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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

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

Phase Denoising (PD)

Phase Unwrapping (PU)

Estimation of

Estimation of

(wrapped phase)

Absolute Phase Estimation

slide3

Applications

  • Synthetic aperture radar/sonar
  • Magnetic resonance imaging
  • Doppler weather radar
  • Doppler echocardiography
  • Optical interferometry
  • Diffraction tomography
slide4

Absolute Phase Estimation in InSAR (Interferometric SAR)

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

slide5

Mountainous terrain around

Long’s Peak, Colorado

Interferogram

slide6

Differential Interferometry

Height variation

7 mm/year

-17 mm/year

slide7

Magnetic Resonance Imaging - MRI

Wrapped phase

Intensity

Interferomeric Phase

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

Data density:

Prior (1st order MRF):

clique set

clique potential (pairwise interaction)

non-convex

convex

Enforce smoothness

Enforce piecewise smoothness

(discontinuity preserving)

Bayesian Approach
phase unwrapping path following methods

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

[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

while success == false

then

success == true

PUMA (Phase Unwrapping MAx-flow)

Finds a sequence of steepest descent binary images

puma convex priors

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
results18
Results ( )

Convex priors does not preserve discontinuities

slide19

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

Non-increasing property

Majorizing nonsubmodular terms

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

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

slide21

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

Interferogram 

MM

QOBOP

QPBOI

Results

absolute phase pu denoising

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

Noise

High phase rate

Major degradation mechanism in PU and APE

slide30

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

proposed algorithm

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

for t=0:tmax

success == false

while success == false

then

success == true

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

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

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

Acknowledgements

Gonçalo Valadão

Yuri Boykov

Vladimir Kolmogorov

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