Two ways to calculate the modulation transfer function (MTF) of radiographic screen-film systems from the measured square wave response function (SWRF) data were investigated with an interactive curve fitting software. The measured SWRF data obtained by digitising a radiographic image of a bar pattern test object were fitted to a curve, and the fitted curve was used to calculate the MTF. Satisfactory MTF was obtained by using 12 terms in the calculation. A second version of the calculation included a correction for the normalization at 0.25 cycles/mm of the SWRF data. Measurements from a screen-film combination showed that the MTF of the first version was higher than the second by an average amount of 0.02 units for spatial range 0-3.5 cycles/mm, and on average the MTF of the first version was higher than the second by 10%. Both the SWRF data fitting and the MTF calculation were done within an interactive curve fitting software which made the calculation relatively easy to perform

Wan Hassan, Wan Muhamad Saridan

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W. M. S. W. Hassan

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Calculation of the Modulation Transfer Function from the Square Wave Response Function data with an interactive curve fitting

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1Physica Medica - Vol. XVIII, N. 1, January-March 2002Calculation of the Modulation Transfer Function from the SquareWave Response Function data with an interactive curve fittingsoftwareW.M.S.W. Hassan*Department of Biomedical Physics and Bioengineering, University of Aberdeen, Foresterhill, Aberdeen AB25 2ZD (UK)Manuscript received: May 19, 1999; revised September 27, 2001Accepted for publication: December 31, 2001AbstractTwo ways to calculate the modulation transfer function (MTF) of radiographic screen-film systems from the measured square waveresponse function (SWRF) data were investigated with an interactive curve fitting software. The measured SWRF data obtainedby digitising a radiographic image of a bar pattern test object were fitted to a curve, and the fitted curve was used to calculatethe MTF. Satisfactory MTF was obtained by using 12 terms in the calculation. A second version of the calculation included acorrection for the normalization at 0.25 cycles/mm of the SWRF data. Measurements from a screen-film combination showed thatthe MTF of the first version was higher than the second by an average amount of 0.02 units for spatial range 0-3.5 cycles/mm,and on average the MTF of the first version was higher than the second by 10%. Both the SWRF data fitting and the MTF calculationwere done within an interactive curve fitting software which made the calculation relatively easy to perform.KEYWORDS: Medical radiography, modulation transfer function, screen-film system, square wave response function.Technical Note1. IntroductionThe modulation transfer function (MTF) is com-monly measured for an evaluation of the spatialresolution properties of screen-film systems [1].There are at least three methods to measure the MTFnamely, the slit, the square wave response function(SWRF), and the edge spread function methods.In the SWRF method, a periodic square wavepattern of varying spatial frequency is used as anobject and the contrast of the resulting image isdetermined experimentally. The ratios of the imagecontrast to the object contrast versus spatial frequen-cy give the square wave response and these can beused to estimate the MTF using an equation quotedin a paper by Coltman [2]. We expand the equationup to the 15th term to obtainR u r u r u r u r u r r ua f a f a f a f a f a f a f= + - + +LNM -p433557711111313- - + + +r u r u r u r u r u15151717191921212323a f a f a f a f a f- + - - OQPr u r r r u r u2929313137374141a f a f a f a f (1)where r(u) and R(u) are the square wave responsefunction and the MTF respectively, and u is spatialfrequency.Equation (1) implies that the obtainable modula-tion transfer factor values are restricted by thesquare wave response factors of a limited numberof spatial frequencies available on the square wavepattern test object. This is further worsened by thefact that modulations of the highest spatial frequen-cies available in the test pattern cannot be deter-mined due to noise interfering with the already smallmodulations. A way to deal with part of theseproblems is by fitting the measured square waveresponse factors to an analytic function, and usingthe resulting analytical function as the SWRF inEquation (1) to calculate the MTF. Morishita et al.