The Modification of Simple Images by Fourier Transform Manipulation

Transcription

1 Rchester Institute f Technlgy RIT Schlar Wrks Theses Thesis/Dissertatin Cllectins The Mdificatin f Simple Images by Furier Transfrm Manipulatin Nel Chavez Fllw this and additinal wrks at: Recmmended Citatin Chavez, Nel, "The Mdificatin f Simple Images by Furier Transfrm Manipulatin" (1979). Thesis. Rchester Institute f Technlgy. Accessed frm This Thesis is brught t yu fr free and pen access by the Thesis/Dissertatin Cllectins at RIT Schlar Wrks. It has been accepted fr inclusin in Theses by an authrized administratr f RIT Schlar Wrks. Fr mre infrmatin, please cntact ritschlarwrks@rit.edu.

2 TE MODIFICATION OF SIMPLE IMAGES BY FOURIER TRANSFORM MANIPULATION by Nel Chavez A thesis submitted in partial fulfillment f the requirements fr the degree f Bachelr f Science in the Schl f Phtgraphic Arts and Sciences in the Cllege f Graphic Arts and Phtgraphy f the Rchester Institute f Technlgy June, 1979 Signature f the Authr Phtgraphic Science and Instrumentatin Certified by Thesis Adviser

3 (,-izs*** Abstract Knwledge f hw an image is changed by a given mdificatin f its Furier transfrm is imprtant when attempting t find slutins t prblems in image prcessing. Several types f filters were used t mdify the Furier transfrms f an edge and a bartarget. The images were recnstructed frm the mdified transfrms t determine the change prduced by a given mdificatin. Varius regins f the Furier transfrm determine specific characteristics f the image. The amplitude f the lw frequencies cntrls the frm and average level f the image, while the amplitude f the high frequencies effects sharp transitins and nise. 11

4 TABLE OF CONTENTS Page Abstract ii Acknwledgements 42 Intrductin 1 Thery 2 Cnclusins 40, 41 References 43, 44 Bibligraphy 45 in

5 INTRODUCTION Any bject which can be represented by a functin in the spatial dmain can als_be represented by a crrespnding functin in the frequency dmain. The frequency dmain representatin f the bject defines the frequency and amplitude f the varius sinusids f which it is cmpsed. The tw representatins f the bject are related thrugh the Furier transfrm. By cmputing the Furier transfrm f the bject in the spatial dmain, the bject is separated int its sinusidal cmpnents. The spatial dmain representatin f the bject can be recn structed frm its sinusidal cmpnents by cmputatin f the inverse Furier transfrm. When the relative amplitude f varius frequencies f the Furier transfrm f the bject are changed, the bject is mdified upn recnstructin. The bject is changed, because the Furier transfrm frm which it is being re cnstructed has been mdified. Varius mdificatins in the Furier transfrm f an bject prduce different mdi ficatins in the bject when recnstructed. Many prblems in image prcessing can be slved using this and ther prperties f the Furier transfrm.

6 la INTRODUCTION, cntinued The purpse f this wrk is t mdify simple images utilizing the prperties f the Furier transfrm, t determine the character and magnitude f its Furier transfrm and tbecme familiar with varius aspects and techniques f digital image prcessing.

7 Thery A great number f prblems in image prcessing require the applicatin and make use f the unique prperties f the Furier transfrm. Mst available mathematical treatments f the Furier transfrm deal with cntinuus functins. wever, in mst if nt all cases invlving the detectin and recrding f a signal, infrmatin is cllected and stred as discrete data. That is, a cntinuus signal is specified by physical 2 measurements made at regular intervals f time r space. Since very few prcesses r events can be accurately r even apprximately specified by exact mathematical relatinships, discrete data must be used. If a functin is represented by a series f values taken at discrete intervals, its Furier transfrm cannt be cmputed by nrmal cntinuus methds. An alternative means is required. The discrete Furier transfrm is a special case f the cntinuus Furier transfrm, and is the nly means whereby the Furier transfrm can be cmputed frm discrete data. With the capability f tday's cmputers t stre and prcess large quantities f data the use and applicatin f the discrete Furier transfrm in image prcessing is fast and efficient. A gd way t understand the discrete Furier transfrm is t start with a cntinuus functin and its

8 , \M- Furier transfrm, and mdify them s that they are in a frm which a cmputer may stre and prcess them. Let 3 a cntinuus functin be represented by h(x) and its Furier transfrm by (f), where f is specified as sme frequency in cycles/mm. h(x) (f) Figure 1 Since cmputers are nly capable f string discrete data, the functin h(x) must be sampled. Sampling is accmplished by multiplying h(x) by a series f delta functins separated by a distance Ax. If s(x) is the sampling functin, then (,j, a j A--CO,-... j Equatin 1 The Furier transfrm f the sampling functin is S(f), which is als a series f delta functins, this time separated by a distance /Ax in the frequency dmain, 5(x) S(f) >*j AX. Figure 2

