Fixed-length two-dimensional modulation coding for imaging page-oriented optical data storage systems

Fixed-length two-dimensional modulation codingfor imaging page-oriented optical data storage systems

Dhawat E. Pansatiankul and Alexander A. Sawchuk

We present a comprehensive discussion of modulation coding for page-oriented optical data storage�PODS� systems that write and read data in a two-dimensional �2-D� bit image format. We give several2-D mathematical models for these systems, including two-photon optical data storage systems. Usingthese models, we describe the nature of intersymbol interference �ISI� in imaging PODS systems and findthat its characteristics are different from ISI in conventional serial magnetic and optical data storagesystems. To overcome the ISI in these imaging PODS systems, we present what is, to our knowledge,a novel 2-D modulation coding scheme. We also present many examples of fixed-length 2-D modulationcodes with diverse properties. Finally, we analyze and compare the bit-error rate performance of thesecodes. © 2003 Optical Society of America

OCIS codes: 210.0210, 210.4680, 200.3050, 070.6020.

1. Introduction

New multimedia and Internet services such as high-definition television �HDTV�, real-time archival dig-ital video�audio, and future haptics �touch-related�applications increasingly demand enormous amountsof data storage capacity and massive data transferrates. Improvements in conventional serial, singlechannel data storage technologies—magnetic harddisk, compact disk �CD�, and digital versatile disk�DVD�—have managed to keep pace with such re-quirements. However, strong evidence indicatesthat these planar or quasi-planar data storage tech-nologies are approaching fundamental limits thatmay be difficult to overcome.1 An alternative candi-date for next-generation data storage systems ispage-oriented optical data storage �PODS� technol-ogy. By using this volumetric data storage technol-ogy, a system with very large data storage capacity��1 Tbits�cm3� and very high data transfer rate ��1Gbits�s� is theoretically achievable.2,3

Figure 1 is a block diagram of a typical data storage

The authors were with the Signal and Image Processing Insti-tute, the University of Southern California, Los Angeles, Califor-nia 90089-2564 when this work was done. A. Sawchuk’s e-mailaddress is [email protected]. D. E. Pansatiankul is now withTRW Inc., Redondo Beach, California 90278.

Received 29 June 2002; revised manuscript received 17 Septem-ber 2002.

0003-6935�03�020275-16$15.00�0© 2003 Optical Society of America

system. These systems generally have two levels ofdata coding: modulation coding and channel coding.Modulation coding accepts digital data as input andprovides signal encoding for writing on the storagemedium. The modulation decoder reverses this pro-cess for readout. Channel encoding and decodingoperate on digital signals and provide additional er-ror detection and correction through the use of paritychecks and interleaving. The overall performance ofa data storage system in terms of bit-error rate�BER�, storage capacity, bit density, size, pitch, andother parameters depends on modulation coding,channel or error-correcting coding, and many otherphysical factors.

Unlike conventional data storage systems, wherethe recording and retrieving processes are performedserially, the recording and retrieving processes ofPODS systems are performed in parallel, implyingthat conventional modulation codes such as the (2,7) runlength-limited variable-length code in IBMdisk storage systems,4 eight-to-fourteen modulation�EFM� code in CD,5 and EFMPlus code in DVD6 maybe inappropriate for PODS systems. For PODSsystems, several technologies exist includingholography,7–9 spectral-hole burning,10–12 and two-photon absorption.13–16 While holography requirescoherent recording of a two-dimensional �2-D� signalrelated to a transform of the original 2-D binary dataarray, in this paper we concentrate on systems withtwo-photon or other mechanisms that incoherentlywrite and read 2-D arrays of data stored as imagesrather than as transforms.

10 January 2003 � Vol. 42, No. 2 � APPLIED OPTICS 275

Even though there has been considerable studyof modulation coding for holographic PODSsystems,17–23 none, to our knowledge, has been inves-tigated for PODS systems that use 2-D imaging tech-nology. In general, modulation codes designed forholographic PODS systems are not applicable to im-aging PODS systems owing to their distinguishabledifferences in recording�retrieving mechanisms.This paper contains a comprehensive analysis of aclass of fixed-length 2-D modulation codes suitablefor imaging PODS systems. The codes are calledfixed-length 2-D codes because they map one-dimensional �1-D� input data blocks of fixed size to2-D output data blocks of fixed size. The codes de-veloped here are extremely general and are applica-ble to any page-oriented incoherent optical recordingor readout system. A more comprehensive analysisof specific systems will follow in later papers.

In Section 2 the image formation process in PODSsystems that use two-photon absorption technologyis particularly described, leading to various math-ematical models for typical imaging PODS systems.Using these mathematical models, we present inSection 3 the nature of intersymbol interference�ISI� in imaging PODS systems and show that itscharacteristics are different from ISI in conven-tional data storage systems. To overcome the in-herent ISI in imaging PODS systems, we propose inSection 4 what is to our knowledge a novel 2-Dmodulation coding scheme and present a number ofexamples of fixed-length 2-D modulation codes withdiverse properties. The general form of thesecodes is also described in this section. In Section 5,the BER performance of these 2-D modulation codesis examined and compared. The conclusions arepresented in Section 6.

2. Mathematical Models of Imaging PODS Systems

In two-photon PODS systems data recording is per-formed using two propagating light beams. Onebeam, carrying the information to be stored, is im-

aged onto a desired plane in the volumetric mediumwhile a second beam, specifying the location, is si-multaneously focused throughout the same plane.The former is called the information beam whereasthe latter is called the addressing beam. With suf-ficient photon energies, a 2-D array of data marks�or spots� can be recorded on a plane �or page� wheretwo beams intersect.15 The typical number of datamarks per page is 103 or more, perhaps as many as106, and, generally, a very large number of datapages are stored in the medium. To retrieve therecorded data, the proper data page is illuminatedby one readout beam, which is identical to theaddressing beam used to record that page. Thisillumination results in a fluorescence pattern thatis imaged onto a charge-coupled device �CCD� cam-era or a 2-D integrated optoelectronic detector ar-ray. Figures 2 and 3 show the recording andretrieval of information in a two-photon PODS sys-tem, respectively.

