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Rohester Institute of Tehnology RIT Sholar Works Theses Thesis/Dissertation olletions 5-1-1980 omparison of the derivative-transform edge gradient analysis tehnique to tatian's method of edge gradient analysis, varying random noise, trunation interval and sampling internal on analytial edges Robert LaFlesh Follow this and additional works at: http://sholarworks.rit.edu/theses Reommended itation LaFlesh, Robert, "omparison of the derivative-transform edge gradient analysis tehnique to tatian's method of edge gradient analysis, varying random noise, trunation interval and sampling internal on analytial edges" (1980). Thesis. Rohester Institute of Tehnology. Aessed from This Thesis is brought to you for free and open aess by the Thesis/Dissertation olletions at RIT Sholar Works. It has been aepted for inlusion in Theses by an authorized administrator of RIT Sholar Works. For more information, please ontat ritsholarworks@rit.edu.

omparison OF THE DERIVATIVE-TRANSFORm EDGE GRADIENT ANALYSIS TEHNIQUE TO TATIAN'S method OF EDGE GRADIENT ANALYSIS, VARYING RANDom NOISE, TRUNATION INTERVAL AND SAmPLING INTERVAL ON ANALYTIAL EDGES by Robert B~ LaFlesh A thesis submitted in partial fulfillment of the requirements for the degree of Bahelor of Siene in the Shool of Photographi Arts and Sienes in the ollege of Graphi Arts and Photography of the Rohester Institute of Tehnology may, 1980 Signature of the Author Photographi Siene and Instrumentation ertified by Thesis Adviser Aepted by A Suoervisor, Undergraduate Rese8rh

G- 91,(0lO~ OMPARISON OF THE DERIVATIVE-TRANSFORM EDGE GRADIENT ANALYSIS TEHNIQUE TO TATIAN'S METHOD OF EDGE GRADIENT ANALYSIS, VARYING RANDom NOISE, TRUNATION INTERVAL AND SAMPLING INTERVAL an ANALYTIAL EDGES by Robert B. LaFlesh Sumitted to the Photographi Siene and Instrumentation Division in partial fulfillment of the requirements for the Bahelor of Siene degree at the Rohester Institute of Tehnology ABSTRAT The following work has been omputer simulated. A umulative gaussian step response and the step response of a photographi emulsion 1 were taken through the derivative-transform edge gradient analysis tehnique and Tatian's method of analysis. Random noise levels, trunation intervals and sampling intervals on the analyti edges were varied to determine their influene on eah tehnique. The variane and means of the alulated m.t.f.s were then statistially tested for no differene of the two tehniques. The exat noise free M.T.F. was also alulated and ompared to the M.T.F.s alulated by the two tehniques. The results show a statistial differene in the two tehniques at low frequenies. This differene was deemed not signifiant from a pratial standpoint beause the magnitude of the differenes between the M.T.F.s was small ompared to the atual magnitude of the M.T.F.s at low frequenies. Also the two tehniques produed M.T.F.s of high devia-~tions from the exat noise free transform at high frequenies.. ii

AKNOWLEDGEMENTS The author wishes to extend his graditude to his thesis advisor Mr. Roland 'JJ. Porth of the Xerox orooration for suggesting the original topi of study and for the numerous interesting and informative talks of imag ing siene that lead to the ulmination of this thesis, Also the author wishes to thank Dr. Newburg of the Rohester Institute of Tehnology for his mathematial aid in the thesis work. Finally a speial thanks to Mr. Dan Siems and Mr. Tim Wilson, students of the Rohester Institute of Tehnology, for their omputer expertise whih was extremely helpful while programming. m

TABLE OF ONTENTS LIST OF FIGURES...vi LIST OF TABLES ix INTRODUTION 1 THEORETIAL BAKGROUND 3 HYPOTHESES, 11 PROEDURE 12 DISUSSION..2 5 REOMMENDATIONS 37 SUMMARY 38 REFERENES 40 APPENDIX A: GRAPHIAL DATA OUTPUT 42 APPENDIX B: MAIN PROGRAMS 67 FDATA 68 GADATA 6 9 DERIV 70 DTRANS, 71 TAT IAN 73 MAIN1 7 5 MAIN2,76 MAIN2P 77 MAIN3 78 iv

FMAIN3 8 0 DIFF 82 TEST VAR 8 3 TESTMEAN 84 APPENDIX : GRAPHING PROGRAMS 85 GRAPH2.86 GRAPHDIFF 8 8 FGRAPH1T 90 FGRAPH2T 92 FGRAPH3T 95

LIST OF FIGURES FIGURE 1: EDGE UJITH 100:1 SIGNALtNOISE RATIO FIGURE 2: EDGE WITH 100:10 SIGNALrNOISE RATIO FIGURE 3: GRAPHIAL DATA GENERATING PROEDURE FIGURE 4: GRAPHIAL DATA GENERATING PROEDURE FIGURE 5: m.t.f. OMPARISON OF GAUSSIAN SPRD. FTN. M.T.F.=AVERAGE OF 1000 M.T.F.S 10:1 (SIGNALtNOISE) 5.333 ( SAMPLES/SIGMA ) FIGURE 6: M.T.F.OMPARISON OF GAUSSIAN SPRD. FTN. M.T.F.sAVERAGE OF 1000 M.T.F.S 10:1 (SIGNAL :NOISE) 1. 333 (SAMPLES7SIGMA ) FIGURE 7: DIFFERENE OF M.T.F. (-)ONTINUOUS TRANSFORM OF GAUSSIAN SPRD. FTN. VARYING S I GNAL:NOISE(DERV. -TRANSFORM) FIGURE 8: DIFFERENE OF M.T.F. (-)ONTINUOUS TRANSFORM OF GAUSSIAN SPRD. FTN. VARYING SIGNAL:NOISE(TATIAN) FIGURE 9: M.T.F. OF GAUSSIAN SPRD. FTN. VARYING SIGNALrNOISE RATIO (TATIAN ) FIGURE 10: VARIANE OF GAUSSIAN SPRD. FTN. VARYING SIGNALrNOISE RATIO (TATIA N ) FTGURE 11: M.T.F. OF GAUSSIAN SPRD. FTN. VARYING SIGNALrNOISE RAT 10 (DERV. -TRANSFORM) FIGURE 12: VARIANE OF GAUSSIAN SPRD. FTN. VARYING SIGNALrNOISE RATIO (DERV. -TRANSFORM) FIGURE 13: M.T^F. OF GAUSSIAN SPRD. FTN. VARYING SAMPLING INTERVAL(TATIAN) FIGURE 14: VARIANE OF GAUSSIAN SPRD. FTN. VARYING SAMPLING INTERVAL(TATIAN) FIGURE 15: M.T.F. OF GAUSSIAN SPRD. FTN. VARYING SAMPLING INTERVAL (DERV. -TRANSFORM) vi

FIGURE 16: VARIANE OF GAUSSIAN SPRD. FTN. VARYING SAMPLING INTERVAL (DERV. -TRANSFORM) FIGURE 17: M.T.F. OF GAUSSIAN SPRD. FTN. VARYING TRUNATION INTERVAL (TAT IAN ) FIGURE 18: VARIANE OF GAUSSIAN SPRD. FTN. VARYING TRUNATION INTERVA L (TATIAN ) FIGURE 19: M.T.F. OF GAUSSIAN SPRD. FTN. VARYING TRUNATION INTERVAL (DERV --TRANSFORM) FIGURE 20: VARIANE OF GAUSSIAN SPRD. FTN. VARYING TRUNATION INTERVAL (DERV- -TRANSFORM) FIGURE 21: M.T.F. OF FRIESER SPRD. FTN. VARYING SIGNALrNOISE RATIO (TATIAN ) FIGURE 22: VARIANE OF FRIESER SPRD. FTN. VARYING SIGNALrNOISE RATIO (TATIAN ) FIGURE 23: M.T.F. OF FRIESER SPRD. FTN. VARYING SIGNALrNOISE RATIO (DERV. -TRANSFORM) FIGURE 24: VARIANE OF FRIESER SPRD. FTN. VARYING SIGNALrNOISE RATIO (DERV.-TRANSFORM) FIGURE 25r M.T.F. OF FRIESER SPRD. FTN. VARYING SAMPLING INTERVAL(TATIAN) FIGURE 26: VARIANE OF FRIESER SPRD. FTN. VARYING SAMPLING INTERVAL(TATIAN) FIGURE 27: M.T.F. OF FRIESER SPRD. FTN. VARYING SAMPLING INTERVAL (DERV. -TRANSFORM) FIGURE 28: VARIANE OF FRIESER SPRD. FTN. VARYING SAMPLING INTERVAL (DERV. -TRANSFORM) FIGURE 29: M.T.F. OF FRIESER SPRD. FTN. VARYING TRUNATION INTERVAL (TATIAN ) FIGURE 30: VARIANE OF FRIESER SPRD. FTN. VARYING TRUNATION INTERVAL (TATIA N ) FIGURE 31: M.T.F. OF FRIESER SPRD. FTN. VARYING TRUNATION INTERVAL (DERV.-TRANSFORM) VII

