Phase Unwrapping

1. Phase Unwrapping Kevin Burnett Jon Marbach

2. Phase Unwrapping • Introduction • Residues • Quality Maps • Masks • Noise Filtering • Phase Unwrapping

3. Introduction • 1D Phase

4. Introduction 2D Wrapped Phase 2D Unwrapped Phase

5. Introduction • 2D phase unwrapping is evaluating a line integral • C is any path in the domain D connecting the points r0 and r • is the phase • is the phase gradient • Phase unwrapping amounts to integrating gradients in 2 Dimensions (along the Time axis) r0 r

6. Introduction Conditions for Path Independence • is an exact differential • , which means that F is the gradient of a single-valued scalar function • , which means that the integral around every simple closed path is zero • , which means that the curl of F is identically zero if, and only if, the cross derivatives are equal

7. Phase Unwrapping • Introduction • Residues • Quality Maps • Masks • Noise Filtering • Phase Unwrapping

8. Residues • If we were to follow two different paths from one pixel to the another we could end up with two different answers • Inconsistencies that cause the differences are caused by structures referred to as residues • A residue is located inside a “loop” of four pixels where the integral of the phase derivatives is not zero but -2π to 2π

9. Residues Residue Detection • are the instantaneous phase values • Residues are calculated by • None-zero values are considered residues • The sign of the residue is referred to as the polarity

10. Orientation…Time slice

11. Orientation…Phase from Time slice

12. Residues

13. Residues • Salt Intrusion – Clean unwrapping in this area is very unlikely

14. Phase Unwrapping • Introduction • Residues • Quality Maps • Masks • Noise Filtering • Phase Unwrapping

15. Quality Maps • Define the quality of each pixel of the given phase data • These values are necessary for guiding some path-following algorithms • Types of Quality Maps • Correlation • Not derivable from phase data alone • Pseudo-Correlation • Phase Derivative Variance • Maximum Phase Gradient

16. Quality Maps Pseudo-Correlation • is the instantaneous phase values • Designed to mimic the correlation map when the magnitudes of the complex valued SAR images are unknown • Computed with a 3x3 window along the Time axis • In practice, this is not a good estimator of phase quality

17. Quality Maps

18. Quality Maps Phase Derivative Variance • is the partial derivatives (wrapped phase differences) • is the average of the partial derivatives • Computes the local sample variance from the partial derivatives of the phase data • Computed with a 3x3 window along the Time axis • Indicates the badness of data but values are negated to create a quality map • Generally the most reliable measure of phase quality when the correlation map is not available

19. Quality Maps

20. Quality Maps Maximum Phase Gradient • is the partial derivatives (wrapped phase differences) • Computes the points of maximum phase change • Calculated using a 3x3 window along the Time axis • Indicates the badness of data but values are negated to create a quality map

21. Quality Maps

22. Phase Unwrapping • Introduction • Residues • Quality Maps • Masks • Noise Filtering • Phase Unwrapping

23. Masks • A mask is a quality map whose pixels takes only 0 or 1 • The challenge of defining a good mask • Setting the threshold so the total number of masked pixels is small • Masking out the lowest-quality pixels and residues • Masking out 10% of the pixels seems to be ideal

24. Masks Mask Creation • Automatic threshold • Create a 10,000 bin histogram with values from 0 to 1 • Quality values are adaptively remapped so 5% of the values lie between 0.0-0.1 and 5% lie between 0.9-1.0 • A minimum is determined by looking for a decrease then increase in the each bin’s value while traversing left to right • Increase must be “substantial” to avoid local minima • The threshold is placed at that minimum • Our data does not always come in the “U” shape that this relies on so a manual method for applying a threshold will be needed

25. Masks Pseudo-Correlation Histogram * Values calculated from the entire volume, not a single slice

26. Masks 13.9% of values removed by threshold

27. Masks Phase Derivative Variance Histogram * Values calculated from the entire volume, not a single slice

28. Masks 27.1% of values removed by threshold

29. Masks Maximum Phase Gradient Histogram * Values calculated from the entire volume, not a single slice

30. Masks 21.4% of values removed by threshold

31. Phase Unwrapping • Introduction • Residues • Quality Maps • Masks • Noise Filtering • Phase Unwrapping

32. Noise Filtering • Increases signal-to-noise ratio • Reduces the number of Residues • Filtering will usually be done on the signal • Phase is a property of signal

33. Phase Unwrapping • Introduction • Residues • Quality Maps • Masks • Noise Filtering • Phase Unwrapping

34. Phase Unwrapping Path-Following Methods • A branch-cut must be placed between residues of opposite polarity to avoid path-dependent results • n! possible ways to pair up branch cuts • Path-Following Methods • Branch Cut Algorithm • Quality-Guided Path Following • Mask Cut Algorithm • Minimum Discontinuity

35. Phase Unwrapping Branch Cut Algorithm • Based on the path-following algorithm of Goldstein, Zebker, and Werner • Generally this is what happens… • Iterate through all pixel values • Add branch cuts between a Positive and Negative Residues • Keep track of Balanced Residues • Integrate on any path that does not cross a branch cut to unwrap the phase

36. Phase Unwrapping Other Path-Following Methods • Quality-Guided Path Following • Relies on Quality Maps • Is able to unwrap some types of phase data where Goldstein’s algorithm fails • Mask Cut Algorithm • Hybrid technique based on Branch Cut and Quality Guided Path Following • Minimum Discontinuity • Minimizes the discontinuities in the unwrapped surface

37. Phase Unwrapping Minimum-Norm Methods • Un-weighted Least-Squares • Weighted Lease Squares • Minimum LP-Norm