CA2353720A1

CA2353720A1 - Method for unwrapping 2-dimensional phase signals

Info

Publication number: CA2353720A1
Application number: CA002353720A
Authority: CA
Inventors: Brendan J. Frey; Ralf Koetter
Original assignee: Individual
Current assignee: Individual
Priority date: 2001-07-25
Filing date: 2001-07-25
Publication date: 2003-01-25

Abstract

Many imaging techniques, including magnetic resonance imaging and interferometric synthetic aperture radar, produce phase-wrapped 2-dimensional signals (images and surfaces). In a phase-wrapped signal, the original signal is measured modulus a known wavelength. We describe a new, computationally efficient algorithm for removing this type of distortion from 2-dimensional signals.
Using the phase-wrapped signal as input, the algorithm infers the relative integer shifts between neighboring signal values by propagating belief vectors across a richly connected network that describes the relationships between the shifts. Belief propagation is generally not guaranteed to converge to the correct answer in networks with cycles. However, we find experimentally that propagating shift beliefs across networks for phase unwrapping produces excellent estimates of the unwrapped signal.

Description

FIELD OF THE INVENTION
The present invention relates to a method of signal processing of phase images, and more particularly the present invention relates to a method of phase unwrapping for obtaining original, unwrapped values from the wrapped phase images.
BACKGROUND OF THE INVENTION
Phase unwrapping is an easily stated, fundamental problem in signal processing.The signal is measured modulus a known wavelength, which we take to be 1 without loss of generality. Fig. 1 B shows the wrapped, 1-dimensional signal obtained from the original signal shown in Fig. 1A. Fig. 1C shows a wrapped surface from Sandia National Laboratories, New Mexico, recorded using interferometric synthetic aperture radar.
The objective of phase unwrapping is to estimate the original signal from the wrapped version, using knowledge about which signals are more probable a priori. The best solution to phase unwrapping in 2-dimensional topologies has not yet been found. Between any two points in the image (see above), there is a large number of paths and after the image is unwrapped, each of these paths must satisfy the prior assumptions. In fact, if phase unwrapping is cast as a "minimum L° norm problem" in integer programming, it turns out to be NP-hard.
Approaches to solving the phase unwrapping problem include least squares estimates (these are not MMSE estimates), integer programming methods, and branch cut techniques.
It would be very advantageous to provide a method of phase unwrapping which avoids the problems discussed above.
SUMMARY OF THE INVENTION
Many imaging techniques, including magnetic resonance imaging and interferometric synthetic aperture radar, produce phase-wrapped 2-dimensional signals (images and surfaces). In a phase-wrapped signal, the original signal is measured modulus a known wavelength. The present invention discloses a new, computationally efficient method for removing this type of distortion from 2-dimensional signals. Using the phase-wrapped signal as input, the method infers the relative integer shifts between neighboring signal values by propagating belief vectors across a richly connected network that describes the relationships between the shifts. Belief propagation is generally not guaranteed to converge to the correct answer in networks with cycles, however in the present invention it has been experimentally verified that propagating shift beliefs across networks for phase unwrapping produces excellent estimates of the unwrapped signal.
BRIEF DESCRIPTION OF THE DRAWINGS
The method of phase unwrapping will now be described, by way of example only, reference being had to the accompanying drawings, in which:

