Modulation transfer function cascade model for a sampledIR imaging system
Luigi de Luca and Gennaro Cardone
The performance of the infrared scanning radiometer (IRSR) is strongly stressed in convective heat transferapplications where high spatial frequencies in the signal that describes the thermal image are present. Theneed to characterize more deeply the system spatial resolution has led to the formulation of a cascade modelfor the evaluation of the actual modulation transfer function of a sampled IR imaging system. The model can
yield both the aliasing band and the averaged modulation response for a general sampling subsystem. For aline scan imaging system, which is the case of a typical IRSR, a rule of thumb that states whether the combined
sampling-imaging system is either imaging-dependent or sampling-dependent is proposed. The model is
tested by comparing it with other noncascade models as well as by ad hoc measurements performed on a
commercial digitized IRSR.
1. Introduction
A. General Statement
The application to some specific convective heattransfer problems boosted the need to enhance ther-mal images obtained by means of a fully computerizedinfrared scanning radiometer (IRSR).1 ,2 In fact, thesystem performance is strongly stressed in the pres-ence of relatively high temperature gradients in thethermal image, such as those encountered in the visu-alization of G6rtler vortices on a surface in hypersonicflow,
3 ' 4 in the cooling of a plate by impinging jets,5 or inthe detection of transition and/or separation regions ofthe fluid flow on models tested in wind tunnels.6 Adeeper characterization of the system capabilities isneeded, mainly from the measurement point of view.
One of the main parameters characterizing thequantitative performance of an IRSR is the modula-tion transfer function (MTF) which may be regardedas a measure of its spatial resolution. Assessing thespatial resolution of an IR imaging system is a crucialpoint since the system itself generally exhibits a rela-tively poor MTF compared with other imaging sys-tems, such as CCD or TV cameras.
The authors are with University of Naples, Faculty of Engineer-ing-DETEC, I-80125 Naples, Italy.
In a digitized IR system, the possible MTF degrada-tion caused by the sampling process has to be investi-gated since, for a sampled system, in general the shift-invariant (or isoplanatic) assumption is no longervalid, where isoplanatism means that the output imagedoes not depend on the location of the input object. Infact, for a sampled system the MTF definition is notrigorously applicable because the MTF itself is a func-tion of the shift variance introduced by sampling.
Wittenstein et al.7 showed that the MTF approachto spatial resolution analysis can be extended to sam-pled systems if the condition of isoplanatism is rede-fined, i.e., if the MTF is defined within the limitedrange of the spatial frequencies where it is indepen-dent of sampling shift. Park et al.8 suggested that ameaningful sampled system response can be calculat-ed by averaging over an ensemble of point source in-puts randomly located relative to the sampling grid.This approach yields an averaged MTF, which ac-counts for the combined effects of image formationand sampling. But it does not allow for an evaluationof the aliasing effects, which depend on all possibleshifts of the sampling grid. Knowledge of the aliasingeffects may be dramatically important in certain usesof IR imaging systems in convective heat transfer, inthe presence of relatively high spatial variations oftemperature in the thermal image.
The common approach of Refs. 7 and 8 is to definethe MTF of a sampled imaging system as consisting ofsampling the impulse response and analyzing the re-sult in the frequency domain by the discrete Fouriertransform. The main result is that the combined ef-fects of imaging and sampling are accounted for, toyield a noncascade model of the MTF.
To evaluate the actual MTF for a sampled imagingsystem a cascade approach is proposed in this paper.The particularity of the proposed model consists of thefact that it can yield both the aliasing band and theaveraged modulation response of the sampling processalone. The results of the present model are comparedwith those of the noncascade models. Use in a com-mercial digitized IR imaging system is also discussed.
sine wave amplitude,sampled maximum value,sampled minimum value,averaged sampling modulation response,averaged sampled system MTF,impulse response,dimensionless frequency f = v/f8,sampling frequency,dimensionless Nyquist frequency,lower limit SMR as defined by Eq. (6),lower limit SMR,limiting sampled system MTF,maximum aliasing error,modulation transfer function,optical transfer function,sampling modulation response,system response to a slit aperture,slit response function,sampled system MTF,upper limiting SMR as defined by Eq. (5),upper limiting SMR,slit aperture,angular coordinate,duty cycle percentage,spatial frequency,nondimensional spatial coordinate = ,shape parameter,degraded shape parameter,sampling shift.
