Gradual and randombinarization of gray-scale holograms

Eryi Zhang, Steffen Noehte, Christoph H. Dietrich, and Reinhard Manner

A new method called gradual and random binarization to binarize gray-scale holograms, based on aniterative algorithm, is proposed. The binarization process is performed gradually, and the pixels to bebinarized are chosen randomly. Errors caused by this operation are spatially diffused. A comparisonwith other established methods based on error diffusion, direct binary search, and iterative stepwisequantization shows that the gradual and random binarization method achieves a very good compromisebetween computational complexity and reconstruction quality. Optical reconstructions are presented.

1. Introduction

Computer-generated holograms 1CGH’s2 have beeninvestigated intensively in recent years owing to theirwide application range and their advantages in termsof flexibility, accuracy, size, weight, and cost. Appli-cations can be found, e.g., in optical informationprocessing, in which CGH’s are used as filters togenerate a required wave front,1 or in optical neuralnetworks, in which they are used to accomplishcomplex synaptic interconnections.2 CGH’s can alsobe used as complex optical elements.3 Nowadaysworkstations provide enough computing power togenerate large CGH’s. A hologram of 1024 3 1024pixels can be generated in a few minutes.Two steps are necessary to produce CGH’s. The

first step is to determine the hologram transmissionthat maps a given input to a desired output. In thesecond step this transmission is exposed onto a photo-material, e.g., a photographic plate or film, by use ofan appropriate output device. We concentrate hereon binary Fourier transform holograms and on thereconstruction of an intensity distribution I1x, y2 in aspecified region, which we call a reconstruction re-gion.A complex signal u1x, y2 5 0u1x, y2 0exp3 jf1x, y24 of

magnitude 0u1x, y2 0 5 3I1x, y241@2 and random phasef1x, y2 is synthesized. As in experimental hologra-phy, the hologram transmission is determined by the

The authors are with the Department of Computer Science,University of Mannheim, Mannheim D-68131, Germany.Received 6 September 1994; revised manuscript received 26April

1995.0003-6935@95@265987-09$06.00@0.

r 1995 Optical Society of America.

interference pattern 0U1µ, n2 1 R1µ, n2 0 2, whereU1µ, n2 5 0U1µ, n2 0exp3 jF1µ, n24 5 FT5u1x, y26 is theFourier spectrum of the object and R1µ, n2 5

exp31j2p1x0µ 1 y0n24 is a plane reference wave; 1x0, y02determines the position of the first-order diffractionpattern in the reconstruction. In computer hologra-phy this interference pattern can be encoded as

H1µ, n2

50U1µ, n2 0cos3F1µ, n2 2 2p1x0µ 1 y0n24 2 Bs

1 2 Bs, 112

where U1µ, n2 is normalized to 1 and Bs 5

min5 0U1µ, n2 0cos3F1µ, n2 2 2p1x0µ 1 y0n246 is a biasfactor that ensures 0 # H1µ, n2 # 1. The values 1x, y2and 1µ, n2 are all discrete values of the form 1xdx, ydy2and 1µdµ, ndn2; dxdµ and dydn fulfill the samplingtheorem. Thus pixels are indexed by 1x, y2 and 1µ, n2,respectively.Hologram distribution 112 is analog valued. The

production of the corresponding hologram is nor-mally difficult because photomaterials as well asoutput devices usually suffer from nonlinearities.Besides, it is required that the output device be ableto expose gray values. These problems are avoidedin binary holograms that have only two values. Theproduction and the reproduction of binary hologramsis especially simple.4 For exposure, even plotters orlaser printers can be used.5 Additionally, binaryholograms have lower noise sensitivity.6 However,one has to take care of the binarization error becausethis error can cause enormous errors in the recon-struction with respect to the original object. Manydifferent methods have been proposed to minimizethis error. They can be divided into two groups:

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noniterative methods and iterative methods. Hardclipping and error diffusion7–12 belong to the former,whereas direct binary search13–15 and iterative step-wise quantization16,17 belong to the latter. The newmethod proposed here, called gradual and randombinarization 1GRB2 of gray-scale holograms, also be-longs to the latter. It binarizes gray-scale hologramsgradually and randomly, based on an iterative algo-rithm.For reconstruction a common Fourier-transform

geometry is used; that is, the produced binary holo-gram is located at the front focal plane of the systemand is illuminated by a collimated wave parallel to theoptical axis. The reconstruction appears at the backfocal plane. It contains the positive and negativefirst orders, i.e., the original signal u1x 2 x0, y 2 y02and its conjugate u*321x 1 x02, 21 y 1 y024, the zerothorder, i.e., the dc peak in the center, and higherorders.The following two sections 1Sections 2 and 32 de-

scribe in detail the new method and the binaryholograms generated by it. A comparison with othermethods is made in Section 4. Optical reconstruc-tions are presented in Section 5, followed by conclu-sions in Section 6.