FIGURE 1: (A) A 1-dimensional signal. (B) The phase-wrapped version of the signal in (.~), «-here the vaveleligth is ~1. (C) ~ wrapped surface (2-dimensional signal) from Sandia \ational Laboratories, \'ew Mexico.
FIGURE 2: Positive a~-direction shifts (arro«~s labeled a) and positi~'e y-direction shifts (ar-rov's labeled G) betv'een neighboring IIIeasllI'eIIleIItS lIl a 2 X 2 patCll Of pOlIltS (lnarlveCl by X'S).
FIGURE 3: (:~) .a graphical model that describes the zero-curl COIlStt'alllt5 (blacl: discs) hetv'~en neighboring shift variables (white discs). 3-element probability-vectors (E.c's) on tile relative shifts between neighboring variables (-l, 0, or +1) are propagated across the llCt\~'OI'lC: (I3) COllstl'Illllt-t0-Slllft vectors Fll'e COlllpllteCl fI'CrIIl II1CO1111I1g Shlft-t0-COllStralIlt \'eC-t01'S; (C) ~hlft-t0-CUlIStI'illllt \'eCt01'S ilI'C COIIIpllteCl from incoming COIIStI'alIlt-t.0-Slllft ~'(:Ct01'S;
(D) Estilllates of the marginal probabilities of tile shifts given the data are computed by COlIblllllg 111COllllllg COIIStI'allt-t0-Slllft vectors.
F1GURE :l: after 10 iterations of belief propagation using the phase-v'rapped surface from Fig. 1C, hard decisions were made for the shift. ~-ariables and the resulting shifts were inte-grated to produce this unwrapped surface.
FIGL'IZ,E 5: (A) ~ surface; ranging in height from -6 to 22. (B) The mean squared re-COIIStI'uC't10I1 CrrOI' versus the v'avelength used to vrrap the surface in (:~), for 10 iterations of belirf propagation follon'ed by least sduares intebration and for a standard least squares Ilcthod (10, 11~.
FIGURE G: a) Tlle grat>hical model underlying tile probability propagation algorithms of section 3: b) .-~ scllelnatlc dravc-lIg showing how messages are CUInblIled in probability prop-alg71tI0I1. Tile messages are being sent to«'ards all directions.
FIGL-13E i: Tile graphical models underl,vinb the probability propagation algorithllls of Sccriolls '? and 3.

