Reverse Distance Transformation Research Articles

On the computation of the 〈3, 4, 5〉 curve skeleton of 3D objects

An algorithm to compute the curve skeleton of 3D objects in voxel images is presented. The skeleton is stable under isometric transformations of the object, since the algorithm is based on the use of the 〈3, 4, 5〉 weighted distance transform, which is a good approximation of the Euclidean distance transform. The 〈3, 4, 5〉 weighted distance transform is used both to identify suitable anchor points, and to efficiently check object voxels according to their distance to the background. The curve skeleton is symmetrically placed within the object, is topologically equivalent to the object, is unit-wide and provides a satisfactory representation of the object. Though the size of the object reconstructed from the curve skeleton via the reverse distance transformation is not as thick as that of the input, shape information is mostly retained by the skeleton, since all regions perceived as significant in the input can still be found in the reconstructed object.

Efficient shape representation by minimizing the set of centres of maximal discs/spheres

Efficient shape representations are important for many image processing applications. Distance transform based algorithms can be used to compute the set of centres of maximal discs/spheres, that represents a shape. This paper describes a method that reduces this set, under the constraint that the shape can be exactly reconstructed using the reverse distance transformation. The reduced set can be used in the same ways as the “standard” set, e.g. for efficient storage, segmentation into parts of different thickness, shape manipulation, and skeletonization, all in 2D and 3D.

Well-Shaped, Stable, and Reversible Skeletons from the (3,4)-Distance Transform

The skeleton of a digital pattern is extracted from the distance transform of the pattern, computed according to a quasi-Euclidean distance function. The pattern can be quite faithfully reconstructed by applying the reverse distance transformation to the skeleton, since this includes almost all the centers of the maximal discs of the pattern. The shape characterizing the discs is rounded, due the use of a quasi-Euclidean distance function, so that skeleton stability under pattern rotation is greatly favored. A pruning step, which makes it possible to simplify the structure of the skeleton without losing significant information, and a beautifying step, which is aimed at reducing the jaggedness possibly affecting some skeleton branches, are added to the process to improve the well-shapedness of the resulting skeleton.

A Width-Independent Fast Thinning Algorithm

The skeleton of a digital figure can often be regarded as a convenient alternative to the figure itself. It is useful both to diminish drastically the amount of data to be handled, and to simplify the computational procedures required for description and classification purposes. Thinning a digital figure down to its skeleton is a time-consuming process when conventional sequential computers are employed. The procedure we propose allows one to speed up the thinning transformation, and to get a well-shaped skeleton. After cleaning of the input picture has been performed, the pixels of the figure are labeled according to their distance from the background, and a set, whose pixels are symmetrically placed with respect to distinct contour parts of the figure, is found. This set is then given a linear structure by applying topology preserving removal operations. Finally, a pruning step, regarding branches not relevant in the framework of the problem domain, completes the process. The resulting skeleton is a labeled set of pixels which is shown to possess all the required properties, particularly those concerning connectedness, topology, and shape. Moreover, the original figure can almost completely be recovered by means of a reverse distance transformation. Only a fixed and small number of sequential passes through the picture is necessary to achieve the goal. The computational effort is rather modest, and the use of the proposed algorithm turns out to be more advantageous the greater the width of the figure to be thinned.