September1969 MULTIPLE-BEAM INTERFEROMETRY BY HOLOGRAPHY 1175

Holographic Compensation of Motion Blur by Shutter Modulation*

OLOF BRYNGDAHL

IBM Research Division, Los Gatos Laboratory, P. 0. Box 66, Los Gatos, California 95030

AND

ADOLF LOHMANN

Dept. of A pplied Electro physics and Visibility Laboratory SIO,University of California, La Jolla, California 92037

(Received 14 March 1969)

Ordinarily, in photographic recording, the shutter is opened for a finite time which-can be described by arectangular shutter function (= flux as a function of time). Such a shutter function, in connection with linearimage motion, creates a degradation due to motion blur. If the shutter modulates the flux by a functioncos2 (t2), a recording is obtained that has the properties of an incoherent one-dimensional hologram. Hence,compensation of motion blur is achieved by reconstructing a sharp image from the hologram. This methodcan be modified in order to cope with harmonic vibrational or random translational motion.INDEX HEADINGS: Holography; Image formation.

To record a moving image on a photographic plate,the exposure time must be sufficiently short to avoidmotion blur. However, if such a short exposure isincompatible with the exposure energy-density require-ment of the photographic emulsion, motion blur isunavoidable. Several data-processing methods areknown for compensating after-the-fact the effects ofmotion blur and have been demonstrated experiment-ally, for example by Harris.'

We wish to present a new method for after-the-factcompensation of motion blur. During the recordingof a moving image, the transmittance of the shutter istime-modulated, for example by a rotating polarizer incombination with a fixed polarizer. The shutter trans-mittance function S(1) is chosen such that the motionblurred image has essentially all of the features of ahologram, although the object radiation might becompletely incoherent. The conversion of the motion-blurred image into a sharp image is similar to recon-structing an image from a hologram.

In this paper we treat principally the compensationof uniform linear motion; however, methods for com-

* Work supported in part by ARPA.'J. L. Harris, Sr., J. Opt. Soc. Am. 59, 497A (1969); 56, 569

(1966).

pensating harmonic and random translational motionwill also be discussed.

RECORDING PROCESS

If an object such as in Fig. 1(a) is moved laterallywith constant velocity during the finite recording timeT, the image will be blurred as shown in Fig. 1(b).The shutter transmittance function S(t) which isresponsible for this type of blur has the shape of arectangle [Fig. 2(a)]. Our goal is to restore this image.Conceptually, that would appear easy to do by coveringeach wide line with a cylinder lens, provided that thewide lines do not overlap. When this combination isilluminated by a plane wave, the vertical lines willappear sharp in the rear focal plane of the cylinderlenses. However, at that plane, the horizontal lineswill be unsharp due to defocusing. A final image inwhich both horizontal and vertical lines appear sharpcan be formed by an astigmatic lens system, which inits simplest form consists of two crossed cylinderlenses of different focal powers and/or longitudinalpositions.

2 G. L. Rogers, Nature 166, 237 (1950); D. Gabor, Proc. Roy.Soc. (London) A197, 454 (1949).

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OLOF BRYNGDAHL AND ADOLF LOHMANN

-)

(a) ( b) ( C )

FIG. 1. Linear-motion blur. (a) object; (b) ordinary blur;(c) modulated blur.

While this scheme, which applies cylindrical lensletsto restore blurred image detail, may appear simple, itdoes not lend itself to implementation. First, making thelenslets would be difficult. If photographic bleachingmethods were employed, it would be hard to obtainlenslets of the same focal power for the various featuresin the image. Secondly, the lenslets could not be madefor image details that have overlapping blurs. Thelenslet problems can be overcome, however, by usingFresnel-zone-pattern lenses instead. These are createdby modulating the shutter transmittance during theinitial exposure, by a function S(t) as in Fig. 2(b), toproduce asymmetric Fresnel zone plates over the widthof the blur. We could continue to describe this schemeas suggested so far, but now, because of similarity,it becomes advantageous to use the terms and conceptsof holography.

When a conventional Fresnel hologram is formed,the light from each object point x', y' is converted intoa Fresnel-zone pattern cos2 {7r[(x-x') 2 + (y-y') 2 ]l/a2},where the scale factor a depends on the wavelengthand the geometry of the setup. If the geometry ischosen such that only off-center portions of the Fresnel-zone patterns are recorded on the hologram, whichusually is achieved by tilting the reference wave,3

separation of the reconstructed image from the twinimage and from the zero-order light is possible.

Somewhat different are "one-dimensional Fresnelholograms."' When they are formed, the light fromeach object point x', y', is converted into a one-dimen-sional Fresnel-zone pattern 6(y-y') cos2[7r(x-x')2/a2],which is the same as a horizontal strip of an ordinaryFresnel-zone pattern. Again, the centers of these one-dimensional Fresnel patterns should not fall onto theholograms (so that it is possible to suppress the twinimage). When Fresnel holograms are formed, anymethod that converts the object points into Fresnel-zone patterns can be used and coherent illuminationmay not be necessary.4. 5 For compensation of motionblur we now propose to convert an object into a one-

3E. N. Leith and J. Upatnicks, J. Opt. Soc. Am. 52, 1123(1962).