2. Iterative Algorithm and Constraints

Iterative algorithms for the computation of hologramsconsist generally of the following steps18,19 1algorithm02:

1i2 Fourier transform the original signal u1x, y2 50u1x, y2 0exp3 jf1x, y24 intoU11µ, n2:

U11µ, n2 5 FT5u1x, y26.

1ii2 Encode the complex spectrum Uk1µ, n2 into thehologram Hk1µ, n2 for the kth iterative cycle. Here Cdenotes the encoding, e.g., of Eq. 112:

Hk1µ, n2 5 C3Uk1µ, n24, k 5 1, 2, 3, . . . .

1iii2 Impose the constraints L in the hologramdomain:

Hk1µ, n2L= H8k1µ, n2.

1iv2 Inverse Fourier transform H8k1µ, n2 into h8k1x, y2that contains the signal u1x, y2:

h8k1x, y2 5 IFT5H8k1µ, n26.

1v2 Impose the constraints l in the object domain:

h8k1x, y2l= hk111x, y2.

1vi2 Fourier transform hk111x, y2 into Uk111µ, n2.Go to 1ii2.

The constraints imposed in steps 1iii2 and 1v2 depend

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on the requirements to the hologram and to thereconstruction. Steps 1i2–1vi2 are repeated until thesignal contained in h8k1x, y2 of step 1iv2, which is recon-structed byH8k1µ, n2, has the required accuracy.GRB uses such an iterative algorithm for the

generation of binary holograms. The produced bi-nary hologram should reconstruct the original signal0u1x, y2 0 in the reconstruction region as well as possible.We choose the first-order diffraction pattern in thereconstruction as the reconstruction region. The con-straints in algorithm 0 are therefore determined asfollows: The constraint L in the hologram domain isa threshold that transforms the gray-valued holo-gram into a binary one. The constraint l in the objectdomain is a replacement; i.e., the reconstructed mag-nitude in the reconstruction region is replaced by theoriginal one if both are different. Without loss ofgenerality we assume that the hologram has N 3 Npixels and the reconstruction region w has M 3 M10 , M , N2 pixels.To assess the reconstruction quality quantitatively,

we use the mean-square error 1MSE2. Suppose fxy tobe the original signal and gxy to be the reconstructionof the hologram; then the MSE is computed by

MSE 51

M2 oxy[w

0 fxy 2 f

sf2gxy 2 g

sg02

,

where

f 51

M2 oxy[w

fxy, g 51

M2 oxy[W

gxy,

sf2 5

1

M2 oxy[w

0 fxy 2 f 02, sg2 5

1

M2 oxy[w

0 gxy 2 g 02.

The sum extends over to the entire region w. TheMSE is independent of the actual value of g and anyadditive constant.Thus algorithm 0 can be rewritten in more detail in

the following fashion 1algorithm 12:

1i2 Fourier transform u1x, y2 5 0u1x, y2 0 exp3 jf1x, y24intoU11µ, n2:

U11µ, n2 5 FT5u1x, y26.

1ii2 EncodeUk1µ, n2 intoHk1µ, n2 by Eq. 112 for the kthiterative cycle, k 5 1, 2, . . . .

1iii2 Impose a constraint on the hologramHk1µ, n2:

H8k1µ, n2 5 50 if Hk1µ, n2 , Tthreshold

1 otherwise.

1iv2 Inverse Fourier transform H8k1µ, n2 into h8k1x, y2:

h8k1x, y2 5 IFT5H8k1µ, n26.

1v2 Calculate the MSE of 0h8k1x, y2 0 in region w inrespect to the original signal 0u1x, y2 0 . If it is notsufficiently smaller than in the last cycle, terminate

the iteration. Otherwise, impose the constraint ofthe object domain; correct pixel values of 0h8k 1x, y2 0 inregionw to form hk111x, y2:

0hk111x, y2 05 50u1x, y2 0@ck if 0h8k1x, y2 0fi 0u1x, y2 0@ck

0h8k1x, y2 0 otherwise,

where ck is a scaling factor calculated by

ck 5

o1x, y2[w

0u1x, y2 0 0h8k1x, y2 0

o1x, y2[w

0h8k1x, y2 02.

