image reconstruction in low-field mri - a super-resolution...
TRANSCRIPT
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Image Reconstruction in Low-Field MRIA Super-Resolution ApproachDelft University of Technology
Merel de Leeuw den Bouter
June 14, 2017
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MRI scanners
• Big
• Very expensive
• Problematic in developingcountries
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MRI scanners
• Big
• Very expensive
• Problematic in developingcountries
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Hydrocephalus in the developing world
• Hydrocephalus• 400.000 newborns per year• 79% in developing countries• Limited or no access to required healthcare
• Goal: develop low-cost, portable MRI scanner
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Hydrocephalus in the developing world
• Hydrocephalus• 400.000 newborns per year• 79% in developing countries• Limited or no access to required healthcare
• Goal: develop low-cost, portable MRI scanner
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Partners
• LUMC
• Pennsylvania State University
• Mbarara University of Science and Technology
• CURE Children’s Hospital of Uganda
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Outline
1 MRI
2 Prototypes
3 Super-resolution
4 Minimization problem
5 Conjugate gradient method
6 Simulations
7 Dataset
8 Conclusion
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How does MRI work?
• Human body: ∼ 62%hydrogen atoms
• H-density ⇒ intensity
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How does MRI work?
• Spin
• Random directions
• B0 ⇒ net magnetic moment
• Radiofrequency pulse
• Induces signal
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Conventional vs low-field MRI
Conventional MRI
• Superconducting magnets
• Strong, homogeneous magnetic field
• High signal-to-noise ratio
• Fourier Transform
S(t) =
∫∫object
I (x , y)e−i(γGx tx+γGyTpey) dx dy
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Conventional vs low-field MRI
Conventional MRI
• Superconducting magnets
• Strong, homogeneousmagnetic field
• High signal-to-noise ratio
• Fourier Transform
Low-field MRI
• Permanent magnets
• Weaker magnetic field withinhomogeneities
• Low signal-to-noise ratio
S(t) =
∫∫object
I (x , y)ω(x , y)e−t/T∗2 (x ,y)e−iγ∆B(x ,y) dx dy
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Low-field MRI
S(t) =
∫∫object
I (x , y)ω(x , y)e−t/T∗2 (x ,y)e−iγ∆B(x ,y) dx dy
discretize======⇒s = W x
+ e
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Low-field MRI
S(t) =
∫∫object
I (x , y)ω(x , y)e−t/T∗2 (x ,y)e−iγ∆B(x ,y) dx dy
discretize======⇒
s = W x
+ e
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Low-field MRI
S(t) =
∫∫object
I (x , y)ω(x , y)e−t/T∗2 (x ,y)e−iγ∆B(x ,y) dx dy
discretize======⇒s = W x
+ e
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Low-field MRI
S(t) =
∫∫object
I (x , y)ω(x , y)e−t/T∗2 (x ,y)e−iγ∆B(x ,y) dx dy
discretize======⇒s = W x + e
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Prototype
LUMC
• Configuration of permanentmagnets
• Inhomogeneities ⇒ spatialencoding
PSU
• Same components
• Inverse Fourier Transform
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PSU prototype
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Super-resolution
• Several low-resolutionimages
I ShiftedI Rotated
• ⇒ Obtain high-resolutionimage
Figure: Using LR images to obtain an HR image. Source:Van Reeth et al. (2012).
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Super-resolution: acquisition model
• x: HR image
• {yk}Nk=1: set of LR image observations
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Super-resolution: acquisition model
• x: HR image
• {yk}Nk=1: set of LR image observations
Figure: The general acquisition model. Source: Van Reeth et al. (2012).
• ⇒
yk = DkBkGk
x
+ vk
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Super-resolution: acquisition model
• x: HR image
• {yk}Nk=1: set of LR image observations
Figure: The general acquisition model. Source: Van Reeth et al. (2012).
• ⇒
yk = DkBk
Gkx
+ vk
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Super-resolution: acquisition model
• x: HR image
• {yk}Nk=1: set of LR image observations
Figure: The general acquisition model. Source: Van Reeth et al. (2012).
• ⇒
yk = DkBk
Gkx
+ vk
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Super-resolution: acquisition model
• x: HR image
• {yk}Nk=1: set of LR image observations
Figure: The general acquisition model. Source: Van Reeth et al. (2012).
• ⇒
yk = Dk
BkGkx
+ vk
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Super-resolution: acquisition model
• x: HR image
• {yk}Nk=1: set of LR image observations
Figure: The general acquisition model. Source: Van Reeth et al. (2012).
• ⇒
yk =
DkBkGkx
+ vk
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Super-resolution: acquisition model
• x: HR image
• {yk}Nk=1: set of LR image observations
Figure: The general acquisition model. Source: Van Reeth et al. (2012).
• ⇒
yk =
DkBkGkx + vk
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Super-resolution: acquisition model
• x: HR image
• {yk}Nk=1: set of LR image observations
Figure: The general acquisition model. Source: Van Reeth et al. (2012).
• ⇒ yk = DkBkGkx + vk
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Super-resolution: acquisition model
• yk = DkBkGkx + vk
• yk = Akx + vk
• y =
y1
y2...yN
,A =
A1
A2...
AN
, v =
v1
v2...vN
• y = Ax + v
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Super-resolution: acquisition model
• yk = DkBkGkx + vk
• yk = Akx + vk
• y =
y1
y2...yN
,A =
A1
A2...
AN
, v =
v1
v2...vN
• y = Ax + v
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Super-resolution: acquisition model
• yk = DkBkGkx + vk
• yk = Akx + vk
• y =
y1
y2...yN
,A =
A1
A2...
