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Multi-source Absolute Phase Estimation: A Multi-precision Approach Based on Graph Cuts. José M. Bioucas-Dias Instituto Superior Técnico Instituto de Telecomunicações Portugal. Graph Cuts and Related Discrete or Continuous Optimization Problems - IPAM 08. Phase Denoising (PD).

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Multi source absolute phase estimation a multi precision approach based on graph cuts l.jpg

Multi-source Absolute Phase Estimation:A Multi-precision Approach Based on Graph Cuts

José M. Bioucas-Dias

Instituto Superior Técnico

Instituto de Telecomunicações

Portugal

Graph Cuts and Related Discrete or Continuous Optimization Problems - IPAM 08


Slide2 l.jpg

Phase Denoising (PD)

Phase Unwrapping (PU)

Estimation of

Estimation of

(wrapped phase)

Absolute Phase Estimation


Slide3 l.jpg

Applications

  • Synthetic aperture radar/sonar

  • Magnetic resonance imaging

  • Doppler weather radar

  • Doppler echocardiography

  • Optical interferometry

  • Diffraction tomography


Slide4 l.jpg

Absolute Phase Estimation in InSAR (Interferometric SAR)

InSAR Problem: Estimate 2- 1 from signals read by s1 and s2


Slide5 l.jpg

Mountainous terrain around

Long’s Peak, Colorado

Interferogram


Slide6 l.jpg

Differential Interferometry

Height variation

7 mm/year

-17 mm/year


Slide7 l.jpg

Magnetic Resonance Imaging - MRI

Wrapped phase

Intensity

Interferomeric Phase

  • measure temperature

  • visualize veins in tissues

  • water-fat separation

  • mapthe principal magnetic field


Outline l.jpg
Outline

  • Forward problem (sensor model)

  • Absolute phase estimation: Bayesian formulation

  • Computing the MAP estimate via integer optimization

  • Multi-source absolute phase estimation

  • Phase unwrapping

  • Convex and non-convex priors

  • Unambiguous interval increasing

  • Phase unwrapping

  • Convex and non-convex priors




Bayesian approach l.jpg

Data density:

Prior (1st order MRF):

clique set

clique potential (pairwise interaction)

non-convex

convex

Enforce smoothness

Enforce piecewise smoothness

(discontinuity preserving)

Bayesian Approach


Maximum a posteriori estimation criterion l.jpg

posterior density

  • Phase unwrapping:

Maximum a Posteriori Estimation Criterion


Phase unwrapping path following methods l.jpg

Assume that

Then

PU ! summing over walks

Phase Unwrapping: Path Following Methods

Why isn’t PU a trivial problem?

Discontinuities

High phase rate

Noise


Phase unwrapping algorithms l.jpg

[Flynn, 97] (exact)! Sequence of positive cycles on a graph

[Costantini, 98] (exact)! min-cost flow on a graph

[Bioucas-Dias & Leitao, 01] (exact)! Sequence of positive cycles on

a graph

[Frey et al., 01] (approx)! Belief propagation on a 1st order MRF

convex

[Bioucas-Dias & Valadao, 05] (exact)! Sequence of K min cuts ( )

non-convex

[Ghiglia, 96]! LPN0 (continuous relaxation)

[Bioucas-Dias & G. Valadao, 05, 07] ! Sequence of min cuts ( )

Phase Unwrapping Algorithms


Puma phase unwrapping max flow l.jpg

while success == false

then

success == true

PUMA (Phase Unwrapping MAx-flow)

Finds a sequence of steepest descent binary images


Puma convex priors l.jpg

  • Related algorithms

[Veksler, 99] (1-jump moves )

[Murota, 03] (steepest descent algorithm for L-convex functions)

[Ishikawa, 03] (MRFs with convex priors)

[Kolmogorov & Shioura, 05,07], [Darbon. 05](Include unary terms)

[Ahuja, Hochbaum, Orlin, 03] (convex dual network flow problem)

PUMA: Convex Priors

  • A local minimum is a global minimum

  • Takes at most K iterations



Results18 l.jpg
Results ( )

Convex priors does not preserve discontinuities


Slide19 l.jpg

PUMA:

Non-convex priors

Ex:

Models discontinuities

Models Gaussian noise

Shortcomings:

  • Local minima is no more a global minima

  • Energy contains nonsubmodular terms (NP-hard)

Tentative suboptimal solutions:

  • Majorization Minimization

  • Quadratic Pseudo Boolean Optimization

    (Probing [Boros et al., 2006], Improving [Rother et al., 2007] )


Slide20 l.jpg

Non-increasing property

Majorizing nonsubmodular terms

Majorization Minimization (MM) [Lange & Fessler, 95]

