Skip to main content
SpringerLink
Log in

Discontinuity Preserving Phase Unwrapping Using Graph Cuts

  • Conference paper
Book cover Energy Minimization Methods in Computer Vision and Pattern Recognition (EMMCVPR 2005)

Part of the book series: Lecture Notes in Computer Science ((LNIP,volume 3757))

Abstract

We present a new algorithm for recovering the absolute phase from modulo-2π phase, the so-called phase unwrapping (PU) problem. PU arises as a key step in several imaging technologies, from which we emphasize interferometric synthetic aperture radar/sonar (InSAR/SAS), magnetic resonance imaging (MRI), and optical interferometry. We adopt a discrete energy minimization viewpoint, where the objective function is a first-order Markov random field. The minimization problem is dealt with via a binary iterative scheme, with each iteration step cast onto a graph cut based optimization problem. For convex clique potentials we provide an exact energy minimization algorithm; namely we solve exactly the PU classical L p norm, with p ≥ 1. For nonconvex clique potentials, it is well known that PU performance is particularly enhanced, namely, the discontinuity preserving ability; however the problem is NP-hard. Accordingly, we provide an approximate algorithm, which is a modified version of the first proposed one. For simplicity we call both algorithms PUMF, for Phase Unwrapping Max-Flow. The state-of-the-art competitiveness of PUMF is illustrated in a series of experiments.

This work was supported by the Fundação para a Ciência e Tecnologia, under the project PDCTE/CPS/49967/2003.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Oppenheim, A.V., Lim, J.S.: The importance of phase in signals. Proceedings of the IEEE 69(5), 529–541 (1981)

    Article  Google Scholar 

  2. Rosen, P., Hensley, S., Joughin, I., Li, F., Madsen, S., Rodriguez, E., Goldstein, R.: Synthetic aperture radar interferometry. Proceedings of the IEEE 88(3), 333–382 (2000)

    Article  Google Scholar 

  3. Jezzard, P., Balaban, R.: Correction for geometric distortion in echo-planar images from B 0 field variations. Magnetic Resonance in Medicine 34, 65–73 (1995)

    Article  Google Scholar 

  4. Pandit, S., Jordache, N., Joshi, G.: Data-dependent systems methodology for noise-insensitive phase unwrapping in laser interferometric surface characterization. Journal of the Optical Society of America 11(10), 2584–2592 (1994)

    Article  Google Scholar 

  5. Ghiglia, D., Pritt, M.: Two-Dimensional Phase Unwrapping. Theory, Algorithms, and Software. John Wiley & Sons, New York (1998)

    MATH  Google Scholar 

  6. Itoh, K.: Analysis of the phase unwrapping problem. Applied Optics 21(14) (1982)

    Google Scholar 

  7. Goldstein, R., Zebker, H., Werner, C.: Satellite radar interferometry: Two-dimensional phase unwrapping. In: Symposium on the Ionospheric Effects on Communication and Related Systems, Radio Science, vol. 23, pp. 713–720 (1988)

    Google Scholar 

  8. Marroquin, J., Rivera, M.: Quadratic regularization functionals for phase unwrapping. Journal of the Optical Society of America 12(11), 2393–2400 (1995)

    Article  MathSciNet  Google Scholar 

  9. Buckland, J., Huntley, J., Turner, S.: Unwrapping noisy phase maps by use of a minimum cost matching algorithm. Applied Optics 34(23), 5100–5108 (1995)

    Article  Google Scholar 

  10. Ghiglia, D., Romero, L.: Minimum L p norm two-dimensional phase unwrapping. Journal of the Optical Society of America 13(10), 1999–2013 (1996)

    Article  MathSciNet  Google Scholar 

  11. Flynn, T.: Two-dimensional phase unwrapping with minimum weighted discontinuity. Journal of the Optical Society of America A 14(10), 2692–2701 (1997)