1vi2 Fourier transform hk111x, y2 into Uk111µ, n2.Go to 1ii2.

Intuitively, one could expect to obtain an improvedreconstruction of H8k1µ, n2 by repeating steps 1ii2–1vi2.However, this is not the case. Because the thresholdin step 1iii2 of algorithm 1 is applied to all hologrampixels, the algorithm converges to an unsatisfactorysolution and stagnates there.16 To avoid the stagna-tion,Wyrowski16 proposed an iterative stepwise quan-tization method based on the gray-value distributionof the hologram. In his method, two variable thresh-olds are used in step 1iii2 instead of a constant thresh-old for all the pixels. At the beginning, Threshold 1is set to 0, whereas Threshold 2 is set to 1. ThenThreshold 1 is increased, and Threshold 2 is de-creased by a step. Pixels having values belowThresh-old 1 are set to zero, and pixels having values aboveThreshold 2 are set to 1. Other pixels having valuesbetween Threshold 1 and Threshold 2 are unchanged.The algorithm is performed iteratively for these twothresholds until the MSE no longer decreases. Thennew thresholds are constructed; i.e., Threshold 1 isincreased, and Threshold 2 is decreased by anotherstep, and the algorithm is started again. This actionis carried out step by step until the two thresholdsbecome equal. Using this method, we achieved animproved reconstruction with a smaller MSE 1seeSection 42. However, this algorithm cannot diffusethe binarization error effectively because normally ahologram is locally correlated. That is to say, thereare always regions in which all pixels have valuesbelow Threshold 1 or above Threshold 2, and they areregionally binarized. Our method described in Sec-tion 3 overcomes this drawback by gradually increas-ing the number of binarized pixels that are chosenrandomly on the hologram.

3. Gradual and Random Binarization Algorithm

Starting from an analog hologram H1µ, n2 describedby Eq. 112 and its reconstruction h1x, y2 containingu1x, y2 in region w, we construct a Fourier-transformpair H1µ, n2 6 h1x, y2. The constraint imposed instep 1iii2 of algorithm 1 is decomposed into twosubconstraints, namely, the number of pixels to bebinarized and the choice of these pixels for the

binarization. Therefore the GRB algorithm is real-ized in two nested iterations: the outer loop deter-mines how many pixels are binarized, and the innerloop chooses these pixels randomly and binarizesthem. The number of binarized pixels increasesgradually with each outer iteration. Figure 1 is the

Fig. 1. Flow chart of the GRB algorithm.

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flow chart of the GRB algorithm, where the outer loopis indexed by i and the inner loop is indexed by k.In the flow chart of Fig. 1 the following two terms

are defined:

c The ith iteration is one outer loop with the ithconstraint that N2@Si pixels should be binarized,where i 5 1, 2, 3, . . . , I.

c The kth iterative cycle is one inner loop tobinarizeN2@Si pixels, where k 5 1, 2, 3, . . . , K.

In the ith iteration, N2@Si pixels in the hologramshould be binarized, where Si is an integer 1Si 9 N22.To determine which pixels are processed in eachiterative cycle k 1k 5 1, 2, . . . , K2 of this iteration, wegenerate N2@Si pseudorandom numbers between 0and N2 2 1 as access addresses. It seems as if thehologramwere randomly divided intoN2@Si subholo-grams, each having Si pixels and in which only onepixel is binarized. This operation is carried outuntil the MSE no longer decreases significantly.Then the 1i 1 12th iteration is started with the con-straint that N2@Si111S i11 , Si2 pixels should be bina-rized.The binarization of pixels in the gray-scale holo-

gram introduces an error with respect to the gray-scale hologram. We call this binarization error inthe hologram. It leads to errors in the reconstruc-tion, whichwe call binarization error in the reconstruc-tion. If an iterative cycle is performed—binarizationof hologram pixels, simulation of the reconstruction,replacement of pixel values in the reconstructionregion, and generation of a new hologram—the errorin the hologram is spread out.20 We say that thebinarization error in the hologram is spatially diffused.After several iterative cycles, the binarization error inthe reconstruction will be distributed to the areaoutside of the reconstruction region. In this way, animproved reconstruction in the reconstruction regionwill be achieved. Gradual binarization avoids thestagnation of the algorithm; random binarizationreduces the likelihood of local error clustering.To speed up convergence, we select the parameters