FIGURE 32 VARIANE OF FRIESER SPRD. FTN. VARYING TRUNATION INTERVAL (DERV. -TRANSFORM) vm

LIST OF TABLES TABLE 1: STATISTIALLY SIGNIFIANT DIFFERENT MEANS TABLE 2r STATISTIALLY SIGNIFIANT DIFFERENT VARIANES TABLE 3: STATISTIALLY SIGNIFIANT DIFFERENT VARIANES ix

INTRODUTION Generating the line spread funtion by taking the numerial derivative of the edge response funtion will 2 magnify the effets of noise. Tatians diret method of redution of the edge response funtion avoids the diff iultly of taking the numerial derivative of a noisy 3 funtion. The influene of noise, in theory, is thought to be quite omplex and was therefore not attempted in 4 this paper. Therefore noise was only used as a blak box system to determine if a differene exist between the final output of Tatian's tehnique and the deriva tive-transform tehnique of edge analysis. The trunation interval and the sampling interval also effet the auray of the M.T.F. due to the in herent nature of the theory of Fourier algebra. By varying these parameters of noise, trunation and sampling intervals, a omparison was made to determine the usefullness of either tehnique. Most edge analysis systems operate on Tatian's tehnique of edge gradient analysis,, But from the theo retial standpoint both methods are the same. There fore there is some question as why most systems operate

on Tatian's tehnique of edge gradient analysis. The purpose of this researh is to determine if a differene exist between the two tehniques while using a omputer simulated analysis.

THEORETIAL BAKGROUND Sine edges our frequently in nature they are readily available for a method of system analysis. The theory of edge analysis begins with an objet. First the objet f(?) is divided into infinitely long ret angles of width d?. The intensity of one of the sub divisions of the objet is f(i)dl. The image of eah of the subdivisions is a line spread funtion. Therefore the line spread funtion h(\) is an image of a line objet formed by a system. If g(x) is the image irradiane, then dg(x) is the image irradiane at the point x due to the subdivisions of the objet at I. Therefore 4}x) =?(*) V.(x-*}<U (D If the point x is held fixed and the irradiane ontributions are summed for all oossible objet sub divisions, the total image irradiane at x will be ob tained. This is aomolished by integrating equation 1 over all possible values of \\. The image irradiane at x, g (x ) is then *(!>H*-T)d5 (2)

The graphial interpretation of this one dimensional onvolution is 4 * *.(*) i-^t. An edge an now be onsidered the objet. For a unit edge f(s) = 0.0 for?<-0.0 and = f(*) 1.0 forl^o.o. Sine f(i) * h(5) = h (I ) * f(l) Sine the edge represents a step funtion X (4) where e(x) is the image irradiane. Graphially e(x} * 'I The edge image irradiane is just the umulative area under the spread funtion I.V\A&E Of D>E. Differentiating equation 4..<<. -Ujtt a*) (5) Equation 5 is one of the basi equations of system The O.T.F. is defined as the Fourier

Transform of the line spread funtion. Given by -oo (6) Substituting equation 5 into equation 6 yields ao dx dx (?j In pratie an edge is sanned on a mirodensitometer whih yields density or transmittane vs. position. This urve is then onverted from density or transmittane vs. position to effetive exposure vs. position using the 7 marosopi response urve. The derivative-transform tehnique of edge analysis makes use of equation 7 in the following way. The num erial derivative of the edge irradiane (effetive exposure) is taken to approximate the line spread funtion of the system. The Fast Fourier transform is then applied whih produes the O.T.F.. Tatian's method of edge analysis makes use of the fat that the numerial derivative of a funtion f(x) in the spatial domain is equal to multiplying the funtions transform F(s), frequeny representation of f(x), by i2-fff. This is represented by 8 V dx (8) where e(x) e -UTT-fx dx, EO (9)

Substituting equation 9 into equation 8 yields Equation 10 is Tatian's derivation of the O.T.F.. In (10) terms of sampled values of the edge response funtion the O.T.F. is given by 10 X L2Trfe0i} -L.2Tr-?y..i 0 (Hi >?0 (n) This applies only where the line spread funtion h(x) is band limited, i.e. its transform vanishes for frequenies greater than a utoff frequeny f ; and the sample spae obeys the ondition t=.-=^f. To obtain a working formula Tatian trunated the infinite limits of equation 11 and added orretion fators. These orretion fators are based on the asymptoti behavior of e(x). 1 -i Sine the transfer fun tion of photographi film is sought only two orretion fators are derived for the working formula. The deriva- 1 2 tion of these orretion fators are as follows It is known that x e(x) ( = /-ao K(1)U (4) whih equals Inverting o the transform of equation 6 yields for a unit edge (12) f(l)=0.0 for?<.0.0

,*.ntti " Substituting equation 13 into equation 12 (13) (14) hanging the order of integration (15) Taking the transform of the step funtion (16) Expanding and making use of the sifting property fm-^t.w-^xfl + lt^f^it-? """ 2TT-f > (17) The integral of an odd funtion over symetri limits equals zero. Therefore (18) Sine T(f) is the transform of a real funtion, T< (f) and T2(f) are even and odd funtions, respetively. Therefore equation 18 an be written Vj r 2TTf + T.fl os 2trf 2TTf u (19) The behavior of e (n ) for large n, (spatial domain), depends on that of T(f) near its origin. 11 Expanding T-j(f) and T2(f) in equation 19 as a power series about

the origin and integrating taking the limit for large n yields e(ne) ss l* T* oy _ 2TT**e 2 T/"^) $or Uf^e n (20) (-no * -xl -Tree),... Sine T.,(f) is an even funtion and T2(f) is an odd funtion all of the oeffiients T '(0), T '''(0),... and Tj'^O),... would be zero if T(f) were an analyti funtion at the origin. The even and odd parts of e(x) are given by (21) '2 Substituting equation 20 into equation 21 yields ea(n-) *, O + Tt (o)/2ttan.i (22) Note that sine analyti edges were ompared in this tehnique for large n

This result is intuitive sine for large -t-n, e (w ) is very lose to one and e(-nfe) is very lose to zero. Equation 11 an be written in its even and odd parts as oo n-\ (24) I. ns, - The values of e (ne) and e2(ne) in equation 24 an be given in terms of atual sampled values only up to some finite value of n, say N. For n larger than N, e.(nfe) and e-,(ne) are given by equation 23. Breaking down equation 24 yields N 1 T(fl.HWt [*!?! t,me-2f* t,m*2^1 (25) Substituting equation 23 into equation 24 and making use of the relationships.v*i a^u^ (26) ^K1 OT IAU = -Siry-^ *^U yields -ppv / W -3TT -P 4. os(n*v^ait^ n..

10 T, 27) Tatian's method is exat when the sum is arried to an infinite san length. The trunation required by Tatian's algorithm(finite san length) produes error. 14

11 HYPOTHESES The purpose of this experiment is to determine if there is no differene between Tatian's method of edge gradient analysis and the derivative-transform tehnique of edge analysis. The fators that will be varied are the level of random noise, trunation interval and the sampling interval of the edge. The hypotheses to be tested are There is no signifiant differene between the varianes of the alulated M.T.F.s using Tatian's vs. the deriv ative-transform edge analysis tehnique and, there is no signifiant differene between the means of the alulated M.T.F.s using Tatian's vs. the derivativetransform edge analysis tehnique. Fisher (F) tests will test for no signifiant diff erene of varianes at various frequenies. T tests will test for no signifiant differene of means at various frequenies.

