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A Practical Algorithm for the Determination of Phase from Image

and Diffraction Plane Pictures

R. W. Gerchberg and W. O. Saxton, Cavendish Laboratory, Cambridge, England

Received 29 November 1971

Abstract

An algorithm is presented for the rapid solution of thephase of the complete wave function whose intensityin the diffraction and imaging planes of an imagingsystem are known. A proof is given showing thata defined error between the estimated function andthe correct function must decrease as the algorithmiterates. The problem of uniqueness is discussed andresults are presented demonstrating the power of themethod.

Inhalt

Ein praktischer Algorithmus zur Berechnung derPhase aus den Intensitaten in Beugungs- undBildebene. Ein Algorithmus zur schnellen Losungder Aufgabe wird mitgeteilt. Wie eine Probe zeigt,nimmt eine definierte Abweichung zwischen der er-rechneten und der wirkliohen Funktion ab, wenn derAlgorithmus iterativ angewendet wird. Das Problemder Eindeutigkeit wird diskutiert. Ergebnisse demon-strieren die Leistungafahigkeit der Methode.

In 1948, Gabor [1948] proposed an experimentalmethod for determining the phase function acrossa wave front. Basically the method involved theaddition of a reference wave to the wave of inter-est in the recording plane. The resulting hologramrecorded was a series of intensity fringes which con-tained enough information to reconstruct the com-

plete wave function of interest. However, in manyreal situations the method has been cumbersome andimpractical to employ. It is interesting that Gabororiginally proposed the method in connection withelectron microscopy and that even now it has notbeen used profitably in that field.

In subsequent years, methods were proposed for in-ferring the complete wave function in imaging exper-iments from intensity recordings which did not em-ploy reference waves (see for example Hoppe [1970],Schiske [1968], Erickson & Klug [1970]). Thesemethods have involved linear approximations andhence have had validity only in the limit of smallphase and/or amplitude deviations across the wavefront of interest. For the most part, they have alsosuffered from an excess of computation and have notgained wide acceptance.

Gerchberg & Saxton [1971] recently proposedthat in many imaging experiments, intensity record-ings of wave fronts can be made conveniently in boththe imaging and diffraction planes. Sets of quadraticequations were developed which defined the phasefunction across a wave in terms of its intensity in theimage and diffraction planes. This method of anal-ysis was not limited to small phase deviations, butagain it required a large amount of computation.

In this paper we put forward a rapid computationalmethod for determining the complete wave function(amplitudes and phases) from intensity recordings inthe image and diffraction planes. The method de-pends on there being a Fourier Transform relationbetween the waves in these two planes and henceconstrains the degree of temporal and/or spatial co-herency of the wave. However, without a required

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degree of coherency in the experimental situation, itwould be profitless to solve for a phase function any-way. The method should prove to be generally usefulin electron microscopy, and under certain conditionseven in ordinary light photography. It is also feltthat it may have exciting implications for crystallog-raphy where only the X-ray diffraction pattern maybe measured. Thus, if one has a crystal which maybe reasonably inferred to be a phase object to X-rayillumination, its image would be contrastless and thisbit of information plus the X-ray diffraction patternwould be sufficient to solve the “phase problem”. So-phisticated methods for phase determination such asheavy atom replacement could be replaced or at leastsupplemented by the method to be described here.

The algorithm

The basic algorithm is an iterative procedure whichis shown schematically as Fig. 1. The input data tothe algorithm are the amplitudes of the sampled im-age and diffraction plane intensity pictures measured.The amplitudes are proportional to the square rootsof the measured intensities. The two sets of dataare accessed once per complete iteration and hencerequire computer storage.

To begin, a random number generator is used togenerate an array of random numbers between πand −π which serve as the initial estimate of thephases corresponding to the sampled image ampli-tudes. These then are multiplied by the respectivesampled image amplitudes and the Fourier Transformof this synthesized complex discrete function is doneby means of the Fast Fourier Transform (FFT) al-gorithm of Cooley & Tukey [1965]. The phasesof the discrete complex function resulting from thistransformation are calculated and combined with thecorresponding sampled diffraction plane amplitudefunction. This function is then Fourier Transformed,the phases of the sample points computed and com-bined with the sampled image amplitude function toform a new estimate of the complex sampled imageplane function and the process is repeated.

In what follows, we will discuss the problems ofsampling and uniqueness and we will show by a sim-

ple proof that the squared error between the am-plitudes of the discrete functions generated by theFourier Transform operation and the discrete set ofamplitudes derived from the measured intensities inthe corresponding image or diffraction plane must de-crease or at worst remain constant with each iter-ation. We will also display some pertinent resultsachieved on modeled problems. However, at thispoint some general remarks about the algorithm arein order.