S1, S2, . . . , Si , . . . , SI to be a geometrical series of11@22 and S1 5 S, SI 5 1. Thus we have altogetherI 5 1 1 log2 S iterations to accomplish. The con-straints for these iterations are the binarization ofpixels N2@S, 2N2@S, . . . , and N2. The number ofiterative cycles in each iteration differs from iterationto iteration, but we limit this number to be not largerthan the number S as a compromise between errorreduction in each iteration and computation time.In practice we can choose S 5 16 or S 5 8.The example given below shows how the algorithm

works and how the binarization errors are diffused tothe neighbors. The object to be reconstructed is thecapital letter F, represented by 30 3 32 pixels of threegray values. It is located at the center of a 128 3 128zero array, as shown in Fig. 21a2. Random phase isused with the amplitude to generate its gray-scalehologram. The hologram coded by Eq. 112 is shown

5990 APPLIED OPTICS @ Vol. 34, No. 26 @ 10 September 1995

in Fig. 21b2 with 1x0, y02 5 132, 322. To binarize thegray-scale hologram 1256 gray values2 of Fig. 21b2, wechoose S 5 8. The reconstruction region w has thesame size as that of the original object. The thresh-old value for the binarization is 0.5. Altogether, I5 11 log2 S 5 4 iterations are required: in the firstiteration 2048 pixels are binarized, in the second 4096pixels, in the third 8192 pixels, and in the fourth16384 pixels 1because of the random selection ofpixels, a pixel may be binarized more than once.This implies the number of actually binarized pixelsis smaller than the number given above2. In eachiteration the kth iterative cycle 1k # S 5 82 is per-formed provided that the reduction of the MSE in thelast two iterative cycles is large enough, here, largerthan 0.001, i.e., d 5 0.001. Figures 31a2–31e2 show thechanges of the hologram after the binarization itera-tions are performed; Figs. 31a82–31e82 are their gray-value distributions.In the first iteration 1i 5 1 and S1 5 82, N2@S1 5

2048 pixels were binarized. Pseudorandom ad-dresses determined which pixels were binarized.After the first iterative cycle 1k 5 12 of this iteration1i 5 12was performed, theMSE caused by the binariza-tion of 2048 pixels was 0.05354. It was reduced bythe subsequent iterative cycles 1k 5 2, 3, . . . ; see Fig.42; i.e., again 2048 pixels were randomly binarized.At the end of the first iteration 1i 5 12, MSE wasreduced to 0.02586; then the reduction became tooslow. The second iteration with i 5 2 and S2 5 4 wasstarted. In this iteration 1i 5 22,N2@S2 5 4096 pixelswere binarized, etc. Figure 4 shows how the MSEchanged during each iteration and with the iterations1i 5 1, 2, 3, 42. Table 1 lists their related values.After the last iteration 1i 5 4, S4 5 12, the threshold

was applied to all pixels to generate the final binaryhologram. Figure 51a2 shows this hologram. Its com-puter-simulated reconstruction is given by Fig. 51b2with MSE 5 1.80 3 1022. The diffraction efficiencyis h 5 6.17%.

4. Comparison with Established Methods

To estimate the performance and the efficiency of theGRB algorithm, we compared it with some estab-lished methods. The comparison focuses mainly ontwo aspects: the computational complexity to gener-

Fig. 2. 1a2 Object 1letter F2 with three gray values and 1b2 itsgray-scale hologram 1256 gray values2.

Fig. 3. Illustration of the binarization process: 1a2 the original gray-scale hologram and 1a82 its histogram; 1b2–1e2 Partly binarizedholograms for S1 5 8, S2 5 4, S3 5 2, and S4 5 1; 1b82–1e82 their gray-value distributions.

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ate a binary hologram and the reconstruction qualityin region w produced by such a hologram. As theobject, we still take the example of Section 3, thecapital letter F. All parameters remain unchanged.The following methods are considered:

c Hard clipping.c Error diffusion.c Direct binary search.c Iterative stepwise quantization.c Gradual and random binarization 1GRB2.

