2d phase unwrapping
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Two-Dimensional
Phase UnwrappingUsing Neural Networks
Wade Schwartzkopf, Thomas E. Milner, Joydeep Ghosh,
Brian L. Evans, and Alan C. Bovik
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Phase-Based Imaging Methods
SyntheticAperture Radar
Mountains
MagneticResonance
Imaging
Knee
Optical DopplerTomography
Blood Vessel
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y(n)+2π
y(n)
y(n)-2π
y(n-1)+π
y(n-1)-π
1-D Phase Unwrapping
• Use information from neighboring pixels
• Errors propagate
Original Signal
0
π
-π
Wrapped Signal
0
π
-π
Unwrapped Signal
0
π
3π
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0.0 0.6π
1.6π 1.2π
2-D Phase Unwrapping
• Possible to use second dimension to recover
• Residues: conflicts in different dimensions
0.0 0.6π
-0.4π 1.2π
0.0 0.6π1.6π 1.2πTwo of the
possible orderings
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Why Neural Networks?
• Fast unwrapping of ODT velocity images– 10 100 × 100 images per second
– Global optimization methods impractical
– Fast local methods necessary
• Learn features in ODT velocity images of skintissue– Sparse images
– All blood flow follows parabolic law
• Goal: Train neural networks to detect phase
jumps in ODT phase images
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Neighboring pixels
Current pixel
Phase Jump Detector
• Feedforward multilayer
perceptron network
• Wrapped values of 5×5
neighborhood of pixels
as inputs to network
• Three outputs for the current
pixel– Positive jump of 2π
– No jump of 2π
– Negative jump of 2π
• Non-iterative, pixel parallel
computation
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Simulated Image
Simulated flow in a blood
vessel without wrapping
Simulated flow in a blood
vessel after wrapping
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Phase Jump Detection
Location of positive (black) and
negative (white) phase jumps
Pixels in the range [π, 3π)
• Unwrap along columns– Top pixel is always zero since no movement at edge of skin
– At positive (negative) jump, add (subtract) 2π to rest of column
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