12 PROEDURE All of the data required for this analysis was 1 5 omputer simulated. The entire program analysis was broken down into many subroutines and mini-programs whih allowed a variety of funtions to be inluded in any one run of the program. It was also onsidered by the experimentor to be the most systemati way of developing the program. Four subroutines by other authors were used. 1 (S They were the Fortran subroutine FOURG by Norman Brenner to alulate Fast Fourier Transforms, subroutines GAUSS 1 7 and RANDU whih together ompute normally distributed 1 fl unorrelated random numbers with a given mean and 1 9 standard deviation and the subroutine NTDR for gen erating the umulative gaussian edge. The omputer program GADATA, to generate the anal yti umulative gaussian urve, was produed first. The umulative gaussian urve is given by '-an t%3 dx This urve was hoosen beause it approximates a um ulative photographi system spread funtion. A normal ized gaussian iuas used so that samdling and trunation

13 interval ould be produed in terms of standard deviation units, <T. A subroutine of Tatian's method, TATIAN, equation 27, was then reated to produe the M.T.F. and Phase angle in radians of the input edge funtion. A subroutine of the derivative-transform method, DTRANS and supporting subroutine DERIV, representing equation 7, within finite limits, were then reated to produe the M.T.F. and Phase in radians of the input edge data. Four main programs were developed next to produe an average M.T.F. of the M.T.F.s alulated by DTRANS and TATIAN while varying levels of random noise, MAIN1; varying sampling interval, MAIN2 and MAIN2P; and varying trunation interval, MAIN3. MAIN1 alulated an average M.T.F. and orrespond ing varianes at the following signalrnoise ratios 100:10 100:5 100:9 100:4 100:8 100:3 100:7 100:2 100:6 100:1 Signalrnoise ratio was defined to be the maximum signal level divided by the standard deviation of the noise, reated by GAUSS. Graphial representations are given

14 in Figure 1 and 2. The systemati flow of generating an average M.T.F. and its varianes for different levels of random noise was as follows An analyti edge with values between zero and one, inlusive, orresponding to +/- 24T, was reated by GADATA. This edge had a samoling interval of 5.333 samples/ty. This edge was then read into MAIN1. A er tain level of random noise that was independent of signal level was then added to the edge* The noisy edge was then taken through both TATIAN and DTRANS to produe two M.T.F.s. This proess, of adding the same level of random noise to the edge and taking it through both TATIAN and DTRANS(with onstant samoling and trunation intervals) was repeated one hundred times to produe two families of urves, one family for eah M.T.F. method. The mean and variane, at different frequenies, of eah family of urves was then determined in MAIN1 to produe an average M.T.F. and orresponding varianes. This entire proedure was repeated for the ten afore-mentioned different levels of random noise. This yielded the first row of Figure 3. MAIN2 and MAIN2P alulated an average M.T.F. and varianes at the sampling interval orresponding to the input edge data sampling interval. The signalrnoise

15 ratio of the edges were 20:1 with a +/- 24 T trunation interval. MAIN2 read five different sets of edge data with sampling interval 5.333 samples/t 4.500 samples/g- 3.458 samples/j 2.417 samples/g- 1. 333 samples/r MAIN2P read only one set of edge data for eah run. Two runs were made with samoling intervals.667 samples/ir.333 samples/v The systemati flow of generating an average M.T.F. and varianes for different sampling intervals was as follows The seven analyti edges of varying sampling interval were reated by GADATA. These edges were read inta their perspetive programs, MAIN2 or MAIN2P. A.05 level of random noise was added over eah of the differently sampled edges. This is representative of a 20:1 signal: noise ratio with +/- 24 <T trunation interval. Eah noisy edge was then taken through both TATIAN and DTRANS to produe forteen M.T.F.s. This proess of adding the same level of random noise to eah differently samoled edge and taking it through both TATIAN and DTRANS

16 (with onstant trunation interval and onstant level of random noise) was repeated one hundred times for eah differently sampled edge to produe forteen families of urves, one family for eah M.T.F. method with eah different sampling interval. The mean and variane, at different frequenies, of eah family of urves was then determined in MAIN2 and MAIN2P to produe an average M.T.F. and orresponding varianes. This proess yielded the seond row of Figure 3. MAIN3 alulated an average M.T.F. and varianes at the following +/- 5.0625 <T trunation intervals?/- 2.8125 T +/-.5625 T?/- 4.6875 T?/- 2.4375 V +/- 4.3125 Q-? /- 2.0625 IT?/- 3.9375 <T +/- 1.6875 <T?/- 3.5625 H +/- 1.3125 T +/- 3.1875 0- +/-.9375 1 The systemati flow of generating average M.T.F.s and varianes for different trunation intervals was as follows An analyti edge with values between zero and one, inlusive, orresponding to +/- 24 <T, was reated by GADATA. The sampling interval of this edge was kept a onstant 5.333 samoles/<t. This original edge was read into MAIN3. A.05 level of random noise was added to

17 the points on the edge inbetween the trunation points. The noisy edge was then taken through both TATIAN and DTRANS to produe two M.T.F.s. This proess of adding the same level of random noise to the edge and taking it through both TATIAN and DTRANS(with onstant sampling interval and onstant level of random noise) was repeated one hundred times to produe two families of urves, one family for eah M.T.F. method. The mean and varianes, at different frequenies, of eah family of urves was then determined in MAIN3 to produe an average M.T.F. and orresponding varianes. This proedure was then repeated for different trunation intervals. The trun ation segment of MAIN3 beame ative after the first average M.T.F. and varianes were determined. The original analyti edge data was trunated by setting ooints outside and inluding the required trunation points to zero and one D.. levels. This proedure represents a trunated edge whih was trunated at exat ly the required points. The trunation interval on the analyti edge dereased in units of +/-.3750 <T after eah average M.T.F. and varianes were determined. The afore-mentioned trunation intervals were thus produed. This proess yielded the third row of Figure 3. The omputer program FDATA, to generate the analyti umulative Frieser urve, was produed next. The

18 umulative Frieser urve is given by This urve was hoosen beause it approximates a umula tive photographi emulsion spread funtion. A normalized Frieser urve was used so that sampling and trunation intervals ould be produed in terms of standard deviation units, <r. The main program MAIN1 read the umulative Frieser analyti edge. This edge had a sampling interval of 5.333 samples/tj-. The systemati flow of generating an average M.T.F. and varianes for different levels of random noise followed the afore-mentioned proedure of MAIN1. This yielded the first row of Figure 4. The main programs MAIN2 and MAIN2P read the umula tive Frieser analyti edges where the sampling intervals were similiar to the previous proedure of MAIN2 and MAIN2P. The systemati flow of generating an average M.T.F. and varianes for different sampling intervals followed the afore-mentioned proedure of MAIN2 and MAIN2P. This yielded the seond row of Figure 4. The main program FMAIN3 was developed to produe an average M.T.F. of the M.T.F.s alulated by TATIAN and DTRANS while varying trunation interval. FMAIN3 alulated an average M.T.F. and varianes at the

->/- 11.0625 19 following trunation intervals of the input edge data?/- 13,8750 Q" +/- 9.3750 V?/_ 4.8750T +/- 13.3125 <T +/- 9.8125 0* +/- 4.3125 r +/- 12.7500 <r >/- 8.2500 ff +/- 3.7500 <T +/- 12.1875 Q +/- 7.6875 T +/- 3.1875 tr +/- 11.6250 <r +/- 7.12 50 r +/- 2.6250 V 0* +/- 6.5625 T +/- 2.0625 <T?/- 10.5000 0" +/- 6.0000 r +/- 1.5000 T +/- 9.9375 <T?/- 5.4375 <T +/-.9375 <T The main program FMAIN3 read the umulative Frieser analyti edge as the original edge data0 The systemati flow of generating average M.T.F.s and varianes for different trunation intervals was the same as that of MAIN3. programs was that FMAIN3 The only differene in the two dereased its trunation interval of the Frieser analyti edge in units of >/-.5625 T after eah average M.T.F. and varianes were determined. This yielded the third row of Figure 4. The number of tunation intervals for both umula tive edges was hoosen on the basis of an upper limit of omputer entral proessing unit time for eah individual run of MAIN3 or FMAIN3. The limit was set to approx imately twenty minutes. The program TESTVAR was developed to test the input varianes at a.1 level of onfidene. The test

20 statisti was the larger sample variane/smaller sample variane. The program output the signifiantly different varianes and the test statisti. The program TESTMEAN was developed to determine the test statisti and degrees of freedom of the input means at a.1 level of onfidene using an approximation of the 20 Fisher-Behrens test A series of graphing programs, for the Zeta plotting system21, to be used for speifi graphing purposes were developed last for data output and data omparison. These programs were GRAPH XGRAPH2T FGRAPH1D GRAPH2 XGRAPH2D FGRAPH2T GRAPHDIFF XGRAPH3T FGRAPH2D XGRAPH1T XGRAPH3D FGRAPH3T XGRAPH1D FGRAPH1T FGRAPH3D A representative group of these programs an be found in appendix of this thesis with their speifi purpose listed in the doumentation of eah program.