    Article  Google Scholar 

  12. Costantini, M.: A novel phase unwrapping method based on network programing. IEEE Transactions on Geoscience and Remote Sensing 36(3), 813–821 (1998)

    Article  MathSciNet  Google Scholar 

  13. Strand, J.: Two-dimensional Phase Unwrapping with Applications. PhD thesis, Department of Mathematics, Faculty of Mathematics and Natural Sciences, University of Bergen (1999)

    Google Scholar 

  14. Dias, J., Leitão, J.: The ℤπM algorithm for interferometric image reconstruction in SAR/SAS. IEEE Transactions on Image Processing 11, 408–422 (2002)

    Article  Google Scholar 

  15. Marroquin, J., Tapia, M., Rodriguez-Vera, R., Servin, M.: Parallel algorithms for phase unwrapping based on markov random field models. Journal of the Optical Society of America A 12(12), 2578–2585 (1995)

    Article  MathSciNet  Google Scholar 

  16. Nico, G., Palubinskas, G., Datcu, M.: Bayesian approach to phase unwrapping: theoretical study. IEEE Transactions on Signal Processing 48(9), 2545–2556 (2000)

    Article  Google Scholar 

  17. Friedlander, B., Francos, J.: Model based phase unwrapping of 2-d signals. IEEE Transactions on Signal Processing 44(12), 2999–3007 (1996)

    Article  Google Scholar 

  18. Fried, D.: Least-squares fitting a wave-front distortion estimate to an array of phase-difference measurements. Journal of the Optical Society of America 67(3), 370–375 (1977)

    Article  Google Scholar 

  19. Ghiglia, D., Romero, L.: Robust two-dimensional weighted and unweighted phase unwrapping that uses fast transforms and iterative methods. Journal of the Optical Society of America A 11, 107–117 (1994)

    Article  Google Scholar 

  20. Chen, C.: Statistical-Cost Network-Flow Approaches to Two-Dimensional Phase Unwrapping for Radar Interferometry. PhD thesis, Stanford University (2001)

    Google Scholar 

  21. Leitão, J., Figueiredo, M.: Absolute phase image reconstruction: A stochastic non-linear filtering approach. IEEE Transactions on Image Processing 7(6), 868–882 (1997)

    Article  Google Scholar 

  22. Dias, J., Leitão, J.: Simultaneous phase unwrapping and speckle smoothing in SAR images: A stochastic nonlinear filtering approach. In: EUSAR 1998 European Conference on Synthetic Aperture Radar, Friedrichshafen, May 1998, pp. 373–377 (1998)

    Google Scholar 

  23. Bioucas-Dias, J., Valadão, G.: Phase unwrapping via graph cuts. Submitted to IEEE Transactions on Image Processing (2005)

    Google Scholar 

  24. Kolmogorov, V., Zabih, R.: What energy functions can be minimized via graph cuts? IEEE Transactions on Pattern Analysis and Machine Intelligence 26(2), 147–159 (2004)

    Article  Google Scholar 

  25. Veksler, O.: Efficient Graph-Based Energy Minimization Method. In: Computer Vision. PhD thesis, Cornell University (1999)

    Google Scholar 

  26. Boykov, Y., Veksler, O., Zabih, R.: Fast approximate energy minimization via graph cuts. IEEE Transactions on Pattern Analysis and Machine Intelligence 23(11), 1222–1239 (2001)

    Article  Google Scholar 

  27. Blake, A., Zisserman, A.: Visual Reconstruction. MIT Press, Cambridge (1987)

    Google Scholar 

  28. Blake, A.: Comparison of the efficiency of deterministic and stochastic algorithms for visual reconstruction. IEEE Transactions on Pattern Analysis and Machine Intelligence 11(1), 2–12 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  29. Hebert, T., Leahy, R.: A generalized EM algorithm for 3-D Bayesian reconstruction from Poisson data using Gibbs priors. IEEE Trans. on Medical Imaging 8(2), 194–202 (1989)