[3],  for example, employed the Fisher [4] equationfor the purpose.In this investigation, the measured SWRF datanormalized at 0.25 cycles/mm were fitted to ananalytical model [5] and the fitted analytical SWRFwas used to calculate the MTF using 5 to 15 termsof Equation (1). The best number of terms to usewas determined by examining the various MTFcurves obtained.It is important to have an accurately measuredsquare wave response factor at the lowest spatialfrequency possible, as the SWRF will be divided bythe factor for normalization. Van Esch [6] estimatedthat using square wave response factor at 0.25 linepairs/mm instead of 0.025 line pairs/mm for the* Current address: Department of Physics, Universiti Teknologi Malaysia, 81310 Skudai, Johor Darul Takzim (Malaysia). E-mail:saridan@dfiz2.fs.utm.my2W.M.S.W. Hassan: MTF Calculation from SWRF datanormalization will introduce systematic positive biasof 4% and 9% for sharp and unsharp screen-filmsystems respectively. For this, another version of thecalculation considered correcting the normalizationof the SWRF data that was done at 0.25 cycles/mm.The purpose of this article is to investigate aspectsregarding the calculation of the MTF from theSWRF data obtained from on site measurement ina hospital setting, using an interactive curve fittingsoftware.2. Materials and MethodsType 53 PTW-Freiburg bar pattern (available fromFacility for the Assessment of X-ray Imaging atLeeds University, The General Infirmary, Leeds)with 19 groups of line pairs of 0.05 mm leadthickness was used for the measurement. Each linepair group has four line pairs, except the first whichhas two. The groups are 0.25, 0.5, 0.6, 0.7, 0.85, 1.0,1.2, 1.4, 1.7, 2.0, 2.4, 2.9, 3.5, 4.2, 5.0, 6.0, 7.0, 8.5,and 10 line pairs per mm. The bar pattern was placedon a Lanex Regular/T Mat G screen-film combina-tion, and the screen-film combination was on a testrig 20 cm from the floor to reduce scatter. Theexposures were made at 80 kVp with a 16 mm thickaluminium filter affixed to the X-ray tube window.The focus-to-film distance was 1.5 m. The set-upattempted to simulate the exit spectrum from humantissue in chest radiography, and the measurementwas done in a hospital X-ray room using the facil-ities in the room. Thus the radiation quality was notsimilar to radiation quality used in standard MTFmeasurement.The image of the bar pattern was scanned witha microdensitometer (Photoscan System P-1000,Optronics International Inc., Chelmsford, Massa-chusetts, USA) with a square aperture of side 12.5$$m, and sampling distance of 12.5 $$m. From theimage matrix, a computer program calculates theSWRF. Approximately 90 adjacent scan traces areaveraged to give a one dimensional profile of thebar pattern image (optical density versus pixel po-sition). The density values are converted to exposurevalues by the characteristic curve of the screen-filmsystem. The modulation at each spatial frequency isdetermined, giving the SWRF. Finally, the SWRFis normalized by the square wave response factor at0.25 line pairs/mm.An MTF calculator was constructed using a curvefitting software which does a non-linear least squarecurve fitting via the Marquardt-Levenberg algorithm(see Appendix). Given a set of measured SWRF data(in the form of a text file containing two columns,with values of spatial frequency in the first columnand values of normalized SWRF in the second), thecalculator first fit the data to a model [5],f u c u d a u bc aa f a f a f= ◊ - + ◊ ++-exp / /p p2 2 2 2 2 11 4 (2)where the parameters to be fitted are a, b, c, and d.The fitted curve was assumed as the SWRF, and theMTF was computed up to any required number ofterms via Equation (1).Another version of the calculation fits the normal-ized SWRF data to a modified equationg u ec u d a u bc aa f a f a f= ◊ ◊ - + ◊ ++LNMMOQPP-exp / /p p2 2 2 2 2 11 4(3)where the parameters to be fitted are a, b, c, d, ande. The analytical SWRF to be used in the MTFcalculation via equation (1) was then taken asf u g uea f a f= (4)This was thought would correct the SWRF that wasnormalized at 