Unwrapping phase images using the sum-product algorithm (loopy probability propagation)

Transcription

1 Unwrapping phase images using the sum-product algorithm (loopy probability propagation) Brendan Frey Electrical and Computer Engineering University of Toronto

2 Collaborators Approximate inference algorithms Ralf Koetter UIUC Kannan Achan Toronto Jodi Moran Waterloo Nemana Petrovic UIUC Synthetic Aperture Radar David Munson Jr. UIUC Lei Ying UIUC Magnetic Resonance Imaging

3 Phase unwrapping: The problem Image on regular grid λ Original surface Phase-wrapped image

4 Phase unwrapping: The problem λ Phase unwrapping: Infer the original image given the phase-wrapped image and prior knowledge

5 λ 1-D phase unwrapping

6 Examples of 2-D wrapped images SAR MRI (from Sandia National Labs)

7 Why 2-D phase unwrapping is hard (Chen and Zebker 2000: NP hard) Path 1 Path 2 Sum of shifts along any two paths must agree N x N image At least N paths between any 2 points C choices for shift between neighboring points Size of cut set = C N

8 Path-finding methods Previous work Branch cuts (Golstein et al 1988) Least squares (Song et al 1995; Jakowatz et al 1996) Integer programming (Costantini 1996; Chen and Zebker 2000) Probabilistic formulations of cost functions (Carballo and Feiguth 1998) Z π M (Dias and Leitao 2001) Claim their algorithm can detect when it finds the MAP solution; empirically order N 1.5 time

9 One common approach Reformulate the problem as inferring the gradient field of the original surface Then integrate the gradient field

10 On gradient fields Sum of gradients around every closed path must be zero A sufficient set of loops is all 2 x 2 loops So there is one zero-curl constraint for every 2 x 2 patch A vector field that satisfies all zero-curl constraints is a gradient field Pixel Gradient field Not a gradient field

11 Least squares method (Greedily) find a vector field v that is a first order approximation to the gradient field ignore zero-curl constraints Introduce zero-curl matrix C Evaluates to zero when applied to a gradient field Find the new solution g (a gradient field) such that Cg = 0 and g v 2 is minimized Vector space of vector fields Initial vector field v g: orthogonal proection of v onto the linear subspace Linear subspace of gradient fields

12 Discrete shift model Represent the vector field by the relative number of shifts between pairs of neighboring points s x (xy) = x-direction shift between the observations φ(xy) and φ(x+1y) s y (xy) = Given the shifts we assume the surface is described by a Gaussian process: P( φ(x+1y) φ(xy) s x (xy) ) exp[-( φ(xy) + s x (xy) - φ(x+1y) ) 2 / 2σ 2 ]

13 Enforcing the zero-curl constraints One potential for each constraint: f(s x (xy)s y (x+1y)s x (xy+1)s y (xy)) 1 if s x (xy)+s y (x+1y)-s x (xy+1)-s y (xy) = 0 = 0 otherwise s x (xy+1) s y (xy) f s y (x+1y) s x (xy)

14 The graphical model (factor graph) Hidden shift variables Observed variables P( φ(x+1y) φ(xy) s x (xy) )

15 Removing the observed variables Hidden shift variables P( φ(x+1y) φ(xy) s x (xy) )

16 Sum-product algorithm in the shift model Messages passed right left up and down iteratively

17 Sum-product algorithm in the shift model Fused messages are used to decode shifts Which are then integrated to unwrap the image P(s x (xy) φ)

18 For data we ve studied the sum-product algorithm takes order N time (using the forward-backward-up-down schedule)

19 Least squares method revisited Recall: v = initial guess at real gradient field LS computes min g g v 2 subect to Cg=0 Equivalent: max g exp[- g v 2 /2σ 2 ] δ(cg) P( v x (xy) g x (xy) ) exp[-(g x (xy)-v x (xy)) 2 /2σ 2 ] Theorem (Freeman and Weiss 2000): In Gaussian networks if the sum-product algorithm converges it finds the MAP means (g)

20 Dynamics of inference in shift model Arrows are ~MAP decisions = curl violation +1 = curl violation -1

21 Robust integration Our algorithm computes marginals which when detected may violate some zero-curl constraints We use LS as post-processor to obtain gradient field Our algorithm Greedy guess LS LS

22 The sum-product algorithm followed by least squares Phase-wrapped image from Sandia National Labs

23 New approach to evaluation Previous work (c.f. Ghiglia and Pritt 1998) has evaluated reconstruction qualitatively A quantitative approach: Acquire ground truth For different wrapping wavelengths (λ) wrap the surface and apply algorithm Plot reconstruction error vs λ We used LS to obtain ground truth for Sandia data

24 Performance on Sandia

25 Performance on Sandia

26 Performance on Sandia

27 Comparison with textbook methods

28 VERY recent results (Nov )

29 BUT sum-product is faster!

30 Current work Beat Z π M More complex surface priors (eg cliff models using line processes) Learn prior from input image using generalized EM Kikuchi free energy minimization Applications in SAR and MRI

31 Product-form variational technique 1 Problem: Posterior is extremely non-product form due to zero-curl constraints Relax zero-curl constraints: Introduce Temperature T ( ) ( ) ( ) = = = + + = = i k k i m i m i l i m n l k i k i k k i k i k k i k k n m l k F i i k i i i k i T k i log log 2 φ φ β φ φ α σ β α β α β β α α Yuck!

32 Product-form variational technique 2 To make posterior closer to product-form choose a different representation Instead of gradients infer absolute heights We haven t been able to get this to work Problems Spontaneous symmetry breaking State space is larger Maybe another representation will work