720 J. Opt. Soc. Am. A/Vol. 4, No. 4/April 1987 0. Sasaki and T. Yamagami
Phase-retrieval algorithms for nonnegative and finite-extentobjects
Osami Sasaki and Taku Yamagami
Faculty of Engineering, Niigata University, Niigata, Japan
Received July 2, 1985; accepted December, 1, 1986We propose new phase-retrieval algorithms that utilize the nonnegativity and the finite extent of objects. Thesquared error in the Fourier domain is minimized with an improved conjugate-gradient method that uses the Kuhn-Tucker theorem to estimate the nonnegative object. The squared error outside the finite support of the object isminimized to estimate the phase of the Fourier transform of the object. This phase-estimation method is combinedwith the improved conjugate-gradient method. The effectiveness of these phase-retrieval algorithms is clearlyshown through computer simulations.
1. INTRODUCTION
The problem of phase retrieval from the modulus of a Fouri-er transform has been discussed often in recent literature.Fienup1 described the method of steepest descent and theconjugate-gradient method with which the squared error inthe Fourier domain is minimized to reconstruct the object byusing the constraints about the object. By using the Kuhn-Tucker theorem, the usual conjugate-gradient method canbe modified for nonnegative signals. 2
In this paper we first apply the improved conjugate-gradi-ent method to the phase-retrieval problem for nonnegativeobjects. In the methods employing error reduction in theFourier domain, the unknown is the object. In the methodbased on the sampling theorem, the phase of the Fouriertransform is taken to be the unknown. Bates,3 Garden andBates,4 Fright and Bates,5 and Bates and Fright6 proposedan algorithm based on the sampling theorem, assuming thatthe value of the Fourier transform at any point was interpo-lated only with two independent values at the adjacent sam-ple points. Arsenault and Chalasinska-Macukow7 and Cha-lasinska-Macukow and Arsenault8 solved the system of non-linear equations obtained from the sampling theorem withthe Newton-Raphson iterative method to estimate thephase of the Fourier transform. In this method, initialphase values close to the solution are required, and thenonnegative constraint of the object is not incorporated.
Next, we estimate the phase of the Fourier transform byminimizing the squared error in the region outside the sup-port of the object with the method of steepest descent. Thisphase-estimation method by error reduction in the objectdomain also uses the sampling theorem in a different way.
Finally, we combine the improved conjugate-gradientmethod with the phase-estimation method.
These new phase-retrieval methods are compared throughcomputer simulations for two-dimensional objects. Nu-merical analysis shows the following points: (1) The im-proved conjugate-gradient method using the Kuhn-Tuckertheorem converges much faster than the usual conjugate-gradient method. (2) Te phase-estimation method is ef-fective for the sampled data of the Fourier modulus, whichproduce a great number of object unknowns. (3) The com-
bined method is the best phase-retrieval method since it isinsensitive to the method used to sample the Fourier modu-lus.
2. ERROR REDUCTION IN THE FOURIERDOMAIN
A. PreliminariesConsider an object that is confined to a square region:
(X, y) = If(x, y), xi, lyl < D/2 (1)
A diffraction-limited optical system yields a band-limitedFourier transform F(u, v) of this object. Let IF(m, n) lbe thesampled modulus at the sample point (mb, nb) (m, n = 0, 1,... , N - 1). The estimate of the object at sample point (a,la) is denoted byg(k, 1), and its discrete Fourier transform isgiven by
N-1G(m, n) = E g(k, ) exp[-j27r(mk + nl)/N]. (2)
k,1=o
We get the following system of nonlinear equations for theunknowns g(k, 1):
IF(m, n)l- G(m, n)l = 0, m,n=0,1,...,N-1. (3)
To solve Eq. (3), we minimize the objective functionN-1
B = N-2 A, [IF(m, n)l - G(m, n)1]2,m,n=O
(4)
which is the squared error in the Fourier domain. Thenumber of the unknowns g(k, ) is (M + 1)2 in the conditionof Da = M.
B. Conjugate-Gradient MethodTo minimize Eq. (4), the conjugate gradient method is used.Here we review the method described by Fienup,' Thepartial derivative of B with respect to g(k, ) is given by
tions of the conjugate-gradient search on the subspace S, wemodify the sets for g(r+i)(k, ) as follows:
g(r+i)(k, 1) = 0 and aBk 1(r+i) > o,
g(r+i)(k, 1) > 0 or aBk l(r+i) < 0,
(k, ) -P,.
