SHAPE ANALYSIS Several qualitative and quantitative techniques have been developed for characterizing the shape of objects within an image. These techniques are useful for classifying objects in a pattern recognition system and for symbolically describing objects in an image understanding system. Some of the techniques apply only to binary-valued images; others can be extended to gray level images. | Digital Image Processing PIKS Inside Third Edition. William K. Pratt Copyright 2001 John Wiley Sons Inc. ISBNs 0-471-37407-5 Hardback 0-471-22132-5 Electronic 18 SHAPE ANALYSIS Several qualitative and quantitative techniques have been developed for characterizing the shape of objects within an image. These techniques are useful for classifying objects in a pattern recognition system and for symbolically describing objects in an image understanding system. Some of the techniques apply only to binary-valued images others can be extended to gray level images. . TOPOLOGICAL ATTRIBUTES Topological shape attributes are properties of a shape that are invariant under rubber-sheet transformation 1-3 . Such a transformation or mapping can be visualized as the stretching of a rubber sheet containing the image of an object of a given shape to produce some spatially distorted object. Mappings that require cutting of the rubber sheet or connection of one part to another are not permissible. Metric distance is clearly not a topological attribute because distance can be altered by rubber-sheet stretching. Also the concepts of perpendicularity and parallelism between lines are not topological properties. Connectivity is a topological attribute. Figure is a binary-valued image containing two connected object components. Figure is a spatially stretched version of the same image. Clearly there are no stretching operations that can either increase or decrease the connectivity of the objects in the stretched image. Connected components of an object may contain holes as illustrated in Figure . The number of holes is obviously unchanged by a topological mapping. 589 590 SHAPE ANALYSIS FIGURE . Topological attributes. There is a fundamental relationship between the number of connected object components C and the number of object holes H in an image called the Euler number as defined by E C - H The Euler number is also a topological property because C
