Computer-generated binary phase holograms havelong been considered an attractive basis for two-dimensional (2D) and three-dimensional (3D) infor-mation display because of the efficiency, robustness,and potential for miniaturization. While an appro-priate device for the rapid dynamic display of suchholograms—the ferroelectric liquid-crystal (FLC) spa-tial light modulator (SLM)—has been available for along time, the computational demands of the holo-gram generation algorithms have until recently pre-cluded the development of a real-time holographicvideo display. However, the advent of the one-stepphase retrieval (OSPR) algorithm1,2 and advancesin field-programmable gate array and application-specific integrated circuit technology have at lastbrought such a display within reach.
However, there are still a number of issues thatshould be addressed before such a display can offer asubjective level of performance to compete with con-ventional 2D technologies, or offer 3D image displayfor general applications. It is well known that anydiffractive element (such as a binary phase hologram)that imparts a purely real modulation on an incidentwavefront produces a conjugate image in the replayfield (RPF), thereby reducing the usable display areaand optical efficiency by half.3 Although it is straight-forward to generate quaternary phase hologramsthat would solve this problem, such holograms cannotbe displayed on the inherently binary ferroelectricdevices available. A second problem arises from thefact that the viewing angle � of a holographic displayvaries inversely with pixel size. As a result, smalldisplay pixels are required to achieve a wide field ofview, and so an inordinate amount of bandwidth isrequired to display a large image.
In this paper we demonstrate that one binary pix-elated phase mask of resolution M � M, identical inphysical size to that of the P � P pixel SLM but ofgreater resolution, can be used to increase the view-ing angle of the overall system by a factor of approx-imately F � P�M. Furthermore, this can readily becombined with the phase mask required for conjugateimage removal to form a quaternary phase mask thatwe show can be used to ameliorate the two aforemen-
E. Buckley ([email protected]), A. Cable, and N. Lawrence arewith Light Blue Optics, Limited, St. John’s Innovation Centre,Cowley Road, Cambridge CB4 0WS, UK. T. Wilkinson is with theDepartment of Engineering, Cambridge University, TrumpingtonStreet, Cambridge CB2 1PZ, UK.
Received 28 March 2006; revised 21 May 2006; accepted 21 May2006; posted 30 May 2006 (Doc. ID 69419).
tioned problems with present binary holographic dis-play systems in one step. We show that, for a givennumber of on points in the RPF, increasing the view-ing angle by a factor of 2 incurs a signal-to-noise ratio(SNR) penalty of approximately 3 dB. However, fur-ther increases in viewing angle by increasing thephase-mask resolution results in a negligible addi-tional SNR drop. The same technique allows an in-crease in the number of addressable points in theRPF from P � P to M � M; although this improve-ment comes at the expense of additional RPF noisecompared with a true M � M pixel hologram, the per-ceptual effect is automatically attenuated through useof the OSPR algorithm, which reduces RPF noise vari-ance by displaying multiple subholograms per frameand exploiting ocular temporal integration. Hence theuse of the superresolution phase mask in conjunctionwith OSPR to generate multiple subholograms perframe effectively allows the exchange of spatial fortemporal bandwidth in a holographic display.
2. Conjugate Image Removal Using a BinaryPhase Mask
It was first suggested in the context of optical corre-lators4 that the conjugate image manifest in the RPF,caused by the purely real �0, �� modulation impartedby a binary phase ferroelectric SLM, could be sup-pressed by using a spatially random binary pixelatedphase mask. The principle of operation is described inFig. 1. The phase mask puv, which consists of a pix-elated pattern of the same pitch as the hologrampixels, is placed in close contact and aligned tothe hologram pattern huv displayed on an SLM. If thephase-mask pixels impart phase modulation in theset �0, ��2� rad and the SLM pixels retard the inci-dent light by �0, �� rad, then the effective phase pat-tern of Fig. 1(c) results, with the pixels imparting anet modulation in the set �0, ��2, �, 3��2�. This pro-vides the extra degree of freedom required to sup-press the conjugate image, despite the fact that theSLM itself is binary.
