Noise reduction in diffusion tensor imaging data using bayesian methods
View Patent ↗A method and computer program product for reducing noise in diffusion tensor imaging data utilizes an optimization function performed on voxels or pixels of the image. The optimization function includes a neighborhood relevance component that considers diffusion tensor matrices from neighboring pixels or voxels. The neighborhood relevance component is modeled as a Markov Random Field. The method includes an iteration that seeks convergence of an Expectation Maximization algorithm performed on the diffusion tensor matrices.
1. A method for reducing noise in diffusion tensor imaging data, the data comprising a matrix of elements, each element represented by a diffusion tensor matrix α, the method comprising the steps of:
for each element of the imaging data:
computing a version of an optimization function for the matrix α, the optimization function including a neighborhood relevance component for the selected element; and
computing a new matrix α using the optimization function;
repeating the steps until convergence of the imaging data.
2. The method of claim 1 , wherein the step of computing a version of an optimization function for the matrix α further comprises diagonalizing the matrix α.
3. The method of claim 1 , wherein the step of computing a version of an optimization function for the matrix α further comprises computing eigenvalues and eigenvectors of the matrix α.
4. The method of claim 1 , further comprising the step of:
repeating the steps of computing a version of the optimization function and computing a new matrix α until convergence for the element.
5. The method of claim 1 , wherein the step of repeating the steps until convergence of the imaging data comprises repeating the steps until a sum of differences between the new matrix α and a previous value of the matrix α is greater than a threshold value.
6. The method of claim 1 , wherein the neighborhood relevance component of the optimization function is modeled as a Markov Random Field.
7. The method of claim 1 , wherein the neighborhood relevance component of the optimization function is computed by placing a maximum a posteriori constraint on the matrix α.
8. The method of claim 7 , wherein a prior probability defining the α constraint has the distribution:
1
Z
ⅇ
-
U
MRF
(
α
^
s
❘
α
^
s
N
)
wherein Z is a normalization constant, and:
U
MRF
(
α
^
s
❘
α
^
s
N
)
=
∫
v
m
(
α
s
,
α
v
)
ⅆ
v
wherein m(α s , α v ) is a distance metric between estimated tensor parameters at volume locations s and v.
9. The method of claim 8 , wherein m emphasizes similarity of direction, considering all directions.
10. The method of claim 1 , wherein the optimization function further comprises a maximum likelihood estimate (MLE) based on measurements of the subject element.
11. The method of claim 10 , wherein the MLE is given by:
α MLE=( B′Σ −1 B ) −1 ( B′Σ −1 ) x
wherein B is a combined matrix summarizing a measured attenuating effect of magnetic gradients in a plurality of directions, Σ is a covariant matrix of a multivariate Gaussian noise, and x is a matrix of the form:
x=Bα+η
wherein η is noise having a distribution N(0, Σ).
12. The method of claim 11 , wherein the noise η is modeled as white Gaussian noise.
13. The method of claim 1 , wherein the diffusion tensor imaging data is collected during measurement of brain white matter.
14. The method of claim 1 , wherein the elements are pixels.
15. The method of claim 1 , wherein the elements are voxels.
16. A computer program product comprising a computer readable recording medium having recorded thereon a computer program comprising code means for, when executed on a computer, instructing said computer to control steps in a method for reducing noise in diffusion tensor imaging data, the data comprising a matrix of elements, each element represented by a diffusion tensor matrix α, the method comprising the steps of:
for each element of the imaging data:
computing a version of an optimization function for the matrix α, the optimization function including a neighborhood relevance component for the selected element; and
computing a new matrix α using the optimization function;
repeating the steps until convergence of the imaging data.
17. The computer program product of claim 16 , wherein the step of computing a version of an optimization function for the matrix α further comprises diagonalizing the matrix α.
18. The computer program product of claim 16 , wherein the step of computing a version of an optimization function for the matrix α further comprises computing eigenvalues and eigenvectors of the matrix α.
19. The computer program product of claim 16 , wherein the method further comprises the step of:
repeating the steps of computing a version of the optimization function and computing a new matrix α until convergence for the element.
20. The computer program product of claim 16 , wherein the step of repeating the steps until convergence of the imaging data comprises repeating the steps until a sum of differences between the new matrix α and a previous value of the matrix α is greater than a threshold value.
21. The computer program product of claim 16 , wherein the neighborhood relevance component of the optimization function is modeled as a Markov Random Field.
22. The computer program product of claim 16 , wherein the neighborhood relevance component of the optimization function is computed by placing a maximum a posteriori constraint on the matrix α.
23. The computer program product of claim 22 , wherein a prior probability defining the a posteriori constraint has the distribution:
1
Z
ⅇ
-
U
MRF
(
α
^
s
❘
α
^
s
N
)
wherein Z is a normalization constant, and:
U
MRF
(
α
^
s
❘
α
^
s
N
)
=
∫
v
m
(
α
s
,
α
v
)
ⅆ
v
wherein m(α s , α v ) is a distance metric between estimated tensor parameters at volume locations s and v.
24. The computer program product of claim 23 , wherein m emphasizes similarity of direction, considering all directions.
25. The computer program product of claim 16 , wherein the optimization function further comprises a maximum likelihood estimate (MLE) based on measurements of the subject element.
26. The computer program product of claim 25 , wherein the MLE is given by:
α MLE =( B′Σ −1 B ) −1 ( B′Σ −1 ) x
wherein B is a combined matrix summarizing a measured attenuating effect of magnetic gradients in a plurality of directions, Σ is a covariant matrix of a multivariate Gaussian noise, and x is a matrix of the form:
x=Bα+η
wherein η is noise having a distribution N(0, Σ).
27. The computer program product of claim 26 , wherein the noise η is modeled as white Gaussian noise.
28. The computer program product of claim 16 , wherein the diffusion tensor imaging data is collected during measurement of brain white matter.
29. The computer program product of claim 16 , wherein the elements are pixels.
30. The computer program product of claim 16 , wherein the elements are voxels.