Hard clipping is the simplest and the fastestmethodamong the five methods considered. It scans thegray-scale hologram and binarizes it using a constantthreshold. If N is the size of the hologram, thecomplexity of this algorithm is O1N2 1the time neededto generate the gray-scale hologram is not taken intoaccount2. The reconstruction from such a binaryhologram consists of the desired object and noisewithin and around the object. Figure 61a2 shows thebinarized hologram; Fig. 61b2 its computer-simulatedreconstruction withMSE5 9.783 1022. The diffrac-tion efficiency is h 5 8.97%.The error-diffusion method7 was originally devel-

oped for displaying images. It was later introducedinto holographic applications.8 The algorithm workssequentially; i.e., it processes input data line by line,pixel by pixel. The basic concept can be described asfollows: The first pixel of the gray-scale hologramH1µ, n2, H10, 02, is compared with a given threshold T.If H10, 02 $ T, the output Hout10, 02 is set to 1,otherwise to 0. The error E10, 02 5 Hout10, 02 2

Fig. 4. MSE changes with iterative cycles k 11–7, 8–14, 15–19,and 20–222 in each iteration and with iterations i 11, 2, 3, 42.

Table 1. Changes of the Mean-Square Error in Each Iteration for theExample ALetter F B

Iteration i 1Si2Number of

Binarized PixelsMSE at Begin-ing of Iteration

MSE at Endof Iteration

1 1S1 5 82 2048 0.05354 0.025862 1S2 5 42 4096 0.03573 0.016983 1S3 5 22 8192 0.02116 0.013004 1S4 5 12 16384 0.01575 0.01420

5992 APPLIED OPTICS @ Vol. 34, No. 26 @ 10 September 1995

H10, 02 caused by this operation is diffused to otherunprocessed pixels, which are given the new valueH81µ, n2 5 H1µ, n2 2 WµnE10, 02 at location 1µ, n2, whereWµn are diffusion coefficients. Modifications of thisalgorithm8–12 use the same general scheme and differmainly in the number and the relative positions ofpixels included in the error diffusion. We use themethod proposed by Weissbach and Wyrowski.12This algorithm determines the diffusion weights de-pendent on the location of the reconstruction regionand results in a zero error there. The complexity of

Fig. 5. 1a2 Binary hologram generated by the GRB method pro-posed here and 1b2 its computer simulated reconstruction.

Fig. 6. Same as Fig. 5 but for hard clipping.

this algorithm is also O1N2, where N is the size of thehologram. Figure 71a2 shows the generated binaryhologram; Fig. 71b2 shows its simulated reconstructionwith MSE 5 5.04 3 1022 and h 5 1.09%.Direct binary search13–15 is based on an iterative

procedure to minimize a given error criterion. Itmanipulates the binary state of individual pixels ofthe hologram directly by monitoring the effect on thereconstruction region: the randomly generated ini-tial binary hologram is scanned, and the transmissionvalues are inverted one by one. After each inversion,the MSE in the reconstruction region is computed.If it has decreased, the inversion is retained; other-wise, the pixel is restored to its previous state. Thealgorithm is terminated when no inversions are re-tained during an entire iteration. The complexity ofthis algorithm isO1N7@43log2 N4 3@42, whereN is the sizeof the hologram. The binary hologram generatedwith this method and its computed reconstruction areshown in Figs. 81a2 and 81b2, respectively, with MSE 51.40 3 1022 and h 5 3.97%. Here the algorithm isterminated when MSE # 1.40 3 1022.Iterative stepwise quantization, proposed by

Wyrowski,16 is also based on an iterative algorithm1see Section 22. A stepwise quantization based on thegray-value distribution of the hologram results in animproved convergence. As an example, the binaryhologram generated with this method and its com-puted reconstruction are given in Figs. 91a2 and 91b2,respectively. The complexity of this algorithm isO1N log2 N2, whereN is the size of the hologram. Thesimulated reconstruction shows MSE 5 2.79 3 1022

and h 5 6.45%.Gradual and random binarization 1GRB2 is de-

scribed in Section 3. The generated binary holo-gram and its computed reconstruction are given in

Fig. 7. Same as Fig. 5 but for error diffusion.

Figs. 51a2 and 51b2, respectively. The complexity ofthis algorithm isO1N log2 N2 too, whereN is the size ofthe hologram. Here MSE 5 1.80 3 1022 and h 56.17%.Table 2 summarizes these results.It was shown that the direct binary search holo-

gram achieved the best reconstruction quality.However, the required computing power is in manypractical cases too large. Gradual and random bina-rization 1GRB2 achieved a very good compromise be-tween computational complexity and the recon-

Fig. 8. Same as Fig. 5 but for direct binary search.