Assuming that the distance between any two suc-cessive recorded data pages is large enough thatthere is no interaction between the data pages, thena mathematical model of data retrieval from a re-corded data page in a typical two-photon PODSsystem is shown in Fig. 4. In this mathematicalmodel �i, j� and �x, y� denote an index of a 2-Ddiscrete signal and an index of a 2-D continuoussignal, respectively. Let a�i, j�, whose value is as-sumed to be either zero for a 0 bit or one for a 1 bit,represent the existence of a digital information bit�or data mark� recorded on a data page. Since thereadout intensity level of each data mark is com-posed of the superposition of light from a large num-ber of statistically independent fluorescenceelements, each radiating with a slightly differenttemporal frequency and a random phase, it is clearthat the output of each data mark during the re-trieving process is spatially incoherent. There-fore, ideally, the point-spread function �PSF�, or theimpulse response, of this imaging system can be

Fig. 1. Block diagram of a typical data storage system.

276 APPLIED OPTICS � Vol. 42, No. 2 � 10 January 2003

modeled mathematically as one of the followingfunctions:

h� x, y� �1

bs2 sinc2� x

bs,

ybs� , (1)

where sinc� x, y� �sin��x�

�xsin��y�

�yor

h� x, y� �1

�bc2 �J1�2�� x2 � y2�1�2�bc�

� x2 � y2�1�2�bc�2

, (2)

where J1 is a Bessel function of the first kind, order 1.Equation �1� arises from a diffraction-limited imag-

ing system containing a square aperture, where bs,

which is inversely proportional to the width of theaperture, represents the location of the first zero of thesinc2 PSF. Similarly, Eq. �2� is obtained from adiffraction-limited imaging system containing a circu-lar aperture, rather than a square, where bc is a pa-rameter inversely proportional to the radius of theaperture. Note that the J1

2 PSF is circularly sym-metric, but its zeroes are not equally spaced in radius.It should also be emphasized that the leading terms1�bs

2 in Eq. �1� and 1��bc2 in Eq. �2� are the factors,

simply derived using the Parseval’s theorem, whichnormalize the volumes of the corresponding PSFs toone.

If an imaging system response is composed of thecascade of many small independent aberrations,

Fig. 2. Recording of information in a two-photon PODS system.

Fig. 3. Retrieval of information from a two-photon PODS system.

Fig. 4. Mathematical model of a two-photon PODS system.


then, from the central limit theorem, the overall PSFcan be modeled by a Gaussian function

h� x, y� �1

2�bg2 exp��

x2 � y2

2bg2 � , (3)

where bg denotes the standard deviation. In addi-tion, we note that other PSF models can also be usedto describe the imaging optics of a particular system.For simplicity, the magnification of the imaging sys-tem in the retrieving process is assumed to be one inour mathematical model. Furthermore, the pitchbetween two nearest data marks, equivalently thepitch between two nearest detectors or the pixel pitchof the CCD camera, is set to one so that all otherdistances can be relatively expressed in units of thispitch. We also note that the size of the data mark isassumed to be very small compared with the extent ofthe PSF, implying that the data mark is effectively adelta function. The PSFs in Eqs. �1�, �2�, and �3�thus include the finite size of the recorded data markand the response of the readout imaging system.From the assumptions, the received intensity at eachdetector, or each CCD pixel, of a noiseless retrievalsystem can be modeled by

r�i, j� � a�i, j� � h�i, j�, (4)

where R represents a 2-D discrete convolution oper-ator and h�i, j� is the effective discrete PSF. Becausethe retrieved photons are integrated over the activearea, assumed to be a square, of each detector or CCDpixel, the effective discrete PSF h�i, j� in Eq. �4� isdefined as

h�i, j� � �j��2

j�2

�i��2

i�2

h� x, y�dxdy, (5)

where is a linear fill factor of a detector or a CCDpixel, whose value is between zero and one, and h�x,y� is one of the continuous PSFs in Eqs. �1�, �2�, and�3�.

However, in general, a retrieval system is notnoiseless. It is assumed here that the retrieval sys-tem is noisy, mainly due to the electronics at thedetector plane. Because the noise from electroniccircuits, often called thermal noise, is typically mod-eled as additive white Gaussian noise �AWGN�,24 thereceived intensity of a noisy retrieval system can bedescribed by

r�i, j� � a�i, j� � h�i, j� � w�i, j�, (6)

where w�i, j� is an AWGN, whose mean and varianceare zero and �w

2, respectively. Finally, we assumethat a simple binary threshold decision scheme isapplied at the CCD or detector array, with a prioriknowledge of a threshold value, T, to digitize thereceived intensities. Therefore the detected data,

i.e., the output of the retrieval system, can be mod-eled mathematically as

d�i, j� � �1, if r�i, j� � T0, if r�i, j� � T . (7)

It should be noted that, for the purpose of analysis,it is often helpful to regard the 2-D discrete signals asmatrices. For example, Eq. �6� can be expressed inmatrix form by

R � A � H � W, (8)

where R is an M � N received intensity matrix whoseentry in the pth row and qth column, Rpq, describesthe real-valued intensity at the �p, q�-th detector orthe �p, q�-th CCD pixel, A denotes the recorded M �N digital information data page in which each entry,Apq, is either zero or one, H is the effective PSFmatrix derived from h�i, j� in Eq. �5� with minor mod-ifications of indices so that h�0, 0� is the entry at thecenter of matrix H, and W represents an M � NAWGN matrix whose entries are independent, iden-tically distributed with mean and variance defined inEq. �6�. We also emphasize that the mathematicalmodels described above can be applied not only totwo-photon PODS systems, but also to any PODSsystems that require incoherent recording and re-trieving of the images �not transforms as in holo-graphic PODS systems� of the original 2-D arrays ofdata.