Fig. 1. The improved conjugate-gradient method.
where
g'(k, ) = Jr-[IF(m, n)IG(m, n)/IG(m, n)lI (6)
and 9-1 denotes the inverse discrete Fourier transform.The procedure to obtain g'(k, ) in the Fourier domain isidentical to that in the Gerchberg-Saxton algorithm. Theconjugate gradient direction at the rth iteration is given by
where aBk,l(r) is the value of the partial derivative at the rthiteration and D(°)(k, ) = -Bk,1( 0°) We add a factor of cmultiplying the first term on the right-hand side of Eq. (7)that is arbitrary in the condition D(°)(k, ) = -cBkI(°). InRef. 1, c = 1/2. We seek a better point g"(k, ) along thisconjugate-gradient direction as follows:
gl(r)(k 1) = g(r)(k, 1) + dD(r)(k, 1). (8)
The linear search to find the optimum value dr can be made,for example, by the golden rule method. In Ref. 1 no linearsearch was made for the optimum step size. Only the pixelswithin the support are modified by Eq. (8). A nonnegativeconstraint on the object domain is imposed on g"(k, ) tomake a new estimate as a final step in one iteration asfollows:
C. Improved Conjugate-Gradient MethodWhen the object has a nonnegative constraint, the Kuhn-Tucker theorem gives a characterization of the optimal solu-tion g(k, ) as
g(k, ) > 0, aBkl = 0 for (k, ) e S,
g(k, ) = 0, aBkl > 0 for (k, 1) e P, (10)
ISUP=Q, SnP=,
where the space Q contains all the sample points (k, ) of theunknowns g(k, 1). The improved conjugate-gradient meth-od is illustrated in Fig. 1. For the initial value g(°)(k, 1), all(k, ) belong to the subspace S, and the subspace P is empty.The conjugate-gradient search described in Subsection 2.Bis executed i times for g(r)(k, ) indexed in S. After i itera-
All the sample points where the characteristic of the optimalsolution corresponding to the subspace P is not satisfiedbelong to the subspace S. r + i is replaced with r; then onthe new subspace S the conjugate-gradient search is per-formed so the estimated subspace S will coincide with thesubspace S given by Eqs. (10). The procedure describedabove is repeated again until the value of the objective func-tion reaches a specified small value. The number of itera-tions of the conjugate gradient i is usually taken to be muchsmaller than the number of unknowns.
3. ERROR REDUCTION IN THE OBJECTDOMAIN
A. Phase-Estimation MethodThe phase of the Fourier transform is taken to be the un-known. Denoting the phase of F(m, n) by 0(m, n), we have
The object has a finite support in the space Q and has zerovalues in the space Q as follows:
g(k, ) = 0, (k, 1) Q. (13)
This equation can be considered to be the system of nonlin-ear equations with the unknowns 0(m, n). If the number ofequations in Eq. (13) is more than the number of unknowns,phase retrieval is achieved by solving Eq. (13). In order tosatisfy this condition, the modulus IF(m, n)l must be sam-pled with the interval b < 1/D. Therefore this method alsoutilizes the sampling theorem in the form of Eq. (13). Tosolve Eq. (3) we minimize the following objective functionwith the method of steepest descent:
B' = (1/N) [ [g(k, 1]2, (14)
(k)e Q
where N, is the number of elements of Q.The phase-estimation method by error reduction in the
object domain is illustrated in Fig. 2. By defining
g(r)(k, 1),(k, ) E Q
(k, ) Q(15)
the derivative of B' with respect to 0(r)(m, n) estimated at therth iteration is given by
where R + jI is the Fourier transform of h(r)(k, 1). Thederivation of Eq. (16) is described in Appendix A. Theoptimal estimate 0k(r)(m, n) is found along this direction ofsteepest descent as follows:
Or) = ¢ (r) - dr[ B//&k(r)]. (17)
0. Sasaki and T. Yamagami
722 J. Opt. Soc. Am. A/Vol. 4, No. 4/April 1987
Object Constraintsr+l) Nonnegativity
Finite Supportg(r+1)
Fig. 2. The phase-estimation method.