0.25 cycles/mm by extrapolating thefitted curve to zero frequency and normalizing theSWRF by the extrapolated SWRF value at thefrequency.3. Results and discussionA set of SWRF data points measured from thescreen-film combination was obtained. These SWRFdata did not represent optimized data usually usedfor conventional MTF measurement, but represent-ed data obtained by our measurement on site in ahospital X-ray room. As we only wanted to inves-tigate the calculation aspect of the MTF, these datawere enough for our purpose. Figure 1 and its closeup (Fig. 2a and b) show the set of measured SWRFdata points, its fit [f(x)], and the various MTF curvescomputed by the calculator. For the curve fitting, thefinal fit parameters were a = 0.146203, b = 7.70645,c = 0.0153656, and d = 14.1664. No seeds constrain-ing strategy for the curve fitting work was attempt-ed, and we are not clear how the strategy and theproblem of overparameterization might affect theresult. This needs further investigation. In this workonly convergence as offered by the software andvisual inspection of the data against the fitted curvewere used as criteria for a good fit. The MTF curvesobtained by using different number of terms inEquation (1) are mtf5(x), mtf6(x), ... , mtf15(x),where they are the MTF calculated up to the 5th, 6th,…, 15th  term of the equation respectively, and x isthe spatial frequency u. We used x instead of u inthe software environment because the softwareaccepts only x as the independent variable. Figure2a and b clearly show that the MTF curves at lowspatial frequencies fluctuate with the number ofterms used to calculate them.Table I gives the MTF values at zero frequencyfor different number of terms used in Equation (1).3Physica Medica - Vol. XVIII, N. 1, January-March 2002Fig. 1 – A set of measured SWRF data points, its fit [f(x)], and the MTF curves computed bythe calculator using 5 to 15 terms of Equation (1) [mtf5(x), mtf6(x), ..., mtf15(x)].The normalized MTF at zero frequency is unity,therefore the curve which has zero frequency MTFvalue that is closest to unity is the best curve. TableI also gives the ranking in terms of the value whichis closest to unity. Hence, the best curve is mtf12(x),followed by mtf14(x), mtf10(x), mtf6(x), mtf15(x),mtf13(x), mtf11(x), mtf7(x), mtf9(x), mtf5(x) andmtf8(x). Thus when 5 to 15 terms of Equation (1)were considered, a satisfactory estimate of the MTFcould be obtained by using 12 terms.An implication of the above result is that asystematic error might be introduced if one uses asmall number of terms and ‘normalizes’ the MTFobtained via Equation (1) by dividing it with thezero frequency MTF, i.e., the normalized MTF istaken as R(u)/R(0). For example, if 9 terms ofEquation (1) are used, the value of mtf9(0) is0.95608 and the systematic error is a positive in-crease in the MTF of (1-0.95608)/0.95608×100% =4.6% at all spatial frequencies. It is clear thatEquation (1) must be used in its absolute form,division by the zero frequency MTF must be avoid-ed. When a small number of terms are used, onlythe MTFs at spatial frequencies less than about 0.1cycle/mm are prone to error.In Figure 3 the measured SWRF data points, itsfit according to Equation (2) [f(x)], and the MTFcomputed using the calculator up to the 12th term[mtf12(x)] are shown once again for clarity. A curvedescribed by Equation (2) is inherently normalizedat zero spatial frequency (its value is 1.0 at zerospatial frequency). Thus the fitted curve f(x) runsunder the normalized SWRF point (0.25, 1.0). Theseare results of the first version of the calculation.Figure 4 shows the same set of measured SWRFdata points, its modified fit according to Equation(3) [g(x)], its corrected SWRF according to Equa-tion (4) [f(x)], and the MTF computed using thecalculator up to the 12th term [mtf12(x)]. For thismodified fit, the final fit parameters were a =5.99709, b = 5.10619, c = -2.32603, d = 5.21272,and e = 1.07624. Here the fitted curve g(x) runs overpoint (0.25, 1.0) and has a value greater than 1.0 atzero frequency. It is not clear whether it is valid toextrapolate the SWRF by assuming a model ofSWRF as per Equation (3), and it is less clear howto attack this problem. The ‘corrected’ SWRF [f(x)]was obtained by dividing g(x) with e. These areresults of the second version of the calculation.Figure 5 compares the MTF obtained via the firstand the second versions of the calculation.  