Figure 8: ~An illustration how the graphical models of Sections 2 and 3 discussed hereinafter can be combined into a larger graphical model.
DETAILED DESCRIPTION OF THE INVENTION
1. GENERAL
A sensible goal in phase uwvrapping is to infer tile number of relative wrappings, or integer ''shifts", between every pair of neighboring measurements. Positive shifts correspond to an increase in the number of wrappings in the direction of the z or y coordinate, whereas negative shifts correspond to a decrease in the number of wrappings in the direction of the x or y coordinate. After arbitrarily aSS1gI11Ilg all abSOlute nLIIllbel' of wrappings to one point, the absolute number of wrappings at any other point can be determined by summing the shifts along a path connecting the two points. To account for direction, when taking a step against the direction of the coordinate, the shift should be subtracted.
Integrating the shifts in this fashion for all points produces an unwrapped signal.
~-~'hen neighboring signal values are more likely closer together than further apart a priori, 1-dimensional signals Call be unwrapped optimally in tlllle that is linear in the signal length.
For evel.s- pair of neighboring measurements, the shift that makes the umvrapped values as close together as possible is chosen. For esanlple, the shift between 0.4 and 0.5 would be 0, whereas the shift between 0.9 and 0.0 would be -1.
For 2-dimensional signals, there are many possible 1-dimensional paths between any two points. These paths should be examined in combination, since the sum of the shifts along every such path should be equal. Viewing the shifts as state variables, the cut-set between any two points is eal)onential in the size of the arid, making exact inference \P-hard (2).
2 Infering the "shifts"
Denote the :r-direction shift at (.r, y) by a,(:r, y) and the y-direction shift at (x, y) by 6(x. y) 215 SI10 '«'I1 111 Fig. 2. If the sum of the shifts around every short loop of :I shifts (e.g., n.(:r. y) + L(:r + 1, y) - a(x, y -~ 1) - !~(:r, y) in Fig. 2) is zero, then perturbing a path will not change the s11I11 Of the shifts along the path. So, a valid Set Of shifts S =
{o(x, y), b(x, y) ;
:r = 1, . . . . _V - 1; y = 1. . . . , ~lI - 1~ in an :V' x _lI signal must satisfy the constraint a(x, y) + ~(~~ + i, y) - a(x, y + 1) - 6(:~, y) = o, (1) > >.
for x = l, . . . . N - 1, y = l, . . . , l1-1 - 1. Since a(x, y) + b(x + 1, y) -, a(x, y + 1) - b(x, y) is a discrete measure of curl at (x, y), we refer to Eq. 1 as a "zero-curl constraint", reflecting the fact that the curl of a gradient field is 0.
In this way, phase unwrapping is formulated as the problem of inferring the most probable set of shifts subject to satisfying all zero-curl constraints.
We describe a new phase unwrapping algorithm that is inspired by our recent work showing that the breakthrough "turbodecoding algorithm" (3~ and related iterative decoders (4) are instances of the belief propagation algorithm in networks with cycles (5, 6, 7~. It is well-known that tire belief propagation algorithm (a.k.a. sum-product algorithm, probability pr'Opagat10I1) is exact in graphs that are trees (8~, but it has been discovered only recently that it can produce excellent results in graphs with many cycles. In fact, for networks with more tlia.n one cycle, the only case in which convergence is known to give the correct solution is linear Gaussian networks (9J. Even in linear Gaussian networks, belief propagation may not converge at all and when it does, although the means are correct, the variances are often overconfident.
~Ve assume that given the set of shifts, the unwrapped surface is described by a low-order Gaussian process. The joint distribution over the shifts S = {a.(x, y), b(x, y) : x = 1, . . . , N-l; ;~ = l, . . . , IlI - 1 } and the wrapped measurements ~ _ {ø(x, y) : 0 <_ ø(x, y) < l, x =
l, . . . , N; y = 1, . . . , llI} can be expressed in the form N~-r m-r P(S, ~) a ~ ~ S(a(:r, y) + b(x + 1, y) - a(x, y + 1) - b(x, y)) x=1 ~=1 ( (ø(a: + l, y) - ø(x, y) - a(x~ y))2J
=r y=r n~ m-i ~ (ø(x~ :J + 1) - ø(x~ J) - b(x, y))21.
e~1' -=i .gym 2Qa The zero-curl constraints are enforced by 8(~), which evaluates to 1 if its argument is 0 and evaluates to 0 otherwise. ~t'e assume the slope of the surface is limited so that the unknown shifts take on the values -l, 0 and 1. Q~ is the variance between two neighboring measurements in the unwrapped signal, bat we find that in practice it can be estimated directly from the wrapped signal.
Phase unwrapping consists of making inferences about the a's and b's in the above probability model. For example, the marginal probability that the x-direction shift at (x, y) is k given an observed wrapped signal ~, is p(a(x, y) _ ~~~) a ~ P(s, ~). (3) S:a(x,y)=k For an N x lLI grid, the above sum ha.s roughly 32"'~~ terms and so exact inference is intractable.
The factorization of the joint distribution in Eq. 2 can be described by a graphical model, as S110wI1 111 Fig. 3A. In this graph, each white disc sits on the border between two measurements (marked by x's), and corresponds to either an x-direction shift (a's) or a y-direction shift (b's). Each black disc corresponds to a zero-curl constraint (~(~) in Eq. 2), and is connected to the ~1 shifts that it COIlstralllS to sum to 0.
Belief propagation computes messages (3-vectors denoted by Ec) which are passed in both directions on eveyv edge in the network. The elements of each 3-vector correspond to the allowed values of the neighboring shift, -l, 0 and 1. Each of these 3-vectors can be thought of as a probability distribution over the 3 possible values that. the shift can take on.
Each element in a constraint-to-shift message summarizes the evidence from the other 3 shifts involved in the constraint, and is computed by averaging the allowed configurations of evidence from the other 3 shifts in the constraint. For example, if ~.cl, E.c2 and ~3 are 3-vectors entering a constraint as shown in Fig. 3B, the outgoing 3-vector, ~4, is computed using ~~.li = ~ ~ ~ ~(~ + l - i -.7)E~ljE~zkf~at~
j=-1 k=-1 l=-1 and then normalized for numerical stability. The other 3 messages produced at the constraint a.re computed in a similar fashion.
Shift-t.o-constraint messages arc computed by weighting incoming constraint-to-shift mes-sages with the likelihood for the shift. For example, if ~1 is a 3-vector entering an x-direction shift as shown in Fig. 3C, the outgoing 3-vector, E.cz is computed using ~2i = Eli P~~~~ (~(x '~- 1v y) - ~(2~ y) - L)2~2~2~~ c7 and then normalized. Messages produced by y-direction shifts are computed in a similar fashion.