We calculate g(r)(k, 1) from IF(m, n)IexpLj0(r)(m, n)] and getthe new object g(r+l)(k, ) by imposing the object constraintsof nonnegativity and finite support on g(r)(k, ) as follows:
g(r+l)(k 1) - g(r) g(r) 0 and ( 1) - (18)10, g(r) <0 or (k,l1) eQ
The next estimate, (r+l)(m, n), is the phase of G(r+l) thatis the Fourier transform of g(r+l). Replacing r + 1 with r, theprocedure described above is repeated until the value of theobjective function reaches a specified small value.
B. Combination with the Method of Error Reduction inthe Fourier DomainThe methods of error reduction in the Fourier domain esti-mate the values of the object within the finite support.They ignore the values of the object outside the finite sup-port that are reduced to zero by the phase-estimation meth-od. Therefore a combination of the methods of error reduc-tion in the object and Fourier domains would provide thebest phase-retrieval method.
One iteration of the phase-estimation method by errorreduction in the object domain is performed, and the newestimate, g(k, 1), given by Eq. (18), is obtained. After thesets S and P are established for g(k, 1), one iteration of theconjugate-gradient search is performed on the set S. A newestimate, K(m, n), is obtained from g(k, 1), which is estimatedwith the improved conjugate-gradient method of i = 1.Thus the phase-estimation method and the improved conju-gate-gradient method are performed alternately.
4. NUMERICAL RESULTS
We designate the methods described in Sections 2 and 3Methods I-IV:
Method I is the conjugate-gradient method (Subsection2.B).
Method II is the improved conjugate-gradient method(Subsection 2.C).
Method III is the phase-estimation method (Subsection3.A).
Method IV is a combination of Methods II and III (Sub-section 3.B).
We compare phase retrieval by these four methods for twodifferent images that are situated to a square, each of whosesides is 16a. The images, A and B, of a 32 X 32 point gridcalculated through interpolation are shown in Fig. 3. Thedimensions of the finite supports of the images are D = 8a
(a)
(b)Fig. 3. (a) Image A and (b) Image B with 16 X 16 samples. Thevalues on the 32 X 32 point grid are calculated through interpola-tion.
0. Sasaki and T. Yamagami
Vol. 4, No. 4/April 1987/J. Opt. Soc. Am. A 723
-210
-3 1 0 \
10
-5 10
-61 0
Fig. 5. Values of theiterations for image A.
I
------- II
~~IV
20 40iterations k
60
objective functions versus the number of
~~~(b)Fig. 4. The moduli of Fourier transforms of (a) image A and (b)image B.
Table 1. Numbers of Unknowns NU and EquationsNE for Images A and B
Image A Image BMethod NU NE NU NE
I and II 64 256 100 256III 70 192 54 156
and D = 10a, respectively. The sampling interval in Fourierdomain is 1/16a. Hence the moduli of the Fourier trans-forms of images A and B are sampled at intervals of 1/2D and5/8D, respectively, on a 16 X 16 point grid, as shown in Fig. 4.For image A, the finite support of the modulus is nearlyequal to a 16 X 16 sample grid. On the other hand, for imageB, it is smaller than a 16 X 16 sample grid, and the zerosampled values are added.
First, it is assumed that the finite support of the image isknown exactly beforehand. In Methods I and II, the num-ber of unknowns of the image B is larger than that of theimage A. Table 1 gives the number of unknowns, NU, andthe number of equations, NE, in the various methods. In
-21 0
-410
-510
-61 0
Fig. 6. Values of theiterations for image B.
20 40iterations k
objective functions versus
60
the number of
(a)
0. Sasaki and T. Yamagami
724 J. Opt. Soc. Am. A/Vol. 4, No. 4/April 1987
the direction of steepest descent. This linear search is per-formed in the two steps: As the first step, the region thatcontains a minimum value of the objective function is soughtby calculating its values with a fixed interval along the
Table 2. Numbers of Unknowns NUNE for Image A'
and Equations
Method NU NE
I and II 169 256III 70 87
12
Fig. 7. Image A reconstructed by the improved conjugate method.
1 6
Fig. 9. Values of theiterations for image A'.
20 40iterations k
objective functions versus
60
the number of
Fig. 8. Image B reconstructed by the combination method.
