1. Copyright © 2012 Elsevier Inc. All rights reserved.. Chapter 10 Boundary Pattern Analysis

2. Copyright © 2012 Elsevier Inc. All rights reserved.. FIGURE 10.1

3. 10.2 Boundary Tracking Procedures Before objects can be matched with their boundary patterns, means must be found for tracking systematically around the boundaries of all the objects in an imahe. It is necessary to ensure that we Never reverse direction, Know when we have been round the whole boundary once, and Record which object boundaries have been encountered. Copyright © 2012 Elsevier Inc. All rights reserved.. 3

4. 10.3 Centroidal Profiles The substantial matching problems that occur with 2D template matching make it attractive to attempt to locate objects in a less demanding search space. Perhaps the most obvious such scheme uses an (r, θ) plot. Centroidal profile: a polar coordinate system is set up relative to the centroid of the object and the object boundary is plotted as an (r, θ) graph. See Fig. 10.2 Copyright © 2012 Elsevier Inc. All rights reserved.. 4

5. FIGURE 10.2 Copyright © 2012 Elsevier Inc. All rights reserved.. 5

6. Matching: Next, the 1D graph so obtained is matched against the corresponding graph for an idealized object of the same type. Since objects generally have arbitrary orientation, it is necessary to “slide” the idealized graph along that obtained from the image data until the best match is obtained. Copyright © 2012 Elsevier Inc. All rights reserved.. 6

7. 10.4 Problems with the Centroidal Profile Approach 1. Any major defect or occlusion of the object boundary can cause the centroid to be moved away from its true position. Copyright © 2012 Elsevier Inc. All rights reserved.. 7

8. 2. The (r, θ) plot will be multivalued for a certain class of object. This has the effect of making the matching process partly 2D and leads to complication and excessive computation. Copyright © 2012 Elsevier Inc. All rights reserved.. 8

9. 3. It will be difficult to obtain an accurate centroidal profile for the region near the centroid of elongated objects such as spanners or screwdrivers. Copyright © 2012 Elsevier Inc. All rights reserved.. 9

10. 10.5 The (S, Ψ) Plot The (S, Ψ) graph has provided particularly popular since it is inherently better suited than the (r, θ) graph to situations where defects and occlusions may occur. In addition, it does not suffer from the multiple values encountered by the (r, θ) method. (S, Ψ) graph does not require prior estimation of the centroid or some other reference point since it is computed directly from the boundary, in the form of a plot of the tangential orientation Ψ as a function of boundary distance s. Copyright © 2012 Elsevier Inc. All rights reserved.. 10

11. The graph has a Ψ value that increases by 2π for each circuit of the boundary, i.e., Ψ(s) is not periodic in s. One way to tackling this problem is to make a comparison with the shape of a circle of the same boundary length P. Thus an (s, ΔΨ) graph is plotted. Thus, the graph is completely 1D and is also periodic. Matching should now reduce to sliding the template. Copyright © 2012 Elsevier Inc. All rights reserved.. 11

12. Another way in which the problem of non- periodic Ψ(s) can be solved is by replacing Ψ by its derivative d Ψ/ds. The increase of 2 π after each circuit of the boundary is eliminated: d(Ψ+2 π)/ds= d Ψ/ds. Note that d Ψ/ds is actually the local curvature function κ(s) (see Fig. 10.4). Many workers take the (s, κ) and expand as a Fourier series, then use the Fourier descriptors as a feature vector. Copyright © 2012 Elsevier Inc. All rights reserved.. 12

13. FIGURE 10.6 Copyright © 2012 Elsevier Inc. All rights reserved.. 13