21 '-4. 00-2. 00 0. 00 2. 00 STRNDRRD DEVIATION UNITS 4. 00 FIGURE 1 : EDGE WITH 100:1 SIGNAL:N0ISE RATIO

22 '-4. 00-2.00 0.00 2.00 STANDRRD DEVIATION UNITS 4. 00 FIGURE 2: EDGE WITH 100:10 SIGNAL:N01SE RATIO

23 UMULATIVE GAUSSIAN EDGE TEHNIQUE DERIVATIVE- TATIAN TRANSFORM VARYING SIGNAL: 100: 10 100:5 100: 10 100:5 NOISE RATIO 100: 9 100:4 100: 9 I00r4 (5.333 samples/or) 100: 8 100:3 100: 8 I00r3 ( +/- 24 (Ttrunation ) 100: 7 100:2 100' 7 100:2 100' 6 100:1 100 6 100:1 VARYING SAMPLING 5.333 samples/tt 5.333 samples/r INTERVAL 4.500 samples/* 4.500 samples/* (20:1 signal :noise ) 3.458 samples/* 3.458 samples/r ( +/- 24 (Ttrunation ) 2.417 samdles/ff 2.417 samples/* 1.333 samples/tt 1.333 samples/*.667 samples/*.333 samples/^.667 samples/*.333 samples/*?/- VARYING TRUNATION 5.0625 T 5.0625 *?/- +/r INTERVAL 4.6875 * +/- 4.6875 * t- (20:1 signal :noise ) */- 4.3125 +/r 4.3125 * 0*?/- (5.333 samples/*) 3.9375 * 3.9375 +/r */- */- 3.5625 * 3.5625 T?/- 3.1875 <T?/- 3.1875 T " +/- 2.8125?/- 2.8125 <T 0" +/- 2.4375 +/- 2.4375 r +/- 2.0625 *?/,- 2.0625 T?/- 1.6875 * +/r 1.6875 *?/- 1.3125 r?/- 1.3125 r +^- +/-.9375 * *.9375 +/-?/-.5625.5625 * * FIGURE 3: GRAPHIAL DATA GENERA TING PROEDURE

24 UMULATIVE FRIESER EDGE TEHNIQUE DERIVATIVE- TRANSFORM TATIAN VARYING SIGNAL: NOISE RATIO (5.333 samoles/*) (+/- 24 <r trunation ) 100r10 100:9 100:8 100:7 100:6 100:5 100:4 100:3 100:2 100:1 100:10 100:9 100:81 100:7 100:6 100:5 100:4 100:3 100:2 100:1 VARYING INTERVAL SAMPLING (20:1 signal:noise ) ( 24 +/- T trunation ) 5.333 4.500 3.458 2.417 1.333 samples/r samples/* samples/* samples/* samples/* 5.333 samples/* 4.500 samples/* 3.458 samples/* 2.417 samples/* 1.333 samples/*.667 samples/*.667 samples/*.333 samples/*.333 samples/* VARYING TRUNATION INTERVAL (20:1 signal :noise ) (5.333 samples/r)?/- 13.8750 *?/- 13.3125 *?/- 12.7500 <r?/- 12.1875 * +/-?/- 11.0625 11.6250 * *?/- 10.5000 *?/- 9.9375 T?/- 9.3750 *?/- 8.ei25 <r?/- 8.2500 ** +/- 7.6875 *?/- 7.1250 * +/- 6.5625 *?/- 6.0000 *?/- 5.4375 *?/- 4.8750 *?/- 4.3125 *?/- 3.7500 * +/- 3.1875 <r?/- 2.6250 V?/- 2o0625<r?/-! I :: t- i 3.8750 * 3.3125 * 2.7500 r 2.1875 * 1.6250 * 1.0625 0.5000 * 9.9375 * 9.3750 * 8.8125 tr 8.2500 r 7.6875 * * 7.1250 6.5625 <T *" 6.0000 5.4375 r 4.8750 r 4.3125 * * 3.7500 3.1875 * 2.6250 r 2.0625 *?/- 1.5000 * 1.5000 +/-.9375 *.9375 * FIGURE 4: GRAPHIAL DATA GENERATING PROEDURE

25 DISUSSION The results of the test on the means and varianes are tabulated in Tables 1,2 and 3. The variability of obtaining M.T.F.s from Tatian's tehnique and the derivative-transform tehnique while varying trunation interval on both the Frieser and the Gaussian umulative edge urves, at the sampling interval of 5.333 samples/* and signalrnoise ratio of 20:1, are the same. The Fisher-Behrens test of the means has shown that there is no differene in alulating M.T.F. values from Tatian's tehnique and the derivativetransform tehnique while varying the trunation interval on both the Frieser and Gaussian umulative edge urves, at the sampling interval of 5.333 samples/* and 20:1 signal:noise ratio. This sampling interval plaes approx imately thirty two points on the edge where the edge ontains 99.72^ of its unit area. The variability of obtaining M.T.F.s from Tatian's tehnique and the derivative-transform tehnique while varying the signal:noise ratios on both the Frieser and the Gaussian umulative edge urves, at the sampling interval of 5.333 samples/tt and trunation interval of

26 +/- 24<T are not the same at low frequenies. This differene in the varianes in the two tehniques an possibly be attributed to how eah tehnique handles the noise. The different varianes oured at the low frequenies of approximately.0208 through.1458 yles/r. This orresponds to an error free M.T.F. in the range of 1.0 through.65. Tatian's tehnique produed lower varianes at these frequenies by an approximate 1 :2 ratio. Figures 5 and 6 demonstrate the differene in the varianes of the two tehniques at low frequenies. It an also be noted that as the level of random noise dereases the higher of the low frequenies, of different variability, began to have the same variability. This adds to the idea that the differene between the var ianes of the two tehniques exist beause of how eah tehnique handles the noise. The higher the noise the more variability between the two tehniques. The Fisher- Behrens test of the means has shown that there is no differene in alulating the M.T.F. values from Tatian's tehnique to the derivative-transform tehnique while varying the signalrnoise ratio on both the Frieser and the Gaussian umulative edge urves, at the samoling interval of 5.333 and trunation interval of +/- 24T. The variability of obtaining M.T.F.s from Tatian's

27 tehnique and the derivative-transform tehnique while varying the sampling interval on both the Frieser and the Gaussian umulative edge urves, at a 20:1 signal: noise ratio with a +/- 24T trunation interval, are not the same at low frequenies. The Fisher-Behrens test of the means has shown that there is a differene in alul ating M.T.F. values, at low frequenies, from Tatian's tehnique and the derivative-transform tehnique while varying the sampling interval on both the Frieser and the Gaussian umulative edge urves, at a 20:1 signal:noise ratio with a +/- 24 T trunation interval. This differ ene in the means oured at the sampling interval of 3.458 samples/r for both the Frieser and the Gaussian umulative edge urves. Sine this oured in the low frequenies it is believed to be attributed to how eah tehnique handles the noise. This is based on the signifiantly different varianes that were observed in the low frequenies during the test of the varianes. The deviation of Tatian's and the derivativetransform tehnique from the ontinuous noise free transform is graphed in Figures 7 and 8, respetively. These represent varying signal:noise ratios on the umulative gaussian edge. General remarks are also to be noted in the graph ial data. An inrease in the magnitude of random

28 flutuations ours with inreasing frequeny. An in rease in flutuations ours with inreasing sampling interval at onstant trunation interval. Also servers trunation auses deformation of the output M.T.F.. This graphial data output is loated in appendix A. A very important topi that must be disussed for these results to apply to the pratial world is normalization. Normalization, after transformation of the line spread funtion, was not required in this study of the derivative-transform tehnique, beause the original edge data was assumed to run from zero to one D.. levels. It is important to note that the prenormalization of the edge data is not neessary for the derivative transform tehnique in atual pratie, as long as the resulting M.T.F. is properly normalized. Pre-normalization of the original edge data is manditory in Tatian's tehnique. Sine there is no entral ordinate value of the M.T.F. produed by Tatian's tehnique, it is not possible to properly normalize the data after the M.T.F. is obtained, by division of eah value by the entral ordinate M.T.F. value. Therefore in pratie the edge data must be pre-normalized to run from zero to one D.. levels. The deision is left to the reader to sarifie the variability of Tatian's tehnique for the ease involved in not having to

29 pre-normalize the edge data for the derivative-transform tehnique. Most of the differenes between the means and varianes, in the data, oured at low frequenies rel ative to the M.T.F. 's folding frequeny values. The signifiantly different varianes were approximately.005 or less in most ases. The signifiantly different means were approximately equal to 1.0. From the pratial standpoint the error produed by using either tehnique is relatively small when the atual magnitude of the M.T.F. values are ompared to the magnitude of the statistially signifiant variane values. Therefore these statistially different mean M.T.F.s,, at low frequenies, are pratially the same.

