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Tomographic Image Reconstruction by Total-Variation Minimization Zhifei Zhang Zhifei Zhang @ EECE, UTK 1
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
min𝒙>𝟎
𝟏
𝟐𝑨𝒙 − 𝒚 𝟐
Zhifei Zhang @ EECE, UTK 5
Drawbacks of FBP and LS
LS:FBP:• Need enough projections
• Sharp edge or noise
• Fit noise (over-fitting)
• 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 ⋯69 ⋯ ⋯⋮ ⋯ ⋯ 𝑛×𝑛
124 100 ⋯69 ⋯ ⋯⋮ ⋯ ⋯ 𝑛×𝑛
𝑻𝑽 =
𝒊,𝒋=𝟏
𝒏
𝑰𝒊𝒋 − 𝑰𝒊,𝒋+𝟏 + 𝑰𝒊𝒋 − 𝑰𝒊+𝟏,𝒋
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
argmin𝒙>𝟎
𝟏
𝟐𝑨𝒙 − 𝒚 𝟐 + 𝝀 ∙ 𝑻𝑽 (𝛌 > 𝟎)
ProjectionSystem model Penalty parameter
WANTED
𝑻𝑽 = 𝒊=𝟏
𝒎
𝒋=𝟏
𝒏
𝛁𝒙𝒊𝒋 𝟐𝛁𝒙𝒊𝒋 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
• Image reconstruction by TV minimization
• Make image more smooth and edge clear
• Robust to noise and need less projections
• It may be tricky to set the TV parameter 𝝀(get balance between removing noise and preserving details)
Zhifei Zhang @ EECE, UTK 13
Thank You!
• Hansen, Per Christian, and Jakob Heide Jørgensen. "Total Variation and Tomographic Imaging from Projections." 36th Conference of the Dutch-Flemish Numerical Analysis Communities.
• Dahl, Joachim, et al. "Algorithms and software for total variation image
reconstruction via first-order methods." Numerical Algorithms 53.1 (2010): 67-92.
References