4 L. Mertz, Transformations in Optics, (John Wiley & Sons, Inc.,New York, 1965).

1 0. Bryngdahl and A. Lohmann, J. Opt. Soc. Am. 58, 625(1968).

dimensional Fresnel hologram during the photographicexposure by moving the object with constant velocityv in x direction while modulating the temporal trans-mittance through the lens, by a function like cos2 (12).

A detailed form of the shutter-transmittance func-tion S(Q), which converts the moving object point6(x-vt-x')6(y-y') into the motion-blurred pointimage v-¾6(y-y') cos1[7r(x-x')2/a2] within x'<x<x'+2vT is

S(t) = cos2[7r(vt/a)2] in 0< t< 2T; 0, otherwise.

This shutter function S(t) [see Fig. 2(b)] is sufficientfor recording a one-dimensional hologram. It is boundedbetween zero and one, and its time integral is the sameas for the ordinarv rectangular shutter function [Fig.2(a)] of duration T:

2T

1 T S(t)dt = T; if (2vt/a) 2 is an integer.

When the convolution with the moving object point6(x-vt-x') is formed, the shutter function S(t) pro-duces the proper Fresnel blur

J S(t)3(x-vt-x')dt= 1/v cos 2E7r(x-x')

21a2 ]

in x'<x<x'+2vT.

It contains only one side of the Fresnel pattern, allowingthe twin image to be eliminated in the holographicreconstruction process.

COMPENSATION PROCESS

Characteristically, a Fresnel zone plate will concen-trate a portion of incident, coherent light into a focalpoint. This is the essential effect of the blur-compensa-tion process. When the Fresnel-modulated record isilluminated by a coherent plane wave, each motion-blurred point image a(y-y') cos2i7r(x-x')s/a2] actslike a horizontal strip of a cylinder lens of focal length1f= a2/2X with cylinder axis vertical. An astigmatic lenssystem consisting of two cylinder lenses, as shown inFig. 3, is needed to form the sharp output image. Thelens L1 with horizontal cylinder axis focuses the blurredpoint images from the blurred record into the outputplane [Fig. 3(a)]. The other cylinder lens L2 with

s(t)

() 'I

t.) (ib)

FIG. 2. Shutter functions. (a) ordinary shutter function; (b) shutterfunction suitable for holographic motion compensation.

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September1969 HOLOGRAPHIC COMPENSATION OF MOTION

vertical axis [Fig. 3(b)] fulfills two functions. In itsrear focal plane, the light that forms the real image, thetwin image, and the zeroth order, are separated laterally.A screen or knife edge in this plane rejects the unwantedportions. Furthermore, this second cylinder lens L2forms an image of the Fresnel-focus plane (at distancef behind the motion-blurred record) onto the outputplane.

The operations of the two cylinder lenses are simplein concept but difficult in practice, because cylinderlenses are usually less corrected than spherical lensesof comparable price. Hence, in the actual experiment,we used a strong spherical lens for the bulk of thefocusing job, and a weak cylinder lens for properastigmatism. The Figs. 4(a)-(d) show (a), the originalobject (b), the record with linear-motion blur (c), therecord with spatially modulated motion blur, and (d) thereconstruction from the motion-compensation process.The most critical part of the experiment was to recordthe blurred image on the linear portion of the photo-graphic amplitude-transmittance vs exposure curve.For this purpose, a small amount of uniform pre-exposure is needed.

RESOLUTION OF THE COMPENSATED IMAGE

The blurred point image or linear Fresnel pattern hasa width 2vT, and a focal length f = a2/2X. Considered asa lens, it has an F-number F= f/2vT= a2/4XvT. Whensuch a lens is illuminated with a coherent plane wave,the point image at a distance f behind the lens willhave a half-width of XF=a2/4vT. This is the bestpossible resolution we can expect in the output plane.On the other hand, it is sufficient to make the F numberof the Fresnel zone pattern less than the F number ofthe camera objective used for recording the blurredimage.

(a)

(b)

ff

-f2

FIG. 3. The holographic compensation process. (a) the horizontalcylinder lens Li images the blurred record into the output plane.(b) the linear Fresnel pattern focuses the light at a distance f,from where it is imaged by the vertical cylinder lens L2 into theoutput plane.

(a) (b)

( c ) (d)

FIG. 4. (a) The object; (b) the object with ordinary motionblur; (c) the recording where the motion blur is spatially modu-lated into one-dimensional Fresnel-zone pattern; (d) recon-structed image from (c). With the lenses used a larger horizontalthan vertical magnification was obtained.

For judging the success of our compensation method,we compare the resolution XF= a2/4vT with the resolu-tion vT we would get in an ordinary motion-blurredimage, exposed during the interval T, while the objectis moving with velocity v. The ratio XF/vT= (a/2vT)2

1/N indicates the improvement of resolution. Atthe same time, N indicates the number of maxima ofthe shutter function S(t) during the exposure 0<t<2TEsee Fig. 2(b)]. Yet another relation exists betweenthe resolution XF and the structure of the shutterfunction S(t). The width 31 of the narrowest maximumis about bt=a2/4v 2T [Fig. 2(b)]. Hence, the totalwidth 5x of the narrowest maximum of the linearFresnel pattern or blurred point image is 6x= v6t= a2 /4vT, which is the same as the resolution XF in thecompensated final image.

