Robustness of Recognition:

In the next experiment, we study the recognition accuracy when a zero mean random noise is added to the position of the synthetically transformed shapes related by affine homographies. The highest two singular values for different maximum noise levels are shown in Table 1, for both cases when the image positions are real numbers and when the image positions are discretized. The ratio of the highest to the next singular values does suffer, but there was still more than an order of magnitude separation between the top two even with a noise of 20% in the positions of the boundary points.

Table 1: Impact of noise on singular values.

	Real		Discrete
Noise	Singular Values		Singular Values
Level	Highest	Next	Highest	Next
0	247476	0.00187	213036	73.0211
0.5%	232918	63.6448	229286	124.335
3%	211296	356.347	228500	483.168
5%	208896	839.34	209417	1233.88
10%	193925	1424.26	197214	2069.28
15%	190745	2324.85	176999	3251.64
20%	180199	3887.51	166523	4931.72

Clearly, the recognition is excellent in all cases with the degradation in performance along expected lines. In the next experiment we demonstrate the performance on a real projective homography. Figure 4 shows three different views of the logo of the International Institute of Information Technology.

Figure 4: Three different views of IIIT's logo.

The ratio of the highest singular value to the next highest singular value of the $\Theta$ matrix for various combinations of views is presented in Table 2. When the $\Theta$ matrix was constructed for all the three views the two highest singular values were 1.02679e+06 and 2878, i.e. the rank of the matrix can be considered to be 1.

Table 2: Ratio of highest singular value to the second highest singular value of the matrix of $\kappa$ measures for different combinations of views shown in figure 4.

Views	a	b	c
a	-	431.048	505.847
b	431.048	-	292.71
c	505.847	292.71	-