Linear transform for diffusion MRI
Samples of a Fourier transform of a signal may be received that are distributed in a generally spherically-shaped pattern on a surface of at least one sphere. The samples may be assembled to form a profile function having a domain that is a surface of at least one sphere. Information indicative of at least one property of the signal may be determined by applying a spherical linear transform to the profile function. The spherical linear transform may use more than just an equator of the profile function for each of multiple orientations.
1. A non-transitory, tangible, computer-readable storage medium containing a program of instructions that cause a computer system running the program of instructions to:
receive samples of a Fourier transform of a diffusion MR signal that are distributed in a generally spherically-shaped pattern on a surface of at least one sphere;
assemble the samples to form a profile function having a domain that is a surface of at least one sphere;
determine information indicative of at least one property of the signal by applying a spherical linear transform to the profile function, wherein the spherical linear transform uses more than just an equator of the profile function for each of multiple orientations; and
use the determined information to generate images or maps.
2. The storage medium of claim 1 wherein the samples of a Fourier transform include diffusion MRI data and the signal is a diffusion function.
3. The storage medium of claim 2 wherein the program of instructions causes the computer system to calculate a response function of the spherical linear transform.
4. The storage medium of claim 3 in which the calculated response function accounts for finite sampling of the sphere.
5. The storage medium of claim 4 in which the calculating the response function uses re-gridding and a Fast Fourier Transform.
6. The storage medium of claim 3 wherein the program of instructions causes the computer system to select the spherical linear transform based on an anticipated accuracy of the information that results from application of the selected spherical linear transform.
7. The storage medium of claim 2 wherein the program of instructions causes the computer system to evaluate the spherical linear transform using a computational algorithm based on a spherical harmonic representation of a transform kernel.
8. The storage medium of claim 2 wherein the at least one determined property of the diffusion function includes a constant solid-angle orientation distribution function.
9. The storage medium of claim 2 wherein the spherical linear transform includes a Funk-Radon and Cosine Transform or a weighted sum or integral of multiple Funk-Radon and Cosine Transforms.
10. The storage medium of claim 2 wherein the determining the at least one property of the diffusion function includes augmenting the spherical linear transform with nonspherical or nonlinear processing.
11. The storage medium of claim 1 wherein the program of instructions causes the computer system to determine how accurately the information indicates the at least one property by calculating a response function of the spherical linear transform.
12. The storage medium of claim 11 in which the calculated response function accounts for finite sampling of the sphere.
13. The storage medium of claim 12 in which the calculating the response function uses re-gridding and a Fast Fourier Transform.
14. The storage medium of claim 11 wherein the program of instructions causes the computer system to select the spherical linear transform based on an anticipated accuracy of the information that results from application of the selected spherical linear transform.
15. The storage medium of claim 1 wherein the at least one determined property of the signal include moments or weighted moments of the signal or its radial projections.
16. The storage medium of claim 1 wherein the program of instructions causes the computer system to evaluate the spherical linear transform using a computational algorithm based on a spherical harmonic representation of a transform kernel.
17. The storage medium of claim 1 wherein the spherical linear transform includes a Funk-Radon and Cosine Transform or a weighted sum or integral of multiple Funk-Radon and Cosine Transforms.
18. The storage medium of claim 1 wherein the determining the at least one property of the signal includes augmenting the spherical linear transform with nonspherical or nonlinear processing.
19. An MRI system comprising:
an MRI machine that generates diffusion MRI data;
a computer system; and
a computer-readable storage medium containing a program of instructions that cause the computer system running the program of instructions to:
receive samples of a Fourier transform of a signal that are distributed in a generally spherically-shaped pattern on a surface of at least one sphere;
assemble the samples to form a profile function having a domain that is a surface of at least one sphere;
determine information indicative of at least one property of the signal by applying a spherical linear transform to the profile function, wherein the spherical linear transform uses more than just an equator of the profile function for each of multiple orientations; and
generates images or maps using the determined information,
wherein the samples of a Fourier transform include the diffusion MRI data and the signal is a diffusion function.