A two-step process for removing noise from polygonal shapes is presented in this chapter. A polygonal shape is represented as its turning function and then a nonlinear diffusion filter and triangle method is applied. In the first step, several different nonlinear diffusion filters are applied to the turning function that identify dominant vertices in a polygon and remove those vertices that are identified as noise or irrelevant features. The vertices in the turning function which diffuse until the sides that immediately surround them approach the same turning function are identified as noise and removed. The vertices that are enhanced are preserved without changing their coordinates, and they are identified as dominant ones. In the second step, the vertices that form the smallest area triangles are removed. Obtained experimental results demonstrate that the proposed two-step process successfully removes vertices that should be dismissed as noise while preserving dominant vertices that can be accepted as relevant features and give a faithful description of the shape of the polygon. In experimental tests of this procedure successful removal of noise and excellent preservation of shape is demonstrated thanks to appropriate emphasis of dominant vertices.
Top1. Introduction
The comparison of two different shapes has always had significant theoretical and practical importance in computer vision (Arkin, 1991), (Floriani, 2007), (Kimmel, 2004). Model-based recognition is concerned with comparing a shape, stored as a model in the database, with a shape found to exist in an image. If these two shapes are close to being the same, then a vision system should report a match and return a measure of how good that match is. The focus of this chapter is on the pre-processing stage of shape comparison which deals with extraction of dominant vertices and noise removal. When a polygon is found in an image it should be identified. First, found polygon is simplified, the dominant vertices are extracted and it is de-noised. This procedure will significantly accelerate the comparison phase and make it more efficient. The idea is to find a polygon’s equivalent representation in its turning function space based on coordinates of its vertices. Then in the turning function space de-noising is applied through the application of a nonlinear diffusion filter and it is converted back to vertex coordinate space with extracted and identified dominant vertices of the original polygon without any loss of its shape.
Standard representation of a polygon is done by describing its boundary with a circular list of vertices where each vertex is given as a pair of its coordinates. Instead of applying de-noising method and dominant vertices identification in this domain an alternative space is used to represent the analyzed polygon. Each polygon is represented by its turning function 
that measures the angle of the counterclockwise tangent as a function of arclength s. 
is the angle that the tangent at the reference point makes with some reference orientation associated with the polygon (such as the x-axis) and keeps track of the turning that takes place, increasing with left-hand turns and decreasing with right-hand turns. The curvature function for a curve which represents the first derivative of the turning function is frequently used as a shape signature which can be found in the literature (Hong and Tan, 1988), (Hong, 1988), (Schwartz, 1984), (Wolfson, 1987), (O'Rourke, 1985). The definition, in which 
is the angle of the tangent line at the reference point (Hong and Tan, 1988), leads to a simple correspondence between a shift of 
about the origin and a rotation of the polygon. Representation of planar curves and particularly polygons as the function of arc length has been used by a number of other researchers in computational geometry (Schwartz, 1984), (Wolfson, 1987), (O'Rourke, 1985) and computer vision (Ballard, 1982).