Results and Discussions

We conducted a number of experiments to affirm the validity of the formulations in the previous section. Extensive experimentations were carried out on synthetic images, natural images with simulated transformations and real natural images. In the rest of this section, we demonstrate the performance of the proposed schemes with quantitative results. For simulation of views, transformations were applied on a reference view and then the boundary representations were shifted by random amounts in each view to simulate lack of correspondence. In experiments on real images, objects of interest were segmented out and their boundaries were sampled to 1024 boundary points. The ranks of the matrices $\Theta$, $\Theta^{\prime\prime}$ and $\Theta ^{\prime \prime \prime }$ were determined using the Singular Value Decomposition algorithm, wherein the number of non-zero singular values gives the rank of the matrix. When the boundary representation is in the form of integer coordinates, discretization introduces quantization noise that make the rank constraint an approximation of the true one derived, but nonetheless enforceable. To verify whether a matrix has an approximate rank $r$, we give the ratio of $r$th to $(r+1)$th singular values (arranged in descending order). This ratio is high if the matrix has an approximate rank of $r$. Also all the following singular values are very small in magnitude.

Table 1: Results on Dinosaur Images. Ratios of the relevant consecutive singular values for $\Theta$ and for $\Theta ^{\prime \prime \prime }$ are shown

	Dinosaur 1	Dinosaur 2	Dinosaur 3	Dinosaur 4
Dinosaur 1	--	43176.5, 504.423	23988.5, 322.283	35453.9, 439.72
Dinosaur 2	43176.5, 504.423	--	25733.7, 312.512	35352.6,322.338
Dinosaur 3	23988.5, 322.283	25733.7, 312.512	--	17548,137.258
Dinosaur 4	35453.9,439.72	35352.6,322.338	17548,137.258	--

In the first example, we considered four views of a dinosaur as in Figure 1(a), (b), (c) and (d). These views are related by affine transformations. One may observe that the Euclidean measures are no longer preserved under these transformation. However, the rank constraints allow us to recognise a dinosaur image given another view. Ranks of $\Theta$ and $\Theta ^{\prime \prime \prime }$ were computed. Ratio of the $r$th and $(r+1)$th singular values are used to verify the ranks. All constraints provided the ratio to be much more than 100 in all cases. The ratios of singular values for each pairs of images for the invariant and magnitude constraints are arranged in Table 1. In all cases, the $r$th singular value ($r$ = 1 for $\Theta$ and $r$ = 3 for $\Theta ^{\prime \prime \prime }$) was found to be greater than the $(r+1)$th one by a factor of $100$ or $1000$.

Figure 2: Three views of the logo of our institute

Now, we demonstrate the performance when a zero mean random noise is added to the position of the synthetically transformed shape for an affine homography. The two singular values of interest of matrix $\Theta$ for different noise levels for real(without quantization) and discrete (integers) boundary representations are shown in Table 2. This ratio does deteriorate with noise, however, there was still more than an order of magnitude separation between them even with a noise of 20% in the positions of the boundary points.

Table 2: Impact of noise on singular values

	Real		Discrete
Noise	Singular Values		Singular Values
Level	Highest	Next	Highest	Next
0	247476	0.0019	213036	73.02
0.5%	232918	63.65	229286	124.34
3%	211296	356.35	228500	483.17
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

Table 3: 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 2.

Views	a	b	c
a	-	431.0	505.8
b	431.0	-	292.7
c	505.8	292.7	-

The recognition is clearly very good in all cases with the degradation in performance along expected lines. We have achieved recognition between two planar shapes under the assumption that the homography between them has a specific form, without knowing the correspondence between points.

Though the theory was primarily developed for affine homographies, the rank constraints are practically valid for images under projective transformation. The logo of the International Institute of Information Technology was imaged from various viewing positions. These images are known to be related by projective homographies. Three views are shown in Figure 2. Ratios of the two highest singular values of the $\Theta$ matrix for various combinations of those views are given in Table 3. All pairs are clearly recognisable and the ratios are more than 250 in all cases.