9 The sampled functin is specified as ^)sm-/t WZ_ (/(x-aax) A'- Equatin 2 By definitin ffx}<f(x.-x,) --fi%.w(x-x*). Equatin 3 then Equatin 4 The sampled functin is nw a series f delta functins f varying amplitude. 5 In the frequency dmain smething quite different takes place. By the cnvlutin therem we knw that the Furier transfrm f the prduct f tw functins is the cnvlutin f their Furier transfrms. In this case yihmsq.))* (f)*s(*) Equatin 5 Since S(f) is a series f delta functins, the result f the cnvlutin f (f) with S(f) is t make (f) a peridic functin with a perid f 1/Ax h(x)s(x) (f )*S(f ) iltf, Figure 3 f

10 ' It is f curse impssible t take an infinite number f samples, therefre sampling must ccur ver a finite interval. The sampled functin h(x)s(x) must be truncated. Truncatin is accmplished by multiplying the sampled functins by a rectangular functin. Let the truncating functin be t(x) where t(x) = Equatin 6 The Furier transfrm f the truncating functin is T(f) where T(f) is a sine functin whse first zer ccurs at 1/X. ' tm \T(f)\ X./j X./A Figure 4 f The truncated sampled functin is the prduct f the sim plified functin and the truncating functin and is specified as h(x)s(x)t(x)= [2_ h(k*x)cf(k-k*x)\ t(*) Equatin

11 If there are N sample intervals within the truncatin interval, then the limits f the summatin becme finite and -\ h(x)s(x)t(x) = Z_ J,(A4X}cT/x-A&x) h Equatin 8 Because the truncated sampled functin is the prduct f the sampled functin and the truncating fuflctin, the Furier transfrm f the sampled functin is cnvlved with T(f) s that F[h(x)s(x)t(x)]= [(f)*s(f)] *T(f) Equatin 9 The result f Equatin 9 is t prduce "ringing" in the Furier transfrm f the truncated sampled functin h(x)s(x)t(x) [(f )*S(f )]*T(f ) I ttf.. Figure 5 The "ringing" r rippling cnvlutin in Equatin 9 is due t the fact that the invlves a sine functin. 6

12 7 The side-lbes f the sine functin create "ringing" in the resulting functin. "Ringing" is an inherent errr prduced by the cmputatin f the discrete Furier transfrm. T minimize the errr due t "ringing", it is imprtant t use a truncatin interval as wide as pssible. The reasn fr this lies in the fact that as the width f the rectangle functin used t truncate the functin increases, the width f the rectangle functin used t truncatethe functin increases, the width f its Furier transfrm, the sine functin, decreases and mre clsely y impulse functin. apprximates an One final peratin is required t make the Furier transfrm pair suitable fr prcessing by cmputer. Frm Figure 5, it can be seen that the riginal functin is nw represented by a finite series f delta functins f varying amplitude. The Furier transfrm f the functin hwever, is a cntinuus peridic functin. The transfrm must nw be sampled s that it als is represented by a series f discrete values. Again, sampling is perfrmed by multiplying the transfrm by a series f delta functins. The sampling functin is specified as A(f). The tranfrm f the sampling functin is A(x) which is defined as Afx)= X / \yr/\>) Equatin 10 0 r'-

13 where X is the separatin f the delta functins in the spatial dmain. A(x) A(f ) X- Figure 6 1/X, Since the transfrm is beine multiplied bv the sampling: functin A ( i"). cnvlved with A(x). then the sampled truncated functin must be 8 That is. r r 1 ihrx,)srxn,rx,)i*a(x)= Z_ /i(aax)s(x'aax)\* XZ- -^fr-rx) r=h. Equatin 11 which reduces t, XZT 2_ }>(k4x,)cf(x-aax-rx). Equatin 12 As a result f this cnvlutin, the sampled truncated functin becmes peridic as shwn by Figure 7.