Some example simulated results are shown in Fig.2. A conjugate-symmetric image Fxy is shown in Fig.2(a), which results from the replay of a binary phasehologram huv where Fxy � ��huv� and ��·� is theFourier transform. In Fig. 2(a), however, a hologramhas been designed that accounts for the presence ofthe phase mask puv, giving rise to a RPF Fxy �
��huvpuv� that is devoid of the conjugate image. Thiscomes at the expense, however, of additional RPFnoise.
We performed a series of simulations to investigatethe effect of hologram resolution and number of onpoints in the target RPF on the SNR, with and with-out a phase mask. It was found that, for a givennumber of on points (500 points with random spatialdistribution), doubling the resolution in both direc-tions increases the SNR by around 6 dB. Further-more, for a given hologram resolution, doubling thenumber of on points decreases the SNR by �3 dB. Inboth cases, the utilization of a phase mask can beseen to cause an additional degradation in SNR ofapproximately 3 dB. These relationships are shownin Fig. 3.
We have previously demonstrated the OSPR algo-rithm, which exploits the large temporal bandwidthof a ferroelectric liquid-crystal-on-silica SLM to ame-liorate the perceptual significance of the constantSNR degradation caused by the presence of the phasemask. Hence the use of a phase mask to remove theconjugate image, thereby doubling the number of ad-dressable points in the RPF, is well suited to videodisplay applications. This result motivates the inves-tigation of the possibility of applying this phase-masking technique to ameliorate the other majorproblem with holographic display, namely, that oflimited viewing angle.
3. Viewing Angle Increase Using a SuperresolutionPhase Mask
As noted in Section 2, the inverse relationship be-tween viewing angle � and pixel size � means that, to
Fig. 1. Effective phase pattern formed by hologram and phase-mask pixels when aligned and placed in close contact: (a) hologram patternhuv, (b) phase-mask pattern puv, (c) effective pattern puvhuv.
Fig. 2. RPFs resulting from (a) a binary phase hologram huv and(b) a phase mask and hologram combination huvpuv.
achieve a display with a wide field of view, a hologramwith a small feature size is required. More specifi-cally, the viewing angle � of a hologram illuminatedby coherent light of wavelength � varies inverselywith pixel size according to the equation
� � 2 arctan�
2�. (1)
It is therefore desirable to reduce the pixel size as faras possible, since if the pixels are sufficiently small,then subsequent demagnification optics are effectivelyobviated. For example, at a wavelength of 532 nm,1.2 �m pixels would give rise to a viewing angle of 30deg and a RPF size of 0.44 m � 0.44 m at 1 m fromthe SLM. For a hologram of fixed size this can clearlybe achieved by increasing the resolution, makingeach pixel smaller. However, this becomes increas-ingly difficult to achieve for dynamically addressabledevices due to electrical bandwidth, materials, andfabrication constraints. Indeed ferroelectric liquid-crystal-on-silicon devices, which switch sufficientlyfast to allow display of multiple holographic sub-frames per image frame for the purposes of noisereduction, are fundamentally compromised at smallpixel sizes due to the limitations imposed by the useof FLC material in conjunction with complementarymetal-oxide semiconductor backplane technology. Tomaintain both high fill factor and small pixels, the
switching voltage must scale with respect to pixel sizeto avoid field breakdown across the silicon backplane.However, to switch sufficiently rapidly, FLC materialrequires relatively large switching voltages (approx-imately 15 V) and it is therefore difficult to reduce thepixel size below 10 �m. Even if new low-voltage FLCmaterials became available, a large number of smalldisplay pixels would require an inordinate electricalbandwidth since, for the same pixel geometry, a dis-play just 20 mm � 20 mm in size would require ap-proximately 3 � 108 pixels. Assuming one hologrampattern per monochrome video frame, then using thisdevice to display a frame-sequential full color videostream at 25 frames�s would require a sustaineddata transfer rate of around 23 Gbits�s—well beyondcurrent technology both in terms of storage and elec-trical bus design.