At any step in the message-passing process, the messages on the edges connected to a shift variable can be combined to produce an approximation to the marginal probability for that shift, given the observations. For example, if ~cl and ~.ez are the 3-vectors entering an ~-direction shift as shown in Fig. 3D, the approtimation is ~'(~(x~ y) = z~~) _ (l~nfizt)~(~ E~-yf~z~)~ (fi) i Given a wrapped signal, the variance Qz is estimated directly from the wrapped signal, the shift beliefs are initialized to uniform distributions, and then shift beliefs are propagated across the graph in an iterative fashion. Different message-passing schedules are possible, ranging from full~-~ parallel, to a "forward-backward-up-down"-type schedule, in which mes-sages are passed across the network to the right, then to the left, then up and then down.
For an N x 111 grid, each iteration takes C~(NIII) scalar computations.
APPENDI1 C shows an embodyment of the described procedure in a matlab program.
After belief propagation converges (or, after a filed number of iterations), estimates of the marginal probabilities of the shifts given the data are computed, and the most probable val~ie of each shift variable is selected. The resulting configuration of the shifts can then be integrated to obtain the unwrapped surface. If some zero-curl constraints remain violated, a robust integration technique, such as the one described in the remainder of this documents should be used.
3 Unwrapping the Surface In this part of the document we consider the problem of obtaining the unwrapped heights, de-noted by ~Y from the wrapped values ~~ and a probability distribution over the shift variables s.
The idea is to again use an iterative procedure operating on a graphical model depicted in Figure 6. Each variable in a grid of variables denotes on unwrapped height value y(x, y).
Lit .,.r (:r, ;y) denote the difference between ~l(:r, y) and ~n(x -f- 1, ,y), i.e. '~,~y) _ ~'~(x + 1; y) -~;~(.r, y). Similarly-, we define ,,y = y(x, y + 1) - ~n(;n,, y). The defined variables satisfy the obvious relations %~~(x, y) - ~%(x + 1, y) +~si(x, y) = 0 (7) and Jy(x, y) - zJ(x, y + 1) + ~si(x, y) = 0. (g) In order to derive a succinct algorithm, we assume that each variable ~%(x, y) is a random variable distributed according to a Gaussian distribution with mean rn~(x, y) and variance o~~(:r, y). We moreover assume, that ;,~(x, y) is a Gaussian random variable with mean rra~,r (a, ;y) and varlaIlCe Qtr (x, y). Similarly, we assume that Jy (x, y) is a random variable with mean ~n~y (x, y) and variance Q~y (x, y).
These Gaussian distributions can be combined into a joint distributions leading to the as-signment of the following joint. probability desnsity to the random variables ~, 0~ and ~y:
N-i nr-r p(~, ~a~'y) °~ rj rj s(~~(x, ~) - r(x + 1, y) +'~(x, y))s(~y(x~ y) -~(x~ y + 1) + ~(x, ~)) =1 y=i ('~(x~ ?l) - ~n.~,(x,, y))2J
CAP -2Q~(x~ y)2 x-1 J-1 N n/-1 ~ (,,~,(x, ~l) - 'p~~.r (x~ 2/))21 , ) J-a-1 y-1 2Q'~cZ(.L, y) N nI-i 1 ~ (!,y(~r, y) - mJb (x, y))21. 9 eh ) -2Q~ 2(x, y) ( ) V'e can obtain an estimate of the mean and the variance of the variables :~~(x,y) and ~y(x, y) in di$erent ways. V'e exemplify this computation for the variable ,~~(x, y). The computations for _'~~(x, ;~) are analogous. Assume we have two neighboring wrapped values ~(a;, ;~) and ~5(x + 1, y). Given the shift variable a(x, y) we find the relation ~(x + 1, y) -~(:x, y) _ ~~(:z~, y) + a(x, y). Hence we can compute the mean of '~,~(x, y) as nz~z (x, y) _ ~(x + 1~ y) - ~(x, y) - ma(x> y)~