II. Sampling Process Model
The spatial resolution of an imaging system is gener-ally defined in the frequency domain by the opticaltransfer function (OTF) which is the Fourier trans-form of the impulse response. The MTF is the nor-malized (to have unit value for a constant signal) mag-nitude of the OTF.
The model proposed herein is based on the consider-ation that for a linear system the MTF may also beregarded as the modulation response to an opticalinput represented by sinusoids of different spatial fre-quencies.9 This interpretation allows one to assumethat imaging and sampling are cascaded; as a conse-quence, the modulation response of the sampling pro-cess alone may be obtained by sampling sinusoids ofdifferent spatial frequencies and by accounting for allthe possible shifts of the sampling grid.
For the sake of simplicity the following analysis islimited to the 1-D case. In particular, for imagingsystems based on a scanning mechanism, the presentanalysis is applicable to the along scan direction. It is
-A - | X CAMIN
Sampling gridFig. 1. Key sketch for the sampling process model.
also assumed that the spatial coordinate system isnormalized to have a unit sampling interval. Thiscorresponds to normalizing spatial frequency v withrespect to sampling frequency /8, i.e., f = v/f, where f isthe nondimensional spatial frequency. In particular,it shows that the nondimensional Nyquist frequency,fN, is 0.5.
Figure 1 is the key sketch for understanding theproposed sampling process model. The sampling of asingle period of a sinusoid (full line) is shown, havingamplitude A and spatial frequency f, performed byassuming a certain sampling shift so and a certain dutycycle percentage . Shift parameter so defines therelative position of the sine wave with respect to thesampling grid. It is conventionally assumed that (p (inabsolute value) is the smallest distance of the two gridpoints adjacent to the sine wave's positive crest, mea-sured from this crest. Since the sampling process isintrinsically periodic, the variation range of so is [-1/2,1/2], the positive values being taken (as in Fig. 1) forgrid points on the right-hand side of the crest. Dutycycle parameter # is the sampling interval fraction(centered at the generic grid point) over which thesignal integration is performed, thus generating thesampled value (solid circle).
By defining Amax and Amin as the maximum andminimum sampled values, respectively, the samplingmodulation response (SMR) is introduced, which isdefined as
SMR(fps3) = Amax Ami. (1)
Note that SMR generally depends on f, samplingshift s, and duty cycle percentage 3. By consideringFig. 1 and the definition of SMR [Eq. (1)], we obtain
Note that two other curves (c and d) are also plottedin Fig. 3, passing by the relative maxima of theULSMR curve and the (cuspidal) relative minima ofLLSMR, respectively. The values of frequency f cor-responding to such maxima and minima are found byimposing the condition that an integer number of sam-ple intervals is included in the half-period 1/(2f), i.e.,
f 1 '2n ' (4)
where n is any natural number.For frequencies given by Eq. (4) and so = 0, two
subsequent sinusoid crests (positive and negative) arecaptured and the SMR maximum value equal tosinc(4/f) is obtained. For s = 1/2 such crests lie on themiddle point of the sample interval and, therefore, thecorresponding SMR relative minimum is uncovered,which is equal to sinc(7rf3) cos(7rf). The aliasing errormay be ultimately represented, in a first crude evalua-tion also by the region delimited by the two dashedcurves c and d in Fig. 3 which, in a more general case,are, respectively, given by
is introduced to evaluate the contribution to SMRfrom the sine wave's negative crest (Amin) since in thegeneral case the sampling interval is not an integersubmultiple of the wave half-length.
For ten different equally spaced values of so (-1/2 <so < 1/2), Figs. 2(a) and 2(b) show the subsets of thefamily of the SMR curves, respectively, for / = 0 (spotsampling) and / = 0.5. The envelope of SMR curves,represented by the two characteristic archs curves,yields a measure of the aliasing effect which, in thepresent context, is the modulation degradation intro-duced by the sampling alone. It should be noted thatthe relative maxima of the upper envelope curve areless than unity for = 0.5. This is a result of theaveraging process which is present for / > 0.
Figure 3 depicts enveloped curves a and b whichrepresent the upper and lower limits of the aliasingerror in Fig. 2(b). In the following the a and b curvesare referred to as upper limiting SMR (ULSMR), andlower limiting SMR (LLSMR), respectively.
ulsmr(f,3) = sinc(7rtf),
llsmr(f,fl) = sinc(7rf3) cos(7rf).