30 TABLE 1: STATISTIALLY SIGNIFIANT DIFFERENT MEANS UMULATIVE GAUSSIAN EDGE VARYING SIGNAL: NOISE RATIO (5.333 samples/d) ( +/- 24 T trunation ) VARYING SAMPLING INTERVAL(samples/r) (20:1 signal :noise ) (+/- 24 j- trunation ) NONE 3.458 a.0208 and.0417 yles/* VARYING TRUNATION INTERVAL(+/-Q" ) (20:1 signal :noise ) (5.333 samples/r) NONE UMULATIVE FRIESER EDGE VARYING SIGNAL: NOISE RATIO NONE (5.333 samples/r) (?/- 24 trtrunation ) VARYING SAMPLING INTERVAL(samples/<r) (20:1 signalrnoise) (+/- 24 * trunation ) 3.458 @.0208 and.0417 yles/** VARYING TRUNATION INTERVAL^/- *) (20:1 signal:noise ) (5.333 samples/*) NONE

31 TABLE 2: STATISTIALLY SIGNIFIANT DIFFERENT VARIANES UMULATIVE GAUSSIAN EDGE VARYING SIGNAL: NOISE (5.333 samples/*) (?/- 24 T trunation ) 100:10 @.0208 through.1458 yles/fr 100:9 @.0208 through.1458 yles/* 100:8 a.0208 through.1458 yles/* 100:7 @.0208 through.1458 yles/v 100:6 a.0208 through.1458 yles/* 100:5 d.0208 through.1458 yles/^ 100:4 @.0208 through.1458 yles/r 100:3 I.0208 through.1458 yles/* 100:2 Q.0208 through.1250 yles/* 100:1 a.0208 through.1250 yles/* VARYING SAMPLING INTERVAL(samples/*) (20:1 signal :noise ) (+/- 24 trunation ) 5.333 @.0208 through.1458 yles/* 4.500!,,0208 through.1250 yles/* 3.458.0208 through.1250 yles/* 2.417 9.0208 through.1042 yles/* 1.333 a.020b through.0625 yles/r.667 tl.0208 and 90417 yles/*.333 ta.0208 yles/ VARYING TRUNATION INTERVAL^/- * ) (20:1 signalrnoise) (5.333 samples/*) NONE

32 TABLE 3: STATISTIALLY SIGNIFIANT DIFFERENT VARIANES UMULATIVE FRIESER EDGE VARYING SIGNAL: NOISE RATIO (5.333 samples/r) (+/- 24 (Ttrunation ) 100:10 a.0208 through 100:9 a.0208 through 100:8.0208 through.1458 yles/*.1458 yles/*,1458 yles/* 100:7 ti.0208 through.1458 yles,* 100:6 8.0208 through.1458 yles/r 100:5 (a.0208 through.1458 yles/<r 100:4 @.0208 through.1458 yles/* 100:3 t.0208 through.1250 yles/* 100:2 e.0208 through.1250 yles/* 100:1 @.0208 through.1250 yles/* VARYING SAMPLING INTERVAL(samples/T) (20:1 signalrnoise) (+/- 24 (Ttrunation ) 5.333 1.0208 through.1458 yles/* 4.500 d.0208 through.1250 yles/* 3.458 ti.0208 through.1250 yles/* 2.417 ta.0208 through.1042 yles/* 1.333 9.0208 through.0833 yles/r.667 @.0208 and.0417 yles/*.333 @.0208 yles/* VARYING TRUNATION INTERVAL(+/~ <T) (20:1 signal :noise ) (5.333 samples/*) NONE

33 O o * * * TRTIRN TEHNIQUE = 2 SIGMA (TATIAN) DERIVATIVE TRANSFORM TEHNIQUE = 2 SIGMA (DERV. -TRANSFORM) ^.0000 0 1250 0.2500 0.3750 0.5000 FREQUENY (YLES/SIGMA) 0. 6250 FIGURE 5: Ml OMPARISON OF GAUSSIRN SPRD. FTN.F. MTF OF 1000 M.T.F.S =AVERAGE 10:1 (SIGNAL-NOISE) 5. 333 (SAMPLES/S I GMA)

34 o in * -= * * TATIAN TEHNIQUE = 2 SIGMA (TATIAN) DERIVATIVE TRANSFORM TEHNIQUE = 2 SIGMA (DERV. -TRANSFORM) O. 0000 0.1250 0.2500 0.3750 0.5000 FREQUENY (YLES/SIGMA) 0. 6250 M f F OMPARISON OF GAUSSIAN SPRD. FTN. M. T. F. =AVERAGE OF 1000 M.T.F.S 10:1 (SIGNALrNOISE) 1. 333 (SAMPLES/SIGMA)

35 W Y Z X + o X + 100 100 100 100 100 100 100 100 100 100 10 (SIGNAL 9 (SIGNAL 8 (SIGNAL 7 (SIGNAL 6 (SIGNAL 5 (SIGNAL 4 (SIGNAL 3 (SIGNAL 2 (SIGNAL 1 (SIGNAL NOI NQI NOI NOI NOI NOI NOI NOI NOI NQI E E E E E E E E E E RR TIO) RR TIO) RR TIO) RR TIO) RR TIO) RR TIO) RR TIO) RR TIO) RR TIO) RR TIO) TRUNATION") t 0 1250 0.2500 0.3750 0.5000 0.6250 FREQUENY (YLES/SIGMA) r DIFFERENE OF M.T.F. (-) ONT I NUOUS TRANSFORM OF GRUSSIRN SPRD. FTN. VARYING SIGNAL-NOISE (DERV. -TRANSFORM)

36 Y Z X X + A o 100: 10 (SIGNRL 100:9 (SIGNRL 100:8 (SIGNRL 100:7 (SIGNRL 100:6 (SIGNRL 100:5 (SIGNRL 100:4 (SIGNRL 100:3 (SIGNRL 100:2 (SIGNAL 100:1 (SIGNAL NOISE NOISE NOISE NOISE NOISE NOISE NOISE NOISE NOISE NOISE RATIO) RATIO) RATIO) RATIO) RATIO) RATIO) RATIO) RATIO) RATIO) RATIO) 0. 0000 0.1250 0.2500 0.3750 0.5000 FREQUENY (YLES/SIGMA) 0. 6250 DIFFERENE OF M.T.F. (-) ONT I NUOUS TRANSFORM OF GAUSSIAN SPRD. FTN. VARYING SIGNAL:NOISE (TATIAN)

37 REOMMENDATIONS A possible avenue for future work is an analyti study of Tatian's tehnique ompared to the derivativetransform tehnique whih would provide the reason for the statistial differene in M.T.F.s at low frequenies, Another prospet ould be formed by omparing the two tehniques using signal dependent noise whih re sembled that of a photographi edge.

38 SUMMARY Two subroutines, TATIAN and DTRANS, were built to be the fundamental data generating soures. TATIAN determined the M.T.F. of the input urve by Tatian's method of edge gradient analysis. This onsisted of taking an F.F.T. of the original edge data, multiplying by i2tff and then applying Tatian's orretion fators. DTRANS determined the M.T.F. of the input urve by the derivative-transform tehnique whih onsisted of taking the numerial derivative of the original edge data to approximate the line spread funtion, and then performing an F.F.T. whih produed the M.T.F. A series of main programs, MAIN1, MAIN2, MAIN2P, MAIN3, FMAIN3, produed average M.T.F.s and orresoonding varianes. The statistial tests, of the mean (TESTMEAN ), and varianes (TESTVAR), were then run to analyze the data and test the hypotheses. The variability of the alulated M.T.F.s using Tatian's tehnique vs. the derivative-transform edge analysis tehnique were differant at low frequenies. Also the alulated mean M.T.F.s using Tatian's tehnique vs. the derivative-transform tehnique were different at

39 low frequenies when a sampling interval of 3.458 samples/j was tested. This sampling interval orresponds to approximately 21 points on the edge where 9 9.72$ of the unit edge exist.