For thespatial frequency range 0-3.5 cycles/mm, the MTFof the first version was higher than the second byan average amount of 0.02 units, and on average theMTF of the first version was higher than the secondby 10%. Results of Morishita et. al. [3], obtainedfrom four institutions by the SWRF method are alsoshown in the figure. Their results are higher thanours, and this might be due to different X-rayradiation qualities used in the measurement. Furtherstudy is needed to validate our work by determiningthe MTF using radiation qualities of the standardMTF measurement and comparing it with publishedMTF results.Droege and Morin [7] have suggested a methodto determine the MTF of CT scanners that reliesupon the measurement of the standard deviation ofthe pixel values within the image of cyclic barpatterns. This method has not been extented toconventional screen-film radiography due to the factthat relatively large and many different areas of filmhave to be digitized by the microdensitometer before4W.M.S.W. Hassan: MTF Calculation from SWRF dataFig. 2 – A close up of Figure 1 showing the MTF curves at low spatial frequencies fluctuatewith the number of terms used in the calculation. For convenience, the first 5 curves [mtf5(x),mtf6(x), ..., mtf9(x)] are shown in (a), while the remaining [mtf10(x), mtf11(x), ..., mtf15(x)]are in (b). The best result was obtained by using 12 terms of Equation (1) [mtf12(x)].the standard deviation of pixel values of the areascould be calculated. With the advent of CCD filmscanners a study in this direction is worth trying,however, the typical resolution of a CCD filmscanner is around 50 microns as opposed to 10microns for a microdensitometer. Furthermore, inType 53 PTW-Freiburg bar pattern, the bars ofdifferent line pairs per mm are arranged one afteranother. This makes it very difficult to select an areaat the high frequency end of the pattern for thecalculation of the standard deviation.We believe the first version is appropriate whenone has the square wave response factor at a rela-tively low spatial frequency, for example at 0.025cycles/mm, obtained from the bar pattern test object.The second version would be useful in one needsto correct the SWRF data due to unavailability ofa SWRF value at a low enough spatial frequencyfrom the bar pattern test object.4. ConclusionsTwo ways to calculate the MTF of a radiographicscreen-film combination from the measured SWRFa)b)5Physica Medica - Vol. XVIII, N. 1, January-March 2002Table I – Results of the calculation using different number of terms in Equation (1).Number of terms used in Modulation transfer funtion Ranking of modulationEquation (1) value at zero transfer funtion value atzero frequency that isclosest to unity5 mtf5(0) = 1.07372 10th6 mtf6(0) = 1.0133 4th7 mtf7(0) = 0.960942 8th8 mtf8(0) = 0.914742 11th9 mtf9(0) = 0.956079 9th10 mtf10(0) = 0.993479 3rd11 mtf11(0) = 1.02763 7th12 mtf12(0) = 1.00054 1st13 mtf13(0) = 1.02588 6th14 mtf14(0) = 1.00465 2nd15 mtf15(0) = 0.985497 5thFig. 3 – The results of the first version of the calculation showing the set of measured SWRFdata points, its fit [f(x)], and the MTF computed by the calculator using 12 terms of Equation(1) [mtf12(x)].data in a hospital environment were investigated. Themeasured SWRF data were fitted to a curve and thefitted curve gave the analytical SWRF to be used inthe calculation of the MTF up to 15 terms. The MTFcurves at low spatial frequencies (less than about 0.1cycle/mm) fluctuated with the number of terms usedin the calculation, and the best result was obtained byusing 12 terms. A second version of the calculationcorrected for the normalization of the SWRF data thatwas done at 0.25 cycles/mm. The results from meas-ured SWRF data a screen-film combination showedthat the MTF of the first version was higher than thesecond by an average amount of 0.02 units for spatialfrequency 0-3.5 cycles/mm, and on average the MTFof the first version was higher than the second by10%. Both the SWRF data fitting and the MTF cal-culation were done within an interactive curve fittingsoftware which made the calculation relatively easyto perform.AcknowledgementsThe author is grateful to Dr. P.F. Sharp for adviceon some part of the work, and to Universiti Teknolo-gi Malaysia for granting a study leave.6W.M.S.W. Hassan: MTF Calculation from SWRF dataFig. 