Two-Dimensional Phase Two-Dimensional Phase UnwrappingUnwrapping

Joint works (one each) withJoint works (one each) withGregory DardykGregory Dardyk

Reuven Bar-YehudaReuven Bar-YehudaIddit ShalemIddit Shalem

Phase Unwrapping ApplicationPhase Unwrapping Application Used in coherent signal processing Used in coherent signal processing

applications, such as SAR, MRI.applications, such as SAR, MRI.

All coherent signal applications are based on All coherent signal applications are based on a single signal property known as ‘phase’.a single signal property known as ‘phase’.

The actual phase cannot be extracted from The actual phase cannot be extracted from the physical signal the physical signal

All we can get is the ‘All we can get is the ‘wrapped phasewrapped phase’ ,i.e. ’ ,i.e. the phase values forced into the interval (-the phase values forced into the interval (-ππ,,ππ] by a modulo 2] by a modulo 2ππ operation. operation.

SAR interferometry

•Two images of the same scene are acquired from slightly different locations

•The images are coregistered into spatial alignment with one another

•The pixel-wise phase differences are proportional to terrain elevations

The Aim of UnwrappingThe Aim of Unwrapping φφ((xx)) – real phase, unknown – real phase, unknown

ΨΨ((xx)) = = φφ((xx)) + 2 + 2ππkk((xx)) – known wrapped phase, – known wrapped phase, where where kk((xx)) is the unknown integer function is the unknown integer function that forces that forces – – ππ< < ΨΨ< < ππ

The aim of unwrapping – reconstruct The aim of unwrapping – reconstruct φφ from from ΨΨ

The task is easy if The task is easy if

• There are no discontinuities in the wrapped There are no discontinuities in the wrapped phase phase ΨΨ

• The resolution is sufficientThe resolution is sufficient

Easy exampleEasy example

Harder exampleHarder example

Wrapped image and the Wrapped image and the unwrapped elevation model unwrapped elevation model

(Etna volcano region)(Etna volcano region)

Classical Unwrapping MethodsClassical Unwrapping Methods

Path-following methods Path-following methods

Minimum-norm methodsMinimum-norm methods

Path-following algorithmsPath-following algorithms

Perform integration of the discrete Perform integration of the discrete gradients (wrapped differences) gradients (wrapped differences) along pathsalong paths

ProblemProblem: need to pick paths that : need to pick paths that avoid problematic areas where the avoid problematic areas where the data is inconsistentdata is inconsistent

Minimum-norm algorithmsMinimum-norm algorithms

More global approachMore global approach

Minimize the distance between Minimize the distance between φφ and the discrete gradient estimated and the discrete gradient estimated from the values of the wrapped from the values of the wrapped function function ΨΨ..

Minimum Minimum LLpp-norm functional-norm functional

We require the grid-based solution We require the grid-based solution φφ(i,j)(i,j) to to minimize the discrete functional minimize the discrete functional

where where ΔΔxxi,ji,j and and ΔΔyy

i,ji,j are the wrapped are the wrapped differences of differences of ΨΨ, and , and pp is the norm is the norm parameter.parameter.

M

i

N

j

M

i

N

j

pyjijiji

pxjijiji

hJ0

1

0

1

0 0,,1,,,,1

Wrapped difference approximationWrapped difference approximation

We assume that if We assume that if ((i,ji,j)) and and ((k,lk,l)) are are adjacent points, then adjacent points, then ||φφi,ji,j –– φφk,lk,l|<|<ππ

Then, Then, ΔΔxxi,ji,j==WW((ΨΨi+1,ji+1,j – – ΨΨi,ji,j)) andand

ΔΔyyi,ji,j==WW((ΨΨi,j+1i,j+1 – – ΨΨi,ji,j)), ,

where where WW is the wrapping operator is the wrapping operator defined by defined by WW((ffi,ji,j) = ) = ffi,ji,j + +22ππkki,ji,j withwith

integerinteger kki,ji,j chosen so that chosen so that WW((ffi,ji,j))(-(-

ππ,,ππ]]

The linear caseThe linear case

ForFor p=2p=2 this results in the discrete Poisson this results in the discrete Poisson equation:equation:

This equation can be easily solved by This equation can be easily solved by Multigrid methods as well as others.Multigrid methods as well as others.

The reconstruction is often unsatisfactory The reconstruction is often unsatisfactory because of the exaggerated effect of because of the exaggerated effect of outlying values.outlying values.

)()()2()2( 1,,,1,1,,1,,1,,1

yji

yji

xji

xjijijijijijiji

The nonlinear caseThe nonlinear case

Better reconstructions may be found Better reconstructions may be found using using p<2p<2

However, for such values of However, for such values of pp the the equations are nonlinear, often with equations are nonlinear, often with nearly discontinuous coefficientsnearly discontinuous coefficients

For For pp<1 <1 the problem is no longer the problem is no longer convex. But convex. But pp -> 0 -> 0 generally yields generally yields the best results. This problem is NP-the best results. This problem is NP-hard.hard.

Some numerical resultsSome numerical results

P = 1.5

P = 2

P = 1 and P=0

Wrapped image

Another exampleAnother example

P = 2

Original image Wrapped image

Wrapped, view from above

Noisy dataNoisy data

P=1.5 P=0

SummarySummary

We can obtain reconstructions efficiently even We can obtain reconstructions efficiently even with small with small pp using (semi)-classical nonlinear using (semi)-classical nonlinear multigrid; (with G. Dardyk). But multigrid; (with G. Dardyk). But pp<1<1 requires a requires a continuation from continuation from pp=1=1..

For For pp -> 0 -> 0, we have a factor-2 approximation , we have a factor-2 approximation algorithm, based on the local-ratio technique; algorithm, based on the local-ratio technique; (with R. Bar-Yehuda).(with R. Bar-Yehuda).

Currently working on a discrete multi-scale Currently working on a discrete multi-scale algorithm; (with I. Shalem).algorithm; (with I. Shalem).