14 llfft [h(x)s(x)t(x)]*a(x) [(f)*s(f)3* T(f) ]A(f) llu~. Ill-n Ih I f T,f JI 1 t Figure 7 The Furier transfrm pair is nw in a frm where bth are represented by a series f a discrete values, and bth are peridic functins. The sampled functin is peridic and is specified by Equatin 13 By definitin the Furier transfrm f a peridic functin is Equatin which is a sequence f equidistant delta functins f varying 9 magnitude. Ntice that the Furier transfrm f a peridic functin is nt cntinuus. Cn represents the Furier cefficients f the functin. In this case the Furier transfrm f h1(x) is 7)---<* Equatin 15

15 -L. 10 where f is the distance between the frequencies where the functin exists and fyi /h Equatin 16 As stated earlier there are N number f samples in the inter val Xn, and each sample is separated frm the adjining samples by a distance Ax, therefre X,r /V-U qrij f'1/a/ix. Equatin 17 & Equatin 18 Cn is the Furier cefficients f the functin and is defined as X-6*/* ^r>~~ r -i**"*/*- I /'/ Xy.Jfx c/x. Equatin 19 Substituting fr h (x) Ct' %/, XlL[5_ h(km)su-a/.x-rxi Equatin 20 c ex. One shuld ntice that the perid ver which Equatin 20, is evaluated has been shied s that the ends f the perid lie between adjining samples. The reasn fr this will be explained later. Since the integral in Equatin 20 is evaluated ver ne perid, r=0 and the summatin with respect t r drps ut f the integral t give

16 _ &1 " Equatin 11 Cn' / Z_ /.(had </(x~j,ax) t dx. 21 Since h(kax) is a cnstant with respect t x, the sum matin f h(kax) is brught utside the integral t give X.-M* k-. *XA Equatin 22 By definitin X, f/)(f(-x.)(j* /fx) Equatin 23 therefre Cn'-L^ h(ktx) A* C dx. Equatin 24 Since Xn=NAx, -0 then Cfi'.Urmk/tJ 2_/;fA4X)e Equatin 25 h'o Substituting Cn back int Equatin 15 (the Furier transfrm f the peridic functin h (x)), we get /y- (m) YLYLh(kAX) cf(f-nlm)), 71-- A--. li Equatin 26 By definitin

17 Equatin 12 d[f-7i[nax))z0 {r mm, Equatin 27 and the summatin with respect t n drps ut since the value f the delta funcitn is zer everywhere except at f= j$~^ when it has the value f 1.0, and A/- 1 b^tx). 28 Equatin 28 is the discrete Furier transfrm which in mre general frm is stated as ^WAXl'2 fikaxjt k- 7/ j Equatin 29 where n=0, 1, 2, 3, 4,... is the number f samples. fix is the sampling 10 interval and N Earlier it was graphically demnstrated that the discrete Furier transfrm is a peridic functin. T shw this, replace n by r in Equatin 29 s that b ("AAuJ z 2 f (*MI. k- ' Equatin 30

18 ' _Zttk 13 It was shwn that the perid f the discrete Furier trans frm was equal t the interval ver which the functin was sampled XQ and that there were N samples within that interval Let r=r+n ^- -Uirrk/tt L^iktxjt e. A-0 Equatin 31 By definitin - CScTfA <L = COs{lTrk)-L SL7i(VTh) Equatin 32 = 1-0 fr k is any integer. Therefre N'1 Ll Uljl/Axj ' L flaaxj, Equatin 33 and G( Mtx) ~ G>[Nax). Equatin 34 The discrete Furier transfrm is a peridic functin where 11 ne perid is cmpsed f N samples. In a similar manner the inverse Furier transfrm can be derived. The inverse Furier transfrm is defined as

19 - 14 <r[k ax) cv Z_ b I a/m J Equatin 35 The discrete Furier transfrm requires that the func tin being transfrmed, must be sampled at equidistant inter vals. The fact that the discrete Furier transfrm is a perid functin impses a restrictin n tl^e rate at which the functin must be sampled in rder t minimize errr. Given that a functin h(x) is cntinuus ver all space and its Furier transfrm, (f), takes n the value f zer aer sme finite frequency fc, that functin is defined t 12 be bandlimited. That is (f) has a finite width f 2fc. It was shwn previusly that sampling a functin caused its Furier transfrm t becme peridic. If a functin is sampled at intervals f Ax, then it s Furier transfrm becmes a peridic functin with a perid f /Ax h(x) S(x) (f )*S(f )..tr Mf*. V AX Figure 8

20 .Ll 15 If the sampling rate is such as t cause the perid f the transfrm f the sampled functin t becme less than 2f, then the repetitins f (f) will begin t verlap. This is knwn as aliasing. h(x)s(x) (f )*S(f ) T ax f Figure 9 The values at the pint f verlap is the sum f the tw repetitins, and the Furier transfrm is crrupted. In rder t prevent aliasing the sampling rate must be Ax< 2fc, Equatin 36 fr band-limited functins. 13, 14 In mst cases the functin under examinatin is nt band-limited, which means its Furier transfrm is cntinuus If the Furier transfrm is cntinuus, then aliasing will always ccur when cmputing the discrete Furier transfrm. When cnvlving a cntinuus functin with a series f equally spaced delta functins, verlap will take place n matter hw far apart the delta functins are spaced. In