(5)
(6)
The averaged SMR (ASMR) (over the shift domain)is defined by the integral
rl/2ASMR(,3 = I SMR(f,<o,3dq',
-1/2(7)
which may be reduced by substituting Eqs. (2a) and(2b) into it. The following relationship is thus ob-tained:
ASMR(f,g) = sinc(rf1) sinc(rf), (8)
which is represented by curve e in Fig. 3.A comparison between the ASMR curves obtained
for several values of duty cycle percentage /3 is given inFig. 4. As expected, the ASMR decreases by increas-ing the duty cycle percentage.
Under the assumption that imaging and samplingsubsystems are cascaded, the product of any SMR bythe imaging subsystem MTF yields the corresponding
Fig. 4. ASMR curves for various duty cycle percentage values. Fig. 5. Maximum aliasing error for ,B = 0.5 and three values of the
nondimensional group aof.
sampled system modulation transfer function(SSMTF) of the combined imaging and sampling sys-tem. In particular, the total averaged degradationfrom imaging and sampling is given by the averagedsampled system MTF:
ASSMTF(ff) = ASMR(ff) X MTF(f). (9)
For spot sampling at the Nyquist frequency, it isASSMTF (fN) = 2MTF(fN)/r.
Park et al.8 studied spot sampling only and defined asample-scene phase-averaged system MTF (ASMTF).As a consequence of the fact that the average wasperformed in the complex field, they found that theASMTF modulus vanishes at fN. But by computingthe average of the moduli of the system OTF corre-sponding to each value of the sampling grid shift ac-cording to Ref. 7, it may be proved that such an averageis equal to 4MTF(fN)/7r, which is double the ASSMTFof the model proposed herein. Note that for f = fN themodel in Ref. 7 gives an aliasing error equal to2MTF(fN), which in turn is double that of the presentone.
For relatively low frequencies substantially goodagreement is found with the model of Ref. 8.
11. Response of a System with a Gaussian MTF
The typical MTF of a line scan imaging system (and,in particular, of an IR imaging system) may be as-sumed to be Gaussian:
MTF(v) = exp(-7r2 a2v2), (10)
where v is the spatial frequency and a is the shapeparameter.
By using sampling frequency fs Eq. (10) may be putin a more convenient form:
MTF(f = exp[-7r2 (af5)2 f2 ]. (11)
A convenient measure of the MTF degradation pro-duced by the sampling process may be represented bythe maximum aliasing error (MAE) defined by
The solid lines in Fig. 5 show the maximum aliasingerror as a function of dimensionless frequency f for j =
0.5 and three values of the nondimensional parametercif8. Note that a is a parameter pertinent to the imag-ing subsystem while f. corresponds to the samplingone. Dashed lines show the values of the llsmr in Eq.(6). It must be emphasized that the MAE is the abso-lute maximum aliasing error due to the sampling pro-cess normalized over the response at zero spatial fre-quency.
By using Eq. (6) for llsmr, it may be shown that,within an approximation of the order of (f) 4124(which is fairly well verified for f < 0.25), the MAEpeaks are obtained at frequencies given by
(13)1= [1+ t9)/6 + [ + ]1/2
The MAE peak decreases by increasing the groupof8 , i.e., for relatively high values of of8, the MTF of theimaging subsystem does not appear to be degraded bythe sampling. But for relatively low values of of8 themodulation response of the composite imaging-sam-pling system is dominated by the sampling process.For the sake of simplicity an operative criterion isintroduced. In fact, for a computed value (qfj)cr corre-sponding to a prefixed maximum allowable MAE andgiven a sampled imaging system which is characterizedby a certain oh, value, the system itself may be consid-ered as imaging-dependent or sampling-dependent ac-cording to whether Ufs (fs)cr or afs < (afs)cr, respec-tively.
The curves shown in Fig. 6 depict the variation of theMAE peak as a function of the group crfs for three fvalues, obtained by using the values of llsmr in Eq.(12). The curves corresponding to the arches LLSMR(not reported) practically coincide with those in Fig. 6,except in the 0.5 < af, < 1 range.