3. 2-D Intersymbol Interference in Imaging PODSSystems

From our mathematical model a PSF defines the im-age intensity of each data mark at the detector plane.Figures 5 and 6 illustrate, for example, the continu-ous PSFs in Eq. �1� with bs 1.5 and Eq. �3� with bg 0.6, respectively. The x and y axes, as mentioned inthe previous section, are in units of the data markpitch or, equivalently, the detector or CCD pixelpitch. It is assumed in Figs. 5 and 6 that only onedata mark, i.e., only one 1 bit, is recorded at �x, y� �0, 0� �of the data plane� and hence the correspondingdetector or CCD pixel is located at the same coordi-nate �of the detector plane�. Suppose the linear fillfactor is one, the active area of a detector or a CCDpixel is, then, a square, whose width and height isequal to one, centered at �x, y� �0, 0� as shown inFigs. 5 and 6. Note that the light originating from adata mark in Figs. 5 and 6 is not completely imagedonto the corresponding detector or CCD pixel. Someof the light is incident onto the local neighbor detec-tors or CCD pixels. This phenomenon is referred toas ISI or, more specifically, the 2-D ISI in which lightfrom a data mark interferes spatially with its localneighbors. In fact, the 2-D ISI results from the in-herent lowpass nature of the imaging system. Ad-ditionally, of particular importance to consider is thefact that the 2-D, or spatial, ISI in two-photon orother imaging PODS systems is somewhat differentfrom the 1-D, or temporal, ISI in today’s commercialdata storage systems, such as CD and DVD. We


note that there is no time-dependent filtering in theimaging PODS systems due to scanning of a single-channel signal on a detector as in the conventionalsystems.

It should be clear from the mathematical modelthat the amount of light from a data mark that isincident onto the neighbor detectors or CCD pixels,

i.e., the 2-D ISI, depends on both the PSF and thelinear fill factor, , of the retrieval system. Indeed,for a given value of , the relative degree of theeffects of 2-D ISI can be simply indicated by thevalue of bs, bc, or bg. As bs, bc, or bg increases, theamount of light from a data mark that is imagedonto the corresponding detector or CCD pixel de-

Fig. 5. sinc2 PSF with bs 1.5.

Fig. 6. Gaussian PSF with bg 0.6.


creases. Consequently, with the fact that the vol-ume of each PSF is normalized to one, theremaining light from the same data mark that isincident onto the neighbor detectors or CCD pixels,i.e., the 2-D ISI, increases. However, for a givenvalue of bs, bc, or bg, although the amount of lightfrom a data mark that is integrated by the corre-sponding detector or CCD pixel increases with in-creasing the linear fill factor, the amount of theremaining light from the same data mark that isintegrated by the neighbor detectors or CCD pixels,i.e., the 2-D ISI, also increases.

Figures 7 and 8 depict more convenient ways tovisualize the 2-D ISI in imaging PODS systems. As-suming that a retrieval system is noiseless and itsimaging system contains a square aperture, Fig. 7�a�shows a 1-D slice through the x axis of the imageintensity that originates from two data marks re-corded at �x, y� ��1, 0� and �x, y� �1, 0�. In other

words, Fig. 7�a� shows the sum of h�x 1, 0� andh�x � 1, 0� at the detector plane, where h�x, y� is fromEq. �1� with bs 1.5. Clearly, the tails of the imageintensity from each data mark �each 1 bit� interferewith others �other 0 and 1 bits�. In addition, Fig.7�b� shows the corresponding received intensity,which is determined from Eqs. �4� and �5� with 1,of each detector or CCD pixel that lies on the x axis,i.e., r�i, 0�. Because only two data marks or two 1bits are recorded, the digitized outputs of all detectorsor CCD pixels, except the ones located at �i, j� ��1,0� and �i, j� �1, 0�, are ideally expected to be 0 bits.However, in Fig. 7�b�, the output of the detector orCCD pixel at �i, j� �0, 0� is ambiguous because itsintensity level is approximately half of the �1 bit�intensity levels at �i, j� ��1, 0� and �i, j� �1, 0�.Depending on a predetermined threshold value T, thedetector or CCD pixel at �i, j� �0, 0� may improperlymake a decision leading to an incorrect output. It is

Fig. 7. �a� h�x 1, 0�, h�x � 1, 0�, and their sum �sinc2 PSF with bs 1.5�, �b� r�i, 0� �sinc2 PSF with bs 1.5 and 1�.

Fig. 8. �a� h�x 1, 0�, h�x � 1, 0�, and their sum �Gaussian PSF with bg 0.6�, �b� r�i, 0� �Gaussian PSF with bg 0.6 and 1�.


important to emphasize that if the total number ofdata marks or 1 bits around the 0 bit at �x, y� �0, 0�is more than two, then the effects of 2-D ISI on that0 bit is even worse. Figure 8 illustrates the similarresults for the case of Gaussian PSF with bg 0.6.

4. Fixed-Length 2-D Modulation Codes for ImagingPODS Systems

This section contains four subsections. We pro-pose in the first subsection what is to our knowledgea novel 2-D modulation coding scheme to mitigatethe effects of spatial ISI in imaging PODS systems.In the second subsection we present an example ofa 2-D modulation code, derived from the proposed2-D modulation coding scheme, and its performanceimprovement. A general form of typical 2-D mod-ulation codes is then described in the third subsec-tion. The last subsection shows several examplesof fixed-length 2-D modulation codes with variousproperties.

A. Basic Principles of the 2-D Modulation CodingScheme

Obviously, from Section 3, the simplest way to avoid2-D ISI due to the imaging system in any two-photon or other imaging PODS system is to sepa-rate the data marks and, consequently, the activeareas of the detectors or CCD pixels as far asneeded. However, since the data mark pitch is in-creased, the utilization of the medium decreases.Moreover, because the detector pitch or the CCDpixel pitch is increased while the size of the activearea is fixed, the linear fill factor decreases.

Instead of increasing the data mark pitch or de-tector pitch, it is suggested that modulation codingbe applied to reduce the effects of 2-D ISI and,perhaps, increase the useful data storage capacityof an imaging PODS system. As discussed in theprevious section, 2-D ISI is regarded as the lightfrom a data mark or a 1 bit that spatially interfereswith other local neighbors, especially the 0 bits.Hence intuitively it is desirable to record the data inan imaging PODS system such that any 0 bit is notsurrounded by too many 1 bits. In contrast, any 1bit is preferred to be surrounded by many 0 bits,although 1 bits can be recorded closely. As an ex-ample, with the same total number of 1 bits, Eq. �9�

is the pattern of data that can relax the effects of2-D ISI more than the one in Eq. �10�.

0 0 0 0 0 00 0 0 0 1 00 1 1 0 0 00 1 1 0 0 00 1 1 0 0 10 0 0 0 0 0

, (9)

0 0 0 0 0 00 0 0 0 0 00 1 1 1 0 00 1 0 1 0 00 1 1 1 0 00 0 0 0 0 0.

(10)

While the maximum number of 1 bits around anysingle 0 bit in Eq. �10� is eight, there are, at most,three 1 bits surrounding any single 0 bit in Eq. �9�.The maximum value of the total amount of lightfrom every 1 bits that interfere with any specific 0bit in Eq. �9� thus should be less than that in Eq.�10�. Note that the light from a 1 bit does notdegrade the intensity signals of other neighbor 1bits; in fact, it adds to the intensity signals of those1 bits.

Alternatively, because the 2-D ISI can be viewedas a phenomenon that arises from the lowpass na-ture of the imaging system, the recorded data isexpected to have low spatial frequency componentsas much as possible. In other words, it is prefera-ble that the recorded data has a minimum of highspatial frequency components so that the lowpasscharacteristics of the imaging system do not signif-icantly affect the recorded information.25 In gen-eral, because it is known that any 1 bit locallyinterferes with other 0 bits in imaging PODS sys-tems, a 3 � 3 data block with high spatial frequencycontent is defined as one having a 0 bit in the middlesurrounded by many 1 bits. Similarly, a 3 � 3 datablock with lower spatial frequency content is de-fined as one having a 0 bit in the middle surroundedby fewer 1 bits. For instance, the patterns of datain Eq. �11� are considered to have higher spatialfrequency components compared with the ones inEq. �12�.

1 1 1 0 1 0 0 0 0 0 1 0 0 1 1 1 0 1 0 11 0 1 0 0 0 1 1 1 1 0 0 0 1 0 1 1 0 1 01 1 0 1 0 1 0 1 1 0 0 1 0 1 1 1 0 1 0 10 0 1 0 0 0 1 1, 1 1 1 0, 0 0 0 0, 1 0 1 0

, (11)

0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 1 1 10 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 1 1 10 0 0 0 0 0 1 0 0 1 0 0 0 1 1 0 1 1 1 10 0 0 0, 0 0 0 0, 0 0 0 0, 0 0 0 0, 1 1 1 1

. (12)


In conclusion, to reduce the effects of 2-D ISI intwo-photon or other imaging PODS systems, we pro-pose what is to our knowledge a novel 2-D modulationcoding scheme. Its main principle is to map a 1-Dinput sequence of information into a 2-D output datain such a way that the spatial frequency content ofthe output pattern, i.e., the number of 1 bits arounda 0 bit in any 3 � 3 block of data, satisfies the nec-essary low spatial frequency constraints to achievethe required system performance.

B. Example of a 2-D Modulation Code and ItsPerformance Improvement

This subsection describes a particular example of a2-D modulation code, derived from our 2-D modula-tion coding scheme, to illustrate possible performanceimprovements. This 2-D modulation code performsthe simple, but nontrivial, input–output mapping26

�a0 a1 a2 a3 a4 a5 a6�3 �0 a1 0 0a0 X a3 a5

0 a2 0 00 a4 0 a6

� .

(13)

Suppose that there is a lengthy sequence ofinformation bits so that each 1-D block of seveninformation bits is mapped into a 2-D block of4 � 4 bits. If the total number of 1 bits in�a0 a1 a2 a3� is greater than two, then X is 1;otherwise X is 0. Figure 9 shows histograms of thereceived intensities at the detector array or CCDcamera when 0 bits and 1 bits are randomly page-wise recorded in the medium. It is assumed thateach data page contains 120 � 120 bits, 100 pagesof data are recorded, each detector or CCD pixel hasa unit linear fill factor, and the retrieval system,containing a square aperture with bs 1.5, is noise-less, i.e., no AWGN is considered. Clearly, the his-togram of the received intensities that originate

from 0 bits overlaps with that from 1 bits; this,indeed, is a consequence of the effects of 2-D ISI.Therefore it is not possible to find an explicitthreshold value for a binary threshold decisionscheme that always gives correct results. How-ever, if the proposed 2-D modulation coding schemeis applied, the effects of 2-D ISI are expected to bereduced. Figure 10 shows the histograms of thereceived intensities at the detector array or CCDcamera after the mapping in Eq. �13� is applied.As before, we assume that each data page contains120 � 120 bits, 100 pages of data are recorded, eachdetector or CCD pixel has a unit linear fill factor,and the retrieval system, containing a square aper-ture with bs 1.5, is noiseless, i.e., the AWGN is notconsidered. It is evident that, after applying themapping, there is no overlap between the histogramof the received intensities that originate from 0 bitsand that from 1 bits. Hence it is ideally possible tofind a threshold value for a binary threshold decisionscheme such that the decisions are always correct.

There are several important aspects of Fig. 10 thatare worth discussing. First, note that the mapping inEq. �13� must be applied to a randomly generated in-put sequence of 630,000 information bits to obtain 100data pages of 120 � 120 bits to be recorded. Thus theamount of the useful information is 43.75% of the totalamount of the recorded data. Alternatively, note thatthe mapping in Eq. �13� maps seven input bits intosixteen output bits, the fraction of useful information is7�16 � 0.4375�. The overhead is the cost of lesseningthe effects of 2-D ISI. In addition, it should be notedthat the total number of counts in the histogram of thereceived intensities that originate from 0 bits, as ex-pected from Eq. �13�, is much higher than that in thehistogram of the received intensities that originatefrom 1 bits. The extra 0 bits in Eq. �13� are used tomitigate the effects of 2-D ISI; to be more precise, theyare padded in such a way that, in any block of 3 � 3 bitsof each recorded data page, the maximum number of 1bits around any single 0 bit is six. Two examples of

Fig. 9. Histograms of the received intensities that originate fromrandomly page-wise recorded 0 and 1 bits, assuming that 100 datapages of 120 � 120 bits are recorded and that a noiseless retrievalsystem is characterized by the sinc2 PSF with bs 1.5 and 1.

Fig. 10. Histograms of the received intensities that originatefrom 0 and 1 bits after applying the mapping in Eq. �13�, otherwisesame as Fig. 9.


such extreme cases are illustrated in Fig. 11. Thesquares locate the 3 � 3 blocks of recorded data withthe highest spatial frequency content, i.e., the maxi-mum number of 1 bits around any single 0 bit. FromFig. 11, the highest spatial frequency content may becontained in any 2-D block of 4 � 4 bits, i.e., any outputblock of the mapping, itself, or it may arise after any2-D block of 4 � 4 bits is recorded with a specific set ofneighbor 2-D blocks of 4 � 4 bits.

In practice, when the retrieval system is noisy, i.e.,when the random effects of AWGN are included, thetails of the histograms always overlap, no matterwhether the 2-D modulation coding scheme is appliedor not, which leads to an inexplicit threshold value.Nevertheless, the proposed 2-D modulation codingscheme is believed to significantly improve the systemperformance with the idea of lowering thedeterministic-like effects of 2-D ISI. Furthermore, weshow in Figs. 12 and 13 the histograms of the receivedintensities before and after applying the mapping in

Eq. �13�, when the retrieval system is assumed to becharacterized by the Gaussian PSF with bg 0.6, andother assumptions remain the same as in Figs. 9 and10. In this particular example, the overlapping areaof the histograms of the received intensities that orig-inate from randomly page-wise recorded 0 and 1 bits inFig. 12 is very large, intuitively, because of a relativelylarge value of bg, which implies a very high degree of2-D ISI. Hence as shown in Fig. 13, while the map-ping in Eq. �13� cannot overcome the effects of a veryhigh degree of 2-D ISI, it results in a smaller overlap-ping area, i.e., reduced 2-D ISI effects. Based on our2-D modulation coding scheme principles, to move thehistograms in Fig. 13 further apart, i.e., to furtherrelieve the effects of 2-D ISI, the highest spatial fre-quency content allowed, i.e., the maximum number of1 bits allowed around any single 0 bit in any 3 � 3 datablock in the recorded data pages therefore must bereduced; instead of Eq. �13�, a new mapping is re-quired.

Finally, to simulate the received intensities at thedetector array or CCD camera, it should be empha-sized that although the continuous PSFs, h�x, y�, inEqs. �1�, �2�, and �3� spread infinitely, the effectivePSF matrix, H, in Eq. �8� is truncated to a finite-sizematrix in the actual implementation to reduce thecomputational complexity in the simulation proce-dure. This truncation does not strongly affect thesimulation results because it is known from Section 3that most of the light from a data mark is incidentonto the corresponding detector or CCD pixel andonto its local neighborhood. In detail, we define theeffective PSF matrix, H, in a way such that it is thesmallest possible K � K matrix, where K is an oddnumber, that contains at least 90% of the total lightoriginating from a data mark. For example, if a re-trieval system is characterized by the sinc2 PSF with

Fig. 11. Examples of a 3 � 3 block of recorded data with thehighest spatial frequency content after mapping by Eq. �13�.

Fig. 12. Histograms of the received intensities that originatefrom randomly page-wise recorded 0 and 1 bits, assuming that 100data pages of 120 � 120 bits are recorded and that a noiselessretrieval system is characterized by the Gaussian PSF with bg 0.6 and 1.

Fig. 13. Histograms of the received intensities that originatefrom 0 and 1 bits after applying the mapping in Eq. �13�, otherwisesame as Fig. 12.


bs 1.5 and 1, the corresponding effective PSFmatrix is

in which it contains 90.99% of the total light origi-nating from a data mark. Noticeably, approxi-mately 35.07% of the light is imaged onto thecorresponding detector or CCD pixel �at the center ofthe matrix�.

Similarly, if a retrieval system is characterized bythe Gaussian PSF with bg 0.6 and 1, theeffective PSF matrix is

�0.0385 0.1168 0.03850.1168 0.3544 0.11680.0385 0.1168 0.0385

, (15)

in which it contains 97.53% of the total light origi-nating from a data mark and the corresponding de-tector or CCD pixel receives approximately 35.44% ofthe total. Notice that the sizes of the effective PSFmatrices are varied depending on the characteristicsof the interested retrieval systems.

C. General Form of 2-D Modulation Codes

As discussed earlier, the recorded data pages in animaging PODS system should satisfy certain spatialfrequency constraints to achieve the required systemperformance. Because for a given imaging PODSsystem, the highest spatial frequency content, i.e.,the maximum number of 1 bits around a 0 bit in anydata block of 3 � 3 bits in the recorded data pageimplies the degree of 2-D ISI effects, the general formof the 2-D modulation codes, derived from our 2-Dmodulation coding scheme, should include informa-tion about the highest spatial frequency content thatis allowed in any recorded data page.

We also know from the properties of the PSFs inEqs. �1�, �2�, and �3� that the amount of interferencelight originating from a data mark incident onto eachof the corresponding four nearest-neighbor detectorsor CCD pixels is always greater than the amount ofinterference light originating from the same datamark incident onto each of the other neighbor detec-tors or CCD pixels �the effective PSF matrices in Eqs.�14� and �15� also support this fact�. Moreover, be-cause the effects of 2-D ISI are considered to be local,we suggest that the general form of typical 2-D mod-ulation codes contain information about the maxi-mum number of 1 bits in the four nearest-neighbor

positions around any 0 bit in the recorded data pageand the maximum number of 1 bits in the next four

nearest-neighbor positions around the same 0 bit.Let us label the neighbor positions around a 0 bit in

a data block of 3 � 3 bits by the compass directions:north �N�, northeast �NE�, east �E�, southeast �SE�,south �S�, southwest �SW�, west �W�, and northwest�NW�, as shown in Eq. �16�, where, similar to the ter-minology in morphological image processing, the fournearest-neighbor positions, i.e., N, E, S, and W, arereferred to as the four-connected neighbor positions,and the NE, SE, SW, and NW positions are referred toas the eight-connected neighbor positions.27

NW N NEW 0 E

SW S SE(16)

An arbitrary 2-D modulation code for an imagingPODS system that maps k bits of 1-D input informa-tion into a 2-D output block of m � n bits �shown inFig. 14�, therefore, can be described by

�m, n; k; �, ��, (17)

where � is the maximum number of 1 bits in thefour-connected neighbor positions of any 0 bit and � isthe maximum number of 1 bits in the eight-connectedneighbor positions of the same 0 bit. It is necessaryto note that � and � are not only the constraints ineach 2-D m � n output block itself, but they are alsothe constraints after the 2-D output blocks are placedside-by-side in a 2-D array that forms a tiling of therecorded data page.

We also emphasize that because the effects from 1bits in the four-connected neighbor positions of a 0 bit

Fig. 14. �m,n;k;�,�� 2-D modulation code.

�0.0000 0.0001 0.0008 0.0031 0.0008 0.0001 0.00000.0001 0.0004 0.0032 0.0120 0.0032 0.0004 0.00010.0008 0.0032 0.0241 0.0920 0.0241 0.0032 0.00080.0031 0.0120 0.0920 0.3507 0.0920 0.0120 0.00310.0008 0.0032 0.0241 0.0920 0.0241 0.0032 0.00080.0001 0.0004 0.0032 0.0120 0.0032 0.0004 0.00010.0000 0.0001 0.0008 0.0031 0.0008 0.0001 0.0000

� , (14)


are stronger than those from 1 bits in the eight-connected neighbor positions, then the ��, �� in Eq.�17� is defined such that, after applying the �m, n; k;�, �� 2-D modulation code, it is possible for the re-corded data page to contain a 0 bit that has more than� 1 bits in the eight-connected neighbor positions aslong as the number of 1 bits in the four-connectedneighbor positions is less than �. For instance, if the�m, n; k; 3, 1� 2-D modulation code is applied, then arecorded data page that contains two 1 bits in thefour-connected neighbor positions of a 0 bit and four1 bits in the eight-connected neighbor positions of thesame 0 bit still satisfies the given constraints: � 3and � 1. Assume that �̂ is the number of 1 bits inthe four-connected neighbor positions of any 0 bit and�̂ is the number of 1 bits in the eight-connected neigh-bor positions of the same 0 bit. In Fig. 15, the opencircles graphically depict all possible combinationsof ��̂, �̂�, and the open squares depict all combina-tions of ��̂, �̂� that satisfy the constraints of the �m,n; k; 3, 1� 2-D modulation code. For this particular2-D modulation code, any valid ��̂, �̂� satisfies thelinear inequalities �̂ � 3 and

�̂ � �3�̂ � 10. (18)

These constraints can be expressed in general forany �m, n; k; �, �� 2-D modulation code. Let �̂ and �̂be the number of 1 bits in the four-connected andeight-connected neighbor positions of any particular0 bit, respectively. The ��̂, �̂� allowed in the re-corded data page must satisfy �̂ � � and

�̂ � �� 4��̂ � �4� � �� , (19)

where �̂, �̂ � �0, 1, 2, 3, 4�.28

We note that many essential issues of Eq. �17�should be discussed. First, the code rate of 2-D mod-ulation code, which is defined as the fraction of theuseful input information in the output, is equal tok��mn�. The code rate is, in fact, one of the criteriaused to compare the performance of 2-D modulationcodes. Second, the size of each data page recorded inan imaging PODS system, M � N, is typically a pos-itive integer multiple of m and n, i.e., M um andN vn, where u, v � �1, 2, 3, . . .�, although this is nota necessary condition. Third, the block size of a 2-Dmodulation code, specified by the general form in Eq.�17�, is fixed. In other words, every 2-D output blockof the �m, n; k; �, �� 2-D modulation code has an equalsize, m � n. Thus, 2-D modulation codes that can bedescribed by Eq. �17� are generally called the fixed-length 2-D modulation codes. However, it is alsopossible to have 2-D modulation codes with variableblock sizes.29 In such a situation, the general formin Eq. �17� requires modifications.

D. Examples of Fixed-Length 2-D Modulation Codes

We present in this subsection some examples of fixed-length 2-D modulation codes with different blocksizes, code rates, and ��, �� constraints.

1. �4,4;7;2,4� 2-D Modulation CodeA 2-D modulation code that maps a 1-D block of seveninformation bits into a 2-D block of 4 � 4 bits ispresented in Eq. �13�; using the general form de-scribed in the previous subsection, it can be called the�4,4;7;2,4� 2-D modulation code. We note that thecode rate of this 2-D modulation code is equal to 7�16� 0.4375� and that a few examples of its highestspatial frequency content, which is � 2 and � 4,are shown in Fig. 11.

2. �4,4;8;2,2� 2-D Modulation Code

�a0 a1 a2 a3 a4 a5 a6 a7�3

�a0 a2 0 0a1 a3 0 00 0 a4 a6

0 0 a5 a7

� . (20)

The �4,4;8;2,2� 2-D modulation code in Eq. �20� mapsa 1-D block of eight information bits into a 2-D blockof 4 � 4 bits.30 This 2-D modulation code has a coderate of 0.5, and its highest spatial frequency contentcontains two 1 bits in the four-connected neighborpositions of a 0 bit and the other two 1 bits in theeight-connected neighbor positions of the same 0 bit.Figure 16 shows several examples of the possiblehighest spatial frequency content after applying the�4,4;8;2,2� 2-D modulation code. Note that the �4,4;8;2,2� 2-D modulation code should outperform the�4,4;7;2,4� 2-D modulation code because the formerhas a lower highest spatial frequency content and ahigher code rate.

Fig. 15. Points marked by a small square box are the possiblecombinations of ��̂, �̂� that satisfy the constraints of the �m,n;k;3,1�2-D modulation code.


3. �3,3;6;3,2� 2-D Modulation Code

�a0 a1 a2 a3 a4 a5�3 �a0 0 a3

a1 0 a4

a2 0 a5

. (21)

The �3,3;6;3,2� 2-D modulation code in Eq. �21� mapsa 1-D block of six information bits into a 2-D block of3 � 3 bits. The code rate of this 2-D modulation codeis 2�3 � 0.6667�. With this code, the highest spatialfrequency content allowed is three 1 bits in the four-connected neighbor positions of a 0 bit and two 1 bitsin the eight-connected neighbor positions of the same0 bit. Examples of the possible highest spatial fre-quency content after applying the �3,3;6;3,2� 2-Dmodulation code are shown in Fig. 17. It is clearthat the �3,3;6;3,2� 2-D modulation code should not beable to relax the effects of 2-D ISI as much as the�4,4;7;2,4� or �4,4;8;2,2� 2-D modulation code becauseof the higher spatial frequency content allowed.However, the �3,3;6;3,2� 2-D modulation code has ahigher code rate, i.e., a higher fraction of useful in-formation, therefore implying that there exists atrade-off between the highest spatial frequency con-tent allowed and the code rate.

5. BER Performance Analysis of 2-D ModulationCodes for Imaging PODS Systems

In Section 4 we show many histograms of the receivedintensities, both before and after applying a 2-D mod-

ulation code at the detector array or CCD camera ofseveral noiseless imaging PODS systems. Referringto Fig. 10, a �4,4;7;2,4� 2-D modulation code can totallyovercome the effects of 2-D ISI, i.e., there is no overlapbetween the histogram of the received intensities thatoriginate from 0 bits and that from 1 bits after apply-ing such a 2-D modulation code. Because there is nooverlap, any intensity value that lies in the gap be-tween the histograms can be used as a threshold valuefor the binary threshold decision scheme that is ap-plied at the CCD or detector array. In fact, any in-tensity value that lies in that gap can be considered asan optimal threshold that always gives correct deci-sions and, consequently, results in a zero BER, i.e., theprobability that an error will occur is equal to zero.

However, as discussed in Section 2, an imagingPODS system is generally not noiseless. Assumethat it is noisy, primarily, because of the electronicsat the detector plane. In this case, the effects ofAWGN are taken into account and hence the histo-gram of the received intensities that originate from 0bits always overlaps with that from 1 bits, regardlessof any 2-D modulation coding. This, indeed, leads toa nontrivial optimal threshold value for the binarythreshold decision scheme and a nonzero minimumBER. In this section we describe analytically how toobtain such an inexplicit optimal threshold to achievea minimum BER for a noisy imaging PODS system.

Because the counts, i.e., the number of occurrences,of the received intensities in any histogram shown inSection 4 are accumulated in a finite number ofequally distributed intensity bins �in our simulations,the bin size is always set to 1�200 of the normalizedintensity�, the effects of AWGN in a noisy imagingPODS system, then, can be simply included by con-volving each intensity bin in the histogram with aGaussian distribution function, whose mean andvariance are zero and �w

2, respectively. Let us de-fine c0,l as the number of counts in the lth bin, whosecenter is located at the intensity value �0,l, of thehistogram of the received intensities that originatefrom 0 bits. Analogously, c1,l is defined as the num-ber of counts in the lth bin, whose center is located atthe intensity value �1,l, of the histogram of the re-ceived intensities that originate from 1 bits. Assum-ing that the total number of counts in the histogramof the received intensities that originate from 0 bits isC0 and that the total number of counts in the histo-gram of the received intensities that originate from 1bits is C1, i.e., C0 � �

lc0,l and C1 � ¥

lc1,l, we can

derive the BER of a noisy imaging PODS system withintensity threshold value T as the following:

BER Prob�error� �

Prob�error�0 bit is recorded� �

Prob�0 bit is recorded�

� Prob�error�1 bit is recorded� �

Prob�1 bit is recorded�. (22)

Fig. 16. Examples of a 3 � 3 block of recorded data with thehighest spatial frequency content after applying the �4,4;8;2,2� 2-Dmodulation code.

Fig. 17. Same as Fig. 16 except after applying the �3,3;6;3,2� 2-Dmodulation code.


It is straightforward from a binary threshold deci-sion scheme, shown in Eq. �7�, that the conditionalprobability of error given that a 0 bit is recorded isequivalent to the conditional probability that the re-ceived intensity, r, is greater than the thresholdvalue, T, given that a 0 bit is recorded. Therefore

Prob�error�0 bit is recorded�

� Prob�r � T�0 bit is recorded� (23)

�1C0

�l

c0,l��T

� 1

�2��w

exp��r � �0,l�

2

2�w2 �dr (24)

�1

2C0�

lc0,l erfc�T � �0,l

�2�w� , (25)

where erfc is the complementary error function de-fined as31

erfc�s� �2

�� s

�

exp��t2�dt. (26)

Similarly, the conditional probability of error giventhat a 1 bit is recorded is equivalent to the conditionalprobability that the received intensity, r, is less thanthe threshold value, T, given that a 1 bit is recorded.Thus

Prob�error�1 bit is recorded�

� Prob�r � T�1 bit is recorded� (27)

�1C1

�l

c1,l��

T 1

�2��w

exp��r � �1,l�

2

2�w2 �dr (28)

�1

2C1�

lc1,lerfc��1,l � T

�2�w� . (29)

Inserting Eqs. �25� and �29� into Eq. �22�, we have

BER Prob�error�

� Prob�0 bit is recorded� �

� 12C0

�l

c0,lerfc�T � �0,l

�2�w�

� Prob�1 bit is recorded� �

� 12C1

�l

c1,lerfc��1,l � T

�2�w� . (30)

Clearly, if the input information bits are recordedwithout applying any 2-D modulation code, then

Prob�0 bit is recorded� � Prob�1 bit is recorded�

� 0.5. (31)

However, if a 2-D modulation code is applied, thenthe probability that a 0 bit is recorded is generally notequal to the probability that a 1 bit is recorded.Based on our 2-D modulation coding scheme, theprobability that a 0 bit is recorded is often much

higher than the probability that a 1 bit is recorded.Table 1 lists such probabilities for all 2-D modulationcodes described in Section 4. Last, we evaluate Eq.�30� repeatedly for each threshold value T. Athreshold value that results in a minimum BER is,hence, an optimal threshold for the binary thresholddecision scheme of a noisy imaging PODS system.

To illustrate the BER performance of our 2-D mod-ulation codes, we show, for examples, in Figs. 18 and19 many plots of the minimum BER as a function ofthe standard deviation of the AWGN for two noisyimaging PODS systems. We assume, for simplicity,in every simulation that each recorded data pagealways contains 120 � 120 bits, the linear fill factor,, of each detector or CCD pixel is always set to one,and, as defined in Section 4, the effective PSF matrixused in Eq. �6� or �8� contains at least 90% of the totallight originating from a data mark. We note furtherthat all optimal threshold values used to calculate theminimum BERs in our simulations are obtained fromthe numerical procedure described earlier, providedthat 100 pages of data are recorded.

In Fig. 18, we assume that the retrieval systemcontains a square aperture and can be modeled by thesinc2 PSF with bs 1.5. Apparently, if there is nomodulation code applied �i.e., the raw case�, the val-ues of the BER are always very high, even with a verylow value of the AWGN standard deviation. This, infact, implies the strong effects of 2-D ISI in an imag-ing PODS system. On the contrary, when a 2-Dmodulation code is applied, it is clear that the valuesof the BER at the small values of the AWGN standarddeviation improve significantly. We note, however,that the BER performance improvement reliesheavily on the characteristic of each 2-D modulationcode. As illustrated in Fig. 18, the �3,3;6;3,2� 2-Dmodulation code cannot improve the BER perfor-mance as much as the �4,4;7;2,4� and �4,4;8;2,2� 2-Dmodulation codes can. It is, indeed, because the�3,3;6;3,2� 2-D modulation code has higher spatialfrequency constraints than the �4,4;7;2,4� and �4,4;8;2,2� 2-D modulation codes; more specifically, the �3,3;6;3,2� 2-D modulation code has a higher value of �than the �4,4;7;2,4� and �4,4;8;2,2� 2-D modulationcodes. It should also be recalled from Section 4 thatthe spatial frequency constraints of the �4,4;8;2,2�,�4,4;7;2,4�, and �3,3;6;3,2� 2-D modulation codes are��, �� 2,2�, ��, �� 2,4�, and ��, �� 3,2�, respec-tively, where we define � as the maximum number of1 bits in the four-connected neighbor positions of any0 bit in a recorded data page and � as the maximum

Table 1. List of Probabilities That a 0 Bit Is Recorded and That a 1 BitIs Recorded for Various 2-D Modulation Codes

2-D modulationcode Prob�0 bit is recorded� Prob�1 bit is recorded�

�4,4;7;2,4� 195�256 � 0.7617� 61�256 � 0.2383��4,4;8;2,2� 3�4 � 0.75� 1�4 � 0.25��3,3;6;3,2� 2�3 � 0.6667� 1�3 � 0.3333�


number of 1 bits in the eight-connected neighbor po-sitions of the same 0 bit.

In addition, in Fig. 19, we consider the retrievalsystem that can be modeled by the Gaussian PSFwith bg 0.6. It is shown in Section 4 that, withthis particular imaging PODS system, the degree of2-D ISI is very high; thus, we expect the very poorBER performance, if the effects of AWGN are takeninto account. From Fig. 19, it is obvious that onlythe �4,4;8;2,2� 2-D modulation code can substantiallyimprove the BER performance. We emphasize thatthis is because the �4,4;8;2,2� 2-D modulation codehas relatively low spatial frequency constraints, i.e.,

��, �� 2,2�, compared with ��, �� 2,4� and ��, �� 3,2� of the �4,4;7;2,4� and �3,3;6;3,2� 2-D modula-tion codes, respectively.

6. Conclusions

This paper discusses the appropriate modulation cod-ing for any PODS systems that incoherently recordand retrieve images of 2-D arrays of data. In Section2, we have presented a number of 2-D mathematicalmodels for various imaging PODS systems based onthe assumption that the distance between any twosuccessive recorded data pages is large enough suchthat there is no interaction between the data pages.

Fig. 18. BER performance of �4,4;7;2,4�,�4,4;8;2,2�, and �3,3;6;3,2� 2-D modulationcodes in a noisy imaging PODS systemmodeled by sinc2 PSF with bs 1.5.

Fig. 19. Same as Fig. 18 except as mod-eled by Gaussian PSF with bg 0.6.


With these 2-D mathematical models, one can inves-tigate the inherent ISI of imaging PODS systems andits effects. In fact, by using our 2-D mathematicalmodels, we have illustrated in Section 3 two exam-ples of imaging PODS systems that have differentdegrees of ISI. Because the ISI in imaging PODSsystems is different from the ISI in conventional datastorage systems �the former is 2-D and spreads overthe spatial domain, whereas the latter is 1-D andspreads over the time domain�, the conventional mod-ulation coding may not be suitable for imaging PODSsystems.

In Section 4 we have proposed what is to our knowl-edge a new 2-D modulation coding scheme that canrelieve, or even eliminate, the effects of 2-D ISI inimaging PODS systems. Because ISI in imagingPODS systems is mainly due to the light from a datamark or a 1 bit that spatially interferes with otherlocal neighbors, especially the 0 bits, on the samerecorded data page, the fundamental principle of our2-D modulation coding scheme is to limit the numberof 1 bits around any 0 bit on each recorded data page.We have also described a general form of 2-D modu-lation codes derived from the proposed 2-D modula-tion coding scheme. Using this general form, we candifferentiate the 2-D modulation codes and approxi-mately compare their performance. In addition, wehave presented several examples of fixed-length 2-Dmodulation codes, with various properties andstraightforward encoding�decoding procedures, forimaging PODS systems. In Section 5 a numericalprocedure that is used to find the optimal thresholdvalue for a binary threshold decision scheme appliedat the detector array of a noisy imaging PODS systemand the corresponding minimum BER has been de-scribed. We have also shown the BER performanceof our fixed-length 2-D modulation codes for somespecific imaging PODS systems. In other work wedescribe variable-length 2-D and three-dimensionalmodulation coding techniques for imaging PODS sys-tems.29,30

Furthermore, we note that adaptive 2-D modula-tion coding for imaging PODS systems, in which theinput-output mapping table can be updated corre-sponding to the incoming input information, has beenexplored recently.32 However, although the fractionof useful information in the recorded data �i.e., thecode rate� may be higher after applying the adaptive2-D modulation coding scheme, its encoding�decod-ing procedures are expected to be very complicatedcompared with our 2-D modulation codes. Addition-ally, it should be emphasized that while the primaryobjectives of conventional modulation coding are toreduce the effects of 1-D ISI and to recover the clockinformation from the recorded data, the main goal ofour 2-D modulation coding is only to reduce the ef-fects of 2-D ISI. Unlike conventional data storagesystems, in which the clock information is usuallyembedded in the recorded data, we assume that theclock information in imaging PODS systems is sup-plied separately in the clock channels. Finally,there is no need to maintain an average signal level

balance of the 0 and 1 bits on each data page in theimaging PODS systems, as required in the holo-graphic PODS systems.

This research was supported by DARPA �DefenseAdvanced Research Projects Agency� through the AirForce Research Laboratory under Cooperative Agree-ment No. F30602-98-3-0226, and by the IntegratedMedia Systems Center, a National Science Founda-tion Engineering Research Center, under Coopera-tive Agreement No. EEC-9529152, with additionalsupport from the Annenberg Center for Communica-tion at the University of Southern California.

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Fixed-length two-dimensional modulation coding for imaging page-oriented optical data storage systems