3 Results

Ideally we would like the methods for measuring corner properties to work reliably over a range of conditions: varying orientations, subtended angles, degrees of noise, etc. To assess the accuracy and robustness of the new methods described here and also some of the previous ones given in [ 14 ] they have been extensively tested on synthetic data. Idealised corners were generated for 13 different subtended angles ( as images, and then averaged and subsampled down to . Different levels of Gaussian noise ( ) were then repeatedly added to create a total of 13000 test images. Some examples are shown in figure  4 with and respectively. The contrast between corner foreground and background was kept constant at 200, so that the images contained signal to noise ratios ranging from 40 to . Note that unless otherwise stated all tests use a window size of and the corner properties were measured at the ideal location: (20,20).

   
Figure 4: Sample test corner images

Table  1 shows the error rates of each method averaged over all 13000 test images. The tests were carried out twice. The first time the corner properties were measured at the known true location of the corner (20,20). The second time the Kitchen-Rosenfeld [ 6 ] detector was applied to the images, and the properties were measured at the corner that was detected closest to the true location.

3.1 Bluntness

Now we look at the effectiveness of the methods for measuring bluntness. Each of the three methods assumes a different model of corner rounding. We cannot therefore generate synthetic blunt corners using any of these models since this would favour the corresponding bluntness measurement method. Instead we blunten the corners by cropping the apex by a straight edge. For each degree of cropping 1000 examples of corners containing various amounts of noise were generated for testing. Results are shown in figure  5 . Both the kurtosis and hyperbola fitting methods appear to do well over a range of degrees of cropping, displaying a fairly linear behaviour. The circle fitting method does not fare so well as it breaks down under large amounts of cropping, losing its linear response, and showing substantial variance. We can quantify the linearity of the methods using Pearsons's correlation coefficient. For the kurtosis method we test against cropping, while for the other two we simply test a and r respectively. The coefficients are 0.98364, 0.98507, and 0.93383, verifying that the first two methods are superior to the third.

   
Figure 5: Measuring bluntness

3.2 Cusps

To test the measurement of concavity and convexity of cusps one synthetic symmetric example was generated of each, and Gaussian noise added as before to provide two sets of 1000 test images. The results of testing the method on test images of each corner type are shown in figure  6 a. For convenience the spatial axis has been scaled which also causes the angles to be scaled. Even at low SNRs the concave and convex corner types can be reliably discriminated. For comparison the method was also tested on 1000 examples of a noisy straight corner. The measured angle is close to zero, allowing it to be confidently classified as a straight edge.

   
Figure 6: Effect of noise on measurement accuracy for boundary curvature