To start the algorithm, we use a random numbergenerator to arrive at a set of phase angles from a uni-form distribution density between π and −π. This isnot necessary in every case and indeed there is ev-ery reason to suppose that an educated guess at thecorrect phase distribution would lessen the compu-tation time required for the process to achieve anacceptable squared error. However, one initial phasedistribution which will cause the algorithm to fail,is to have all phases equal to a constant when theintensity pattern in both the image and diffractionplane is centro-symmetric. This is because the phasedistribution will not change under Fourier Transfor-mation in this case. If one studies the algorithm asit is shown in Fig. 1, it may be realized that basi-cally the mechanism relies on the fact that a changein the amplitude distribution alone in one domainof the Fourier Transform will result in changing boththe amplitude and phase distributions in the oppositedomain.

Figure 1: Schematic drawing of phase determiningalgorithm

The fact that the algorithm is rapid computation-

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ally rests on using the Fast Fourier Transform (FFT)algorithm of Cooley and Tukey to do the transforma-tions. This algorithm has reduced the time requiredto compute finite Fourier transforms by the fraction(log2N)/N , where N is the number of points under-going transformation. To give some idea of the timerequired to solve the phase problem on a large mod-ern digital computer (ICL-ATLAS 2), we have beenable to solve for the phase of a picture on a 32 by 32grid (1024 points) in under 80 seconds. Thirty sevenFourier transforms were required during this time toreduce the squared error from 73.0 to 0.01.

The computer storage required depends of courseon the density of points necessary for adequate sam-pling of the picture. In the example just given, 2048words of storage were needed, to store the functiongenerated by the FFT block (see Fig. 1) because thenumbers are complex. Another 2048 words were re-quired to store the reference amplitudes in the imageand diffraction planes. Therefore, the total requiredstorage was 4096 words.

The Uniqueness Problem

The information that is used to solve the phase prob-lem is the measured intensity distribution in boththe image and diffraction planes of the image form-ing system. But to start, it is clear that the solutionon this basis will not be unique. By adding a constantbut arbitrary phase to any function whose intensityin the image and diffraction planes is the same asthe measured intensities, we generate a new functionwhose intensities in both planes will also be the sameas those which were measured. Thus only relativephases of the solution functions are meaningful. An-other case of inherent ambiguity occurs when bothintensity distributions (diffraction and image planes)are centro-symmetric. The complex conjugate func-tion to any solution found in this case will also be apossible solution. There may be more inherent am-biguities but thus far these are the only two that wehave encountered in the trials we have completed. Inmost applications, these ambiguities are tolerable.

Proof that the Algorithm ErrorMust Decrease

We define the squared error as the sum of the squareddifferences between the amplitudes of points in eitherthe image or diffraction plane (based on intensitymeasurements) and the amplitudes of those pointscalculated by the FFT in our algorithm (see Fig. 1).When the squared error is zero, we have found thecorrect phases of the points in both planes and wehave a solution to the phase problem. Referring toFig. 1, the energy (defined as the sum of the squaredamplitudes of the finite discrete function) of the func-tion which undergoes transformation is the same foreither image or diffraction plane data. This is a con-sequence of Parseval’s theorem and the fact that theimage function is the Fourier Transform of the diffrac-tion plane function. Also by Parseval’s theorem, theenergies of the two functions immediately on eitherside of the FFT block in Fig. 1 are equal. Now con-sider the operation of the algorithm at two samplepoints, one from the diffraction plane and one fromthe image plane. The situation is shown in Fig. 2.The reference amplitudes of the two points, basedon the measured data are t1 and t2 The FFT of thediffraction plane function estimate (immediately af-ter the FFT block in Fig. 1) yields the vector g1 atthe image plane point. The algorithm corrects thisvector in amplitude but retains its phase by addingthe vector c1 to g1 to yield h1 the new image pointestimate. All points in the image plane are similarlycorrected and the function is transformed to yield h2in the diffraction plane. Fig. 2b shows an arbitrarypoint in the diffraction plane. The h2 vector is thesum of g2 and c2 (the transforms of g1 and c1 re-spectively). By Parseval’s theorem the sum of thesquared amplitudes of c1 in domain 1 must equal thesum of the squared amplitudes c2 in domain 2. But :∑

all points

|c1|2 , squared error

and from Fig. 2 it is clear that the correction vectord2 at each point must be less than or at most equalto c2 in amplitude. Hence the squared error mustdecrease or remain constant with each pass through

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the FFT in the algorithm.

Figure 2: The action of the algorithm on a) an ar-bitrary point in the image plane and b) an arbitrarypoint in the diffraction plane. “t1” (“t2”) is the mea-sured amplitude at the arbitrary image (diffraction)plane point. After FFT g1 is the vector at point in“a”. “c1”is correction vector added. “h1”is new esti-mate. “h1”is then transformed to yield “b”. In “b”d2is correction vector added to h2 to form new estimatewhich then transformed to complete one iteration orloop of algorithm.

