Computing and analysing convex deficiencies to characterise 3D complex objects

Abstract

Entities such as object components, cavities, tunnels and concavities in 3D digital images can be useful in the framework of object analysis. For each object component, we first identify its convex deficiencies, by subtracting the object component from a covering polyhedron approximating the convex hull. Watershed segmentation is then used to decompose complex convex deficiencies into simpler parts, corresponding to individual cavities, concavities and tunnels of the object component. These entities are finally described by means of a representation system accounting for the shape features characterising them.

Introduction

Digital objects in binary images can often be analysed by resorting to their decomposition into simple parts, e.g. nearly convex regions characterised by constant or monotonically changing width, and by following the structural approach according to which the description of the object is given in terms of the description of its parts and of their spatial relations. The description of simple parts can be given in terms of global shape properties (e.g. elongation and compactness) and geometric properties (e.g. volume and surface). However, an object can be complex without being easily decomposable into meaningful simple parts. This is the case for the example in Fig. 1, where a brick-shaped object consisting of one connected component is shown. The object includes a concavity, visible on top of the object, a tunnel, crossing the object in the middle, and a cavity on the bottom of the object, visible only in the cross-section of the object. In this case, object description can be achieved if also the shape of the complement of the object, the background, is investigated. In fact, object and background play dual roles and it is, then, of interest to analyse the portions of the background in correspondence with object cavities, concavities and tunnels, respectively.

The number of cavities in each object component is easy to compute by means of local operators. In fact, cavities are the components of the background that are completely enclosed by the object.

Local concavities along the border of the object can be identified based on the number, and possibly the configuration, of object voxels in the neighbourhood of background voxels. However, global concavities, as the one on top of the object in Fig. 1, are more difficult to identify.

In turn, according to the formal definition [1], a tunnel exists whenever a path can be found within the object that cannot be deformed to a single voxel. Unfortunately, this definition does not allow a computationally convenient way to check the existence of tunnels. In this paper, we prefer an intuitive definition and interpret a tunnel as a canal passing through the object and characterised by at least two exits, as it is the case for the tunnel crossing the object in Fig. 1. We point out that, according to this interpretation, we classify as a cavity the torus-shaped entity shown in Fig. 2, where two cross-sections of an object are given. In fact, this entity is completely enclosed by the object and does not include any exit. This entity would instead be classified as a tunnel, if the formal definition given in Ref. [1] was taken into account.

To analyse entities such as individual cavities, concavities and tunnels, we compute the convex hull of the object [2], or actually a suitable approximation of it [3]. In this way, the portion of the background to be investigated is limited to the convex deficiencies CDs, i.e. the difference between the convex hull and the object. In turn, the CDs are regions of the background adequately representing cavities, concavities and tunnels. In principle, a bounding box could be used as a rough approximation of the convex hull. However, a bounding box would only limit the size of the portion of the background to be investigated, but would not provide useful hints for object analysis. In fact, the difference between the bounding box and the object seldom originates components that faithfully represent the portions of the background corresponding to concavities and tunnels.

In simple cases, the obtained CDs can be seen as directly corresponding to individual entities. For example, refer to the brick with hollows shown in Fig. 1, whose convex hull is just the brick itself without hollows, and to the relative CDs, shown in Fig. 3.

In general, in presence of CDs having complex shape, as those corresponding to tunnels articulated in a number of branches or concatenations of tunnels and concavities, single CDs do not directly correspond to individual entities. For example, see Fig. 4, where a concatenation of tunnels and concavities close to each other is shown. In this case, a single, complex, CD is originated. Complex CDs should be decomposed into the individual constituting entities and a more detailed analysis of the so obtained regions should be accomplished.

Using the convex hull as a tool for shape analysis in 2D and 3D images has already been suggested [4]. There, only cavities and concavities were taken into account. In the 2D case, a more sophisticated analysis of complex concavities was performed. Complex concavities were defined as corresponding to non-convex CDs. The convex hull of the CDs of the object was computed and the corresponding new convex deficiencies were detected, by using the same process as that applied to the original object. In this way, for each complex concavity of the object, its metaconcavities were also identified. This process was repeated until all convex deficiencies were actually convex regions. A metaconcavity tree was then built, whose nodes were the concavities and the metaconcavities of the object. In principle, that method can be extended to 3D, but its cost would be rather high. In this paper, we take into account tunnels, besides cavities and concavities, and prefer to decompose complex CDs into the individual constituting entities, instead of resorting to the metaconcavity tree. In particular, purpose of this paper is to present a computationally convenient, general method to identify and analyse cavities, concavities and tunnels.

The paper is based on recently achieved results [5], [6], and is organised as follows. Some notions are introduced in Section 2. An algorithm to compute a covering polyhedron approximating the convex hull of an object, and to obtain the CDs is briefly illustrated in Section 3. An efficient way to decompose complex CDs into the constituting entities, based on the detection of watersheds in a distance image, is described in Section 4. In Section 5, an algorithm to transform each obtained entity into a compact representation by means of topological erosion is outlined. Finally, some concluding remarks are given in Section 6.

To show the performance of our method, we will use only synthetic images, built by means of the composition of simple solid object or background components. Both of these synthetic examples allow us to show in a simple manner the various steps of the process, and because our work is mainly basic research. Though we do not have yet in mind a specific application, the results obtained on the many complex synthetic objects we considered, indicate that the method could provide a useful tool for shape analysis in 3D voxel images.