Multibaseline gradient ambiguity resolution to support Minimum Cost Flow phase unwrapping Marie Lachaise, Richard Bamler, ...
Outline  <ul><li>TanDEM-X mission </li></ul><ul><li>Multibaseline gradient-based phase unwrapping  </li></ul><ul><li>Maxim...
TanDEM-X mission requirement & processing concepts <ul><li>Requirement :  </li></ul><ul><ul><li>Spatial resolution: 12m x ...
<ul><li>interferometric phase modeled as: </li></ul><ul><li>gradient estimate: </li></ul><ul><li>gradient  pdf : </li></ul...
Some gradient pdfs 0 2  -2  1 2 3 Gradient in azimuth (rad) pdf 0 2  -2  0.5 1 1.5 Gradient in range (rad) pdf
Which additional information can we use ?  curl(i,k) =    i (i,k)+   k (i+1,k) -   i (i,k+1)-   k (i,k)=0 (i,k) (...
The zero curl constraint <ul><li>The constraint or the prior is: the  zero curl constraint </li></ul><ul><li>Joint probabi...
Maximum A Posteriori and energy minimization Data energy or data penalty Compatibility between  neighboring variables
Graphical model Gradient estimates  Observable variable node Function node Pdf (hidden | observation) Zero-curl constraint...
Message passing <ul><li>Sum-product algorithm (a.k.a. belief propagation) allows to find an approximate cost labeling (bel...
Message update: from constraint node to gradient node m 4 Gradient node Constraint node m 1 m 2 m 3
Messages update: from gradient node to constraint node m 4 m 5 Gradient node Constraint node
MAP computation m 4 m 5 Gradient node Constraint node
Gradient log pdf and speed up Range  (i-direction)  Azimuth  (k-direction)   Pixel 1 Pixel 2 0 2  -2  Gradient [rad] 0 2...
Message update: Forward -6.3 0 6.3 0 E 0 =0.9+0.5+0.8=2.2 0 E -1 =0.9+0.5+0.0=1.4   i (i,k)  +    k (i+1,k)  -    i ...
Message update: MAP 0.3 -2.7 0.2 -3.1 0.0 3.0 0.2 0.9 0.4 0 0.5 0 0.8 0.8 0 0.6 0.0 0.7 0.7 0 1 0.1 0.3 0 0.9 0.4 0 0.9 1....
Results: Test site “ footprint” south from Salar de Arizaro (Argentina)
Results: Unwrapped Gradients with MAP Unwrapped gradient in Azimuth Unwrapped gradient in Range
Results: Remaining residues and MCF results Residues and branch-cuts from MCF Unwrapped phase
Results: MCF results, comparison with single baseline Single baseline phase unwrapping (MCF) Multibaseline  gradient-based...
Results: The unwrapped phase
Conclusions <ul><li>Multibaseline gradients unwrapping is used to disambiguate the gradients. The range of search is thus ...
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FR3.L09 - MULTIBASELINE GRADIENT AMBIGUITY RESOLUTION TO SUPPORT MINIMUM COST FLOW PHASE UNWRAPPING

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  • For multichannel, where different noisy scaled wrapped measures are available the ML function of the phase measure exhibits a „unique“ peak or at least „less ambiguous“ maxima. MLE eliminates or at least mitigates the ambiguity problems related to the wrapping operator but it has the disadvantage to amplify the noise contribution during disambiguation process.
  • The max product BP algorithm works by passing messages around the graph defined by the four connected image grid.
  • FR3.L09 - MULTIBASELINE GRADIENT AMBIGUITY RESOLUTION TO SUPPORT MINIMUM COST FLOW PHASE UNWRAPPING