4 – The results of the second version of the calculation showing the set of measured SWRFdata points, its modified SWRF fit [g(x)], its corrected SWRF [f(x)], and the MTF computedby the calculator using 12 terms of Equation (1) [mtf12(x)].Fig. 5 – A comparison of the MTF obtained from the first and second versions of the calculation.The MTF from the second version was smaller than the first version by an average value of0.02 units. Results of Morishita et al. [3], obtained from four institutions by the SWRF methodare also shown in the Figure.AppendixThe MTF calculator uses an interactive curve fitting software (Gnufit version 1.2, by C. Grammes, availablefrom WWW site http://src.doc.ic.ac.uk/packages/gnuplot, file to download: gft12dos.zip. Newer version mightalso be used, gp371cyg.zip). The calculator was implemented as follow:1. Prepare a normalized SWRF data in the form of a computer ASCII file (call it swrf.dat) consisting 2 columns,the first column (x column) consists of the spatial frequency values, the second column (y column) consistsof the SWRF values.7Physica Medica - Vol. XVIII, N. 1, January-March 20022. Load the Gnufit program.3. Define the following functions and variables:f(x)=(c*exp(-pi**2*x**2/d)+a/(1.+4.*pi**2*x**2/b**2))/(c+a)g(x)=e*((c*exp(-pi**2*x**2/d)+a/(1.+4.*pi**2*x**2/b**2))/(c+a))m5(x)=f(x)+f(3.*x)/3.-f(5.*x)/5.+f(7.*x)/7.+f(11.*x)/11.m9(x)=m5(x)-f(13.*x)/13.-f(15.*x)/15.-f(17.*x)/17.+f(19.*x)/19.m12(x)=m9(x)+f(21.*x)/21.+f(23.*x)/23.-f(29.*x)/29.mtf12(x)=m12(x)*pi/4.a = 0.34b = 4.42c = 1.54d = 260.0e = 1.0In these functions x stands for the spatial frequency in cycles/mm. Thus f(x) represents f(u) in Equation(2), or r(u) in Equation (1). Furthermore, mtf12(x) represents R(u) to the 12th term. The initial guessesof the fit parameters were taken from Boone and Seibert [5]. Other values might work as well.4. Fit the SWRF data:fit f(x) ‘swrf.dat’ via a, b, c, dIf the operation fails to converge, redefine one or any number of the variables to new values, and repeat4. The values of the variables can be displayed by the command show variables. For the second versionof the calculation usefit g(x) ‘swrf.dat’ via a, b, c, d, e5. Plot the SWRF data and its fit:plot ‘swrf.dat’, f(x)If you are satisfied with the fit, do 6. Otherwise redefine one or any number of the variables to new values,and repeat 4. For the second version of the calculation the command isplot ‘swrf.dat’, g(x)6. Plot the MTF:set xrange [0:3.5]plot mtf12(x)Numerical value of the MTF at any spatial frequency can be obtained by the print command. To get theMTF at 2 cycles/mm, type:print mtf12(2)REFERENCES[1] ICRU (International Commission on Radiation Units andMeasurements). Modulation Transfer Function of Screen-Film Systems. Bethesda (Maryland) 1986. Report 41.[2] Coltman JW. The specification of imaging properties byresponse to a sine wave input. J Opt Soc Am 1954: 44;468-471.[3] Morishita J, Doi K, Bollen R, Bunch PC, Hoeschen D,Sirand-rey G, Sukenobu Y. Comparison of two methodsfor accurate measurement of modulation transfer func-tions of screen-film systems. Med Phys 1995: 22; 193-200.[4] Fisher MG. MTF, noise power and DQE of radiographicscreens. Photo Sci Eng 1982: 26; 31-41.[5] Boone JM, Seibert JA. An analytical edge spread func-tion model for computer fitting and subsequent calcula-tion of the LSF and MTF. Med Phys 1994: 21; 1541-1555.[6] Van Esch R. Modulation transfer function evaluation ofscreen-film systems by the grating method. In: BIRReport 18. Technical and physical parameters for qualityassurance in medical diagnostic radiology. Edited by:Moores BM, Stieve FE, Eriskat H, Schibilla H. Eds. TheBritish Institute of Radiology. London 1989; 110-112.[7] Droege RT, Morin RL. A practical method to measurethe MTF of CT scanners. Med Phys 1982: 9; 758-760.

Calculation of the modulation transfer function from the square wave response function data with an interactive curve fitting software

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Calculation of the modulation transfer function from the square wave response function data with an interactive curve fitting software

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