[Rother et al., 05] ! similar approach for alpha expansion moves


Slide21 l.jpg

Interferogram

no. of nonsubmodular terms

iter

us

MM

QOBOP

QPBOI

QPBOP

MM

QPBOI

1

590/0

2,5 e-2

590/0

590

326/0

1,0 e-2

2

326/0

410

263/0

1,0 e-2

263/0

271

3

154/0

6,0 e-3

154/0

179

4

123/0

4,0 e-3

123/0

141

5

94/0

4,0 e-2

6

94/0

117

88/0

2,5 e-3

88/0

91

7

57/15000

1,0e-3

57/15000

57

8

T

1 s

120 s

2 s

Results


Slide22 l.jpg

Interferogram

MM

QOBOP

QPBOI

Results


Slide23 l.jpg

Multi-jump version of PUMA

Jumps 2 [1 2 3 4]


Absolute phase pu denoising l.jpg

PUMA + dyadic scaling

then

  • Unary terms may be non-convex

    Compute using the algorithm [Darbon, 07] for 1st order

    submodular priors (complexity )

Absolute Phase (PU + Denoising)

  • Related algorithms: [Zalesky, 03], [Ishikawa, 03],

    [Ahuja, Hochbaum, Orlin, 04]



Slide29 l.jpg

Noise

High phase rate

Major degradation mechanism in PU and APE


Slide30 l.jpg

Use more than one observation with different frequencies

Two sources

We can infer

  • noise is an issue

  • unwrap phase images with range larger than

Multi-source Absolute Phase Estimation



Computing the map estimate l.jpg

  • Phase v-unwrapping:

Computing the MAP estimate


Proposed algorithm l.jpg

Optimization: Non-convex data term + TV

Exact solution: Levelable functions [Darbon, 07],

[Ahuja et al, 04], [Zalesky, 03], [Ishikawa, 03],

(takes time)

2. Run PUMA in a multiscale fashion with the schedule:

  • scale v ! v-unwrapping]

  • scales ! denoising

Proposed Algorithm


Absolute phase 1 pu v pu denoising l.jpg

for t=0:tmax

success == false

while success == false

then

success == true

Absolute Phase (1-PU+v-PU + Denoising)




Future directions l.jpg
Future Directions

  • High order interactions

  • Denoise (first) + Unwrap

    • Local adaptive models (collaboration with

      Vladimir Katkovnik, Tampere University of Technology)

  • Huge images (ex: 10000£10000)


Concluding remarks l.jpg
Concluding Remarks

  • Addressed discontinuity preserving phase unwrapping

    and phase denoising methods based on integer

    optimization

  • Addressed multi-source absolute phase estimation

  • Introduced the concept of v-phase unwrapping

  • Introduced a new algorithm for multi-source absolute

    phase estimation based on integer optimization


Slide41 l.jpg

References

  • J. Dias and J. Leitao, “The ZM algorithm for interferometric image

    reconstruction in SAR/SAS”, IEEE Transactions on Image processing,

    vol. 11, no. 4, pp. 408-422, 2002.

  • J. Bioucas-Dias and G. Valadao, “Phase unwrapping via graph cuts",

    IEEE Transactions on Image processing, vol. 16, no. 3, pp. 698-709, 2007.

  • V. Kolmogorov and A. Shioura, “New algorithms for the dual of the convex cost

    network flow problem with applications to computer vision", Technical Report,

    June, 2007

  • J. Darbon, Composants logiciels et algorithmes de minimisation exacte d’energies

    dedies au traitement des images, PhD thesis, Ecole Nationale Superieure des

    Telecommunications, 2005.

  • Y. Boykov and Vladimir Kolmogorov, An Experimental Comparison of

    Min-Cut/Max-Flow, IEEE Transactions on Pattern Analysis and Machine

    Intelligence, September 2004.


Slide42 l.jpg

References

  • C. Rother, V. Kolmogorov, V. Lempitsky, and M. Szummer,

    “Optimizing binary MRFs via extended roof duality”, in IEEE Conference

    on Computer Vision and Pattern Recognition (CVPR), June 2007.

  • E. Boros, P. L. Hammer, and G. Tavares. Preprocessing of unconstrained

    quadratic binary optimization. Technical Report RRR 10-2006,

    RUTCOR, Apr. 2006.

  • J. Darbon and M. Sigelle, “Image restoration with discrete constrained total

    variation Part II: Levelable functions, convex and non-convex cases”, Journal of

    Mathematical Imaging and Vision. Vol. 26 no. 3, pp. 277-291, December 2006.

  • B. Zalesky, “Efficient determination of Gibbs estimators with submodular energy

    functions”, arXiv:math/0304041v1 [math.OC] 3 Apr 2003.


Slide43 l.jpg

Acknowledgements

Gonçalo Valadão

Yuri Boykov

Vladimir Kolmogorov


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