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

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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).

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

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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)

Phase Unwrapping (PU)

Estimation of

Estimation of

(wrapped phase)

Absolute Phase Estimation

Applications

- Synthetic aperture radar/sonar
- Magnetic resonance imaging
- Doppler weather radar
- Doppler echocardiography
- Optical interferometry
- Diffraction tomography

Absolute Phase Estimation in InSAR (Interferometric SAR)

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

Mountainous terrain around

Long’s Peak, Colorado

Interferogram

Differential Interferometry

Height variation

7 mm/year

-17 mm/year

Magnetic Resonance Imaging - MRI

Wrapped phase

Intensity

Interferomeric Phase

- measure temperature
- visualize veins in tissues
- water-fat separation
- mapthe principal magnetic field

- 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

Forward Problem: Sensor Model

Simulated Interferograms Images of

Data density:

Prior (1st order MRF):

clique set

clique potential (pairwise interaction)

non-convex

convex

Enforce smoothness

Enforce piecewise smoothness

(discontinuity preserving)

posterior density

- Phase unwrapping:

Assume that

Then

PU ! summing over walks

Why isn’t PU a trivial problem?

Discontinuities

High phase rate

Noise

[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 ( )

while success == false

then

success == true

Finds a sequence of steepest descent binary images

- 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)

- A local minimum is a global minimum

- Takes at most K iterations

Convex priors does not preserve discontinuities

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] )

Non-increasing property

Majorizing nonsubmodular terms

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

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

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

Interferogram

MM

QOBOP

QPBOI

Results

Multi-jump version of PUMA

Jumps 2 [1 2 3 4]

PUMA + dyadic scaling

then

- Unary terms may be non-convex
Compute using the algorithm [Darbon, 07] for 1st order

submodular priors (complexity )

- Related algorithms: [Zalesky, 03], [Ishikawa, 03],
[Ahuja, Hochbaum, Orlin, 04]

Multi-source Absolute Phase Estimation

Noise

High phase rate

Major degradation mechanism in PU and APE

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

Two sources

- Absolute phase estimation:

- Phase v-unwrapping:

- 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

for t=0:tmax

success == false

while success == false

then

success == true

High phase rate + noise

Parabolic surface

- High order interactions
- Denoise (first) + Unwrap
- Local adaptive models (collaboration with
Vladimir Katkovnik, Tampere University of Technology)

- Local adaptive models (collaboration with
- Huge images (ex: 10000£10000)

- 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

References

- J. Dias and J. Leitao, “The ZM 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.

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.

Acknowledgements

Gonçalo Valadão

Yuri Boykov

Vladimir Kolmogorov