It was previously noted that a meaningful measureof combined imaging and sampling degradation is thequantity ASSMTF defined by Eq. (9). By neglectingterms of the order of (7rf)4 /100 (which is practicallypossible for f values ranging from 0 to 0.5), it can beshown that ASSMTF may again be given by a Gauss-ian relationship:
IV. Applications to the Sampled AGEMA Thermovision880 IR System
The criterion introduced above, as well as the ex-pression for the degraded shape parameter [Eq. (15)]of the ASSMTF, are valid in imaging systems whoseMTF is Gaussian, which is generally the case for a linescan system. The main results of the previous sectionsare now applied to the AGEMA Thermovision 880 IRsystem.
It should be recalled that the system response to aninput slit aperture (SRSA) is given by the convolutionproduct 0
SRSA(x,w) = rect (-) * D(x), (16)
where w is the slit aperture and x is the angular coordi-nate.
Under the assumption of a Gaussian impulse re-sponse,
D(x) = exp ] (17)
the slit response function (SRF), which is the systemmaximal response (normalized to the value corre-sponding to an infinite width slit) at x = 0, is
W/2,
f exp(-_ 2)dtSRF(w) = X = erf - (18)J exp(-_ 2)dt 2
where t = x/a. It follows that shape parameter a maybe evaluated from the relationship
w0 (19)
0- 0.96'
where w0 is the slit aperture corresponding to the SRFvalue of 50%.
For the IRSR under discussion, when used with a 70lens, the manufacturer gives as a nominal value wo =
Fig. 7. SSMTF curves of the sampled AGEMA 880 IR system: a,imaging subsystem MTF; b, lower LSSMTF of the combined imag-
ing-sampling system; c, predicted ASSMTF.
0.7 mrad. In the tests we performed, the measuredvalue is 0.73 mrad which, by making use of Eq. (19),leads to a = 0.76 mrad.
The standard AGEMA sampling system, which isbased on the TIC 8000 A/D board, has a duty cyclepercentage of -50% and is characterized by 140 sam-ples/line along the scan direction, which for the 70 lenscorresponds to a sampling frequency offs = 1.5 mrad1.In this case the dimensionless group (rfs is equal to 1.14and, on the basis of the operative criterion proposedabove, the system appears to be sampling dependent.Figure 7, reporting the imaging subsystem MTF (curvea) and the limiting SSMTF [LSSMTF = (MTF) X(LLSMR), curve b] for the investigated system, clearlyshows the aliasing error which is introduced by thesampling. The MAE peak is -14%; the dashed linecorresponds to the llsmr. The ASSMTF (curve c) isalso shown in Fig. 7, where on the abscissa axis thespatial frequency v is reported. Both ASSMTF andLSSMTF curves are cut off at the Nyquist frequencywhich is -0.75 mrad-1.
For 140 samples/line and 3 = 0.5, the shape parame-ter of ASSMTF is found to be a = 0.82. This value,computed on the basis of AGEMA data and on actuallymeasured spatial resolution, is slightly less than themeasured value a = 0.84.11 This small discrepancycan probably be attributed to the electronics effects(not included in the present model) which further de-grade the system's modulation response.
Finally, note that, if the MAE peak value of 5% isdesired, from Fig. 6 we obtained (fs)cr _ 2. It followsthat the minimum spatial fequency that does not de-grade the AGEMA 880 imaging subsystem MTF is fs =2.63 mrad-1, which, as a 70 lens was used, correspondsto a sampling of 246 samples/line. On the other hand,the TIC 8000 digitizer was originally designed to beused in connection with the AGEMA 782 scanner,whose MTF is remarkably poor with respect to that ofthe 880 scanner.
V. Conclusions
Use for convective heat transfer problems requiresthe enhancement of fully computerized IRSRs. To
characterize more deeply the system spatial resolu-tion, a cascade model to predict the actual MTF for alinear sampled imaging system has been developed.The model can yield both the aliasing band and theaveraged modulation response for a general samplingsubsystem.
For a line scan system, which is the case of a typicalIRSR, an operative criterion has been proposed thatstates whether the system itself is imaging-dependentor sampling-dependent. In the latter case it was foundthat the average MTF degradation introduced by thesampling may be approximated with a variation of theshape parameter of the assumed Gaussian-type MTF.
The model has been applied to the AGEMA Ther-movision 880 IR camera connected to a TIC 8000 A/Dsampling board. It has been shown that the standard140 samples/line digitization of the above system issampling-dependent, the maximum aliasing errorpeak being 14%, therefore a higher sampling ratewould be desired.
The authors wish to thank Giovanni M. Carlomagnofor helpful discussions throughout the course of thiswork.
References
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