1、 - 1 - Digital Image Processing 1 Introduction Many operators have been proposed for presenting a connected component n a digital image by a reduced amount of data or simplied shape. In general we have to state that the development, choice and modi_cation of such algorithms in practical applications
2、 are domain and task dependent, and there is no best method. However, it is interesting to note that there are several equivalences between published methods and notions, and characterizing such equivalences or di_erences should be useful to categorize the broad diversity of published methods for sk
3、eletonization. Discussing equivalences is a main intention of this report. 1.1 Categories of Methods One class of shape reduction operators is based on distance transforms. A distance skeleton is a subset of points of a given component such that every point of this subset represents the center of a
4、maximal disc (labeled with the radius of this disc) contained in the given component. As an example in this _rst class of operators, this report discusses one method for calculating a distance skeleton using the d4 distance function which is appropriate to digitized pictures. A second class of opera
5、tors produces median or center lines of the digital object in a non-iterative way. Normally such operators locate critical points _rst, and calculate a speci_ed path through the object by connecting these points. The third class of operators is characterized by iterative thinning. Historically, List
6、ing 10 used already in 1862 the term linear skeleton for the result of a continuous deformation of the frontier of a connected subset of a Euclidean space without changing the connectivity of the original set, until only a set of lines and points remains. Many algorithms in image analysis are based
7、on this general concept of thinning. The goal is a calculation of characteristic properties of digital objects which are not related to size or quantity. Methods should be independent from the position of a set in the plane or space, grid resolution (for digitizing this set) or the shape complexity
8、of the given set. In the literature the term thinning is not used - 2 - in a unique interpretation besides that it always denotes a connectivity preserving reduction operation applied to digital images, involving iterations of transformations of speci_ed contour points into background points. A subs
9、et Q _ I of object points is reduced by a de_ned set D in one iteration, and the result Q0 = Q n D becomes Q for the next iteration. Topology-preserving skeletonization is a special case of thinning resulting in a connected set of digital arcs or curves. A digital curve is a path p =p0； p1； p2； :； p
10、n = q such that pi is a neighbor of pi 1, 1 _ i _ n, and p = q. A digital curve is called simple if each point pi has exactly two neighbors in this curve. A digital arc is a subset of a digital curve such that p 6= q. A point of a digital arc which has exactly one neighbor is called an end point of
11、this arc. Within this third class of operators (thinning algorithms) we may classify with respect to algorithmic strategies: individual pixels are either removed in a sequential order or in parallel. For example, the often cited algorithm by Hilditch 5 is an iterative process of testing and deleting
12、 contour pixels sequentially in standard raster scan order. Another sequential algorithm by Pavlidis 12 uses the de_nition of multiple points and proceeds by contour following. Examples of parallel algorithms in this third class are reduction operators which transform contour points into background
13、points. Di_erences between these parallel algorithms are typically de_ned by tests implemented to ensure connectedness in a local neighborhood. The notion of a simple point is of basic importance for thinning and it will be shown in this report that di_erent de_nitions of simple points are actually
14、equivalent. Several publications characterize properties of a set D of points (to be turned from object points to background points) to ensure that connectivity of object and background remain unchanged. The report discusses some of these properties in order to justify parallel thinning algorithms.
15、1.2 Basics The used notation follows 17. A digital image I is a function de_ned on a discrete set C , which is called the carrier of the image. The elements of C are grid points or grid cells, and the elements (p； I(p) of an image are pixels (2D case) or voxels (3D case). The range of a (scalar) ima
16、ge is f0； :Gmaxg with Gmax _ 1. The range of a binary image is f0； 1g. We only use binary images I in this report. Let hIi be the set of all pixel locations with value 1, i.e. hIi = I 1(1). The image carrier is de_ned on an orthogonal grid in 2D or 3D - 3 - space. There are two options: using the gr
17、id cell model a 2D pixel location p is a closed square (2-cell) in the Euclidean plane and a 3D pixel location is a closed cube (3-cell) in the Euclidean space, where edges are of length 1 and parallel to the coordinate axes, and centers have integer coordinates. As a second option, using the grid p
18、oint model a 2D or 3D pixel location is a grid point. Two pixel locations p and q in the grid cell model are called 0-adjacent i_ p 6= q and they share at least one vertex (which is a 0-cell). Note that this speci_es 8-adjacency in 2D or 26-adjacency in 3D if the grid point model is used. Two pixel
19、locations p and q in the grid cell model are called 1- adjacent i_ p 6= q and they share at least one edge (which is a 1-cell). Note that this speci_es 4-adjacency in 2D or 18-adjacency in 3D if the grid point model is used. Finally, two 3D pixel locations p and q in the grid cell model are called 2
20、-adjacent i_ p 6= q and they share at least one face (which is a 2-cell). Note that this speci_es 6-adjacency if the grid point model is used. Any of these adjacency relations A_, _ 2 f0； 1； 2； 4； 6； 18； 26g, is irreexive and symmetric on an image carrier C. The _-neighborhood N_(p) of a pixel locat
21、ion p includes p and its _-adjacent pixel locations. Coordinates of 2D grid points are denoted by (i； j), with 1 _ i _ n and 1 _ j _ m； i； j are integers and n；m are the numbers of rows and columns of C. In 3Dwe use integer coordinates (i； j； k). Based on neighborhood relations we de_ne connectednes
22、s as usual: two points p； q 2 C are _-connected with respect to M _ C and neighborhood relation N_ i_ there is a sequence of points p = p0； p1； p2； :； pn = q such that pi is an _-neighbor of pi 1, for 1 _ i _ n, and all points on this sequence are either in M or all in the complement of M. A subset
23、M _ C of an image carrier is called _-connected i_ M is not empty and all points in M are pairwise _-connected with respect to set M. An _-component of a subset S of C is a maximal _-connected subset of S. The study of connectivity in digital images has been introduced in 15. It follows that any set
24、 hIi consists of a number of _-components. In case of the grid cell model, a component is the union of closed squares (2D case) or closed cubes (3D case). The boundary of a 2-cell is the union of its four edges and the boundary of a 3-cell is the union of its six faces. For practical purposes it is easy to use neighborhood operations (called local operations) on a digital image I which de_ne a value at p 2 C in the transformed image based on pixel