40 REFERENES 1. H.F.Gilmore, 0. Opt. So. Am. 57,75 (1967) 2. M.E.Rabedeau, J. Opt. So. Am. 59,1309 (1969) 3. B.Tatian, 0. Opt. So. Am. 61, 1223 (1971) 4. Ibid., 1223 (1971) 5. Private ommuniation with R.Porth 6. O.arson, Supplementary leture notes, Rohester Institute of Tehnology, p. 5-7 (1977) 7. 3.. Dainty, R.Shaw, Image Siene, Aademi Press, N.Y. p. 245 8. R. Brae well, The Fourier Transform and Its Applia tions. MGraw-Hill, U.S. (1978) p. 117 9. B.Tatian, 0. Opt. Sa. Am. 55, 1015 (1965) 10. Ibid., p. 1016 11. Ibid., p. 1016 12. Ibid., p. 1014-1017 13. R.Barakat, ft. Houston, 0. Opt. So. Am. 53, 1244(1963) 14. D.Dutton, Applied Optis 14, 515 (1975) 15. Xerox Sigma Nine main frame omputer, DE 11/34 Intelligent Remote Bath Terminal and peripherals, ROSS User omputing enter, Rohester Institute of Tehnology 16. N.Brenner, M.I.T. Linoln Lab, 9/12/68 17. I.B.M. Sientifi Subroutine Pakage 18. Private ommuniation with R.Porth

41 19. I.B.M. Sientifi Subroutine Pakage 20. A.Rikmers, H.Todd, Statistis An Introdution, MGraw-Hill, U.S. p. 119 (1967) 21. Zeta Inremental Plotting System, Zeta 3600s plotter with attahed PDP11/04KA ontroller and a 1600 BPI tape drive, ROSS User omputing enter, Rohester Institute of Tehnology

APPENDIX A: GRAPHIAL DATA OUTPUT 42

- NO 43 X Y Z X + X + A D ;,' r.- Q 100 100 100 100 100 100 100 100 100 100 ONTI 10 9 8 7 6 5 4 3 2 1 (SIGNAL (SIGNAL (SIGNAL (SIGNAL (SIGNAL (SIGNAL (SIGNAL (SIGNAL (SIGNAL (SIGNAL UOUS TR NO IS IS ' NOIS - NO IS. NOIS i NOIS -NOIS i NOIS NQIS NOIS ANSFQ E E E E E E E E E E RM. RA RA RA RA RA RA RA RA RA RA TIO) TIO) TIO) TIQJ TIO) TIO) TIG) TIG) TIO) TIO) 0. 0000 0. 1250 0. 2500 FREQUENY YI 0,3750 0,5000 FS/SIGMA) 0. 5250 FIGURE 9: M. T. F. SIGNAL OF GAUSSIAN SPRD, NOISE RATIO (TATIA ) T N. VARYING

44 X Y Z X X + A 100 : 10 100:9 100:8 100 s 7 100 100 100 100 100 100 6 5 4 3 2 1 (SIGNRL (SIGNAL (SIGNAL (SIGNAL (SIGNAL (SIGNAL (SIGNAL (SIGNAL (SIGNAL (SIGNAL NOISE NOISE NOISE NOISE NOISE NOIS^i_ NOISE NOISE NOISE NOISE RAT RAT RAT RAT RAT RAT HhT RAT RAT RAT I) 10) 10) 10) 10) 10) 10) 10) 10) 10) 0. 0000 0 1250 0,2500 0,3750 0,5000 0. B250 FREQUENY (YLES/SIGMA) FIGURE 10: VARIANE QF GAUSSIAN SPRD. SIGNALRNOISE RATIO (TATIAN) FT V A h Y I N G

. 45 X Y X X + A 100: 10 100:9 100:8 100:7 100:6 100:5 100: 4 100:3 100:2 100: 1 ONTIN iz'..*-' at, 'Q"; (SIGNAL (SIGNAL (SIGNAL (SIGNAL (SIGNAL (SIGNAL (SIGNAL (SIGNAL (SIGNAL (SIGNAL UOUS TR NOI NQI NOI NQI NOI :NOIS :NQIS :NQIS :NOIS :NQIS ANSFQ E E E E E E E E E E RM RA RA RA RA RA RA RA RA RA RA TIO) TIO) TIO) TIO) TIO) TIO) TIO) TIO) TIO) TIO) "0. 0000 0.1250 0.2500 0.3750 0.5000 FREQUENY (YLES/SIGMA) 0. 6250 figure 11: GRUSStrn s p p, Q FTN. VARYING SIGNAL:NQISE RATIO (DERV. -TRANSFORM)

46 X Y X X + A o 100 100 100 100 7 100 6 100 5 100 4 100 3 100 2 100: 1 10 (SIGNAL 9 (SIGNAL 8 (SIGNAL (SIGNAL (SIGNAL (SIGNAL (SIGNAL (SIGNAL (SIGNAL (SIGNAL NOISE NOISE NOISE NOISE NOISE NOISE NOISE NOISE NOISE NOISE RATIO) RATIO) RATIO) RATIO) RATIO) RATIO) RATIO) RATIO) RATIO) RATIO) 333 o 0. 0000 0.1250 0.2500 0.3750 0.5000 FREQUENY (YLES/SIGMA) 0. 6250 FIGURE 12: VARIANE OF GAUSSIAN SPRD. FTN. VARYING SIGNAL:NOISE RATIO (DERV, -TRANSFORM)

47 A + X X m 5. 4 3 2 1 333 500 458 417 333 667 333 ONTINUOUS (SAMPLES/SIGMA) (SAMPLES/SIGMA) (SAMPLES/SIGMA) (SAMPLES/SIGMA) (SAMPLES/SIGMA) (SAMPLES/SIGMA) (SAMPLES/S IGMA) TRANSFORM 0, 0000 0.1250 0.2500 0.3750 0.5000 FREQUENY (YLES/SIGMA) 0. 6250 "T^l^aF GAUSSIAN SPRD. FTN. VARYING SAMPLING INTERVAL (TATIAN)

48 A + X X 5 4 3 2 1 333 500 458 417 333 667 333 (SAMPL (SAMPL (SAMPL (SAMPL (SAMPL (SAMPL (SAMPL ES/S ES/S ES/S ES/S ES/S ES/S ES/S IGMA) IGMA) IGMA) IGMA) IGMA) IGMA) IGMA) SE. o atf 0, 0000 0.1250 0.2500 0,3750 0,5000 FREQUENY (YLES/SIGMA) 0, 6250 FIGURE 14: VARIANE OF GAUSSIAN SPriD, FT SAMPLING INTERVAL (TATIRN) VARYING

49 A T X X a 5. 333 (SAMPLES/SIGMA) 4, 500 (SAMPLES/SIGMA) 3. 458 (SAMPLES/SIGMA) 2, 417 (SAMPLES/SIGMA) QQ 1. 3 O ^ (SAMPLES/SIGMA). 667 (SAMPLES/SIGMA). 333 (SAMPLES/SIGMA) ONTINUOUS TRANSFORM i r1 1_! : o i J i_ j.. + / tl.- -i i -, n i O, 0. 0000 0.1250 0.2500 0.3750 0.5000 FREQUENY (YLES/SIGMA) 0. 6250 F I GURE 1 5 * M T F 'OF GAUSSIAN SPRD. FTN. VARYING SAMPLING INTERVAL (DERV. -TRANSFORM)

50 O A + X X 5. 4. 3. 2, 1. 333 500 458 417 333 667 333 :"-: T ^ F \ (SAMP (SAMP (SAMP (SAMP LES/S LES/S LES/S LES/S (SAMPLES/S (SAMPLES/S (SAMPLES/S IGMA) IGMA) IGMA) IGMA) IGMA) IGMA) IGMA) o "TJ. 0000 0.1250 0.2500 0.3750 0.5000 FREQUENY (YLES/SIGMA) 0. 6250 FIGURE 16: VARIANE OF GAUSSIAN SPRD, FTN, VARYING SAMPLING INTERVAL (DERV, -TRANSFORM)

51 X + ffl. 9375 SIGMA 1. 3125 SIGMA 1. 6875 SIGMA 2, 0625 SIGMA 5, 0625 SIGMA ONTINUOUS TRANSFORM D : -: ; b. oooo 0,1250 0.2500 0.3750 0.5000 FREQUENY (YLES/SIGMA) 0, 6250 M^tV^QF GAUSSIAN SPRD, FTN, VARYING TRUNATION I NTERVAL (TAT I AN)

