How Well is the Original Similarity Measure Approximated by CWDS

Our CWDS similarity measure is evaluated by comparing the distances between the fibers in their original representation (measured by cosine similarity [Eq. (4)]) and the distances between the same fibers as calculated in the sparse representation space using CWDS [Eq. (11)]. The distances were calculated using 10,000 randomly selected fibers from each brain. Figure 3 presents plots of the original cosine vs. CWDS for one of the brains and K = 500.

Mean differences between the two similarity measures were calculated for all fiber-sets and different values of K and T_{0}. The results are presented in Table 2.

Both Fig. 3 and Table 2 show that the differences in similarity measures are very small for all tested values of K and T_{0}, with the smallest being for T_{0} = 7, K = 700.

Fig. 2 Reconstruction of fibers from their sparse representations (arbitrary chosen fiber set). Here K = 700, T0 = 7. Row (1) two views of the same fiber set; blue-original fibers, green— reconstructed. Rows (2), (3), (4) each shows three examples of fibers reconstructed with the error of 1 mm, 2 mm, 3 mm, respectively

Fig. 3 Original vs. CWDS similarities for one of the brains. K = 500; green is an identity line. (a) To = 3, (b) To = 5, (c) To = 7

Table 2 Mean differences between the two similarity measures

Mean(S_{orig}-CWDS)

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To = 5

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К = 500

0.0016(0.08- 10^^{3})

0.0004(0.04- 10^^{3})

0.00013 (0.02 • 10~^{3})

К = 600

0.0015(0.08- 10^^{3})

0.00037 (0.04 • 10~^{3})

0.00013 (0.02 • 10~^{3})

К = 700

0.0014(0.08- 10^^{3})

0.000035(0.03- 10^^{3})

0.000011 (0.01 • 10~^{3})

Table 3 Performance with common dictionary: mean reconstruction errors and mean differences between the two similarity measures

To = 3

To — 5

То = 7

Mean rec. error

6.5(0.23)

3.85(0.18)

2.49(0.12)

Mean (S_{ori/!} - CWDS)

0.0018 (0.07 - 10—^{3})

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Representation of Different Fiber-Sets with Common Dictionary

For this experiment, the coresets of all 15 brains were pooled into one huge mixed set. This is possible due to the fact that the original brains are all preregistered to a common space. The mixed set was downsampled by 3 to speed up the calculation and a common dictionary was learned with K = 500, T_{0} = 3,5,7. The mean reconstruction errors and mean differences between cosine similarity in original space and CWDS in compressed space are shown in Table 3.

Here too, the lowest reconstruction error occurs at T_{0} = 7 and is only slightly higher than the error received for individual dictionaries. The distances differences remain as small as before.