The alternative approach, presented here, is to ex-tend the concept of the phase mask. The phase-masktechnique is extended to demonstrate that a binary�0, �� phase mask prs of resolution M � M, identical inphysical size to that of the P � P pixel SLM but ofgreater resolution, can be used to increase the view-ing angle of the overall system by a factor of approx-imately P�M. This technique increases the number ofaddressable points in the RPF from P � P to M � M,at the cost of introducing additional RPF noise, whichreduces the RPF SNR by 3 dB. However, this SNRdrop is virtually independent of the viewing angleincrease, and, although the RPF noise is greater com-pared with a true M � M pixel hologram, the percep-tual effect can be automatically attenuated by theuse of multiple OSPR subframes. Hence the use ofthe superresolution phase mask in conjunction withOSPR, generating multiple subholograms per frameto reduce noise variance, effectively allows the ex-change of spatial for temporal bandwidth in a holo-graphic display. This approach relaxes the demandsimposed upon the display technology required forhigh-quality 2D and 3D holographic projection.
The OSPR algorithm has previously been pre-sented as a novel method of hologram generation anddisplay, exploiting the human perception of statisti-cal image noise parameters encountered in holo-graphically generated images to generate images ofsubstantially improved quality compared with othercommonly utilized algorithms such as direct binarysearch. The algorithm begins with the specification ofa P � P pixel target intensity image Txy, returning aset of N individual P � P pixel holograms huv
�n�. Eachof the noise fields produced by the N holograms areindependent identically distributed (i.i.d.), and hencewhen N holograms are displayed within the integra-tion period of the eye, ocular temporal averaging oc-curs and the noise variance of the resultant RPFs fallas 1�N.
A modified version of the OSPR algorithm was de-rived to account for the presence of a phase mask andis detailed in Algorithm 1 in Appendix A. Given anM � M target image Txy, the algorithm accounts forthe presence of the superresolution M � M pixel
Fig. 3. Effect of (a) hologram resolution and (b) number of onpixels upon RPF SNR, with and without a phase mask.
phase mask puv to generate 2N distinct P � P pixelholograms huv
�n� where P is an integer multiple of M.When the phase mask and hologram are then placedin close contact and illuminated by coherent light, theresulting RPF Fxy � ��puvhuv� is at the higher reso-lution of M � M, thereby overcoming the conven-tional constraint whereby a P � P pixel hologramcannot be designed to form more than P � P points inthe RPF. This modified algorithm also includes animprovement to generate two holograms exhibitingi.i.d. RPF noise per Fourier transform, in comparisonwith the original algorithm that produced just one.Step 1 of Algorithm 1 forms N targets equal to theamplitude of the supplied intensity target Txy, butwith i.i.d. uniformly random phase. Step 2 computesthe inverse Fourier transform, taking into accountthe presence of the phase mask puv, to produce con-tinuous complex holograms of size M � M. Step 3averages the resultant complex hologram over blocksof size F � F to produce suv
�n�, a set of N averagedcomplex hologram fields of the required size P � P.Two independent holograms are then generated insteps 4 and 5 from each single complex hologram fieldsuv
�n�. Binarization of these holograms is then per-formed in step 6, as per previous implementations ofthe OSPR algorithm; thresholding around the me-dian of muv
�n� ensures that equal numbers of �1 and1 points are present in the holograms, achieving dcbalance (by definition) and also minimal reconstruc-tion error.5 Details of the derivation of the new algo-rithm are given in Appendix A.
4. Binary Phase Mask for Viewing Angle Increase
To demonstrate the efficacy of the modified OSPRalgorithm, a target image was selected and singlebinary �0, �� 256 � 256 holograms were generatedfor use with and without a binary �0, �� 512 � 512phase mask. The simulated RPFs—both of whichhave the same physical dimensions—are shown inFig. 4.
