Visualization of labeled segments cross-contour surfaces - CiteSeerX

Visualization of labeled segments cross-contour surfaces Dani Tost

Anna Puig

[email protected], [email protected] Dept. LLenguatges i Sistemes Informatics, Universitat Politecnica de Catalunya

Abstract

Cross contour surfaces are composed of sets of planar contours. They are the natural output of surface extraction algorithms based on contouring features in parallel image slices of volume models. They are also suitable for the representation of CAD objects with tubular elongated shapes such as pipes and tools. Rendering these surfaces consists of tiling between successive contours, which is mainly a problem of establishing correspondences: between successive contours (branching) and also between vertices of consecutive contours (triangles de nition). Most of the existing algorithms solve these problems by minimizing a distance function between vertices. However, contours are generally composed of segments belonging to dierent semantic regions that should not be mixed during tiling, as for instance, functional regions of the brain and types of terrain in elevation maps. A drawback of the existing distance based approaches is that they may establish correspondences between points of dierent segments. This paper proposes a representation model for surfaces from cross contours composed of labeled segments. In addition, a rendering algorithm of this model is described, that removes undesirable tiles between segments of dierent labels. The proposed method allows the tiling to be done on the y, avoiding thus a double representation of the surface (contours plus triangle mesh). It also allows adaptive levels of resolution in the rendering.

Keywords Surface extraction from Volume Data - Surfaces from contours Labeled regions - Generalized cylinders - Rendering of tubular surfaces.

1 Introduction and previous work Cross contour surfaces are composed of sets of planar contours. They are the natural output of surface extraction algorithms based on contouring features in parallel image slices of volume models. They are also suitable for the representation of CAD objects with tubular elongated shapes such as pipes and tools. To render these objects, triangular surfaces are reconstructed from the contours (tiling) and next visualized. However, the triangular model cannot totally replace the contour model because update operations and queries on the contours should often be done in addition to the rendering. As an example, the rendering of the surface may show errors in the contours attributable to the segmentation process, and therefore it may be necessary to locally modify some of them after the rst rendering. Thereby, most applications handle the two models simultaneously and they recompute the triangular one each time the contours are partially updated. In this paper, we propose a model of cross contour surface that allows a fast tiling to be done during the rendering stage, avoiding thus the double representation (triangles plus contours). The main advantage of this model is that it allows the tiling to be done at dierent levels of resolution without need of handling various faceted models. The reconstruction of surfaces from contours is a classical problem in Computer Graphics [1], [2]. Most publications address two problems related with the tiling: (i) the search of correspondent contours (branching problem) and, (ii) the search of correspondent vertices to form the tiles (correspondence problem). Extensive surveys on these topics can be found in [3], [4]. Generally the triangulation criterion is based on nding the minimum distance between points of successive contours [5], [6]. This criterion is not sucient at the branching or when the contour shapes dier a lot. In these cases, it is sometimes necessary to interpolate intermediate contours [7]. More recently, [8] have proposed to compute an approximate Voronoi diagram (angular Bisector Network) of the contours that gives the proximity information needed to decide the triangulation. Finally, [9] propose to use the medial axis of the structure computed from the distance eld of the each contour that assigns to each point is minimum distance to the contour. These methods are designed for the general case of sections not necessarily perpendicular to an object axis. However, many tubular objects have symmetry axes and the contour sections are actually cross-sections, which makes it possible to design speci c and more ecient algorithms. In addition, they

assume that all the points in a contour belong to the same semantic region, which guarantees the applicability of distance-based criteria. However, this assumption is not true in many applications is which contours are actually composed of adjacent segments of points belonging to dierent semantic regions. For instance, in CAD applications such as the simulation of the construction of drilling tools, a cross contour is composed of dierent segments corresponding to each of the machining operation performed ( uting, gashing, etc). In medical applications, a blood vessel presenting an aneurysm has a contour composed of a normal segment and a deformed, enlarged segment. Tiling between contours without taking into account the segments may lead to erroneous topology of the nal surface such as self intersections between regions. In this paper, a construction algorithm of the contour surface model is proposed that generates the required topological information needed to tile on the y contours composed of labeled segments. The paper is structured into 4 sections: the proposed model is described in Section 2. In Section 3.1, the construction of the model from labeled segments is presented and the rendering algorithm is explained in Section 3.2. Finally, the results are discussed in Section 4, before the conclusions.

1.1 The model From a physical point of view, the set of objects that we want to represent are composed of connected tubular pieces of non-negligible width that do not self intersect. The boundary of the objects is composed of various connected surfaces: (i) the external one, (ii) the internal one, which exists only if the object has a hole in it and, (iii) the head surfaces at the free extremes of the tubular pieces. The external and the internal surfaces are composed of dierent patches connected with at least a C0 continuity order. The patches belong to dierent semantic regions and, therefore, they have dierent optical properties. Color Plates 1 to 4 show three examples of such surfaces: a blood vessel composed of an aneurysm and a normal region; a mandible composed of an in amed area and a normal one; nally, a drilling tool whose external surface is composed of segments corresponding to dierent machining operations; The proposed representation model in structured in two-levels. The top level keeps information on the topological relationships between the dierent components of the object. It is organized as a graph having at the nodes con-

nectivity between pieces, i.e. branching information and, at the edges, the representation of each surface joint. This structure allows topological traversal of the structure to be done. The geometric information is stored only at the edges: the nodes represent merely connectivity between branches. The surface representation at an edge is a generalized cylinder de ned as a sweeping curve plus a set of planar contour curves. In each cross section, a Frenet frame is de ned such that the normal vector of the cross section is the w-axis, the u and v axes are in the section plane and they de ne the orientation of the points in the section. The twisting between cross sections is implicit in the u-v axes. In each cross section, at least one contour curve, the external shape of the surface, is represented. Additionally, a unique hole contour may be de ned. The topological relationships between contours of a same generalized cylinder are implicit: the external contours correspond one with each other and the hole contours also. The contour curves are structured into labeled connected segments. The segments are sorted circularly according to the Frenet frame de ned in the section and each one may have its own twisting angle. Their shape can be circular (cylindrical joints) or not. In the former case, the representation of the geometry is parametric (radius) and the segments are a set of 2D solid angles de ned at the cross section center. Circular contours are stored at points where the skeleton curve has a signi cant gradient or at points where the radius derivative changes so that additional cross sections can thus be interpolated during tiling. On the contrary, the general shape cross sections are sampled at the highest resolution, so that no interpolation between explicit curves should be done during tiling. These sections are represented explicitly, as sets of labeled segments of contour points such that: 

For any given contour composed of contour points, = f = 1 g and given = f = 1 g the set of labeled segments in , for any point 1 = = , it exists a unique segment , 1 = = , such that 2 Given a point of a contour , two unique points and exist such that = ( ) and = ( ) in the contour. C

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