Lohmann et al. Vol. 14, No. 11 /November 1997 /J. Opt. Soc. Am. A 2901

Significance of phase and amplitude inthe Fourier domain

Adolf W. Lohmann,* David Mendlovic, and Gal Shabtay

Department of Physical Electronics, Faculty of Engineering, Tel Aviv University, 69978 Tel Aviv, Israel

Received March 5, 1997; revised manuscript received May 20, 1997; accepted May 22, 1997

We add new thoughts and aspects to the importance of phase and amplitude in the Fourier domain. We showhow very similar objects react radically differently if, in the Fourier domain, either the phase was lost com-pletely or the amplitude was modified to be constant. We also discuss the great influence of symmetry on therelative significance of the Fourier amplitude and of the Fourier phase. We show how changing the value ofone pixel in some objects completely changes the significance of the Fourier phase and amplitude. © 1997Optical Society of America [S0740-3232(97)00111-7]

1. DEFINITION OF THE PROBLEMIt is a widespread belief that the Fourier phase is moreimportant than the Fourier amplitude. Even if one ig-nores the Fourier amplitude, it is still supposedly possibleto recover a fairly good image from the Fourier phasealone. This belief was established several decades ago byKozma and Kelly1 and others. The synthetic-apertureradar community even named the radar signal they re-ceived ‘‘phase history,’’ because they had only to preservethe phase accurately. One of the most famous papers re-garding the importance of the Fourier phase was writtenby Oppenheim and Lim.2 These authors used careful for-mulations when arguing about the validity of their find-ings, but the community largely adopted the plain viewthat the phase is more important than the amplitude.

Our aim is to shed some new light on this proposition.In Section 2 we will define quantitatively how one can an-swer the question of the relative significance of phase andamplitude. In Section 3 we will briefly review some casesthat support the proposition. In Section 4 we mention animportant counterexample. We do not have a completeanswer to the question of which is more important, theFourier phase or the Fourier amplitude. But we willpresent some numerical examples that indicate that thesymmetry properties of the object have some effect. Also,it matters whether the object is real valued or somehowcomplex. A real-valued object causes the Fourier trans-form to be hermite symmetrical. Section 5 is dedicatedto symmetrical objects and Section 6 is dedicated to asym-metrical objects. Some conclusions are drawn in Section7.

2. QUANTITATIVE DESCRIPTION OF THEPROBLEMThe object, which may be complex, is defined as

u~x ! 5 uu~x !uexp@if~x !#. (1)

The associated spatial frequency spectrum is

0740-3232/97/112901-04$10.00 ©

u~n! 5 E u~x !exp~2i2pnx !dx, (2)

u~n! 5 A~n!exp@ia~n!#. (3)

The question is, what is more important, A(n) or a (n)?One way to answer may be called test in isolation. Forexample, the phase factor exp@ia (n)# is ignored. Hence,the remainder of the frequency spectrum is

A~n! 5 vA~n!. (4)

The associated image is

vA~x ! 5 E vA~n!exp~i2pnx !dn. (5)

The other kind of isolation yields

exp@ia~n!#rectS n

Dn D 5 vF~n!. (6)

The rect function confines vF(n) into the bandwidth of theFourier amplitude A(n). The phase-only output is

vF~x ! 5 E vF~n!exp~i2pnx !dn. (7)

In many cases the image qualities of both vA(x) and vF(x)will be obvious, either quite good or quite poor.

To make a more general statement, one might computeensemble averages of normalized correlations:

CA 5

E vA~x !u* ~x !dx

F E uvA~x !u2dx G1/2F E uu~x !u2dx G1/2 , (8)

CF 5

E vF~x !u* ~x !dx

F E uvF~x !u2dx G1/2F E uu~x !u2dx G1/2 . (9)

One can pursue this correlation approach easily by invok-ing the Parseval theorem, which converts those integralsinto the Fourier domain:

1997 Optical Society of America

2902 J. Opt. Soc. Am. A/Vol. 14, No. 11 /November 1997 Lohmann et al.

E v~n!u* ~n!dx

F E uv~n!u2dx G1/2F E uu~n!u2dx G1/2 . (10)

This makes sense because the isolations are introduced inthe spatial-frequency domain. However, we will not con-tinue in that direction because it is difficult to define asensible joint probability for A(n) and a (n). These twoquantities seem to be coupled.

Another approach might be called the test of robust-ness. For example, we falsify the Fourier phase a (n) butkeep the Fourier amplitude intact:

a~n! → a~n!

A~n! → A~n!

uA~n! 5 A~n!exp@ia~n!#. (11)

Similarly, we may test the robustness of the Fourier am-plitude:

a~n! → a~n!