From Fig. 4(a) it can be seen that the viewing angleis limited by the presence of side orders in the RPF.The area inside the rectangle is the first-order dif-fraction pattern determined by the pixel pitch of thehologram; outside the rectangle are the repeatedhigher orders, which cannot be controlled by the ho-
logram alone. When the phase mask is employed asshown in Fig. 4(b), the addressable RPF area, andhence the viewing angle, is doubled in each dimen-sion. However, a drop in SNR is observed due to thepresence of the superresolution phase mask. We havealready shown how SNR is affected by the number ofon points in the RPF and by hologram resolutionwhen no phase mask is present. In Fig. 5 we proceedto show how the SNR is also affected by the phase-mask resolution for a 256 � 256 OSPR-generatedhologram �N � 1� forming a 500-point RPF.
Although a significant decrease in SNR of 3.6 dBresults from the doubling of the viewing angle usinga 512 � 512 phase mask, we find surprisingly thatfurther increases in the viewing angle result in anegligible subsequent decrease in SNR. For example,increasing the viewing angle by 12 times (by using a3072 � 3072 phase mask) results in an SNR drop ofjust 4.1 dB compared with no phase mask at all. ThisSNR degradation of an image reconstructed from ahologram and �0, �� phase-mask combination can becharacterized by examining three different cases cor-responding to no viewing angle increase, where thehologram and phase mask are the same resolution�F � 1� and there are viewing angle increases of twoand greater �F 2�.
When the hologram and phase-mask resolutionsare equal, then F � 1 and there is no increase in theviewing angle. If the hologram is designed usingAlgorithm 1, the hologram pixels will be chosen toaccount for the presence of the �0, �� phase mask andhence there is no fundamental difference in the re-play of a hologram and the replay of a hologram and�0, �� phase-mask combination of the same resolu-tion. Hence, any RPF noise is entirely due to thebinarization of the hologram. When F � 2, however,the viewing angle is doubled and twice as many pixelsin the RPF can be addressed. As a consequence, thenoise energy is doubled compared with the no phase-mask case and a 3 dB reduction in SNR results. Afurther doubling of the viewing angle would require afourfold increase in the phase-mask resolution, caus-ing an additional doubling of noise energy. However,the noise energy increase is accompanied by a further
Fig. 4. Simulated RPFs produced by 256 � 256 holograms (a)without and (b) with a 512 � 512 phase mask.
Fig. 5. SNR variation with a resolution of �0, �� phase maskemployed in conjunction with 256 � 256 pixel holograms.
quadrupling of the RPF addressable area so that theadditional SNR degradation is much less than 3 dB.
To demonstrate the operation of the superresolu-tion phase-mask technique, binary phase diffractiveoptical elements (DOEs) were made to simulate theoperation of a SLM and phase mask. A 512 � 512pixel hologram DOE was calculated using the OSPRalgorithm with N � 1, accounting for the presence ofa spatially random binary phase mask. The pixelsize was 40 �m, so that the active area was approx-imately 20 mm. The spatially random phase-maskDOE was designed to contain 1024 � 1024 pixels,each of size 20 �m, resulting in a viewing angle in-crease of F � 2. Both DOEs were made by e-beametching of a 1.5 mm thick fused-silica substrate ofrefractive index 1.46, with a step height of 0.65 �m tooptimize operation at a wavelength of 532 nm.
The phase-mask and hologram DOEs were placedin close contact and carefully aligned using customoptical mounts so that the pixels were accuratelyregistered. The elements were illuminated using a10 mW green laser coupled into a single-mode fiber,and since the diffraction angle from the hologram andphase-mask combination was small, an objective lensof focal length 100 mm was used to image the result-ant RPF onto a complementary metal-oxide semicon-ductor sensor array of active area 25 mm � 25 mm.The simulated and captured RPFs are shown in Figs.6(a) and 6(b), respectively; the AJC � EB image cor-responds to the first-order diffraction pattern thatwould occur from a hologram with pixel size 40 �m,whereas the EXTRA pattern is due to the increasedviewing angle provided by the 20 �m pixels of thephase mask. If a 512 � 512 pixel hologram had beenused alone, the space occupied by the EXTRA imagewould have contained the overlapping secondary or-ders of Fig. 4(a).