The Squared Error Limit

It is worthwhile to study and understand the actionof the algorithm through circle point figures like thosein Fig. 2. One thing that becomes clear is that fora finite limit to the squared error to exist, the algo-rithm must approach or reach a situation where thecorrection function (c) is colinear with the estimated

function (hh) in both domains 1 and 2 (see Fig. 2).

Thus the estimate (h) and the correction (c) wouldhave the same phase functions in both domains 1 and2 at every point. This is a difficult condition to im-pose on two distinct functions and yet it has beenachieved in every instance where we have forced thealgorithm to fail. The squared error appeared to ap-proach a finite limit other than zero at the same timethat the phase function of the estimate appeared toapproach a limiting function as well. There may be

some characteristic way that the estimate approachesthis limiting condition. We have not examined thisquestion closely but we can say that in the trials wehave run, no obvious characteristic stands out. Thisis plain in the example of an incorrect function gen-erated by the algorithm and displayed as Fig. 3.

Sampling Considerations

The only way that the algorithm has been forced tofail thus far is by inadequate sampling in either ofthe two domains. If a continuous function is sam-pled at a rate which is too “slow”to adequately por-tray its spectrum, the algorithm fails. At this point,we would not attempt to set minimum sampling ratelimits. The problem stems from the fact that sam-pling occurs in both domains and the duration of thefunction must be infinite in at least one domain. Atleast one of the domains will present the functionin a distorted way. The severity of this effect ap-pears to vary with the function. It was a surpriseto find that in the case of the “chirp function”(seeFig. 3) rect(2t) exp(i30πt2), the sampling incrementt = 1/128, was too large. That sampling incre-ment corresponded to a Shannon-Whittaker samplingbandwidth containing better than 99% of the func-tion energy. This has not been the case for every testfunction we have run. However, in every instance ofour testing, the algorithm has succeeded in achievinga valid solution with the square error limit appearingto be zero when the function sampling was taken ata sufficiently high rate.

Typical Test Results

Fig. 3 shows the FFT of the functionrect(2t) exp(i30πt2) and the incorrect answerthat the algorithm appeared to limit on. Fig. 3acompares the amplitudes of the two functions andFig. 3b compares the phases. For purposes of illus-tration, only the results on this family of functionsare given. Aside from this trial, which failed becauseof improper sampling, the other results which will bediscussed yielded curves which were valid solutions of

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the phase problem. The history of this test leadingto the results of Fig. 3 is shown in Table 3.

Figure 3: Point plot of a) amplitude and b) phases ofFFT of sampled function rect(2t) exp(i30πt2). Theincorrect solution found by the algorithm is com-pared to the correct solution. Details of this trialare recorded in Table 1.

Table 2 shows the trial history of the same func-tion but this time the sampling increment was halved.The squared error was recorded only to three decimalplaces and went from a normalized value of 0.538 to0.000 in approximately 65 loops of the algorithm asshown in Fig. 1. The amplitude of the function foundwas of course virtually identical to the correct one.Except for a constant phase shift and the fact thatthe function was complex conjugate to the one usedto create the image, it was virtually perfect. It wasnoted previously that functions which, along withtheir transforms, have centro-symmetric amplitudedistributions as this class does, can produce valid so-lution functions which are complex conjugates.

Table 3 shows the history of a solution run

Table 1:

Table 2:

on a problem with slightly incompatible picturesin the diffraction and image planes. The algo-rithm was used on the amplitudes of the functionrect(2t) exp(i15πt2). The amplitudes of the FFT of

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this function were modified by suppressing all thoseamplitudes less than 1/30th the maximum. Thus asituation was simulated in which the dynamic rangeof the recording medium was in the neighborhood of30 dB limited by noise at the low end. The resultswere very good indeed. The error decreased from0.639 to 0.003 in 31 loops of the algorithm. The en-ergy of the function in the two domains is not thesame and the error limits at this value. The phasesof the points in the solution found are quite closelycorrect (±2◦) for corresponding amplitudes greaterthan 10% the maximum. The phase error becomesas high as (±10◦) at some points of lesser amplitude.

Table 3:

Perusal of the solution histories indicates that theprogress of the algorithm toward a solution becomesless rapid as the error decreases. In this respect theseresults are not characteristic of all the trials we havemade. Some functions have actually moved towardzero squared error at a geometric rate with a factorof around 0.5. R. W. Gerchberg acknowledges thesupport of the United States National Institutes ofHealth by way of Fellowship NIH (1 F03 GM49953-01). W.O. Saxton acknowledges the support of the

Science Research Council.

References[1] J. W. Cooley & J. W. Tukey, Mathematicsof Computation 19 (1965) 297.[2] H. Erickson and A. Klug, Berichte derBunsen-Gesellschaft 74 (1970) 1129.[3] D. Gabor, Nature 161 (1948) 777.[4] R. Gerchberg & W. Saxton, Optik 34 (1971)[5] W. Hoppe, Acta Cryst. A 26 (1970) 414.[6] P. Schiske, 4th European Conference on Elec-tron Microscopy Rome (1968).

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