Methods I and II the sample data for image A in which thefinite support of the Fourier modulus is nearly equal to the16 X 16 sample grid are preferred. The sampled data forimage B, which have the zero sampled values outside thefinite support of the modulus, are adopted for Method IIIsince the number of the unknowns in image B is smaller thanin image A when the unknown is the phase of the Fouriertransform.
Figures 5 and 6 show the values of the objective functionversus the number of iterations for the various methods.The number of iterations k is the number of executions ofthe conjugate-gradient search or the steepest-descentsearch. Hence in Method II the value of k increases by avalue of i per iteration, where i is taken equal to 3. InMethod IV the value of k increases by 2 per iteration. Inthese four methods, the dominant computation burden isthe linear search along the conjugate-gradient direction and Fig. 10. Image A' reconstructed by the combination method.
I
II
IV
0. Sasaki and T. Yamagami
Vol. 4, No. 4/April 1987/J. Opt. Soc. Am. A 725
-21 0
-51 0
-61 0
I , I
20 40iterations k
Fig. 11. Values of the objective functions versuiterations for the image A' at 30-dB signal-to-noise
tion. The results, shown in Figs. 5 and 6, make the followingpoints clear: (1) The improved conjugate-gradient methoddecreases the value of the objective function more rapidly
------- II than the usual conjugate-gradient method does. This dif--- -- - -ference in the convergence speed becomes greater as the
number of unknowns increases. (2) The phase-estimationIV method by error reduction in the object domain is useful
when the sampled data consist of the values of the Fouriermodulus within the finite support and the zero values areadded. (3) The combination method is the best phase-retrieval method since it is not influenced by the methodused to sample the Fourier modulus. Image A reconstruct-ed by the improved conjugate-gradient method and image Breconstructed by the combination method are shown in Figs.7 and 8, respectively.
Next it is assumed that the finite support of the object isnot known beforehand. So we take D = 13a for image A.For this image, A', the numbers of unknowns and of equa-tions for the various methods are listed in Table 2. Thenumber of unknowns in Methods I and II increases greatly,and the number of equations in Method III decreases. Thevalues of the objective function versus the number of itera-tions are shown in Fig. 9. The combination method gives
60 the best result. The image reconstructed by the combina-tion method is shown in Fig. 10.
Finally, the noise is added to the modulus of Fouriers the number of transform, and the sampled data at a 30-dB signal-to-noiseratio. ratio for image A' are formed. Figure 11 shows the values of
the objective function versus the number of iterations. Theimage reconstructed by the combination method is shown inFig. 12.
CONCLUSIONS
In this paper we consider the two different squared errorsthat should be reduced for phase retrieval. Error reductionin the Fourier domain is carried out with the improvedconjugate-gradient method using the Kuhn-Tucker theo-rem, which utilizes the nonnegativity of the object. Thismethod converges much faster than the usual conjugate-gradient method. Error reduction in the object domain forthe finite extent object yields the estimation of the phase ofthe Fourier transform of the object. This phase-estimationmethod can basically be considered to utilize the samplingtheorem. The combination method by error-reduction inboth the Fourier domain and the object domain provides thebest phase retrieval.
Fig. 12. Image A' reconstructed by the combination method at 30-dB signal-to-noise ratio.
search direction. As the second step, within this region apoint where the objective function has a minimum value issought by the golden rule method. The linear search inMethods I-IV requires Fourier transforms of the order of10-20 to calculate the values of the objective function atvarious points. For example, in Method III, 379 Fouriertransforms were performed during 20 interations. There-fore the amount of computation per iteration is similar forthe four methods compared.
The initial values are obtained by generating the values ofthe phase of the Fourier transform with a uniform distribu-
APPENDIX A
Using Eq. (12) and the relation
F[mod(N - m, N), mod(N - n, N)] = F*(m, n),(Al)
we have
ag2(k, l)/ap(m, n) = -(4/N 2 )g(k, )IF(m, n)l
X ImlexpUO(m, n)] Vmk+nll,
(A2)
where W = exp(j2ir/N) and Ia + jb} = b.
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726 J. Opt. Soc. Am. A/Vol. 4, No. 4/April 1987
From Eq. (A2), the derivation of B' with respect to 0(m, n)is given by
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0. Sasaki and T. Yamagami
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