    1. 1. Multibaseline gradient ambiguity resolution to support Minimum Cost Flow phase unwrapping Marie Lachaise, Richard Bamler, Fernando Rodriguez Gonzalez Remote Sensing Technology Institute, DLR
    2. 2. Outline <ul><li>TanDEM-X mission </li></ul><ul><li>Multibaseline gradient-based phase unwrapping </li></ul><ul><li>Maximum A Posteriori with the zero curl constraint as prior </li></ul><ul><li>Message passing </li></ul><ul><li>Results </li></ul>
    3. 3. TanDEM-X mission requirement & processing concepts <ul><li>Requirement : </li></ul><ul><ul><li>Spatial resolution: 12m x 12m </li></ul></ul><ul><ul><li>Absolute vertical accuracy: <10m </li></ul></ul><ul><ul><ul><li>Relative vertical accuracy < 2m </li></ul></ul></ul><ul><li>Year 1: full coverage with smaller baseline </li></ul><ul><ul><li>height ambiguity ~ 45 m good for moderate terrain </li></ul></ul><ul><li>Year 2: repeat with larger baseline </li></ul><ul><ul><li>height ambiguity ~ 30 m gives full accuracy </li></ul></ul><ul><li>robust phase unwrapping by combining both years </li></ul><ul><ul><li>multi baseline PU </li></ul></ul>
    4. 4. <ul><li>interferometric phase modeled as: </li></ul><ul><li>gradient estimate: </li></ul><ul><li>gradient pdf : </li></ul><ul><li>in i-direction: </li></ul><ul><li>in k-direction: </li></ul>Multibaseline gradient-based phase unwrapping
    5. 5. Some gradient pdfs 0 2  -2  1 2 3 Gradient in azimuth (rad) pdf 0 2  -2  0.5 1 1.5 Gradient in range (rad) pdf
    6. 6. Which additional information can we use ?  curl(i,k) =   i (i,k)+   k (i+1,k) -   i (i,k+1)-   k (i,k)=0 (i,k) (i+1,k) (i+1,k+1)   k (i+1,k)   i (i,k)+   k (i+1,k) -   i (i,k+1)-   k (i,k)=0 (i,k+1)   i (i,k)   i (i,k+1)   k (x,y) (x,y+1) Phase pixels Gradient estimates Zero-curl constraint
    7. 7. The zero curl constraint <ul><li>The constraint or the prior is: the zero curl constraint </li></ul><ul><li>Joint probability: </li></ul>
    8. 8. Maximum A Posteriori and energy minimization Data energy or data penalty Compatibility between neighboring variables
    9. 9. Graphical model Gradient estimates Observable variable node Function node Pdf (hidden | observation) Zero-curl constraint check nodes Hidden variable node = unknown true values of gradients = gradient node Partial derivative over range Partial derivative over azimuth Phase value Gradient node Measured gradient Constraint node
    10. 10. Message passing <ul><li>Sum-product algorithm (a.k.a. belief propagation) allows to find an approximate cost labeling (belief) of energy functions </li></ul><ul><ul><li>inexact in graphs with cycles but produces excellent results </li></ul></ul><ul><li>To obtain the MAP, a variant called max-sum is used </li></ul><ul><ul><li>(If used with negative log probabilities -> min-sum) </li></ul></ul><ul><ul><li>1. Initialize gradient nodes to their energy </li></ul></ul><ul><ul><li>2. Update messages iteratively </li></ul></ul><ul><ul><li>3. Receive messages and take the minimum </li></ul></ul>
    11. 11. Message update: from constraint node to gradient node m 4 Gradient node Constraint node m 1 m 2 m 3
    12. 12. Messages update: from gradient node to constraint node m 4 m 5 Gradient node Constraint node
    13. 13. MAP computation m 4 m 5 Gradient node Constraint node
    14. 14. Gradient log pdf and speed up Range (i-direction) Azimuth (k-direction) Pixel 1 Pixel 2 0 2  -2  Gradient [rad] 0 20 40 60 pdf 0 2  -2  Gradient [rad] 0 20 40 60 pdf 0 2  -2  Gradient [rad] 0 20 40 60 pdf 0 2  -2  Gradient [rad] 0 20 40 60 pdf
    15. 15. Message update: Forward -6.3 0 6.3 0 E 0 =0.9+0.5+0.8=2.2 0 E -1 =0.9+0.5+0.0=1.4   i (i,k) +   k (i+1,k) -   i (i,k+1) -   k (i,k) 1.4 0.9 0.4 -3.3 3 9.3 -6.1 0.2 6.5 -6.3 0 6.3 -9.4 -3.1 3.2 0.3 -2.7 0.2 -3.1 0.0 3.0 0.2 curl 1.4 0.9 0.4 1.4 1.9 0.5 0.9 0.4 0 0.5 0 0.8 0.8 0 0.6 0.0 0.7 0.7 0 1 0.1 0.3 0 0.9 0.4 0 0.9 2.0 1.7 1.6
    16. 16. Message update: MAP 0.3 -2.7 0.2 -3.1 0.0 3.0 0.2 0.9 0.4 0 0.5 0 0.8 0.8 0 0.6 0.0 0.7 0.7 0 1 0.1 0.3 0 0.9 0.4 0 0.9 1.4 0.3 1.0 2.0 1.7 1.6 3.4 3.0 2.7
    17. 17. Results: Test site “ footprint” south from Salar de Arizaro (Argentina)
    18. 18. Results: Unwrapped Gradients with MAP Unwrapped gradient in Azimuth Unwrapped gradient in Range
    19. 19. Results: Remaining residues and MCF results Residues and branch-cuts from MCF Unwrapped phase
    20. 20. Results: MCF results, comparison with single baseline Single baseline phase unwrapping (MCF) Multibaseline gradient-based phase unwrapping
    21. 21. Results: The unwrapped phase
    22. 22. Conclusions <ul><li>Multibaseline gradients unwrapping is used to disambiguate the gradients. The range of search is thus very restricted to 3 or 5 cycles. </li></ul><ul><li>The zero-curl constraint (in every interferogram) is used as a prior. </li></ul><ul><li>A graphical model which incorporates the gradients log pdf and this constraint is introduced. </li></ul><ul><li>We propose a method based on probability passing which is a good method for inferring the different gradients. </li></ul><ul><li>Since we just want to find out the right ambiguity, the messages can be reduced to the number of cycles and thus the processing is accelerated </li></ul><ul><li>The prior could be improved by incorporating at least a smoothness criterium </li></ul>
    23. 23. thank you!

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