and aim at construct- ing a discrete gradient that best fits function f . Discrete methods are dimension-agnostic and they have shown practical advantages in computational feasibility over real-size data with respect to PL-methods. Our method, described in Section7.3.2, constitutes a multivariate counterpart of those algorithms.