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How Well Do the Sparse Representations Approximate the Original Fibers

For each of the fiber-sets F, a dictionary D was learned as described in Sect. 2. The dictionary learning was performed on a subset C of the fibers, which constitute a coreset of F. The coreset size was set to 20,000 for all the experiments (reduction by 50), in order to shorten the dictionary building times in Matlab. Dictionary sizes of 500, 600, and 700 atoms were evaluated. The sparsity constraint, T0, was set to: 3, 5,7. The complexity of calculating cosine similarity on the original space is O (Di), Di being the dimension of the original representation. The complexity of CWDS is О (?>;), where D2 is T0. Therefore in order not to increase the computational requirements of CWDS, T0 needs to be kept below */D, or in our case T0 < 1.1 (as Di = 60).

First, we calculate the representation error relatively to the coreset fibers, which were used to train each dictionary. For each original coreset fiber, the reconstructed fiber is found using Eq. (3) and a L2 distance is evaluated between the original and the reconstructed fibers. Figure 1 presents the mean distance for all 15 brains for different values of K and T0. The mean values over all 15 fiber-sets are summarized in Table 1. In addition, we check whether the quality of sparse approximation remains the same for the fibers in the sets which are not part of the coreset. Here the reconstruction error is calculated for 10,000 fibers randomly selected from the full fiber set. Mean values over all sets are summarized in Table 1.

As expected, the reconstruction error becomes smaller for higher T0. T0 = 7 achieves the lowest error which is stable around 2 mm. It is also important to note that the error for random fibers is only a slightly higher than that of the coreset fibers. The difference is of the order of magnitude of the standard deviations. This can be seen as evidence that the coreset is a good enough subset for training the dictionary.

Mean reconstruction errors for coreset fibers; blue-T0=3, purple-T0=5, green-T0=7

Fig. 1 Mean reconstruction errors for coreset fibers; blue-T0=3, purple-T0=5, green-T0=7

Table 1 Mean reconstruction errors in mm and standard deviations (in parentheses)

Mean (1/ —/1)

T0 = 3

Tq — 5

To = 7

Coreset. К = 500

9(0.2)

3.01 (0.15)

1.92(0.13)

Random set. К = 500

5.65(0.24)

3.31 (0.16)

2.18(0.15)

Coreset. К = 600

4.97(0.2)

2.81 (0.15)

1.81 (0.12)

Random set. К = 600

5.28(0.25)

3.09(0.18)

2.05(0.14)

Coreset. К = 700

4.65(0.2)

2.62(0.13)

1.7(0.0.09)

Random set. К = 700

4.97(0.23)

2.91 (0.15)

1.94(0.11)

Figure 2 illustrates the reconstruction quality for one of the fiber sets. The original and reconstructed fibers are shown in row 1. Rows (2-4) show examples of single fibers and their reconstructed versions for different error magnitudes (zoomed in).

 
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