Rebuttal Exp
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1
Experimental Validation of the Unbiasness Property
To validate the unbiased property that we proved in Theorem 1, we manually extracted points on the outer contour of the corpus callosum from nine normal subjects, (as shown Figure 1, indicated by ”o”). The recovered deformation between each point-set and the mean shape are superimposed on the first nine plots in Figure 1. The resulting atlas (mean point-set) is shown in third row of Figure 1, and is superimposed over all the point-sets before and after the alignment. As we described in the paper, all these results are computed simultaneously and automatically. This example clearly demonstrate that our joint matching and atlas construction algorithm can simultaneously align multiple shapes (modeled by sample point-sets) and compute a meaningful atlas/mean shape. We then separate the nine point-sets into three groups, each group contains three point-sets. We can construct three probabilistic atlases from these subsets using our algorithms, and all the point-sets inside each of the subsets are registered. Then we construct a single atlas from these 3 atlases, the resulted hierarchical atlas and the atlas that we constructed non-hierarchically are shown in Fig. 2. Note that we use the same parameter setting as in the previous non-hierarchical experiment. From the figure, it is evident that our algorithm yields an atlas not much different from the atlas constructed from the original nine point-sets in Fig. 1, which confirms that the unbiasness property that we proved in Theorem 1. We also observed that the Jensen-Renyi value computed from the registered density function is in the order of 1e − 6, which is reflective of the statistical similarity between the registered density functions. Point Set 1
Point Set 2
Point Set 3
Point Set 4
Point Set 5
Point Set 6
Point Set 7
Point Set 8
Point Set 9
Before Registration After Registration
Figure 1: Experiment results on nine 2D corpus callosum point-sets. The first two rows and the left image in third row show th deformation of each point-set to the atlas, superimposed with initial point-set (show in ’o’) and deformed point-set (shown in ’*’). Middle image in the third row: The estimated atlas is shown superimposed over all the point-sets. Right: The estimated atlas is shown superimposed over all the deformed point-sets.
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Figure 2: Illustration of the unbiasness property of the constructed atlas. The figure shows the resulting hierarchical atlas superimposed over the atlas constructed from the original nine point-sets in Fig. 1.
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Experimental Validation of Other Transformation Models
To test the validity of our approach to any type of smooth registration transformation, we perform a set of exact rigid registration experiments on both synthetic and real data sets without noise and outliers. Some examples are shown in Figure 3. The top row shows the registration result for a 2D real range data set of a road (which was also used in Tsin and Kanade ECCV04’s experiments). The figure depicts the real data and the registered (using rigid motion). The top left frame contains two unregistered point-sets superposed on each other. The top right frame contains the same point-sets after registration using our algorithm. A 3D helix example is presented in the second row (with the same arrangement as the top row). Initial setup
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Figure 3: Results of rigid registration in noiseless case. ’o’ and ’+’ indicate the model and scene points respectively.
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Experimental Validation of the Unbiasness Property
To validate the unbiased property that we proved in Theorem 1, we manually extracted points on the outer contour of the corpus callosum from nine normal subjects, (as shown Figure 1, indicated by ”o”). The recovered deformation between each point-set and the mean shape are superimposed on the first nine plots in Figure 1. The resulting atlas (mean point-set) is shown in third row of Figure 1, and is superimposed over all the point-sets before and after the alignment. As we described in the paper, all these results are computed simultaneously and automatically. This example clearly demonstrate that our joint matching and atlas construction algorithm can simultaneously align multiple shapes (modeled by sample point-sets) and compute a meaningful atlas/mean shape. We then separate the nine point-sets into three groups, each group contains three point-sets. We can construct three probabilistic atlases from these subsets using our algorithms, and all the point-sets inside each of the subsets are registered. Then we construct a single atlas from these 3 atlases, the resulted hierarchical atlas and the atlas that we constructed non-hierarchically are shown in Fig. 2. Note that we use the same parameter setting as in the previous non-hierarchical experiment. From the figure, it is evident that our algorithm yields an atlas not much different from the atlas constructed from the original nine point-sets in Fig. 1, which confirms that the unbiasness property that we proved in Theorem 1. We also observed that the Jensen-Renyi value computed from the registered density function is in the order of 1e − 6, which is reflective of the statistical similarity between the registered density functions. Point Set 1
Point Set 2
Point Set 3
Point Set 4
Point Set 5
Point Set 6
Point Set 7
Point Set 8
Point Set 9
Before Registration After Registration
Figure 1: Experiment results on nine 2D corpus callosum point-sets. The first two rows and the left image in third row show th deformation of each point-set to the atlas, superimposed with initial point-set (show in ’o’) and deformed point-set (shown in ’*’). Middle image in the third row: The estimated atlas is shown superimposed over all the point-sets. Right: The estimated atlas is shown superimposed over all the deformed point-sets.
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hierarchical atlas non−hierarchical atlas
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−0.05 0.4
0.5
0.6
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0.9
1
1.1
Figure 2: Illustration of the unbiasness property of the constructed atlas. The figure shows the resulting hierarchical atlas superimposed over the atlas constructed from the original nine point-sets in Fig. 1.
2
Experimental Validation of Other Transformation Models
To test the validity of our approach to any type of smooth registration transformation, we perform a set of exact rigid registration experiments on both synthetic and real data sets without noise and outliers. Some examples are shown in Figure 3. The top row shows the registration result for a 2D real range data set of a road (which was also used in Tsin and Kanade ECCV04’s experiments). The figure depicts the real data and the registered (using rigid motion). The top left frame contains two unregistered point-sets superposed on each other. The top right frame contains the same point-sets after registration using our algorithm. A 3D helix example is presented in the second row (with the same arrangement as the top row). Initial setup
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Figure 3: Results of rigid registration in noiseless case. ’o’ and ’+’ indicate the model and scene points respectively.
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