where nad(x, y) denotes the mean of a(x, y).
V'e can estimate the mean of a(x, y) either locally according to a suitable probabilistic model, e.g.
P(a(x, y) _ ~ I ø(x, y), ø(x + 1, y)) ~ exp(- (ø(x + 1, y) 2~~(x, y) + ~)2 ) (10) or we use the probability propagation step and equation 3 for this purpose. We note that Qo is a parameter that should be chosen according to the unwrapping task at hand.
If we choose the local probability model 10 where k is restricted to the a priori equally likely values -l, 0, 1 it is easy to verify that exp(- ~(~(a+i,y)-~(~,y)) ) a~~
I'(~(x, y) _ ~I ø(x, y)~ ø(x + 1, y)) _ exp( 2 0 ) + exp(- (m(~+i,~)o m(~,y)) ) + exp( (d(~+i,;~, o ~(~,y)) ) Za (11) It is straightforward to compute the conditional mean of a.(x, y) as P(a(x, y) = 1~ø(x, ;~), ø(x+
1,'J)) - P(rr(.c, y) _ -1~ø(~L, r), ø(a; + l, y))~
The variance of :~z (x, y) equals the variance of a(x, y) which follows from the equation ø(:~ + 1, y) - ø(x, y) _ ~~(:z, y) + a.(x, y) Hence, we can determine the variance either locally according to equation 10 or we employ again probability propagation and equation 3.
Once va-e know rra~r and o~~r (and hence by similar arguments rn~y and Q~a) we are ready to perform an probabilistic inference algorithm for the graphical model depicted in Figure 6b.
The messages being sent from a variable node ~ are denoted by ;.~~ and the messages from checks to variable nodes ø are denoted by v. The messages are computed according to the rules of probability propagation. In particular for Gaussian networks we can reduce the messages to means and variances. The mean and variance of a message v is computed as:
I (12) w a1, +o~+a~+Q
1 '~'2 "'3 V
z( mwl + mWZ + m~,3 + r~z~, ) 97t" = Q~ z 2 2 2 ~cJ1 ~~":2 ~cJ3 where mw; and Q~= denote mean and variance of the messages wi.
Similarly, the mean and variance of the messages W is computed as ~n~, _ 'm" + m~ ( 14) a ? z ~W = ~" + ~~. (15) In order to perform a probabilistic inference we have to specify a variance Q~(x, y) and a mean ~ia~,~s,,y~. In order to allow a. large variation in ~ we choose Q~(x, ;y) much larger than ~~s, effectively allowing ~(x, y) to range over very large values without penalty. The resulting probability propagation algorithm favours unwrapped surfaces with zJ
values closer to zero.
A second version of belief propagation is obtained by choosing a fixed position (xo, yo) ac-cording to any suitable rule and set ~,u(xo, yo) _ ~(xo> yo) «'hile the corresponding variance is chosen to be a very small number, effectively equaling zero. Hence, ~(xo, yo) is not con-sidered to be a random variable. All other variances Q(x, y), x ~ xo, y ~ -yo can be chosen as a very large number, effectively choosing there as infinity. 't'lre probability propagation procedure takes this position as starting point propagates only messages that.
can be traced back to a message from the variable node representing ~,h(xo, yo).
APPEVDI1 A shows an embodyment of the described procedure in a matlab program.
The particular updating rules used in the setup are in correspondence with prior art and evident from the appendix.
ID

4 Unwrapping the Wrapped Error Let S2 be the unwr apped image obtained by any suitable phase-unwrapping algorithm and in particular the phase unwrapping algorithm of the previous section. If the uwvrapping procedure was successful then the values SZ-~ are all integers. Defining the error component T of an unwrapping procedure as T = S2 - ~modulol we see that we should strive to make T equal to the all zero array.
On the other hand, T itself has the structure of a wrapped surface and we can unwrapp the wrapped values T again in order to obtain a smooth approximation of T. The idea is that the unwrapped surface corresponding to T corresponds to the wrapped surface T.
~~'e construct a sequence of unwrapped surfaces with decreasing rewrapping error according to the following algorithm:
Input: .a two dimensional wrapped image ~~°~; an array ~ t- ~t°>
an unwrapping algorithm .d(~) rendering unwrapped or approximately unwrapped surfaces S2 = rl(~); a maximal number of iterations l.; an array h; a counting variable i <- 0 Iterations: While ~ is not the all zero array and i is less than L
1. C'.ompute S2 = :d(~>).
2. rE-h-S2 3. ~~ ~- (r - ~~°~)modulol i. ~- i. + 1 Output l,.

