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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


Slide9 l.jpg

Forward Problem: Sensor Model


Slide10 l.jpg

Simulated Interferograms Images of


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

  • is submodular: each binary optimization

    has the complexity of a min cut

  • 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


Results l.jpg

Results ( )


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]


Slide28 l.jpg

Multi-source Absolute Phase Estimation


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


Slide31 l.jpg

Two sources


Computing the map estimate l.jpg

  • Absolute phase estimation:

  • Phase v-unwrapping:

Computing the MAP estimate


Proposed algorithm l.jpg

  • Initialization: 1-unwrapp in the interval using

    total variation (TV)

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)


Slide37 l.jpg

High phase rate + noise


Slide38 l.jpg

Parabolic surface


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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