A jagged grid is used as a graphical representation to manage multiple-shell design; each column of data points is a collection of points on a spherical shell. Points in each row may be thought of as collections points around a radial line.
Computation of the cost function is greatly simplified by the use of the metric function S between two “real” points, which is specifically designed to deal with point set that is endowed with antipodal symmetry. The lower triangular matrix shown above is used to keep the values of S.
50 000 random permutations were generated to fill the 12 × 12 grid. The initial cost function values of these 50 000 samples are shown in the histogram that is color-coded in red. The histogram of the final cost function values of these 50 000 samples is shown in blue. Inset shows the magnified version of these two histograms.
The points are generated from the example of a 12 × 12 grid. Every point in the same shell has the same color and each shell is assigned a distinct color and these colors are shown on the right. It can be seen that each set of points with the same color is nearly uniformly distributed on the sphere, which is related to the goal of criterion #1.
The clusters seen here are generated from the example of a 12 × 12 grid. Each row contains points with the same color, and these points are designed to be close together in a form of a cluster so that, when the points are projected to different shells, we would have fulfilled the criterion #2, which is to provide the maximum coverage around each radial line. The problem of the boundary effect in which there might be two neighboring points with distinct colors but are moved to some common shell will not be an issue here because of criterion #1.
A collection of 18 acquisition designs and the corresponding matrix condition numbers. The design matrices were constructed from the three-dimensional basis functions with u = 0.00827, which in turn depends on Δ and D. Here, the diffusivity is chosen to be close to free diffusion of water in the brain. For example, design #8 has (9,18,27,27) points in the (1st, 2nd, 3rd, 4th) shells, respectively. This design has matrix condition number of 38.8 and A-optimal measure of 2.0 × 107. Further, the q-value at the first shell is 25.2 mm−1. Design #18 is the square acquisition design.
The same collection of 18 acquisition designs as in Fig. 6 but the design matrices were now constructed from the basis functions with u = 0.000827. Here, the value of the diffusivity was chosen to be low similar to the case of hindered diffusion.
The condition number of the design matrix of design #18 as a function of the diffusion time.
(A) There are eight rows and each row of ten points is shown here as points with the same hue on the translucent unit sphere. Points in different rows have different hues. (B) There are ten columns and each column has eight points. Each column of eight points is shown here as points with the same hue on the translucent unit sphere. Again, points in different columns have different hues.
(A) The ratios of the electrostatic energy of points in each row of the 2D bit-reversal method (in red or with the highest ratios), of the proposed method (in blue or with the lowest ratios) and of the Golden Mean method (in black or with the medium ratios) to that of the point set of the same size (100 points) generated from the analytically exact spiral scheme. (B) The ratios of the electrostatic energy of points in each column of the 2D bit-reversal method (in red or with the highest ratios), of the proposed method (in blue or with the lowest ratios) and of the Golden Mean method (in black or with the medium ratios) to that of the point set of the same size (128 points) generated from the analytically exact spiral scheme.
Box plots and basic statistics on the Voronoi areas (A) and circumferences (B) generated from the proposed method (analytically exact spiral scheme) and the Golden Mean method.
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