''Ve emphasize that the unwrapping or approximate unwrapping algorithm ~=1 can employ any of the belief propagation procedures described in the previos sections. In particular, it can use the probability propagation algorithms described in section 2. Also we would like to emphasize that any mixture of unwrapping algorithms is possible. An embodyment of a repeated unwrapping procedure is given in APPEMDI1 B.
Combinations The graphical models presented in Sections 2 and 3 can be used in a combined fashion.
Figure 7a and 71 show the schematic graphical models. In particular, we can propagate messages between the graphical models. Figure 8 shows how the graphical models would be joined together. The variables ~, ::~.L, ~1~ and a satisfy the relation ~(2; +
1, y) - ~(.x, y) _ ~.,.(:r, ,y) + a,(x, y) and the corresponding equation involving %1y It was described in section 3 low the mean and variance of :~~. (or .~~) is computed from the mean and variance of cc. SInnilarly, we can compute a probability message for a from the distribution over ,~~
(respectively ,,~). The message being passed to the site representing a(:~, y) is a vector representing a probability distribution over the values that the shift variable can assume.
6 Further Remarks The 512 x 512 surface shown in Fig. 1C wa,s unwrapped i.ising 10 iterations of belief prop-nga.tion a.c:cording to section 1, taking 2 minutes on a G00 lflHz Pentium III
processor (14).
The finial reconstruction is shown in Fig. 4. All main discontinuities in the wrapped surface are umvrapped and details in the surface are preserved. In contrast, a standard least squares method (10, 11) tends to blur details of the surface.
It is difficult to numerically eval~.~ate the performance of belief propagation, since the "ground truth'' is not known. However, the least. squares method can 1>e viewed as least squares integration, based on greedy decisions for the shifts. For the surface shown in Fig. 1C, greedy decisions for the shifts lead to 281 violated zero-curl constraints. In contrast, 10 iterations of belief propagation reduces the number of violated zero-curl constraints to 28.
To evaluate the algorithm on reconstruction error, we compared it with the standard least iz squares method on a variety of synthetic, wrapped surfaces, with grids ranging in size from 30 x 30 to 1000 x 1000. VVe found that the algorithm always converges to reasonable ap-proximations.
Fig. 5A shows a 30 x 30 surface generated by adding 5 Gaussian surfaces with random positions, widths and heights. Fig. 5B shows the reconstruction error versus wrapping wavelength, for 10 iterations of belief propagation followed by least squares integration and for a standard least sduares method (10, 11). Belief propagation performs significantly better than the least squares method.
Shift belief propagation is a simple algorithm derived from a principled probability model describing the wrapped data. , Several heuristic approaches have properties that can be discerned in the belief propagation algorithm. Branch cut techniques (12) use a heuristic to choose pairs of nearby violated zero-curl constraints that should be connected by changing the values of intermediate shift variables. Initially, belief propagation finds a field of probabilistic connections between pairs. after multiple iterations, belief propagation identifies connections between pairs that are most likely a priori. C~uality-guided methods (1) make use of a user-p rovided map or a precomputed map that, identifies which parts of the wrapped signal can be tr ust.ed when determining shifts using local information. Belief propagation adaptivel5-ident,ifies which parts of the signal can be trusted or not trusted, by producing belief vectors that have all mass on one shift value or that have uniformly distributed mass.
Networlc programming methods (13) use discrete search techniques to find probable configurations of the shifts. Belief propagation keeps track of multiple, plausible configurations for each shift variable and examines multiple compatible configurations simultaneously.
/.3 References (1~ D. C. Ghiglia, l~I. D. Pritt, Two-Dimensional Phase Unwrapping. Theory, Algorithms and Software, (John ~~'iley & Sons, New York, 1998).
(2~ C. ~~%. Chen, H. A. Zebker, Journal of tlae Optical Society of Arnerica A
17:3, 401-414 (2000).
(3~ C. Berrou, A. Glaview, IEEE Transactions on Comnaurzications 44, 1261-1271 (1996).
(4~ D. .1. C. ~ZacKay, IEEE Trarzsactioras on Information Theory 45:2, 399-431 (1999).
(5~ N. V'iberg, H.-.~. Loeliger, R. Koetter, European Transactioras on Telecommunications 6, 513-525 (1995).
(G~ B. .1. Fret', F. R. hschischang, Proceedings of tlse 3=lrh Allerton Confere~ace on Cornmu-nication, Control and Computing 1996, \-Ionticello, IL (199G).
( 7 ) R. .1. VIcEliece, Tlairteenth Co~afererace on Uncer~tairzty in Artificial hatelligerace, Provi-dence, RI (1997).
(8J .1. Pearl, Artificial Intelligence 29, 241--288 (198G).
(9~ 1'. V'eiss, B. Freeman, to appear in Neural Computation (2001).
(10~ S. \I.-H. Song, S. Napel; N. .1. Pelc, G. H. Glover, IEEE Transactions ora Image Pro-cessi~ag 4:5, GG7-G7G (1995).
(11) C. t% . .lako«latz, Jr., D. E. ~Vahl, P. H. Eichel, P. A. Thompson, Spotlight-mode Synthetic Aperture Radar: A Signal Process-ing Approach, (Kluwer Academic Publishers, Boston, l J9G).
(12~ R. \I. Goldstein, H. A. Zehl:er, C. L. «'erner, Radio Science, 23:4, 713-720 (1988).
(13) It'I. Costantini, Proceedings of the Fringe '96 Workshop, ES A SP-=106, Zurich, Switzer-land (1996).
(14~ ~~'e thank Nemanja Petrovic for obtaining the results on the image from Sandia \ational Laboratories.
/~

