System and method for determining k-space views and diffusion weighting directions using centroidal voronoi tessellations
Described here are a system and method for generating uniform antipodally symmetric point sets that define diffusion-weighting directions or three-dimensional radial k-space trajectories for magnetic resonance imaging. The point sets are generated robustly and efficiently using a constrained centroidal Voronoi tessellation endowed with a pseudometric. This pseudometric is derived from a measure of the electrostatic energy of points distributed on a hemispherical surface.
1. A method for producing an image of a subject with a magnetic resonance imaging (MRI) system, the steps of the method comprising:
a) determining MRI acquisition parameters by generating a point set that defines the MRI acquisition parameters, each point being computed as a generator of a centroidal Voronoi tessellation that is formed using a pseudometric on a sphere that enforces antipodal symmetry of points distributed on a hemisphere of the sphere, the point set defining points that are substantially uniformly and antipodally distributed on a surface of the sphere;
b) directing the MRI system to acquire k-space data from a subject using the MRI acquisition parameters determined in step a); and
c) reconstructing an image of the subject from the k-space data acquired in step b).
2. The method as recited in claim 1 in which the MRI acquisition parameters determined in step a) include at least one of a k-space trajectory and a diffusion-weighting gradient direction.
3. The method as recited in claim 2 in which the k-space trajectory is a three-dimensional radial k-space trajectory.
4. The method as recited in claim 1 in which step a) includes providing a set of initial generators; determining a centroidal Voronoi tessellation for each initial generator; and iteratively updating the centroidal Voronoi tessellations using the pseudometric.
5. The method as recited in claim 4 in which determining the centroidal Voronoi tessellation for each initial generator includes:
selecting one of the initial generators;
identifying surrounding generators within a specified distance of the selected initial generator; and
computing vertices of the centroidal Voronoi tessellation by computing a convex hull using the selected initial generator and the identified surrounding generators.
6. The method as recited in claim 5 in which the vertices of the centroidal Voronoi tessellation are computed by:
rotating the selected initial generator and the identified surrounding generators such that the selected initial generator lies on a z-axis; and
stereographically projecting the identified surrounding generators onto an x-y plane.
7. The method as recited in claim 5 in which the specified distance is based on a radius of a spherical cap centered on the selected initial generator.
8. The method as recited in claim 1 in which the pseudometric is based on a measure of electrostatic energy of points distributed on the hemisphere of the sphere.
9. The method as recited in claim 8 in which the pseudometric has the following form:
(
1
r
ij
+
1
4
-
r
ij
2
)
-
1
;
wherein r ij is a Euclidean distance between two points on the surface of the hemisphere.
10. The method as recited in claim 1 in which the pseudometric has the following form:
(
1
r
ij
n
+
1
(
4
-
r
ij
2
)
n
/
2
)
-
1
;
wherein r ij is a Euclidean distance between two points on the surface of the hemisphere and n is a positive integer.
11. A method for producing an image of a subject with a magnetic resonance imaging (MRI) system, the steps of the method comprising:
a) providing a look-up table that defines points on a sphere that are substantially uniformly distributed on the surface of the sphere, each point corresponding to a centroid of a centroidal Voronoi tessellation that is defined by a pseudometric that enforces antipodal symmetry on the sphere;
b) determining MRI acquisition parameters using the provided look-up table;
c) directing the MRI system to acquire k-space data from a subject using the MRI acquisition parameters determined in step b); and
d) reconstructing an image of the subject from the k-space data acquired in step c).
12. The method as recited in claim 11 in which the MRI acquisition parameters determined in step b) include at least one of a k-space trajectory and a diffusion-weighting gradient direction.
13. The method as recited in claim 12 in which the k-space trajectory is a three-dimensional radial k-space trajectory.