A~n! → A~n!

uF~n! 5 A~n!exp@ia~n!#. (12)

The question then is to find out whether uA(x) or uF(x) isbetter. Both kinds of deterioration do occur in practice,especially during recording of a Fourier hologram. Anexample of photographic nonlinearity is shown in Eq.(12). Fringe distortion due to a lens that suffers fromgeometrical distortion is described by Eq. (11). An un-even substrate of the hologram will also cause a defect ofthe type shown by Eq. (11).

3. HISTORICAL CASES OF PHASEDOMINANCEKozma and Kelly, around 30 years ago, made a Fourierhologram that served as a matched filter.1 Their photo-graphic development process caused ‘‘hardclipping,’’which made the hologram almost binary. Hence the ef-fective filter amplitude was practically uniform across thefrequency domain. The experiment was quite successful,in spite of the loss of the Fourier amplitude.

The kinoform3 is another example that is directly re-lated to optics in which it is assumed that the phase ismuch more important than the amplitude. Furthermore,a hologram always contains a complex transmissionA(n)exp@1ia (n)# and also A(n)exp@2ia (n)#. The impactsof these two terms can be separated if the spatial carrierfrequency is high enough. There exist projects such asimage deblurring and code translation in which both com-plex transmission terms are needed.4,5 For example, ifone wishes to convert an object u1(x) into an image u2(x),one needs a spatial filter such as

u2~n!

u1~n!5

A2~n!

A1~n!exp$i@a2~n! 2 a1~n!#%. (13)

For doing the opposite, the filter should be

u1~n!

u2~n!5

A1~n!

A2~n!exp$2i@a2~n! 2 a1~n!#%. (14)

The two filter phases are conjugates and hence perfectlypresent in a Fourier hologram. But the two requests fora proper filter amplitude are apparently incompatible.Various compromises were conceivable and were easy toincorporate into a computer hologram.5 The results werequite good. It was possible, for example, to transformEinstein into Galilei, and vice versa, with only onecomputer-generated hologram filter.

Some other projects tried to give more quantitativemeaning to the importance of phase with respect to am-plitude. For example, Pearlman and Gray showedthrough rate-distortion theory that the phase is approxi-mately three times more informative than the amplitudefor data compression in the discrete-Fourier-transformdomain.6 Powers and Goodman showed that one shoulddedicate more bits to the phase rather than the amplitudewhen quantization is introduced.7

4. CASE OF AMPLITUDE SIGNIFICANCEThere exist situations where only the power spectrumuu(n)u2 5 A2(n) is accessible. Fraunhofer diffraction andsecond-order speckle interferometry are examples.Fienup and others8 showed how one can recover a fairlygood image with the help of the well-known Gerchberg–Saxton algorithm and modifications thereof. An impor-tant piece of a priori information is required: The objectshould be known to be real valued and nonnegative. But

Fig. 1. Centrosymmetric object.

Fig. 2. Reconstructions from a centrosymmetric amplitude-onlyobject: peak phase zero amplitude-only reconstruction (upperleft), peak phase zero phase-only reconstruction (upper right),peak phase 90° amplitude-only reconstruction (lower left), peakphase 90° phase-only reconstruction (lower right).

Lohmann et al. Vol. 14, No. 11 /November 1997 /J. Opt. Soc. Am. A 2903

Oppenheim and Lim implied the same a priori knowl-edge, which is often overlooked.

5. ROLE OF OBJECT SYMMETRYOur standard object is shown in Fig. 1. It is centrosym-metric (we shall continue with one-dimensional notationfor the sake of simplicity):

u~x ! 5 u~x 2 x0! 1 u~2x 2 x0! 1 2A0d~x !. (15)

The strength 2A0 of the central peak is as large as2A(0) 5 2uu(0)u. We performed a set of 2 * 3 simula-tion experiments that differed because of phase assign-ments in the object domain:

u~x ! 5 uu~x !uexp@if~x !#. (16)

In the first two experiments the object phase was zero:f(x) 5 0. The difference consisted of a 90° phase shift ofthe central peak in the object domain. We show theamplitude-only and phase-only outputs vA(x) and vF(x)in Fig. 2. When the peak phase is zero, one can see thatthe amplitude-only output yields the desired result,whereas the phase-only output has no information in it.The opposite result is obtained when the peak phase isshifted. The results can be understood if one considershow the two Fourier transforms occupy the complex plane(Fig. 3).

Fig. 3. Complex Fourier domain of symmetrical objects: phasepeak 5 0, phase peak 5 90°.

Fig. 4. Reconstructions from an object with centrosymmetricamplitude and antisymmetric random phase: peak phase zeroamplitude-only reconstruction (upper left), peak phase zerophase-only reconstruction (upper right), peak phase 90°amplitude-only reconstruction (lower left), peak phase 90° phase-only reconstruction (lower right).

