Gunilla Borgefors

Distance transformations in digital images

A distance transformation converts a binary digital image, consisting of feature and non-feature pixels, into an image where all non-feature pixels have a value corresponding to the distance to the nearest feature pixel. Computing these distances is in principle a global operation. However, global operations are prohibitively costly. Therefore algorithms that consider only small neighborhoods, but still give a reasonable approximation of the Euclidean distance, are necessary. In the first part of this paper optimal distance transformations are developed. Local neighborhoods of sizes up to 7×7 pixels are used. First real-valued distance transformations are considered, and then the best integer approximations of them are computed. A new distance transformation is presented, that is easily computed and has a maximal error of about 2%. In the second part of the paper six different distance transformations, both old and new, are used for a few different applications. These applications show both that the choice of distance transformation is important, and that any of the six transformations may be the right choice.

Hierarchical chamfer matching: a parametric edge matching algorithm

The algorithm matches edges by minimizing a generalized distance between them. The matching is performed in a series of images depicting the same scene with different resolutions, i.e. in a resolution pyramid. Using this hierarchical structure reduces the computational load significantly. The algorithm is reasonably simple to implement and is insensitive to noise and other disturbances. The algorithm has been tested in several applications. Two of them are briefly presented. In the first application the outlines of common tools are matched to gray-level images of the same tools, with overlapping. In the second application lake edges from aerial photographs are matched to lake edges from a map, with translation, rotation, scale, and perspective changes. The hierarchical chamfer matching algorithm gives correct results using a reasonable amount of computational resources in all tested applications. >

Distance transformations in arbitrary dimensions

Abstract In many applications of digital picture processing, distances from certain feature elements to the nonfeature elements must be computed. In two dimensions at least four different families of distance transformations have been suggested, the most popular one being the city block/chessboard distance family. The purpose of this paper is twofold: To generalize these transformations to higher dimensions and to compare the computed distances with the Euclidean distance. All of the four distance transformation families are presented in three dimensions, and the two fastest ones are presented in four and arbitrary dimensions. The comparison with Euclidean distance is given as upper limits for the difference between the Euclidean distance and the computed distances.

On Digital Distance Transforms in Three Dimensions

Digital distance transforms in 3D have been considered for more than 10 years. However, not all of the complexities involved have been unravelled. In this paper the complete geometry and equations for 3D transforms based on a 3 × 3 × 3 neighborhood of local distances are given. A new type of valid distance transforms (DTs) have been discovered. The optimal solutions are computed, where optimality is defined as minimizing the maximum difference from the true Euclidean distance, thus making the DTs as direction independent as possible. The well-known ?3, 4, 5? DT is confirmed as the most practical weighted DT, where the distance is set to 3 between neighbors sharing an area, 4 between neighbors sharing an edge, and 5 between neighbors sharing a point.

Combining intensity, edge and shape information for 2D and 3D segmentation of cell nuclei in tissue sections

We present a region‐based segmentation method in which seeds representing both object and background pixels are created by combining morphological filtering of both the original image and the gradient magnitude of the image. The seeds are then used as starting points for watershed segmentation of the gradient magnitude image. The fully automatic seeding is done in a generous fashion, so that at least one seed will be set in each foreground object. If more than one seed is placed in a single object, the watershed segmentation will lead to an initial over‐segmentation, i.e. a boundary is created where there is no strong edge. Thus, the result of the initial segmentation is further refined by merging based on the gradient magnitude along the boundary separating neighbouring objects. This step also makes it easy to remove objects with poor contrast. As a final step, clusters of nuclei are separated, based on the shape of the cluster. The number of input parameters to the full segmentation procedure is only five. These parameters can be set manually using a test image and thereafter be used on a large number of images created under similar imaging conditions. This automated system was verified by comparison with manual counts from the same image fields. About 90% correct segmentation was achieved for two‐ as well as three‐dimensional images.

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 ...

A survey on skeletonization algorithms and their applications

A comprehensive survey of skeletonization algorithms and their applications.Different Skeletonization approaches are summarized.Topology preservation and parallel skeletonization are discussed.A review of multi-scale skeletonization is presented.Applications and performance evaluation of skeletonization are discussed. Skeletonization provides an effective and compact representation of objects, which is useful for object description, retrieval, manipulation, matching, registration, tracking, recognition, and compression. It also facilitates efficient assessment of local object properties, e.g., scale, orientation, topology, etc. Several computational approaches are available in literature toward extracting the skeleton of an object, some of which are widely different in terms of their principles. In this paper, we present a comprehensive and concise survey of different skeletonization algorithms and discuss their principles, challenges, and benefits. Topology preservation, parallelization, and multi-scale skeletonization approaches are discussed. Finally, various applications of skeletonization are reviewed and the fundamental challenges of assessing the performance of different skeletonization algorithms are discussed.

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

Abstract 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.

Analyzing Nonconvex 2D and 3D Patterns

A nonconvex pattern can be analyzed by describing its concavity regions. These can be identified by computing the difference between the convex hull of the pattern and the pattern itself. A suitable approximation of the convex hull can be obtained by repeatedly filling local concavities of the pattern. Parallel and sequential algorithms are proposed, to fill concavities of 2D and 3D patterns. The resulting approximation of the convex hull is a covering polygon or polyhedron, which is either convex or nearly convex. Exact measures of worst remaining concavities and created protrusions are given. The concavity regions are extracted and different features are presented and computed. Hierarchical pattern descriptions are also suggested, based on the use of concavity trees.

On Reversible Skeletonization Using Anchor-Points from Distance Transforms

In many applications thinning of objects is of great interest. We here present a skeletonization algorithm that is based on the idea of iteratively thinning the distance transform of an object layer by layer until either an anchor-point is reached or the connectivity breaks. Our algorithm is general in the sense that any metric and any connectivity can be used. Also, it is based on ideas that are not specific for 2D. The properties of the resulting skeletons are evaluated according to the “Lee?Lam?Suen properties.”