We have shown that successful removal of the conju-gate image can be accomplished with a �0, ��2� ran-dom phase mask and that the viewing angle can beincreased through use of another random superreso-lution phase mask, albeit with pixels imparting phase
modulation in the set �0, ��. Utilization of each maskindividually incurs a SNR penalty of approximately3 dB. A reasonable conjecture is that the use of botha �0, ��2� mask and a �0, �� mask in an optical sys-tem would facilitate simultaneous removal of the con-jugate image and an increase in viewing angle. Sincethe optical system is linear, the total phase modula-tion imparted on the incident wave by both maskstogether is equal to the sum of the phase shifts im-parted by each mask. As a result, we can use a singlerandom quaternary phase mask whose phases liein the set �0, ��2, �, 3��2�. Surprisingly, combiningthese masks into a �0, ��2, �, 3��2� mask to bothdouble the viewing angle and remove the conjugateimage does not give the expected 6 dB penalty, butinstead results in a SNR degradation of only 3.8 dBapproximately. This general result is found to holdindependent of resolution or on-pixel count, as isshown in Fig. 7.
We find, as established with the �0, �� mask, thatfurther increases in viewing angle can be achieved atlittle expense in terms of RPF SNR degradation. Thisis demonstrated for a variety of hologram and phase-mask resolutions in Fig. 8.
Fig. 6. (a) Simulated and (b) measured performance of a 512 �512 pixel hologram used with a 1024 � 1024 pixel phase mask todouble the viewing angle.
Fig. 7. SNR degradation caused by use of different types of phasemask. Case 1, no phase mask; case 2, conjugate image removal usinga �0, ��2� phase mask; case 3, �0, �� superresolution phase maskused to double the viewing angle; case 4, conjugate image removaland viewing angle doubling using a quaternary �0, ��2, �, 3��2�phase mask.
Fig. 8. SNR variation with various hologram and phase-maskresolutions for a RPF with 500 on pixels.
6. Compensation for Additional Noise by One-StepPhase Retrieval
We have shown that use of the OSPR technique toincrease the viewing angle and remove the conjugateimage introduces a fixed but significant drop inSNR of around 3 dB. We propose that, as previouslyshown,1 the perceptual significance of the RPF noisecan be substantially attenuated by generating mul-tiple subframes using the OSPR algorithm withN 1. Ocular temporal averaging is then exploited toyield a significantly reduced perceived noise level,even though the SNR is not improved.
We demonstrate in simulation the perceived de-crease in RPF noise that results for different numbersof subframes N generated by the OSPR algorithm andsubsequently displayed sequentially within a 1�25 stime interval. Figure 9 demonstrates that increasingthe number of subframes N significantly improvesthe perceived quality of the resultant RPF image,despite the fact that the SNR varies negligibly be-tween each of the images.
7. Manufacture of the Quaternary Phase Mask
Although we have shown that the combination of aquaternary phase mask and a binary SLM can be uti-lized in conjunction with OSPR to produce a wide view-ing angle, a conjugate image-free display with a RPF ofhigh perceptual quality, we have so far not addressedthe issues involved in manufacturing such a mask.
Previous efforts in manufacturing random pixelatedphase masks have proceeded by first producing a pre-mask as a binary amplitude pattern, either printed onplastic or as a chrome-on-glass structure. UV-sensitivephotoresist is then spun onto glass and the requiredphase pattern is produced photolithographically using
the premask, exposing the photoresist for the neces-sary duration to form the required phase steps. How-ever, production of a four-phase mask using thistechnique is complicated, first because this requiresvariable amplitude levels on the premask and secondbecause these amplitude levels need to be preciselycalibrated with respect to the exposure process to en-sure that the required depths of the phase step areobtained.