Fig. 9. Same as Fig. 5 but for iterative stepwise quantization.

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struction quality. This result can be seen qualita-tively by comparison of computed reconstructions,e.g., Figs. 91b2 and 51b2. Figure 91b2 is the computedreconstruction from the hologram Fig. 91a2 generatedby the stepwise quantization method, and Fig. 51b2 isthat from the hologram Fig. 51a2 generated by theGRB method. Clearly, for the stepwise quantizationmethod, errors in the reconstruction are mainly dis-tributed to the area around the reconstruction region,whereas for the GRB method errors are uniformlydistributed to the entire reconstruction plane outsideof the reconstruction region. That is, the GRB algo-rithm distributes binarization errors more efficientlythan the stepwise quantization method.

5. Optical Reconstructions

The object to be reconstructed is the emblem of theUniversity of Mannheim with 250 3 340 pixels, asshown in Fig. 10. It was centered on a 1024 3 1024array of zeros. Random phase was assigned to it.The reference wave was exp31j2p1x0µ 1 y0n24 with1x0, y02 5 1256, 2562. Thus the object, the twin, andthe dc peak were seen in the reconstruction. Forcomparison, two binary holograms were produced.One was generated by iterative stepwise quantiza-tion, the other by gradual and random binarization1GRB2. A photoprinter with a resolution of 1625 dpiwas used to expose the calculated binary hologramsonto a film of size 16 mm 3 16 mm. After chemicalprocessing, the binary amplitude holograms wereobtained. A He–Ne laser of wavelength l 5

Table 2. Results of Simulation of Binary Holograms Generated byDifferent Methods and Computational Complexity of These Methods

Methods MSE 110222 h 1%2 Complexity

Hard clipping 9.78 8.97 O1N2Error diffusion 5.04 1.09 O1N2Direct binary search 1.40 3.97 O1N7@43log2 N4 3@42Iterative stepwisequantization

2.79 6.45 O 1N log2 N2

Gradual and randombinarization

1.80 6.17 O 1N log2 N2

Fig. 10. Emblem of the University of Mannheim as the object.

5994 APPLIED OPTICS @ Vol. 34, No. 26 @ 10 September 1995

0.6328 µm was used for the reconstruction. Figures11 and 12 show the optical results.During the optical reconstruction the diffraction

efficiency was measured. The hologram was illumi-nated by a circular laser beam 1B 5 11.6 mm2. Aphoto powermeter with a circular aperture of diam-eter 12 mm was used to measure the energy in thefirst diffraction order. The measured value is actu-ally summed over the circular aperture of the photopowermeter, which is normally larger than the noncir-cular reconstruction region. The absorption of theholographic film and the imaging lens must also betaken into consideration.The measured values listed in the column h8 of

Table 3 show a good agreement of the experimentally

Fig. 11. Optical reconstruction of the binary hologram generatedby iterative stepwise quantization.

Fig. 12. Optical reconstruction of the binary hologram generatedby GRB.

generated data with the computer simulations.Computer-calculated MSE and h are also listed inTable 3. Both the computer simulation and theoptical reconstruction show that the hologram gener-ated by the method proposed here is of high quality.

6. Conclusions

Anewmethod to binarize gray-scale holograms, basedon an iterative algorithm, has been proposed. Usingthis method, we generated an optimized binary holo-gram and demonstrated its optical reconstruction.Because pixels in the gray-scale hologram are intro-duced gradually and binarized randomly, the stagna-tion of the algorithmwas avoided, and amore efficientdiffusion of binarization errors was achieved,20 whichresults in an improved convergence. Compared withother established methods, GRB has shown a moresatisfactory reconstruction with fewer computationaldemands. More tests were done. They show thatthe gradual and random binarization method canalways reduce errors in the reconstruction region.The error-reduction degree is tightly related to thesize of the reconstruction region. In this sense, thisalgorithm is robust.Although the binarization method described here is

considered for Fourier-transform holograms only, it ispossible to extend it to generate any other type ofbinary hologram. Owing to the limitation of space,the theoretical foundation of this method are not bedealt with here but in Ref. 20.

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Table 3. Calculated MSE, h, and Measured h8 from the Reconstruction ofBinary Holograms

Methods MSE 110222 h 1%2 h8 1%2

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0 September 1995 @ Vol. 34, No. 26 @ APPLIED OPTICS 5995