Gabriella Sanniti di Baja

Computing skeletons in three dimensions

Skeletonization will probably become as valuable a tool for shape analysis in 3D, as it is in 2D. We present a topology preserving 3D skeletonization method which computes both surface and curve sk ...

Using distance transforms to decompose 3D discrete objects

Object decomposition into simpler parts greatly diminishes the complexity of a recognition task. In this paper, we present a method to decompose a 3D discrete object into nearly convex or elongated ...

Simplifying curve skeletons in volume images

The curve skeleton of a 3D solid object provides a useful tool for shape analysis tasks. In this paper, we use a recent skeletonization algorithm based on voxel classification that originates a nearly thin, i.e., at most two-voxel thick, curve skeleton. We introduce a novel way to compress the nearly thin curve skeleton to one-voxel thickness, as well as an efficient pruning algorithm able to remove unnecessary skeleton branches without causing excessive loss of information. To this purpose, the pruning condition is based on the distribution of significant elements along skeleton branches. The definition of significance depends on the adopted skeletonization algorithm. In our case, it is derived from the voxel classification used during skeletonization.

Using resolution pyramids for watershed image segmentation

In this paper we build a shape preserving resolution pyramid and use it in the framework of image segmentation via watershed transformation. Our method is based on the assumption that the most significant image components perceived at high resolution will also be perceived at lower resolution. Thus, we detect the seeds for the watershed transformation at a low resolution, and use them to distinguish significant and non-significant seeds at any selected higher resolution. In this way, the watershed partition obtained at the selected pyramid level will include only the most significant components, and over-segmentation will be considerably reduced. Segmentations of the image at different scales will be available. Moreover, since the seeds can be detected at different pyramid levels, alternative segmentations of the image at a given resolution can be obtained, each characterized by a different level of detail.

Surface Skeletons Detected on the D6 Distance Transform

We present an algorithm for extracting the surface skeleton of a 3D object from its D6 distance transform. The skeletal voxels are directly detected and marked on the distance transform within a small number of inspections, independent of object thickness. This makes the algorithm preferable with respect to algorithms based on iterative application of topology preserving removal operations, when working with thick objects. The set of skeletal voxels is centred within the object, symmetric, and topologically correct. It is at most 2-voxel wide (except for some cases of surface intersections) and includes all centres of maximal D6 balls, which makes skeletonization reversible. Reduction to a unit wide surface skeleton can be obtained by suitable post-processing.

Note on the multiscale representation of 2D and 3D shapes

Abstract Binary pyramids in two and three dimensions can be used for multiresolution representation. The “standard” OR and AND pyramids have serious drawbacks, as they distort the shape significantly; therefore they can seldom be used effectively. Here we present alternative approaches to build binary pyramids, aimed at improving shape preservation (and, as far as possible, topology preservation) in lower resolutions. The algorithms are easy to implement and produce good results.

Surface Skeletonization of Volume Objects

Tools for quantitative analysis of volume images are becoming more important, as volume images are becoming more common in a number of application fields, but especially in biomedical tomographic images at different scales. Here we present a method for reducing a volume (3D) object to a surface skeleton. The original object can be recovered from its skeleton. The method is based on the notion of “multiple voxels,” derived from that of “multiple pixels” in the 2D case. It consists of two phases. During the first phase non-multiple voxels are iteratively removed. During the second phase, the remaining set of voxels is thinned to a set of one-voxel thick surfaces and curves. This skeletonization method requires only a small number of local (3×3×3 neighbourhood) operations per voxel, no extra memory and no look-up tables. It is suited both for sequential and parallel implementation. We exemplify the results of the method on a number of 128×128×128 images.

Skeletonizing Volume Objects Part 2: From Surface to Curve Skeleton

Volume imaging techniques are becoming common and skeletonization has begun to prove valuable for shape analysis also in 3D. In this paper, a method to reduce solid volume objects to their 3D curve skeletons is presented. The method consists of two major steps. The first step is aimed at the computation of the surface skeleton, and is an improvement of a previous method. In the second step, the surface skeleton is further reduced to the 3D curve skeleton. Our skeletonization method preserves topology; no disconnections, holes or tunnels are created. It also preserves the general geometry of the object, especially in the case of elongated objects. Resulting skeletons for a number of synthetic and real images are presented.

Characterising 3D Objects by Shape and Topology

Information on the shape of an object can be combined with information on the shape of the complement of the object, in order to describe objects having complex shape. We present a method for decomposing and characterising the convex deficiencies of an object, i.e., the regions obtained by subtracting the object from its convex hull, into parts corresponding to cavities, tunnels, and concavities of the object. The method makes use of the detection of watersheds in a distance image.

Multiresolution representation of shape in binary images

Binary pyramids can be used for multiresolution pattern representation. The two “standard” pyramids schemes, OR- and AND-pyramids, have, however, serious drawbacks, as they distort the shape significantly. In OR-pyramids black pixels spread all over the array due to expansion and merging of close regions. The shape of the original pattern is rapidly blurred. In AND-pyramids narrow regions of the initial pattern may either completely vanish or become disconnected. In both cases, the shape of the pattern is not preserved. Here alternative approaches, aimed at improving shape and topology preservation in binary pyramids, are presented. The first approach combines the OR and AND rules, whereas the second approach uses grey-level images as an intermediate step. The algorithms are easy to implement and produce significantly better results than the ones obtained by OR/AND pyramids.