Improved Visualization and Quantification of 4D Flow
Data using Divergence-free Wavelet Denoising
F Ong, M Uecker, U Tariq, A Hsiao, MT Alley, SS Vasanawala and M Lustig
Electrical Engineering and Computer Sciences, UC Berkeley,
Radiology, Stanford University
4D Flow MRI
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• Measures 3D blood velocity over cardiac cycles
• Provides comprehensive cardiac evaluation – flow
and anatomy – in a single acquisition
• Huge potential for congenital heart defect
• Problems:
– Long scan time (>20 minutes)
– Artifact – motion…
– Low velocity-to-noise ratio
Ref: Markl, et al. JMRI, 2003;17(4);499-506
4D Flow Denoising using Divergence Free Wavelet
Rapid 4D Flow
• Trend today: reduce scan time via undersampling
– Less acquisition time ⇒ low SNR ⇒ low VNR
• Random sampling for CS creates noise-like artifacts
• Flow denoising algorithm is highly desired
Noise-like
artifact
Coronal slice of accelerated 4D flow data
Ref: Hsiao, Vasanawala AJR 2012; 198:250-59
4D Flow Denoising using Divergence Free Wavelet
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This Work
Goal:
• Separate actual flow data from noise/artifacts in the
post processing stage
Utilize:
• Blood flow has zero divergence
• Flow field is sparse in wavelet domain
Approach:
• Use divergence-free wavelet transform for denoising
4D Flow Denoising using Divergence Free Wavelet
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What is Divergence-free?
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What flows in flow out
Blood is incompressible ⇒ Flow is divergence-free
Most divergence in acquired flow are noise/artifacts
Imposing divergence-free conditions removes noise
Not DivFree
DivFree Field
Original Field
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4D Flow Denoising using Divergence Free Wavelet
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Previous Works: Finite Difference Method
• Approximate divergence with finite difference
• Solve for a divergence-free field by inverse Laplacian
Noisy
Ref: Song, Pelc, et al. JMRI, 1993;4(7);587-596
4D Flow Denoising using Divergence Free Wavelet
Divergence-free
Previous Works: Radial Basis Function
• Project data onto divergence free radial basis function (RBF)
with iterative least squares
Ref: Busch, Kozerke ,et. al MRM 2012;69(1):200-10
4D Flow Denoising using Divergence Free Wavelet
Previous Works
• Both methods are effective in denoising*
• Problem:
Enforces flow to be strictly divergence-free
• Acquired blood flow is not strictly divergence-free
because of
– discrete approximation
– turbulent flow
– partial voluming
*Leocher, Wieben, et al. ISMRM 2012;1246
4D Flow Denoising using Divergence Free Wavelet
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Example
Original Field
Analytically
DivFree Flow
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Calculated divergence using finite difference
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4D Flow Denoising using Divergence Free Wavelet
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Projection onto DivFree Radial Basis function
DivFree projection constrained
within interior flow region
Original Field
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DivFree projection on entire
flow field RBF
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Perfect Reconstruction
4D Flow Denoising using Divergence Free Wavelet
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Distortion
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Projection onto DivFree Radial Basis function
• Distortion more apparent in velocity magnitude
DivFree projection constrained
within interior
Div Free Wavelet Denoised
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Div Free RBF Denoised
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DivFree projection on entire
flow field
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Quality heavily depends on segmentation
4D Flow Denoising using Divergence Free Wavelet
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Motivates for “Softer” Treatment
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• Want to enforce mostly divergence-free except for
sparse locations*
Original Field
Analytically
DivFree Flow
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Calculated Divergence using finite difference
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*Tafti and Unser. IEEE Trans. On Image Processing, 2011;20(11); 3163-78
4D Flow Denoising using Divergence Free Wavelet
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Wavelet Denoising
Not Sparse
Sparse in wavelet domain
Can we do this with flow data?
Ref: Donoho and Johnstone. Biometrika, 1994;81(3): 425-55
4D Flow Denoising using Divergence Free Wavelet
Denoised
Divergence-free Wavelet
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• Computational Fluid Dynamics community has been
using wavelets for a while
• In particular, Divergence-free wavelet
– Sparse Representation of flow
– Separates flow into both divergence-free and nondivergence-free components
Ref: Deriaz and Perrier, J. Turbul 2006;1
4D Flow Denoising using Divergence Free Wavelet
How to Construct Divergence-free Wavelets?
• Consider the simple 2D vector field as an example:
• Works because x, y and x2, y2 are related by
differentiation
4D Flow Denoising using Divergence Free Wavelet
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How to Construct Divergence-free Wavelets?
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Linear Spline Wavelet
Quadratic Spline Wavelet
Can be generalized to entire set of 3D DF wavelet
4D Flow Denoising using Divergence Free Wavelet
Divergence-free Wavelet Basis Functions
• Despite the name, spans the complete vector space
Div-Free
Basis functions
Non-divFree
Basis Functions
4D Flow Denoising using Divergence Free Wavelet
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Overview of Divergence-free Wavelet Denoising
• Soft-threshold wavelet coefficients to enforce sparsity
• Soft-threshold non-divFree components with a higher
threshold to enforce divergence-free conditions
– Similar to l1-minimization on divergence*
• Multiscale decomposition and divergence-free enforcement
• Fast - linear complexity, non-iterative
*Tafti and Unser. IEEE Trans. On Image Processing, 2011;20(11); 3163-78
4D Flow Denoising using Divergence Free Wavelet
Proposed Flow Diagram
DivFree1
DivFree2
Non-DivFree
4D Flow Denoising using Divergence Free Wavelet
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Acquired Flow and Wavelet Coefficients
Axial slice of acquired cardiac flow
4D Flow Denoising using Divergence Free Wavelet
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Corresponding non-divFree wavelet coeff.