' ' 52 <!> X + A = = = s:-::jl - 9375 SIGMA 1= 3125 SIGMA 1= 6875 SIGMA 2, 0625 SIGMA 3, 9375 SIGMA 5, 0625 SIGMA 333 tr <* -- J O. 0. 0000 0.1250 0.2500 0.3750 0,5000 FREQUENY (YLES/SIGMA) 0,6250 VF^Ufy&t OF GAUSSIAN SPRD, FT TRUNATION I NTERV AL (T AT I AN) VARYI L-

53 o X + +, 9375 SIGMA 1. 3125 SIGMA 1, 6875 SIGMA 2, 0625 SIGMA 5. 0625 SIGMA ONTINUOUS TRANSFORM 0. 0000 0.1250 0.2500 0.3750 0.5000 FREQUENY (YLES/SIGMA) 0. 6250 r t ijpr 19 M. T. F. OF GAUSSIAN SPRD, FTN, VARYING I INTERVAL "(DERV, -TRANSFORM)

. _i ' 54 O X I T A. 9375 SIGMA 1, 3125 SIGMA 1. 6875 SIGMA 2, 0625 SIGMA 3. 9375 SIGMA 5, 0625 SIGMA 3 3 3 ;/ d r. 0. 0000 0. 1250 0. 2500 0.3750 0.5000 FREQUENY (YLES/SIGMA) 0. 6250 ^PJA^E OF GAUSSIAN SPRD, FTN, VARYING TRUNATION INTERVAL (DERV, -TRANSFORM)

55 x Y Z X X + A a 100 100 100 100 100 100 100 100 100 100 QNTI 10 9 8 7 6 5 4 3 2 1 (SIGNAL (SIGNAL (SIGNAL (SIGNAL (SIGNAL (SIGNAL (SIGNAL (SIGNAL (SIGNAL (SIGNAL UQUS TR. NOISE : NQISE :NQISE : NOISE NOISE :NOISE : NOISE :NOISE :NOISE : NOISE ANSFQR M RAT RAT RAT RAT RAT RAT RAT RAT RAT RAT 10) 10) 10) 10) 10) 10) 10) 10) 10) 10) 'A 0. 0000 0 1250 0,2500 0.3750 0,5000 FREQUENY (YLES/SIGMA) 0. 6250 riure 2i: FRIE5ER sprd. FTN SIGNRL-NOISE RATIO(TATIAN) VARVING

56 Y Z X 4- <L> X + A 100: 100: 100: 100: 100: 100: 100: 100: 100: 100: 10 (SIGNAL 9 (SIGNAL 8 (SIGNAL 7 (SIGNAL 6 (SIGNAL 5 (SIGNAL 4 (SIGNAL 3 (SIGNAL 2 (SIGNAL 1 (SIGNAL NOISE NOISE NOISE NOISE NOISE NOISE NOISE NOISE NOISE NOISE RATIO) RATIO) RATIO) RATIO) RATIO) RATIO) RATIO) RATIO) RATIO) RATIO) n1 fl.es. 0. 0000 0 1250 0.2500 0.3750 0,5000 0.6250 FREQUENY (YLES/SIGMA) F I GU RE 22 * VARIANE OF FRIESER SPRD SIGNALrNOISE RATIO (ThTiAN TN, VARYING

57 X Y Z X o X + A o m 100: 10 (SIGNAL:N0ISE RATIO) 100:9 (SIGNAL:NOISE RATIO) 100:8 (SIGNAL:NQISE RATIO) 100:7 (SIGNAL:NOISE RATIO) 100:6 (SIGNAL: NOISE RATIO) 100:5 (SIGNAL:NOISE RATIO) 100:4 (SIGNAL:NOISE RATIO) 100:3 (SIGNAL: NOISE RATIO) 100:2 (SIGNAL:NOISE RATIO) 100: 1 (SIGNAL:NQISE RATIO) ONTINUOUS TRANSFORM Q, 0.0000 0.1250 0.2500 0.3750 0.5000 FREQUENY."YLES/SIGMA) 0. 6250 FIGURE 23: M.T.F. OF FRIESER SPRD, FTN, VARYING SIGNAL: NOISE RATIO (DERV, TRANSFORM)

58 X Y ~7 X X + A 100: 10 (SIGNAL: NQISF RATIO) 100:9 (SIGNAL-NOISE RATIO) 100:8 (SIGNAL:NOISE RATIO) 100:7 (SIGNAL:NQISE RATIO) 100:6 (SIGNAL:NOISE RATIO) 100:5 (SIGNAL:NOISE RATIO) 100:4 (SIGNAL: NOISE RATIO) 100:3 (SIGNAL:NOISE RATIO) 100:2 (SIGNAL: NOISE RATIO) 100: 1 (SIGNAL:NOISE RATIO) I'D) 0. 0000 0.1250 0.2500 0.3750 0.5000 FREQUENY (YLES/SIGMA) 0. 6250 FIGURE 24: VARIANE OF FRIESER SPRD, FTN. VARYING SIGNAL: NOISE RATIO (DERV, -TRANSFORM)

4, 59 ~z 5, 333 A -= 500 + = 3, 458 X = 2. 417 <!> = 1. 333 + =. 667 X =. 333 m = QNTI (SAMPLES/SIGMA) (SAMPLES/SIGMA) (SAMPLES/SIGMA) (SAMPLES/SIGMA) (SAMPLES/SIGMA) (SAMPLES/SIGMA) (SAMPLES/SIGMA) NUQUS TRANSFORM s: ^.0000 0.1250 0.2500 0.3750 0.5000 FREQUENY (YLES/SIGMA) 0. 6250 FIGURE 25: M T F OF FRIESER SPRD, FTN, VARYING SAMPLING INTERVAL (TATIAN)

60 A + X X 5, 4, 3, 2. 1. 333 500 458 417 333 667 333 (SAMP (SAMP (SAMP (SAMP (SAMP (SAMP (SAMP LES/S LES/S LES/S LES/S LES/S LES/S LES/S IGMA) IGMA) IGMA) IGMA) IGMA) IGMA) IGMA).;oise. o b. oooo 0.1250 0.2500 0.3750 0.5000 FREQUENY (YLES/SIGMA) 0. 6250 FIGURE 26: VARIANE OF FRIESER SPRD, FTN, VARYING SAMPLING INTERVAL (TATIAN)

61 A + X X m 5,333 (SAMPLES/S IGMA) 4, 500 (SAMPLES/S IGMA) 3. 458 (SAMPLES/S IGMA) 2,417 (SAMPLES/S IGMA) 1.333 (SAMPLES/S IGMA). 667 (SAMPLES/S IGMA) q33 (SAMPLES/S IGMA) ON INUOUS TRANS FORM 0. 0000 0 1250 0.2500 0.3750 0.5000 FREQUENY (YLES/SIGMA) 0. 6250 FTTH RF 9 7 MTF 'OF FRIESER SPRD. FTN. VARYING SAMPLING INTERVAL (DERV. -TRANSFORM)

s"g:'l' do 62 o = 5, 333 A 4, 500 + 3. 458 X rr 2, 417 <L> = 1 - * X rr j. 667. 333 (SAMPLES/SIGMA) (SAMPLES/SIGMA) (SAMPLES/SIGMA) (SAMPLES/SIGMA) (SAMPLES/SIGMR) (SAMPLES/SIGMA) (SAMPLES/SIGMA) 0. 0000 0.1250 0.2500 0.3750 0.5000 FREQUENY (YLES/SIGMA) 0. 6250 FIGURE 28: VARIANE OF FRIESER SPRD, FTN, VARYING SAMPLING INTERVAL (DERV, -TRANSFORM)

L--I ' 63 Y Z X + + 1, 2, 2, 8, 13, ON 9375 SIGMA 5000 SIGMA 0625 SIGMA 6250 SIGMA 1875 SIGMA 8125 SIGMA 8750 SIGMA TINUOUS TRANSFORM : i 3..H_ j. O i_ i '. 0. 3 j ^i jnr llj, Oii;"!.-.. O, 0. 0000 0 1250 0. 2500 0.3750 0,5000 FREQUENY (YLES/SIGMA) 0, 6250 MItV9V FRIESER SPRD, FTN, VARYING INTERVAL (TATIAN)

-- - -W I a. 64 Y Z X X + A + +. 9375 1, 5000 2, 0625 2, 6250 3. 1875 6, 0000 8, 8125 11, 6250 13, 8750 SIGMA SIGMA SIGMA SIGMA SIGMA SIGMA SIGMA SIGMA SIGMA o -l LO LU J yz. T i i Q r o / V / 0, 0000 0 1250 0,2500 0,3750 0.5000 FREQUENY (YLES/SIGMA) 0. 6250 F I GURE 30 * VARIANE OF FRIESER SPRD, FT TRUNATION INTERVAL (TATiAN) VARYING