However, we have determined that it is possible toproduce a suitable four-phase mask using just a sin-gle random binary amplitude premask, which can beproduced using conventional processes. We begin bydetermining the exposure times t and t�2 necessaryto form steps of � and ��2, respectively, which can beachieved by examining the surface profile using ascanning surface profilometer (Dektak) for differentexposure times. We then spin photoresist onto glassas before and UV expose the mask for the duration t�.Under red light, so as not to expose the photoresistfurther, we rotate the premask by 90 deg and realignit with the previously exposed pattern. Exposure fora further time of t��2 produces the required four-phase mask. It has been shown in simulation that amask produced in this way, while not truly spatiallyrandom in phase, exhibits a sufficient spatial decor-relation between pixels to ensure that the reconstruc-tion error compared to a truly random quaternaryphase is minimal.
8. Extension to Three-Dimensional Image Display
One of the merits of this approach is the ability, for afixed available display data bandwidth, to substan-tially increase the viewing angle in a holographicdisplay without incurring substantial degradation in
Fig. 9. Simulated RPFs produced by OSPR Algorithm 1 using a quaternary phase mask for different numbers of hologram subframes N.
image quality. Since a very limited viewing angle isone of the major problems with 3D holographic dis-plays created thus far, we believe that this approachconstitutes a significant advance toward an effectivewide viewing angle 3D holographic display that isfeasible with current technology.
We consider the problem of designing a 3D displayof physical size 10 mm � 10 mm providing a viewingangle of around 14 deg, which would require 4096 �4096 pixels of 2.5 �m in size, necessitating a databandwidth of 16 Mbits�frame. Instead, we proposemanufacturing a fixed 10 mm � 10 mm quaternaryphase mask of 4096 � 4096 pixels and employing inconjunction a 10 mm � 10 mm SLM of only 1024 �1024 pixels, each of size 10 �m. Such a display isreadily available from ForthDD displays, part num-ber SXGA-R2-H1. With this architecture, we pre-serve the viewing angle of 14 deg but now require adata bandwidth of only 1 Mbit�frame.
As a demonstration of the efficacy of this approach,we calculated both a single standard 4096 � 4096hologram and also a single 1024 � 1024 hologramusing a 3D variant of OSPR with N � 1 and a 4096 �4096 quaternary phase mask. The 3D scene used inboth cases was a set of 944 point sources that formeda wire-frame cuboid close to the hologram plane.Small sections of the simulated RPFs reconstructedfrom each hologram, obtained by propagating Huy-gens wavelets through a pinhole aperture, are shownin Fig. 10.
As can be seen, Fig. 10(a) is of comparable quality toFig. 10(b), despite the 16-fold reduction in hologrambandwidth, as expected from the results of previoussimulations. While Fig. 10(a) does display an increasednoise level, this can, as before, be compensated for byusing the necessary number of OSPR subframes.
9. Conclusion
We have demonstrated methods, using pixelated bi-nary phase masks, of mitigating the two remainingissues of holographic displays, namely, the presence ofthe conjugate image when employing a real-valuedSLM and the limited viewing angle. The conjugateimage could be suppressed by employing a phase maskwith pixels that imparted phase modulation in the set�0, ��2�, at the expense of a 3 dB drop in RPF SNR.
A novel approach of increasing the viewing anglewas presented, utilizing a superresolution fixed pix-elated phase mask that imparted �0, �� modulation.It was found that, although doubling the viewingangle incurred a 3 dB drop in contrast, the viewingangle could be increased arbitrarily with only a smallfurther decrease in contrast. An increase in viewingangle of 12 times was demonstrated with a resultantSNR decrease of just 4.1 dB.
Finally, it was demonstrated that these improve-ments could be combined into one quaternary phasemask, providing increased viewing angle and con-jugate image suppression for a SNR penalty ofapproximately 4 dB, regardless of RPF content. Thehologram-generating algorithm was able to accountfor the presence of such a mask, so that only one maskis required for all hologram patterns. This technologytherefore provides a dramatic increase in viewing an-gle for 2D and 3D holographic displays without theneed to construct SLMs with large numbers of verysmall pixels or huge electrical bandwidth require-ments.