Denoised Flow and Wavelet Coefficients
Axial slice of denoised cardiac flow
4D Flow Denoising using Divergence Free Wavelet
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Corresponding non-divFree wavelet coeff.
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Look at Same Experiment as before
DF Wavelet with Cycle Spinning:
Non-DivFree
Threshold = 0.16*Max
Div Free Wavelet Denoised
Radial Basis
Function Projection
Div Free RBF Denoised
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4D Flow Denoising using Divergence Free Wavelet
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Results: Acquired (Vena Cava)
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20 cardiac phases
Size = 256x256x164
Resolution~1.15x1.15x1 mm
Undersampled by 4
Reconstructed with L1-SPIRiT
Corrected Phase
Ref: Lustig, Pauly et al. MRM 2010;64(2):457-471, Markl, et al. JMRI, 2003;17(4);499-506
4D Flow Denoising using Divergence Free Wavelet
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Results: Denoised (Vena Cava)
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20 cardiac phases
Size = 256x256x164
Resolution~1.15x1.15x1 mm
Undersampled by 4
Reconstructed with L1-SPIRiT
Corrected Phase
Ref: Lustig, Pauly et al. MRM 2010;64(2):457-471 , Markl, et al. JMRI, 2003;17(4);499-506
4D Flow Denoising using Divergence Free Wavelet
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Results: Acquired (Vena Cava)
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20 cardiac phases
Size = 256x256x164
Resolution~1.15x1.15x1 mm
Undersampled by 4
Reconstructed with L1-SPIRiT
Corrected Phase
Ref: Lustig, Pauly et al. MRM 2010;64(2):457-471, Markl, et al. JMRI, 2003;17(4);499-506
4D Flow Denoising using Divergence Free Wavelet
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Results: Denoised (Vena Cava)
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20 cardiac phases
Size = 256x256x164
Resolution~1.15x1.15x1 mm
Undersampled by 4
Reconstructed with L1-SPIRiT
Corrected Phase
Ref: Lustig, Pauly et al. MRM 2010;64(2):457-471 , Markl, et al. JMRI, 2003;17(4);499-506
4D Flow Denoising using Divergence Free Wavelet
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Results: Acquired (Vena Cava)
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20 cardiac phases
Size = 256x256x164
Resolution~1.15x1.15x1 mm
Undersampled by 4
Reconstructed with L1-SPIRiT
Corrected Phase
Ref: Lustig, Pauly et al. MRM 2010;64(2):457-471, Markl, et al. JMRI, 2003;17(4);499-506
4D Flow Denoising using Divergence Free Wavelet
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Results: Denoised (Vena Cava)
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20 cardiac phases
Size = 256x256x164
Resolution~1.15x1.15x1 mm
Undersampled by 4
Reconstructed with L1-SPIRiT
Corrected Phase
Ref: Lustig, Pauly et al. MRM 2010;64(2):457-471 , Markl, et al. JMRI, 2003;17(4);499-506
4D Flow Denoising using Divergence Free Wavelet
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Results: Acquired (Aorta)
4D Flow Denoising using Divergence Free Wavelet
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Results: Acquired (Aorta)
4D Flow Denoising using Divergence Free Wavelet
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Results: Denoised (Aorta)
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Study 1
• Done on 8 pediatric patients with congenital heart defect
• Measured regurgitant fraction before and after denoising
– Regurgitant fraction is the percentage of blood flowing back that are not
supposed to flow back.
• Only small change in regurgitant fraction before and after
denoising
Appears to preserve flow information
Example of regurgitant flow
4D Flow Denoising using Divergence Free Wavelet
Regurgitant
Fraction
< 5%
> 30%
Change
before and
after
denoising
0.08%
1.88%
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Study 2
• Measured net flow rate difference between aorta and pulmonary
trunk
• Ideally should be zero
• Smaller difference in net flow rate between aorta and pulmonary
trunk after denoising
4D Flow Denoising using Divergence Free Wavelet
Regurgitant
< 5%
> 30%
Net Flow Rate
2.95 L/min
2.21 L/min
Flow Diff
before
denoising
0.395 L/min
1.151 L/min
Flow Diff after
denoising
0.353 L/min
0.926 L/min
Interactive Denoising On-the-fly
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Implemented in CUDA, incorporated into a JAVA visualization software
Runs ~1 sec for a 3D volume for each cardiac phase (~160 MBytes)
DivFree Threshold
NonDivFree Threshold
4D Flow Denoising using Divergence Free Wavelet
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Summary
• Proposed divergence-free wavelet denoising:
– Improves visualization
– Appears to be safe for quantification
– “Soft” processing
– Fast to compute, allows interactive control
4D Flow Denoising using Divergence Free Wavelet
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Thank you!
Hellman Family fund
MikGroup
4D Flow Denoising using Divergence Free Wavelet