65 Y Z X + D. 9375 SIGMA 1. 5000 SIGMA 2. 0625 SIGMA 2, 6250 SIGMA 3. 1875 SIGMA 8, 8125 SIGMA 13. 8750 SIGMA ONTINUOUS TRANSFORM sio;:-: I SE ; ;-' -? '.L.3 0. 0000 0 1250 0.2500 0.3750 0.5000 FREQUENY (YLES/SIGMA) 0. 6250 r t pi ip- 3 1 MTF OF FRIESER SPRD, FTN, VARYING TRUNATION INTERVAL (DERV, -TRANSFORM)

66 Y Z X 4> <r> x + A + 1 2 2, 3, 6, 8. 1 1. 13. 9375 5000 0625 6250 1875 0000 8125 6250 8750 SIGMA SIGMA SIGMA SIGMA IGMA IGMA IGMA IGMA SIGMA TJ. 0000 0.1250 0.2500 0.3750 0.5000 FREQUENY (YLES/SIGMA) 0. 6250 ^A^iVnE OF FRIESER SPRD, FTN, VARYI TRUNATION INTERVAL (DERV, -TRANSFORM) G

APPENDIX B: MAIN PROGRAMS FDATA GADATA DERIV DTRANS TATIAN IYIAIN1 MAIN2 I7IAIN2P IY1AIN3 FffIAIN3 DIFF TESTVAR TESTIYIEAN

.& 68 (^ # # Ai ^ ^ *** WRITTEN *Y ROBERT BRIAN LAFLESH 1979-80 *** *** *-?* ^ si 35; 5} jfe a; ;& *** THIS PROGRAM GENERATES THE UMMULATIVE *** **# frieseti URVE *** *** **=? *** THE DATA IS OUTPUT TO DEVIE 3 *** *** *** P a> Ta X av * -j* *** NBIG = NUM6EK UF DATA POINTS ON EDGE *** *** DELTAX = SAMPLING INTERVAL ON EDGE *** *** D = UMULATIVE FRIESER URVE *** *** XSIGMA = THE EDGE fxist FROM MINUS XSIGMA *** *** TO PLUS XSIGMA *** *** X,DATA,D MUST At DI '.ENSIO'NED TO SIZE ftbig-1 ** *** *** " DIMENSION X(-32:32)-UATA(-32:32),Dt>32:32) NIG=33 X5IGMA=24.0 DELTAX=(XSIGMA*2.0)/FLOATNBIG)-1.0) w = f\big-l DO 10 I=-N,N.2 X(I)=FLOAT(l5*XSIGMA/FL.JATN) 10- DATAI)=0.b*EXP(ASXI))*-1.0D ) 3 Df-N) =DATA-N) UMDIFF=0.0 DO 20 I=-N+2,M,2 DIFP'sABS DATAD-OATAI-2) ) UMDIFF sumdifk+diff 20 DI)=D-N)+UfiDIFF writt3,2), ObLTAX.NHIG, (D(I) =,I -N,iM,2) 2 FORMAT ',F10.7,I5,/, F13.6)) STOP E r-i u

69 10 2 *** WRITTEN BY ROBERT HP IAii LAFLESH 1979-80 *** *** jj.** * -f :>rat> *** THIS PROGRAM GEivER ATES THE UMMULATIVE *** *** GAUSSIAN URVE *** *** ** T * -P f $ *** THE DATA IS OUTPUT TO DEVIE "l *** *** #** J* ^ ajj. & $ $ *** S.6IG = NUMBER UF DATA POINTS ON DGE *** *** DELTAX = SAMPLING INTERVAL Oh EDGE *** *** DATA = UMMULATIVE GAUSSIAw URVE *** *** XSIGMA = THE EDGE EXIST FROM MINUS XSlGfA *** *** TO PLUS XSIGMA *** *** X.DATA MUST BE tlmensioned TO SIZE NBIG-1 *** *** **=> ^T--T-#a()j.*!J*#*^***#**^***#* Xlb) -DATA16) DIMENSION NBIG=17 X3IGMA=3.0 DELT AX= XSIGMA *2. 0 )/ FLOAT (NB 1 0-1.0 J N = rjbig-l DO 10 I=-N,N,2 X ( I ) =FLOAT I ) *XSIGHA/f LOAT N ) ALL MDTR(Xl), DATaU), DENSITY) WRITE I 00, 2) DELTAX, nbig, (DATA D, 1 =-^, N, 2 ) FORMAT ' f,f10.7,ib,/, F13.6)) STOP E:\;D

70 30 sis 3k aft -^? ^ *** WRITTEN 6Y ROBERT BRIAN LAFLESh 1979-80 *** *** THIS SUBROUTINE DETERMINES THE NUMERIAL ***?** DERIVATIVE OF THE I.'-PUT URVE. ***, Jj jjj jf ^ jj; *** Y = (Y)ORUINATE *** *** DELTAX = SAMPLING INTERVAL ON URVE *** *** i,pig = wumeeh OF POINTS ON URVE *** *** SLOPE = rtuhtrial DERIVATIVE URVE *** *$$ #** *** THIS PROGRAM DETERMINES THE NUMERIAL *** *** DERIVATIVE BY TAKING THE POINT BY POINT *** *#* INREMENTAL DELTA Y ) /DELTAX *** **# *#* SUBROUTINE DER I V ( Y, ELT AX, N8IG, SLOPE) DIMENSIUN Y257), SLOPE (257) H=NBIG-1 DO 3 0 1 = 1,M SLOPE I ) = Y 1 + 1 ) -Y I ) ) / OELT AX ONTINUE RETURN END

' 71 **#* ******* ************** ** *********************** *** WRITTEN BY ROBERT BRIAN LAFLESH 1979-30 *** ************************************************'* *** DERIVATIVE TRANSFORM EDGE GRADIENT ANALYSIS*** *** TEHNIQUE *** ***********************************************JJJ f 'p JJ* "> hv itf "i *** THIS SUBROUTINE OMPUTES THE MODULATION *** iff TRANSFER FUNTION AND PHASE OF THE INPUT *** *** EDGE DATA. THE VALUES ARE ALULATED OUT *** *** TO THE FOLDING FREQUENY. THE (X)ABSIA *** *** OORDINATES ARE ALULATED FOR THE *** *** INPUT EDGE DATA SO THAT THE EDGE nata *** *** VS. X)&BSIA OORDINATES AN BE PLOTTED. *** *** THE (F)ABSIA OORDINATES ARE ALULATED *** *** FOR THE OUTPUT DATA SETS SO THAT THE *** *** MODULATION TRANSFER FUNTION VS. FJABSIA *** *** OORDINATES & PHASE VS. (F)ABSIA O- *** *** ORDINATES AN BE PLOTTED. *** * * * * Hr at. ************************************************** *** *** *** DELTAX = SAMPLING INTERVAL ON EDGE(INPUT) *** *** NBIG = NUMBER OF POINTS ON EDGE(INPUT) *** *** DATA = EDGE DATA VALUES ( INPUT ) *** *** X = ALULATED (X)ABSIA OORDINATES (OUTPUT ) ** *** F = ALULATED F)A3SIA OORDINATES (OUTPUT) ** *** XMOD = MODULATION TRANSFER FUNTION (OUTPUT) ** *** PHASE = PHASE IN RADIANS (OUTPUT) *** *** N = NUMBER OF POINTS TO FOLDING FREQUENY *** *** (OUTPUT) *** *** *** Q ************************************************** SUBROUTINE DTRANS ( DELTAX, NBIG, DAT A, X, F, XMOD, PHASE,N ) DIMENSION X257) F(2b7, ).DATA ( 257 ), RESUuT ( 257 ), + TDATA (2,257),TXIMAG(257).XIMAG 257 ),TREEL 257 ) +,WOKK2,257),XMOD(257),PHASE(257) * SET IMAGINARY PART OF INPUT DATA SET TO ZERO DATA XIMAG/257(0.0)/ * ALULATE THE NUMBER OF POINTS OUT TO THE * FOLDING FREQUENY (NJ MsRiBIG-l NsM/2 * ALULATE THE (X)ABSIA OORDINATE VALUES 10 X?l}= (FLOAT N)*DELTAX* -l. n+(floati-l)*deltax) * APPROXIMATE THE LINE SPREAD FUNTION BY TAKING * THE NUMERIAL DERIVATIVE r ALL DERI V( DATA, DELTAX, NBIG, RESULT)