Appendix A. Derivation of Modified One-Step PhaseRetrieval Algorithm
In this Appendix, we detail the modifications tothe original OSPR algorithm (see Algorithm 2) thatenable two output holograms to be formed perFourier-transform operation instead of one. We beginby quoting the original algorithm, which generatessets of N distinct M � M binary phase holograms huv,each producing a RPF with i.i.d. noise that approxi-mates the same target image. The algorithm beginswith the specification of a target intensity image Txy
and proceeds as follows.Step 1 forms N targets Txy
�n� equal to the amplitudeof the supplied intensity target Ixy, but with i.i.d. uni-formly random phase. Steps 2 and 3 shift the inverseFourier transform holograms by a large distance to theright in the complex plane. This has the effect of mak-ing the phase of each point in the holograms verysmall, so that when we take their magnitude in step 4(forcing the phase of every point to zero), we introducepractically no error. However, we note that
lim�→�� � � guv
�n� � � ��guv
�n�� 1; (A1)
so for a large �, we can replace steps 3 and 4 with
muv�n� � � � � �guv
�n�. (A2)
Binarization of the hologram is then performed instep 5 by thresholding about the median of muv
�n� asbefore. However, thresholding � � ��guv
�n� around itsmedian is equivalent to thresholding just ��guv
�n�around its median.
The statistical independence between the real andthe imaginary parts of the discrete Fourier transformcan now be exploited to calculate a second hologrammuv
�n�N� � ��guv�n� and its binarization huv
�n�N� that,
Fig. 10. Simulated RPFs generated from (a) 1024 � 1024 holo-gram with 4096 � 4096 random quaternary phase mask and (b)true 4096 � 4096 hologram.
being uncorrelated with the first hologram huv�n�, gen-
erates a RPF whose noise is also independent fromthe first. As a result, using just a single Fourier trans-form, a pair of binary holograms can be generated—both of which exhibit i.i.d. noise in their respectivereconstructions, satisfying the condition required forthe central limit theorem to apply.
Algorithm 1. Modified OSPR algorithm for thegeneration of P � P pixel holograms, given the pres-ence of a supplied M � M pixel phase mask. When thephase mask and hologram are placed in close contactand illuminated by coherent light, the resulting RPFis at the higher resolution of M � M.
1. Let Txy�n� � �Ixy exp� j�xy
�n��, where �xy�n� is uni-
formly distributed between 0 and 2� and n ��1, . . . , N�, x, y � �1, . . . , M�.
2. Let guv�n� �
1puv
��1�Txy�n��, where puv is the
phase mask and u, v � �1, . . . , M�, with M P.3. Let suv
�n� � �k�1F �l�1
F gFu�k�1,Fv�l�1�n�, where F
� M�P.4. Let muv
�n� � ��suv�n�, for n � �1, . . . , N�2� and
u, v � �1, . . . , P�.5. Let muv
�n�N�2� � ��suv�n�, for n � �1, . . . , N�2�
and u, v � �1, . . . , P�.6. Let
huv�n����1 if muv
�n� � Q, where Q � median �muv�n��
1 if muv�n� Q, and n � �1, . . . , N�
.
Algorithm 2. OSPR algorithm.
1. Let Txy�n� � �Ixy exp� j�xy
�n��, where �xy�n� is uni-
formly distributed between 0 and 2� and x, y ��1, . . . , N�.
2. Let guv�n� � ��1�Txy
�n��, where ��1 represents the2D inverse Fourier-transform operator.
3. Let R be the smallest positive real value suchthat |guv
�n�| � R∀ u, v, n.4. Let muv
�n� � |� � guv�n�|, where � � � with
� �� R.5. Let
huv�n����1 if muv
�n� � Q, where Q � median �muv�n��
1 if muv�n� Q
.
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