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

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Filtered Back Projection (FBP)

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Least Square (LS)

Ax = y

Find x knowing A and y

min𝒙>𝟎

𝟏

𝟐𝑨𝒙 − 𝒚 𝟐

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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

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What can Total Variation Achieve?

Original

No noise SNR=20dB SNR=10dB

LS

TV

Reconstruct

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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

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What is Total Variation (TV)?

If let TV approach zero

Original TV = 0TV = high TV = low

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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

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Comparison (Less Projections)

TV

LS

200 50 20 10 5

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TV

FBP

Comparison (Less Projections)

64 128 256 512

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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)

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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