SOME POSSIBLE MODIFICATIONS

Elimination of Nonlinear Effects

The two most severe experimental difficulties were:(1) The recording of the blurred image had to be linearin terms of amplitude-transmittance vs exposure. (2)The cylinder lenses (Fig. 3) used in reconstruction hadto be reasonably good, because, otherwise, the knifeedge would not only eliminate the zeroth order and thetwin image, but it would also cause shadows to appearin the compensated image. This effect of the presenceof the knife edge is ordinarily used for making theeffects of lens aberrations visible, which is here, ofcourse, undesirable. Both of these problems can prob-

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1 OLOF BRYNGDAHL AND ADOLF LOHIX\ANN

ably be avoided by using the shutter function S(t)=cosE7r(v//a)2 ] not in the interval 0<1<2T, but in2T<I<4T. This corresponds to increasing the tilt ofthe holographic reference wave, which in turn increasesthe separation between zeroth order and the image-forming light. Hence the knife-edge action for suppress-ing the zeroth order would not be aberration sensitive.

Shifting the shutter-function range will also renderthe nonlinear photographic-recording effects harmless.The blurred point image 1+cos[27r(x/a)2] in the off-center region 2vT<x<4vT is almost periodic with alocal spatial frequency beginning at v o=4vT/a2 atx= 2vT and increasing to 2vo at x= 4vT. If this patternis distorted because of photographic nonlinearity, forexample by a quadratic law, other spatial frequencieswill be generated in the range 2vo to 4vo, but the originalfrequency range (vo,2'o) is unchanged. Also, the beatfrequencies that might occur, owing to nonlinear inter-actions of two overlapping point images will not enterthe frequency range (vo,2vo). Hence, it should bepossible to suppress all nonlinear effects by means of afrequency window for (vo,2vo) in the knife-edge plane.

Encoding as Fourier Hologram

Previously, each object point on the blurred imagewas converted into a one-dimensional Fresnel pattern.Hence, the compensation process consisted essentiallyof the known reconstruction process for one-dimensionalFresnel holograms. Instead, a modulated blurred image,which has all of the features of a one-dimensionalFourier hologram, can be produced. In this case, theobject point (x',y') is converted to a strip of a sine-wave pattern with a spatial frequency proportional tox'. This can be achieved by putting a temporal modula-tor, with a temporal frequency proportional to x', intothe object or image plane. The moir6 fringes formed bytwo gratings rotating in opposite directions will dothis.6

Compensation of Lateral Vibration

Assume that each object point (x',y') vibrates hori-zontally: (x'+B sinwt,y'). For those time intervals forwhich I sinwt I- , we can set sinwk :cot. Hence, we havereduced this case again to the case of linear imagemotion by addition of a time mask M(t), which selectsthe intervals (for which sinctt wt). In addition to M(t),the shutter function must provide the Fresnel modula-tion cos2[EN(wt)2

].

I A. W. Lohmann, Appl. Opt. 5, 669 (1966).

Compensation of Random Translational Motion

The whole image may dance at random in twodimensions. Let us assume that the object contains abright point that is somewhat removed from the restof the object. The dancing bright spot is made to fallonto a photocell in the shape of a Fresnel-zone patternwhich controls the shutter. If the dancing bright spothas traveled, after awhile, uniformly over a circulararea of the size of the Fresnel-zone pattern, each objectpoint will have painted its own Fresnel zone pattern.Hence, the blurred image will have all of the features ofa two-dimensional Fresnel hologram. The compensa-tion process consists merely of the reconstruction of thehologram. Separation of the twin image and eliminationof nonlinear effects can be accomplished as described be-fore, by using only an off-center portion of the Fresnel-zone pattern.

CONCLUSIONS

It has been shown in theory and experiment thatlinear-motion blur can be compensated after the fact,by modulating the shutter transmittance like cos2 (t2).This creates small Fresnel-zone-pattern lenses, whichsharpen the final image by means of their diffractivefocal power. Without motion, a conventional sharpimage is obtained. The compensation process can beinterpreted also as a holographic reconstruction process.This reduces the problem of suppressing certain un-wanted terms, simply to the known process of eliminat-ing the zeroth order and the twin image.

The amount and the direction of the linear motiondo not have to be known beforehand. In modified form,our scheme can also cope with vibrations and withrandom motions.

In radar technology, there exists a problem similarto ours. For a fixed-signal peak power a signal form iswanted that contains a large total power and yet yieldsthe same resolution as the shortest possible pulse. Twosolutions are known: a pseudo-random pulse sequenceand a chirp signal, which is the temporal analog of aFresnel pattern. In principle, motion blur could becompensated if the shutter function is pseudo-randommodulated; but the chirp modulation is preferable inoptics, because it is so easy to decode by holographicreconstruction.

ACKNOWLEDGMENT

We would like to thank A. Nafarrate for doing thelaboratory work.

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