Claims

1)A method for two dimensional phase unwrapping where each message in a 2 dimensional array of messages is iteratively modified according to messages that are nearby in a suitable topology 1.3) The method of claim 1 with the array extended to three or more dimensions.

1.5) The method of claim 1 wherein the topology is given by a 2 dimensional grid

2) The method of claim 1 wherein the messages represent the confidence in the number of relative wrappings between adjacent observed values.

3) The method of claim 1 wherein the messages represent the mean and variace of the unwrapped observation.

4) The method of claim 2 wherein additional messages are used to represent discontinuities between neighboring unwrapped values.

5) The method of claim 3 wherein additional messages are used to represent discontinuities between neighboring unwrapped values.

6) The method of claim 1 wherein each message represents the confidence in the number of relative wrappings between adjacent observed values and the mean and variace of the unwrapped observation.

7) The method of claim 6 wherein an additional message is used to represent a discontinuty between neighboring unwrapped values.

8) The method of claim 2 wherein the messages are modified according to observations of the wrapped values.

9) The method of claim 8 wherein the messages are modified so that the confidences are increased according to how well the corresponding number of relative wrappings matches the observed values.

10) The method of claim 1 wherein the messages are modified by a subset of the values of neighboring messages, where the subset satisfies the constraint that the sum of relative wrappings is zero.

11) The method of claim 3 wherein the messages are modified according to the wrapped difference of neighboring observations.

12) The method of claim 8 wherein each message is modified according to messages repre-senting discontinuities between neighboring unwrapped values.

13) The method of claim 11 wherin each message is modified according to messages repre-senting discontinuities between neighboring unwrapped values.

14) The method of claim 1 wherein messages are updated in parallel.

15) The method of claim 14 wherein the parellel updates are implemented on a serial device 18) The method of claim 2 wherein the number of relative wrappings between adjacent observed values is chosen to be the value with the highest confidence in the corresponding message.

19) The method of claim 18 wherein the number of absolute wrappings for each observation is determined by summing the number of relative wrappings along any path from a point of known or assumed height.

20) The method of claim 3 wherein the unwrapped observation is chosen to be the mean of the corresponding message.

21) The method of claim 1 wherein the resulting messages are used as input to another phase unwrapping algorithm.

22) The method of claim 8 wherein the way in which the messages are modified is adapted to all or a subset of the observed values.

23) The method of claim 11 wherein the way in which the messages are modified is adapted

16 to all or a subset of the observed values.