The double object is real and symmetrical. The sameis true for its Fourier transform, which is added to thepeak contribution. In case (a) all complex Fourier ampli-tude values stay on the real axis. Hence the overall Fou-rier phase is zero everywhere. That changes after thepeak phase has been rotated by 90° (Fig. 3b).

Next, we assign a random phase f(x 2 x0) to u(x2 x0) and 2f(2x 2 x0) to u(2x 2 x0). The phase dis-tribution is antisymmetric. The results of this experi-ment are depicted in Fig. 4. In the next experiment therandom phases in the two parts were unrelated. Theamplitude-only and phase-only outputs are presented inFig. 5. Here we can see that even though a randomphase was inserted into the object, we still obtain thesame results.

Perhaps the most dramatic result is that a phase shiftby 90° in a single pixel, such as the peak at x 5 0, can tiltthe balance between phase only and amplitude only com-pletely.

6. ASYMMETRIC OBJECTNow the object contains only one symbol besides the cen-tral peak (Fig. 6):

u~x ! 5 u~x 2 x0! 1 A0d~x !. (17)

We performed a set of 2* 2 experiments that were charac-terized by

Fig. 5. Reconstructions from an object with centrosymmetricamplitude and totally random phase: peak phase zeroamplitude-only reconstruction (upper left), peak phase zerophase-only reconstruction (upper right), peak phase 90°amplitude-only reconstruction (lower left), peak phase 90° phase-only reconstruction (lower right).

Fig. 6. Asymmetric object.

2904 J. Opt. Soc. Am. A/Vol. 14, No. 11 /November 1997 Lohmann et al.

• Peak phase zero or 90°.• u(x 2 x0) phase constant or random.

The outputs are shown in Figs. 7 and 8. The most re-markable result here is the generation of a twin image in

Fig. 7. Reconstructions from an asymmetric object with zerophase: peak phase zero amplitude-only reconstruction (upperleft), peak phase zero phase-only reconstruction (upper right),peak phase 90° amplitude-only reconstruction (lower left), peakphase 90° phase-only reconstruction (lower right).

Fig. 8. Reconstructions from an asymmetric object with totallyrandom phase: peak phase zero amplitude-only reconstruction(upper left), peak phase zero phase-only reconstruction (upperright), peak phase 90° amplitude-only reconstruction (lower left),peak phase 90° phase-only reconstruction (lower right).

Figs. 7 and 8. Again, the phase shift of the central peakhas a major influence on the phase-only and amplitude-only outputs.

7. CONCLUSIONSComputer experiments showed that imposing a change onone pixel of an object may switch the importance of theFourier phase and the Fourier amplitude. We examinedthe influence of symmetry and asymmetry of objects onthe significance of phase and amplitude in the Fourier do-main. It is our conclusion that there is no strict confir-mation of the proposition ‘‘phase is more important thanamplitude.’’ Several object properties such as symmetryand reality play an important role in determining the sig-nificance of phase and amplitude in the Fourier domain.

*Permanent address, Erlangen-Nurnberg University,Physikalisches institut, Erwin Rommelstrasse 1, 91058Erlangen, Germany.

REFERENCES1. A. Kozma and D. Kelly, ‘‘Hard-clipped matched filtering,’’

Appl. Opt. 4, 387 (1965).2. A. V. Oppenheim and J. S. Lim, ‘‘The importance of phase

in signals,’’ Proc. IEEE 69, 529–541 (1981).3. L. B. Lesem, P. M. Hirsch, and J. A. Jordan, Jr., ‘‘The kino-

form: a new wavefront reconstruction device,’’ IBM J. Res.Develop. 13, 150–155 (1969).

4. A. W. Lohmann and H. Werlich, ‘‘Holographic production ofspatial filters for code translation and for image restora-tion,’’ Phys. Lett. 25, 570 (1967).

5. A. W. Lohmann and Ch. Thum, ‘‘Two-way code translationby computer generated holographic filters,’’ Opt. Commun.46, 74–78 (1983).

6. W. Pearlman and R. Gray, ‘‘Source coding of the discreteFourier transform,’’ IEEE Trans. Inf. Theory IT-24, 683–692 (1978).

7. R. S. Powers and J. W. Goodman, ‘‘Error rates in computer-generated holographic memories,’’ Appl. Opt. 14, 1690–1701 (1975).

8. J. R. Fienup, ‘‘Phase-retrieval imaging problems,’’ in Inter-national Trends in Optics, Vol. 1 of International Commis-sion for Optics book series, J. W. Goodman, ed. (Academic,Ann Arbor, Mich., 1991).