Unifying reconstruction and motion estimation in first pass cardiac perfusion imaging
View Patent ↗Methods and a system to unify reconstruction and motion estimation steps in first pass cardiac perfusion MRI include a global objective function that meets data consistency, spatial smoothness, motion and contrast dynamics constraints. The global objective decomposed into simpler sub-problems which include low pass filtering of a deformed object, TV shrinkage, analytical Fourier replacement and an l 2 minimizing problem. A registration tool based on the local cross-correlation similarity measure and enabled to perform both rigid and flexile deformations, is applied. Registration parameters are tuned by rigid, semi rigid and flexible models at different stages of iterations. A system to perform the methods is also disclosed.
1. A method for processing perfusion images to correct motion effects, comprising:
a processor receiving undersampled image data of a dynamic scene of a first-pass cardiac perfusion MRI;
the processor minimizing a single objective function that combines at least three terms, each term depending on image data representing a true dynamic scene, wherein a first term expresses a motion estimation related to the undersampled image data, a second term expresses a temporal constraint and a third term expresses a spatial constraint;
the processor minimizing the objective function by constraining the temporal constraint term and the spatial constraint term and applying a penalizing term that penalizes a violation of the constraining; and
the processor generating data representing a reconstructed perfusion image corrected for motion based on the minimized single objective function.
2. The method of claim 1 , wherein the objective function is expressed as:
|Af−b| 2 2 +γ|(Df) t | 2 2 +μ|Tf|,
wherein
f is a vector representing the true dynamic scene related to the undersampled image data,
A is an operator specifying an undersampling Fourier task,
b is a vector representing the undersampled image data,
D is a deformation operator,
T is a spatial total variation operator,
(.) t is a first order temporal derivative operator,
γ a regularization factor, and
μ is a tuning parameter.
3. The method of claim 1 , wherein the constrained objective function is expressed as:
Af
-
b
2
2
+
λ
g
t
2
2
+
β
1
2
Df
-
g
2
2
+
μ
w
+
β
2
2
Tf
-
w
2
2
,
with Df=g and Tf=w,
wherein:
f is a vector representing the true dynamic scene related to the undersampled image data,
A is an operator specifying an undersampling Fourier task,
b is a vector representing the undersampled image data,
D is a deformation operator,
T is a spatial total variation operator,
(.) t is a first order temporal derivative operator,
γ a regularization factor,
μ is a tuning parameter,
g and w are auxiliary variables, and
β 1 and β 2 are penalty parameters.
4. The method of claim 3 , wherein the constrained objective function is minimized by the processor for the four variables f, D, g and w.
5. The method of claim 4 , wherein the constrained objective function is minimized iteratively by alternately solving one of the four variables while keeping three other variables fixed.
6. The method of claim 5 , wherein f, D and w are fixed and g is solved in a temporal Fourier domain by a numerical implementation of a low-pass filter.
7. The method of claim 5 , wherein w is solved by applying a shrinkage operator.
8. The method of claim 5 , wherein f is solved analytically in a Fourier domain.
9. The method of claim 5 , further comprising:
solving the four variables for a first value of β 1 and β 2 until a stopping criterion is met;
incrementing β 1 and β 2 to a second value; and
solving the four variables for the second value of β 1 and β 2 .
10. The method of claim 3 , wherein β 1 and β 2 are increased in value followed by solving the four variables until the constraints Df=g and Tf=w are met.
11. The method of claim 3 , wherein the reconstructed perfusion image corrected for motion based on the minimized single objective function is determined by Df.
12. A system to process perfusion images to correct motion effects, comprising:
a memory, enabled to store data:
a processor, enabled to execute instructions to perform the steps:
receiving undersampled image data of a dynamic scene of a first-pass cardiac perfusion MRI;
minimizing a single objective function that combines at least three terms, each term depending on image data representing a true dynamic scene, wherein a first term expresses a motion estimation related to the undersampled image data, a second term expresses a temporal constraint and a third term expresses a spatial constraint and minimizing the objective function by constraining the temporal constraint term and the spatial constraint term and applying a penalizing term that penalizes a violation of the constraining; and
generating data representing a reconstructed perfusion image corrected for motion based on the minimized single objective function.
13. The system of claim 12 , wherein the objective function is expressed as:
|Af−b| 2 2 +γ|(Df) t | 2 2 +μ|Tf|,
wherein
f is a vector representing the true dynamic scene related to the undersampled image data,
A is an operator specifying an undersampling Fourier task,
b is a vector representing the undersampled image data,
D is a deformation operator,
T is a spatial total variation operator,
(.) t is a first order temporal derivative operator,
γ a regularization factor, and
μ is a tuning parameter.
14. The system of claim 12 , wherein the constrained objective function is expressed as:
Af
-
b
2
2
+
λ
g
t
2
2
+
β
1
2
Df
-
g
2
2
+
μ
w
+
β
2
2
Tf
-
w
2
2
,
with Df=g and Tf=w,
wherein:
f is a vector representing the reconstructed image data representing the dynamic scene related to the undersampled image data,
A is an operator specifying an undersampling Fourier task,
b is a vector representing the undersampled image data,
D is the deformation operator,
T is a spatial total variation operator,
(.) t is a first order temporal derivative operator,
γ a regularization factor,
μ is a tuning parameter,
g and w are auxiliary variables, and
β 1 and β 2 are penalty parameters.
15. The system of claim 14 , wherein the constrained objective function is minimized by the processor for the four variables f, D, g and w.
16. The system of claim 15 , wherein the constrained objective function is minimized iteratively by alternately solving one of the four variables while keeping three other variables fixed.
17. The system of claim 16 , wherein f, D and w are fixed and g is solved in a temporal Fourier domain by a numerical implementation of a low-pass filter.
18. The system of claim 16 , wherein w is solved by applying a shrinkage operator.
19. A method for processing perfusion images to correct motion effects, comprising:
a processor receiving undersampled image data of a dynamic scene of a first-pass cardiac perfusion MRI;
the processor minimizing a single objective function that combines at least three terms, each term depending on image data representing a true dynamic scene, wherein a first term expresses a motion estimation related to the undersampled image data, a second term expresses a temporal constraint and a third term expresses a spatial constraint; and
the processor generating data representing a reconstructed perfusion image corrected for motion based on the minimized single objective function,
wherein the objective function is expressed as:
|Af−b| 2 2 +γ|(Df) t | 2 2 +μ|Tf|,
wherein
f is a vector representing the true dynamic scene related to the undersampled image data,
A is an operator specifying an undersampling Fourier task,
b is a vector representing the undersampled image data,
D is a deformation operator,
T is a spatial total variation operator,
(.) t is a first order temporal derivative operator,
γ a regularization factor, and
μ is a tuning parameter.