24) The method of claim 11 wherein the way in which the messages are modified is adapted to the most recently computed set of all or a subset of messages.

25) A method for 2-dimensional phase unwrapping that obtains an estimato of the unwrapped signal by probabilistic inference in a graphical model.

26) The method of claim 25 wherein the unobserved variables represent the number of relative wrappings between neighboring observations.

27) The method of claim 25 wherein the unobserved variables represent the unwrapped signal values.

28) The method of claim 26 wherein each unobserved variable is extended to include a value that indicates a discontinuity in the unwrapped signal.

29) The method of claim 27 wherein each unobserved variable is extended to include a value that indicates a discontinuity in the unwrapped signal.

30) The method of claim 25 wherein each unobserved variable represents the number of relative wrappings between neighboring observations and the unwrapped signal values.

31) The method of claim 30 wherein each unobserved variable is extended to include a value that indicates a discontinuity in the unwrapped signal.

32) The method of claim 26 wherein the likelihood of the unobserved variables representing the relative number of wrappings is higher if the corresponding unwrapped observations are closer together.

33) The method of claim 26 wherein the local functions (potentials) favour configurations of neighboring variables that satisfy the constraint that the sum of the relative wrappings around a closed loop is zero.

34) The method of claim 33 wherein among the configurations that satisfy the constraint,

17 the local functions favour some configurations over others.

34.3) The method of claim 33 wherein among the configurations that satisfy the constraint, configurations that are closer to planar are favoured.

34.5) The method of claim 33 wherein the local functions favour some configurations over others, depending on whether or not one of the variables indicates a discontinuity.

35) The method of claim 27 wherein the likelihood of the unobserved variables favours those values that, when wrapped, are close to the observed values.

36) The method of claim 35 wherein the local functions favour configurations such that neighboring unobserved variables are closer together.

37) The method of claim 36 wherein the local functions favour some configurations over others depending on whether or not one of the variables indicates a discontinuity.

38) The method of claim 25 wherein the surer-product algorithm is used to perform probailis-tic inference.

38.3) The method of claim 25 wherein belief propagation is used to perform probailistic inference.

38.4) The method of claim 25 wherein probability propagation is used to perform probailistic inference.

39) The method of claim 25 wherein a mean-field algorithm is used to perform probailistic inference.

40) The method of claim 25 wherein a structured variational technique is used to perform probailistic inference.

41)A computer-readable medium comprising computer-executable instructions for perform-ing two dimensional phase unwrapping by creating a 2-dimensional array of messages and where each message in the 2 dimensional array is iteratively modified according to messages

18 that are nearby in a suitable topology.

41.3) The method of claim 41 with the array extended to three or more dimensions.

41.5) The method of claim 41 wherein the topology is given by a 2 dimensional grid 42) The method of claim 41 wherein the messages represent the confidence in the number of relative wrappings between adjacent observed values.

43) The method of claim 41 wherein the messages represent the mean and variace of the unwrapped observation.

44) The method of claim 42 wherein additional messages are used to represent discontinuities between neighboring unwrapped values.

45) The method of claim 43 wherein additional messages are used to represent discontinuities between neighboring unwrapped values.

46) The method of claim 41 wherein each message represents the confidence in the number of relative wrappings between adjacent observed values and the mean and variace of the unwrapped observation.

47) The method of claim 46 wherein an additional message is used to represent a discontinuty between neighboring unwrapped values.

48) The method of claim 42 wherein the messages are modified according to observations of the wrapped values.

49) The method of claim 48 wherein the messages are modified so that the confidences are increased according to how well the corresponding number of relative wrappings matches the observed values.

50) The method of claim 41 wherein the messages are modified by a subset of the values of neighboring messages, where the subset satisfies the constraint that the sum of relative wrappings is zero.

19 51) The method of claim 43 wherein the messages are modified according to the wrapped difference of neighboring observations.

52) The method of claim 48 wherein each message is modified according to messages repre-senting discontinuities between neighboring unwrapped values.

53) The method of claim 51 wherin each message is modified according to messages repre-senting discontinuities between neighboring unwrapped values.

54) The method of claim 43 wherein the unwrapped observation is chosen to be the mean of the corresponding message.