AN
, v =
v1
v2...vN
• y = Ax + v
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Super-resolution: acquisition model
• yk = DkBkGkx + vk
• yk = Akx + vk
• y =
y1
y2...yN
,A =
A1
A2...
AN
, v =
v1
v2...vN
• y = Ax + v
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Research question
Can super-resolution reconstruction yield images of better qualitythan direct high resolution reconstruction?
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Minimization problem
• y = Ax + v
• v unknown
• Ill-posed problem
• minx
12
||y − Ax||2
+
12
λ||Fx||2
• λ: regularization parameter
• F : prior knowledge about x
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Minimization problem
• y = Ax + v
• v unknown
• Ill-posed problem
• minx
12
||y − Ax||2
+
12
λ||Fx||2
• λ: regularization parameter
• F : prior knowledge about x
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Minimization problem
• y = Ax + v
• v unknown
• Ill-posed problem
• minx
12
||y − Ax||2
+
12
λ||Fx||2
• λ: regularization parameter
• F : prior knowledge about x
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Minimization problem
• y = Ax + v
• v unknown
• Ill-posed problem
• minx
12
||y − Ax||2 +
12
λ||Fx||2
• λ: regularization parameter
• F : prior knowledge about x
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Minimization problem
• y = Ax + v
• v unknown
• Ill-posed problem
• minx
12 ||y − Ax||2 + 1
2λ||Fx||2
• λ: regularization parameter
• F : prior knowledge about x
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Minimization problem
• y = Ax + v
• v unknown
• Ill-posed problem
• minx
12 ||y − Ax||2 + 1
2λ||Fx||2
• λ: regularization parameter
• F : prior knowledge about x
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Different kinds of regularization
• Tikhonov:
minx
1
2||y − Ax||22 +
1
2λ||Fx||22
• Total variation:
minx
1
2||y − Ax||22 +
1
2λ||Fx||1
• Edge-preserving
• F first-order difference matrix
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Different kinds of regularization
• Tikhonov:
minx
1
2||y − Ax||22 +
1
2λ||Fx||22
• Total variation:
minx
1
2||y − Ax||22 +
1
2λ||Fx||1
• Edge-preserving
• F first-order difference matrix
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Different kinds of regularization
• Tikhonov:
minx
1
2||y − Ax||22 +
1
2λ||Fx||22
• Total variation:
minx
1
2||y − Ax||22 +
1
2λ||Fx||1
• Edge-preserving
• F first-order difference matrix
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General problem statement
• Minimization problem of the form
minx
1
2||y − Ax||2 +
1
2λ||x||2R
• Convex problem
• Sufficient condition for optimality:
(ATA + λR)x = ATy
• Conjugate gradient method
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General problem statement
• Minimization problem of the form
minx
1
2||y − Ax||2 +
1
2λ||x||2R
• Convex problem
• Sufficient condition for optimality:
(ATA + λR)x = ATy
• Conjugate gradient method
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General problem statement
• Minimization problem of the form
minx
1
2||y − Ax||2 +
1
2λ||x||2R
• Convex problem
• Sufficient condition for optimality:
(ATA + λR)x = ATy
• Conjugate gradient method
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Conjugate gradient method
• Iterative method
• System of equations Ku = f
• Search directions pk conjugate wrt K (pkKpl = 0, k 6= l)
• uk+1 = uk + αkpk
• ||u− uk ||K = minv∈u0+
span{p0,...,pk−1}
||u− v||K
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Conjugate gradient method
• Iterative method
• System of equations Ku = f
• Search directions pk conjugate wrt K (pkKpl = 0, k 6= l)
• uk+1 = uk + αkpk
• ||u− uk ||K = minv∈u0+
span{p0,...,pk−1}
||u− v||K
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Super-resolution simulations
Phantom (128 x 128 pixels)
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Super-resolution simulations
Shifted
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Super-resolution simulations
Blurred
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Super-resolution simulations
Down-sampled
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Super-resolution simulations
Noise added
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Super-resolution simulations4 low resolution images (32 x 32 pixels)
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Super-resolution simulationsModel solution
Total variation
Tikhonov
Edge-preserving
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Low-field MRI simulations
• s = W x + e
• Angles 0◦, 10◦, ..., 350◦
• Signal-to-noise ratios starting from 0.5
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Low-field MRI simulations
Model solution Direct HR solution (SNR = 10)
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Low-field MRI simulations
Model solution Direct HR solution (SNR = 0.5)
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Low-field MRI simulationsLR images (8× 8 pixels)
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Low-field MRI simulations
SR solution
HR solution
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Low-field MRI simulations
SR solution HR solution
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LUMC dataset
• GoalI Application to real dataI Validation of the model
• 7 T MRI scanner
• Gradient in one direction
• 16 angles
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LUMC dataset
Model solution First 1D projection
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LUMC dataset
Model solution
First 1D projection
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LUMC dataset
Model solution First 1D projection
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LUMC dataset
16 1D projections
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LUMC dataset
Model solution
Result
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LUMC dataset
Model solution Result
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LUMC dataset
Model solution Result
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LUMC dataset
Model solution Result
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Conclusions and further research
• ConclusionsI Super-resolution can yield better resultsI Total variation regularizationI Validation of the measurement model
• Further research
I Apply super-resolution to PSU dataI New LUMC prototypeI Measurements at LUMC with more complicated fieldI Dictionary learning
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Conclusions and further research
• ConclusionsI Super-resolution can yield better resultsI Total variation regularizationI Validation of the measurement model
• Further researchI Apply super-resolution to PSU dataI New LUMC prototypeI Measurements at LUMC with more complicated fieldI Dictionary learning
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LUMC dataset: super-resolutionModel solution
One LR image
HR result
SR result
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