Recognition of shapes by morphological attributed ... - CiteSeerX

Recognition of shapes by morphological attributed relational graphs C. Di Ruberto, G. Rodriguez, L. Casta Department of Mathematics, University of Cagliari, Italy

Abstract. Skeletons represent a powerful tool for qualitative shape matching because they resume, synthesize and help the understanding both of the object shape and of its topology. The aim of this work is mainly on using the potential strength of skeleton of discrete objects in computer vision and pattern recognition where features of objects are needed for classification. In this paper we propose a method to improve the topological skeleton representation of a binary shape. This allows to find a graph model characterized by efficient and effective attributes, leading to a more correct discrimination between shapes by an attributed graph matching algorithm.

Introduction A classical technique in pattern recognition is to work on a contour representation of the object to extract the features to classify it. An alternative approach consists in representing the object by a pattern obtained by thinning it as much as possible. The result of this process is a set of idealised lines which is called the skeleton or medial axis of the input pattern and it is the thinnest representation of the original pattern that preserves the topology, aiding synthesis and understanding. The methods to accomplish this are called thinning or skeletonisation. The detection of end points, junction points and curve points of medial axis is important for a structural description that captures the topological information embedded in the skeleton. The skeleton can be converted into a graph associating the end points and the junctions with the vertices and the curve points with the edges. The graph can then be used as an input to graph matching algorithms. In this paper we introduce a method to approximate the skeleton representation of an object, starting from its characteristic points (end points, junction points and curve points). From the approximated skeleton we build an attributed relational graph to organize in a structured way information about object shape and topology embedded in its medial axis. This structural representation allows the comparison of different objects by graph matching algorithms. [5, 6, 10, 11, 16]. 1 From objects to morphological skeletons Digital skeletons can be used to represent objects in a binary digital image for shape analysis and classification [9, 12]. Most of the existing algorithms to generate digital skeletons produce a non-connected skeleton, which is useless for shape description applications since homotopy is not preserved and characteristic points such as junction points or endpoints in the continuous case are lost. On the contrary, a thinning process guarantees the condition for obtaining one-pixel thick and connected skeletons [13].

A digital set can be skeletonised by using morphological operators so as to preserve these important properties by thinning the set with structuring elements (SE) preserving homotopy, i.e. homotopic SEs. The skeleton can be obtained by thinning the input image with a series of homotopic SEs and their rotations until stability is reached [15] and the object is reduced to a set of one-pixel width connected lines. If the skeleton is considered as a connected graph each vertex can be labelled as an end point or a junction point, while an edge is just made of curve points. In [3] two different new methods to identify end points and to detect junctions using mathematical morphology have been proposed. We recall here the definitions of end points, junctions points and curve points. Definition 1. A point of a one-pixel width digital curve is an end point if it has a single pixel 
 neighbourhood. among its Definition 2. A point of a one-pixel width digital curve is defined as a junction point if it has 
 neighbourhood. more than two curve pixels among its Definition 3. A point of a one-pixel width digital curve is defined as a curve point if it has 
 neighbourhood. two curve pixels among its In this way a skeleton  of an object  can be considered as the union of the end points, the junctions and the curve points of  , i.e.

   "!#$%&'()+*,!-/.102'(354 As a consequence, a skeleton  can be partitioned into 6 branches 7185 , 9: 718CDC4 8A@1B

, i.e.

From the skeleton, considering end points and junctions as vertices and branches as links, we build an attributed relational graph to organize in a structured way information about object shape and topology embedded in its thinned representation. Such a graph can be used as a structural object model allowing the comparison of different objects by means of graph matching algorithms. Unfortunately the skeleton of an object often contains both spurious and extremely rough branches which are due to boundary irregularity. To eliminate short external branches on a skeleton a parametric morphological pruning transformation should be applied. We remind that a pruning transformation removes the end points of an image and proceeds until stability is reached [15], while a parametric pruning of a given size E consists in removing E pixels of each branch of the skeleton, starting from an end point. We propose an alternative and faster algorithm to prune a morphological skeleton. The size E for the pruning depends on the length F of the skeleton. All the pixels belonging to a chain of the skeleton and lying within a distance of less than 5% of F from an end point are removed, i.e. the size E for the pruning is equal to 5% of the skeleton length. To eliminate only the branches with distance E from an end point, we first partition the skeleton into all its branch parts by subtracting the junction points from it. By a morphological reconstruction conditioned to the end points we isolate the branches containing the end points only. Applying an area-open transform of size E we remove all the branches containing fewer pixels than E .

The union of the remaining branches with junction points and the branches that were not reconstructed from end points gives us the filtered pruned skeleton. To prune also short internal branches, we adopted the following additional procedure. First, all the junction points are analyzed, and those clustered within a fixed threshold G are collapsed into their center of mass. Then, all the branches whose end points coincide are eliminated from the skeleton. This procedure obviously introduces non-integer coordinates for some of the junctions, but this is not a problem since, as it will be shown in the next section, we are going to adopt a continuous representation for the digital skeleton. 2 Approximated morphological skeleton To improve the smoothness of the skeleton representation and to reduce the amount of data necessary to describe it we have substituted the discrete morphological skeleton described above by an approximated morphological skeleton, that is a set of continuous spline curves which approximate, in the least squares sense, the branches of the original skeleton. Moreover such a continuous representation allows us to compute more easily and more accurately the weights of the attributed relational graph that will be introduced in the next section. Fixed a discretization H2IKJ=LIMBC=444N=LI$OQP on an interval R IKJS=$I$OQT , a cubic spline curve is a paraB[Z Z metric curve UQIVW XU3Y IVC=2U3Y]\ IV5 , whose components are cubic piecewise polynomial functions in ^ \ R IKJS=LI$O_T [1]. This means that each component of the curve is continuous together with its first and second derivative and that it can be represented by a particular polynomial of degree 3 in each of the intervals R I$8A`1BC=$I$8aT , 9bc;2=44242=LE . 8gZ 8AZ 8AZ For each skeleton branch 7185 , 9de;2=4424M=6 , let f h Y ei h Y =Lj h Y  , k+