    Article  Google Scholar 

  30. Geman, S., Reynolds, G.: Constrained restoration and the recovery of discontinuities. IEEE Trans. Pattern Analysis and Machine Intelligence 14(3), 367–383 (1992)

    Article  Google Scholar 

  31. Bouman, C., Sauer, K.: A generalized Gaussian image model for edge-preserving MAP estimation. IEEE Transactions on Image Processing 2(3), 296–310 (1993)

    Article  Google Scholar 

  32. Lange, K.: Optimization. Springer, New York (2004)

    MATH  Google Scholar 

  33. Greig, D., Porteous, B., Seheult, A.: Exact maximum a posteriory estimation for binary images. Jounal of Royal Statistics Society B 51(2), 271–279 (1989)

    Google Scholar 

  34. Ferrari, P., Gubitoso, M., Neves, E.: Reconstruction of gray-scale images. Methodology and Computing in Applied Probability 3, 255–270 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  35. Wu, Z., Leahy, R.: An optimal graph theoretic approach to data clustering: Theory and its application to image segmentation. IEEE Transactions on Pattern Analysis and Machine Intelligence 15(11), 1101–1113 (1993)

    Article  Google Scholar 

  36. Ishikawa, H.: Exact optimization for markov random fields with convex priors. IEEE Transactions on Pattern Analysis and Machine Intelligence 25(10), 1333–1336 (2003)

    Article  Google Scholar 

  37. Freedman, D., Drineas, P.: Energy minimization via graph cuts: settling what is possible. In: Proceedings of the IEEE Computer Society Conference on Computer Vision and Pattern Recognition - CVPR 2005, San Diego, CA (June 2005)

    Google Scholar 

  38. Rother, C., Kumar, S., Kolmogorov, V., Blake, A.: Digital tapestry. In: Proceedings of the IEEE Computer Society Conference on Computer Vision and Pattern Recognition - CVPR 2005, San Diego, CA (June 2005)

    Google Scholar 

  39. Boykov, Y., Kolmogorov, V.: An experimental comparison of min-cut/max-flow algorithms for energy minimization in vision. IEEE Transactions on Pattern Analysis and Machine Intelligence 26(9), 1124–1137 (2004)

    Article  Google Scholar 

  40. Lim, H., Xu, W., Huang, X.: Two new practical methods for phase unwrapping. In: Proccedings of the 1995 International Geoscience and Remote Sensing Symposium-IGARSS 1995, Firenze, Italy, pp. 196–198 (1995)

    Google Scholar 

  41. Flynn, T.: Consistent 2-D phase unwrapping guided by a quality map. In: Proceedings of the 1996 International Geoscience and Remote Sensing Symposium-IGARSS 1996, Lincoln, NE, vol. 4, pp. 2057–2059 (1996)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Instituto de Telecomunicações, Instituto Superior Técnico, Torre Norte, Piso 10, Av. Rovisco Pais, 1049-001, Lisboa, Portugal

    José M. Bioucas-Dias & Gonçalo Valadão

Authors
  1. José M. Bioucas-Dias
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Gonçalo Valadão
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of CISE, University of Florida, Gainesville, FL, USA

    Anand Rangarajan

  2. University of Florida, 32611, Gainesville, FL, USA

    Baba Vemuri

  3. Department of Statistics, University of California, Los Angeles

    Alan L. Yuille

Rights and permissions

Reprints and permissions

Copyright information

© 2005 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Bioucas-Dias, J.M., Valadão, G. (2005). Discontinuity Preserving Phase Unwrapping Using Graph Cuts. In: Rangarajan, A., Vemuri, B., Yuille, A.L. (eds) Energy Minimization Methods in Computer Vision and Pattern Recognition. EMMCVPR 2005. Lecture Notes in Computer Science, vol 3757. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11585978_18

Download citation

  • DOI: https://doi.org/10.1007/11585978_18

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-30287-2

  • Online ISBN: 978-3-540-32098-2

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation