Detection accuracy in zero-crossing-basedspectrum analysis and image reconstruction

Caesar Saloma and Matthew George Escobido, Jr.

We examine how the accuracy in describing the exact location of a signal crossing affects the quality of thespectrum or the reconstructed two-dimensional image that is computed by a representation of sampledzero crossings. The position of a zero crossing within a Nyquist interval is described by the ratio betweenthe number of clock pulses that have elapsed before the crossing occurred and the total number of clockpulses that could fit within the interval. The pulses scale the Nyquist interval, and the greater their totalnumber, the more accurate the description of the crossing location. In a real zero-crossing detector theability to increase the total number of square pulses contained within the Nyquist interval is limited bythe finite response time of its circuit components [Opt. Lett. 18, 1468 (1993)].

PerspectiveZero-crossing (ZC) sampling has attracted attentionbecause of its conceptual elegance.'-5 The reconstruc-tion of a band-limited signal s(x) from its representa-tion of sampled zero crossings is based on the ideathat s(x) can be interpreted as a 2Mth-degree polyno-mial s(Z), where x is the independent variable, T isthe sampling period, wo = 2r/T, and Z = exp(jwx).To satisfy the Nyquist sampling criterion, we need todetect one zero crossing within a sampling interval A,given by A = 1/2W, where Wis the signal bandwidth.

Signal reconstruction from a set of sampled cross-ings has to resolve several operational issues. Thenecessity of detecting all the 2M roots of s(x) has beensatisfactorily addressed with the technique of sinusoi-dal crossings.- 8 With regard to the handling of alarge number of zero crossings, a more efficientalgorithm that permits faster, and more importantly,error-free, computation of discrete-Fourier-trans-form coefficients has already been tested.9

Recently a single-channel ZC optical spectrum ana-lyzer has been realized.'0 Its dynamic range of sam-

When this research was performed, the authors were with theNational Institute of Physics, University of the Philippines, Dili-man, Quezon City, Philippines. C. Saloma is now with theInformation Optics Section, Department of Optical Materials,Osaka National Research Institute, AIST, Midorigaoka 1-8-31,Ikeda, Osaka 563, Japan.

Received 6 August 1993; revised manuscript received 6 June1994.

0003-6935/94/327617-05$06.00/0.C 1994 Optical Society of America.

pling was limited only by the supply voltages of thesingle comparator that the detector requires. A ZCdetector has the advantage of wider dynamic rangeand simpler hardware as compared with conventionalanalog-do-digital converters.

In realizing a ZC detector an additional technicalproblem exists. The accuracy in locating the zerocrossing within A is limited by the response character-istics of the electronic components comprising thedetector circuit. In practice, a trade-off exists be-tween detection accuracy and the size of the Nyquistinterval A. Inaccuracies in describing the exact posi-tions of the zero crossings are the main source oferroneous frequencies in the computed spectrum ofthe zero-crossing-based optical spectrum analyzer.

To detect a zero crossing within A at an accuracy of1 part in 2B, a fundamental clock frequency of W2B isrequired by the circuit. For example, if an accuracyof 1 part in 4096 is desired within a preselected 1-psNyquist interval, then a 4.096-GHz clock is required.This fundamental frequency is too fast for mostlinear and transistor-transistor logic devices, whichhave typical response times of 200 and 15 ns, respec-tively.'"

We analyze the consequences of location inaccuracyon the quality of the spectra and two-dimensionalimages that are computed or reconstructed from theirrespective data set of sampled zero crossings. Wealso clarify the issue concerning the required accu-racy that a zero crossing must be detected within theNyquist interval in order to obtain a reliable recon-struction.2

10 November 1994 / Vol. 33, No. 32 / APPLIED OPTICS 7617

Effect of Detection Accuracy on the Computed Fourier

Coefficients

The polynomial representation of the band-limitedsignal s(x) can be developed from its Fourier-seriesexpansion:

M

S(Z) = E CmZm , (1)m=-M

where Z = exp(jwox) and cm is the mth Fourier-transform coefficient. As a 2Mth-degree polynomial,s(Z) can be represented in terms of its roots Zi:

2M

S(Z) = c.MZM J(Z Zi)

= c-MZ-m(ao + ajZ + a2Z2 +...., + a2MZ2M), (2)

where the polynomial coefficients am are related tothe Fourier-transform coefficients according to cm =C-MaM+m, where M = -M, -M + 1, . . . , M - 1, M.To within a multiplicative constant, s(x) can bereconstructed by inverse-Fourier transforming theset of coefficients {cm I that are determined from {am}.

The set of 2M roots {Zi } satisfying S(Z) = 0 isobtained from the sinusoid crossings {xi } of s(x) withrespect to a reference sinusoid A(x) by: Z =exp(fjxi). To within a multiplicative constant, thecrossings of s(x) with respect to the sinusoid A(x) = Asin(2rWx) of amplitude A, provide a unique crossingrepresentation of s(x).4,6,8 The sinusoid crossingsrepresent the roots of s'(x) = s(x) + A(x) = 0. Therequired number of 2M sinusoidal crossings {xi} aredetected if A is made at least equal to the largestpossible I s(x) value within the sampling period T =2MA. During implementation 0 the distances be-tween two successive extrema of A(x) are used asmarkers for A. One sinusoidal crossing occurs withineach A.

An efficient way of computing the polynomial coef-ficients am from the roots Zi has been devised inwhich 9

and where A = lOB, T = 2MA = 1OB2M, and B is apositive integer. The value of b, can vary only from1 to 10 B because x, occurs within the first samplinginterval. Choosing B = 0 leads to the trivial case inwhich the location of a sinusoid crossing is uncertainover the entire interval A. The location of anysucceeding crossing xi within the ith Nyquist intervalAi is denoted by (i - )iOB + bi, where 1 < bi < lOB.

The associated uncertainty ex in describing theposition of a crossing xi within a Nyquist interval isgiven by ex = 0.5. The uncertainty e. induces acorresponding uncertainty Ez(i) in the computed valueof Z, where Z = expljw[(i - l)lOB + bi] =exp[jrr(i - 1)/M]exp(jbi /M1OB) and w = rr/M10B.This uncertainty is given by'3 EZ() = jrrZi E/IOBM =±j5TrZi 10-(B+1)/M. The magnitude of Ez(i) decreasesrapidly with an increase in the number of divisions10 B contained within A.

The corresponding uncertainty Es(q) inducedfor sq in Eq. (3) is Es(q) = +q [(Zf'ez(,)) 2 +(Zq- )2 + . .. + (Zq-1 )2]1/2 = + 0(B+1)/M)[(Zq) + (Zq)2 + * + (Zq )2]1/2. The uncertain-ties ES(q) affect the reliability of the various ai values

1.0

IXj1

In-4CGDa

13

1.02

0.37

1.0 -

am = -- sqam-q, (3)M q=1

where sq = (Zl)q + (Z2)q ,..., + (ZN)q and ao = 1.The recursion starts from m = 1 and ends at m =2M = N. The above algorithm has a computationalcomplexity that is of the N2 class.

In the recursive computation of the polynomialcoefficients am with Eq. (3), the roots Zi are subin-dexed according to the order that their correspondingsinusoid crossings xi are detected. During serialdetection with a single-channel ZC detector, ZI andZN are, respectively, computed from the first- and thelast-detected sinusoid crossings of s(x) within thesampling period T.

If we divide each A into 10 B equal parts (squareclock pulses), then the location of the first sinusoidcrossingx, can be denoted by bl, where 1 < b, < 1 B,

0.0 +

4,

-40.EM0

040 +

0.10 +

.20 +

-64.30

.40 M

position(a)

-SIAN -12.0 a 12.90frequency(b)

39.40 64.0

Fig. 1. (a) Interferogram s(x) representing a doublet, (b) itscorresponding spectrum computed by Fourier transforming 128equally sampled data points of s(x).

7618 APPLIED OPTICS / Vol. 33, No. 32 / 10 November 1994

I

that are determined by the recursion relation of Eq.(3), where ao = 1.

The value of a, is calculated with a = -sla0 =-(Z1 + Z2 + * * * + ZN)- Its value has an associateduncertainty Of Ea(l) = Es(q) = [(EZ(1))2 + (EZ(2))2 + ... +(Ez(2M))2]'/ 2 . The value of a2 is given by a2 =-(2-)(sial + 2), and its associated uncertainty isEa(l) = (2-1/2 )[(ajES()) 2 + (SlEa(l))2 + ES(2)2]1/2. Thevalue of the succeeding ith coefficient is a, =(-i-)(slai, + s2ai-2 + * + silal + si). Its calcu-lated value has an associated uncertainty of

Ea(i) = ±(i)-/ 2 [(aj.jEs(())2 + (SlEa(i-1))2

+ (S2E(i)) 2 + (ai-2eS(2))2 + + Es(i)]/ (4)

Consequently each element cm in the Fourier spec-trum of s(x) that is computed by cm = cMaM+m alsopossesses an associated uncertainty given by EC(m) =Ea(M+m)(C-M)1/2 . If CM = 1, then Ec(m) = a(M+m).

It should be noted that the computational errorarises from the uncertainties in locating the crossings.It is caused by the precisional limits of the samplingprocess, not by rounding-off error associated with thefinite precision p that the floating-point arithmeticrepresents in a number in a (postdetection) computa-

1.72

1.s1

GP

ID

1.09

.4.

0.34

0.00

tional process.9"4 The latter is complexity depen-dent and relies on the type of algorithm used incomputing {ail from {Zi}. The precision p, whichindicates the number of significant digits utilized inexpressing a computational result, is much greaterthan d + 1, thus the corresponding uncertainty isusually larger than (5)j0-(d+1). This value repre-sents the difference between l0(d+2) and 10-P, whichis not rounded off to zero when the computer per-forms an operation.

In the preceding analysis we assumed that all thecrossings are detectable regardless of the value of JOB.When the technique of thresholding through activecomparison between the values of s(x) and referencesinusoid A(x) is employed to detect the crossing, it ispossible that a crossing remains undetected for widerdivisions (small B) of the Nyquist interval. In thezero-crossing detector that we constructed, 0 thresh-olding was not employed, and all crossings weredetected as long as s(x) was less than the amplitude ofthe sinusoidal reference for all values of x.

Experiments

We examined the effects ofpossible applications of

1.00

.0 .1.

0.D 4

0.40 .

0.20 .,

GD

U'.

0.00 .4.00 -31.40

(a)

detection accuracy in twozero-crossing sampling,

-12.10 0 12.10frequency(c)

31.40 i4.00

1.07

O.*S 4.

GP

.54

CLIiU

0.64 .1

0.43

0.21

o.o 1-d6;.00 -31.40

II ... II

-Li.10 0 12.frequency

1.00

D00

0.'O

D.A

0.40

0.20

o10 30.40 4.00

(b)

Fig. 2. Computed spectra corresponding to different values of B:inside the Nyquist interval is given by = l0 -B.

0.00-64.00 -33.40 -12.0 0 12.80 39.40 64.00

frequency(d)

(a) B = 1, (b) B = 2, (c) B = 3, and (d) B = 4. Detection resolution

10 November 1994 / Vol. 33, No. 32 / APPLIED OPTICS 7619

l z * X- - l- - - And-XEElEwE wffi All In tElEEl is-and - - Al - ffi-ffi |

W r

- . ... I ...........

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namely, spectral analysis and image reconstruction.The spectrum of a noise-free, synthetic interferogramgiven by s(x) = exp{ - [x(20)'/ 2 ]2)cos(2'rrl5x)cos(2Tr2.5x)was computed from its zero-crossing samplings atvarious degrees of location accuracy. The interfero-gram s(x) represents a doublet with the resonanceslocated at ± 12.5 and ± 17.5 Hz, respectively.

The interferogram s(x) is presented in Fig. 1(a), inwhich T = 1, and shown in Fig. 1(b) is its correspond-ing magnitude spectrum, which is computed by Fou-rier transforming (Cooley-Tukey algorithm) a set of128 equally sampled data points of the doublet inter-ferogram (A = 1/128). A 6-byte floating-point repre-sentation was employed in the computation (p equals11 to 12 significant digits; the value range is (2.9)10-39to (21.7)1038).15

Figure 2 illustrates how the inaccuracy in locatingthe position of a sinusoid crossing within A under-mines the quality of the Fourier spectrum {ci I of s(x)that is computed from a sampled set of 128 sinusoidalcrossings. The various spectra were computed withthe recursive relation outlined in Eq. 93) with ao =C-M = 1. Each spectrtal element ci has an associateduncertainty of EC(i) = Ea(M+ 1), where i is in the range of-64 to 64.

A detection resolution 8 of 10-1 (B = 1) was used toobtain the amplitude spectrum shown in Fig. 2(a),while that in Fig. 2(b) was computed with 6 = 10-2(B = 2). Figures 2(c)-2(d) were determined with 8 =10-3 (B = 3) and 8 = 10-4 (B = 4), respectively. Notethat all the spectra shown are not normalized.

Unlike the spurious frequencies caused by rounding-off errors, which have a low-frequency spectral distri-bution,10,'4 the error frequencies arising from detec-tion inaccuracy exhibit a wide distribution in thecomputed spectrum. The amplitudes of the errorfrequencies weaken quickly with improvement indetection accuracy.

We also examined the effects of detection accuracyon the visual quality of an image reconstructed fromits sampled representation of zero crossings. It hasbeen demonstrated' 6 that a good reconstruction (nor-malized mean square error equal to 10-7) could beobtained when a zero crossing was located with anaccuracy of 1 part in 106 (B = 6).

Figure 3(a) shows an equally sampled image ofhuman chromosomes (100x magnification, N.A. =1.25). The image was obtained with a CCD videocamera (a black and white Tektronix 1001 with pixelsize of 17p1m x 13 .lm) and spatially digitized by avideo frame store board (Tektronix DCS 01 with 127gray scales). The larger chromosomes have circum-ferences that are in the range of 60 to 70 pixels.

Figures 3(b)-3(d) present various reconstuctions ofthe microscope image corresponding to different detec-tion accuracies in locating the zero crossing withinthe Nyquist interval. The interval was set by thepixel size of the video camera. The analytical expres-sion for the microscopic image in Fig. 3(a) wasobtained by linear interpolation of the sampled points.

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Fig. 3. Effect of detection resolution 8 = 10-B on the visualquality of an image reconstruction: (a) 256 x 256 point equallysampled image of human chromosomes, (b) 8 = 10-l, (c) a = 10-2,

and (d) 8 = 10-3.

7620 APPLIED OPTICS / Vol. 33, No. 32 / 10 November 1994

�LAL_ �7X�. I � : .: I LI iw.

The zero crossings were then obtained with a sinu-soid reference signal of frequency 1/512 (T = 1).

Figures 3(b)-3(d) were reconstructed with detec-tion resolutions of equal to 10-1, 10-2, and 10-3,respectively. It can be observed that the visualquality of Fig. 3(d) is already difficult to distinguishfrom the equally sampled image shown in Fig. 3(a).

The inaccuracy in describing the position of acrossing within a Nyquist interval reduces the dy-namic range of the gray-level representation of theimage reconstruction [Fig. 3(b)]. It introduces aminimum gray-level resolving ability that is greaterthan zero. This results in a reduction in the con-trast of the image. The dynamic range of the gray-level representation widens rapidly as detection accu-racy improves [Figs. 3(c) and 3(d)].

Also note that detection inaccuracy does not intro-duce spurious details in parts of the reconstructedimage [Fig. 3(b)] that correspond to sparse regions ofthe original image of Fig. 3(a). The effects of theuncertainties are confined to regions within the chro-mosome images.

Discussion

Experimental results indicate that detection inaccu-racy gives rise to the appearance of error frequenciesin the computed spectrum. The distribution of thesefrequencies covers the entire bandwidth W. It isbroadband because each element cm in the Fourierspectrum of a signal s(x) has an associated uncer-tainty EC(m) that can be computed by use of Eq. (4).The spurious frequencies brought about by theseuncertainties decrease if the number of divisions 1 0 Bwithin the Nyquist interval is increased.

The magnitudes of these error frequencies decreaserapidly with improvement in detection resolution.At 8 = 10-4 the computed spectrum is already visiblynoise-free. A similar conclusion can be made regard-ing the visual quality of an image that is recon-structed from sinusoidal crossings. Error frequen-cies arising from detection inaccuracy are the primaryerror in the output spectrum of a zero-crossing-basedoptical spectrum analyzer.'0

Our results show that even a detection resolutionof 8 = 10-3 (B = 3) yields a set of Fourier coefficients{ci } sufficient for applications in spectral analysis andimage reconstruction. For applications confined toedge detection of image boundaries the accuracy

requirement needed in describing the location of acrossing is even more lenient. Figure 3(a) ( = J0-B)

shows that the relative locations of the edges of thevarious chromosomes are not shifted and, in fact, aremore pronounced.

The authors are grateful for the valuable com-ments of the unknown reviewer.

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