Upload
others
View
2
Download
0
Embed Size (px)
Citation preview
Tomographic Image Reconstruction by Total-Variation Minimization
Zhifei Zhang
Zhifei Zhang @ EECE, UTK 1
Zhifei Zhang @ EECE, UTK 2
Outline
1. Learned in class
2. Total variation
3. Comparison
Zhifei Zhang @ EECE, UTK 3
Filtered Back Projection (FBP)
Zhifei Zhang @ EECE, UTK 4
Least Square (LS)
Ax = y
Find x knowing A and y
minš>š
š
ššØš ā š š
Zhifei Zhang @ EECE, UTK 5
Drawbacks of FBP and LS
LS:FBP:ā¢ Need enough projections
ā¢ Sharp edge or noise
ā¢ Fit noise (over-fitting)
ā¢ Sensitive to outliers
No noise No noiseOriginal SNR=20dB
ReconstructReconstruct
FBP LS
Zhifei Zhang @ EECE, UTK 6
What can Total Variation Achieve?
Original
No noise SNR=20dB SNR=10dB
LS
TV
Reconstruct
Zhifei Zhang @ EECE, UTK 7
What is Total Variation (TV)?
124 100 āÆ69 āÆ āÆā® āÆ āÆ šĆš
124 100 āÆ69 āÆ āÆā® āÆ āÆ šĆš
š»š½ =
š,š=š
š
š°šš ā š°š,š+š + š°šš ā š°š+š,š
TV
Zhifei Zhang @ EECE, UTK 8
What is Total Variation (TV)?
If let TV approach zero
Original TV = 0TV = high TV = low
Zhifei Zhang @ EECE, UTK 9
Total-Variation Minimization
argminš>š
š
ššØš ā š š + š ā š»š½ (š > š)
ProjectionSystem model Penalty parameter
WANTED
š»š½ = š=š
š
š=š
š
šššš ššššš denotes gradient
approximation for pixel ij
Zhifei Zhang @ EECE, UTK 10
Comparison (Less Projections)
TV
LS
200 50 20 10 5
Zhifei Zhang @ EECE, UTK 11
TV
FBP
Comparison (Less Projections)
64 128 256 512
Zhifei Zhang @ EECE, UTK 12
Conclusion
ā¢ Image reconstruction by TV minimization
ā¢ Make image more smooth and edge clear
ā¢ Robust to noise and need less projections
ā¢ It may be tricky to set the TV parameter š(get balance between removing noise and preserving details)
Zhifei Zhang @ EECE, UTK 13
Thank You!
ā¢ Hansen, Per Christian, and Jakob Heide JĆørgensen. "Total Variation and Tomographic Imaging from Projections." 36th Conference of the Dutch-Flemish Numerical Analysis Communities.
ā¢ Dahl, Joachim, et al. "Algorithms and software for total variation image
reconstruction via first-order methods." Numerical Algorithms 53.1 (2010): 67-92.
References