21 16 rder t minimize aliasing in the regin f interest, a cutff frequency must be chsen aer which n significant infrmatin is thught r knwn t exist. By setting this maximum frequency f interest equal t f c, a minimum sampling rate is calculated using the relatinship in Equatin 36. By meeting this requirement aliasing shuld be insignificant thrughut the regin f interest. As a general rule, the higher the sampling rate the lwer the errr due t aliasing and the mre accurate the results will be. Anther cnsideratin when applying the discrete Furier transfrm is the truncatin interval which is used. The interval ver which the functin is sampled, determines the spacing f the frequencies cmputed by the discrete Furier transfrm. If the functin is sampled ver an interval f X0, then the frequencies will be reprted every 1/XQ units. That is Af= 1/Xq. Equatin 37 Frm the width f the truncatin interval, the number f samples required can be calculated frm the fact that XQ =NAx r N= X0/Ax. Equatin 38 Since Af= 1/XQ and XQ = NAx then l=naxaf. Equatin 39

22 17 This defines the relatinship between the number f samples, the sampling rate and the spacing between the reprted fre quencies. Using Equatin 39 and Ax<l/2f then all the parameters fr prper sampling and truncatin can be determined. A final requirement fr cmputing the discrete Furier transfrm is that the end pints f the truncatin functin d nt cincide with the psitin f a sample value. The reasn fr this, is the fact that the sampled truncated functin is peridic, as shwn earlier. If the end pints f the truncatin functin cincides with the psitin f a sampled value then time dmain aliasing will ccur. This results frm the fact that the N"--1 p0int f ne perid will cincide and add t the first pint f the next perid. Therefre the end pints f the truncatin functin shuld always be placed between adjining sample values. 15 When applying the discrete Furier transfrm ne shuld be aware f certain prperties. If a functin is sampled using N number f samples, then the cmputatin f the dis crete Furier transfrm will result in N frequencies. The discrete Furier transfrm prduces a real and an imaginary part. The real part will be a functin symmetric abut the pint n=n/2, while the imaginary part will be an dd functin with respect t the same pint. The mdulus f the transfrm

23 18 is als a symmetric functin abut n= N/2. What is ccurr ing is that the results frm n=< N/2 t N represent the negative frequencies f the transfrm. Because f this, nly N/2 frequencies are actually reprted. Als f inter est is the fact that every frequency 16 is repeated except at f=0. T calculate the discrete inverse Furier transfrm, the real and imaginary abve, therwise errr will result. parts must be in the frm described 17 Frm the mathematical expressin fr the discrete Fur ier transfrm (Equatin 29) it can be shwn that G(tt2 ) 2 results frm N 18 cmplex multiplicatins and N(N-l) cmplex additins. Even when dne by cmputer, cmputatin can becme very time cnsuming, especially if the number f pints is large. Out f the need fr a fast, efficient means f cmputing the discrete Furier transfrm, James W. Cley and Jhn Tukey in 1965 prduced what is nw knwn as 19 the Fast Furier Transfrm (FFT). By means f matrix fact rizatin the FFT is able t reduce the number f cmplex peratins t N lg2 N. Because the number f peratins is reduced, the time and cst f cmputing the Furier transfrm is als reduced. A gd example f this is the cmputatin f the transfrm f 8192 samples which takes three quarters f an hur n an IBM 7094 by cnventinal cmputatin. takes nly Using an FFT algrithm the same calculatin 20 5 secnds n the same machine. The discrete

24 in 19 Furier transfrm is nw almst always cmputed using an FFT algrithm. Experimental - The images used in this wrk were thse f an edge and a bartarget. These images were selected n the basis that they are easily represented as a functin f ne dimensin. They were als selected because the eye views an image as a series f relative differences, transmissin r reflectance. An edge and a bartarget are very simple examples where the differences between adjacent regins are used t define the quality f an image. A Natinal Bureau f Standards standard edge and a United States Air Frce transmissin test target were cntact printed nt Kdak Plus X film, which was prcessed accrding t manufacturer recmmendatins. Kdak Plus X film was selected because it is a fairly fine-grained film capable f prducing a clean, nise-free image. All measurements were made in transmissin. Transmissin was chsen instead f density because transmissin is a fund amental physical quantity. Befre scanning the image with the micrdensitmeter, several parameters had t be calcu lated in rder t prperly set up the instrument s that a linear respnse was achieved. If the micrdensitmeter is nt set up prperly in regard t gemetry, partial cherence,

25 20 depth f fcus, flare, slit alignment and substrate scatter ing, then a nn-linear respnse may result causing the prper interpretatin f results t becme impssible. The thery f linear micrdensitmetry is beynd the scpe f this paper. The parameters used and their means f calculatin are thrughly described in "Micrdensitmeter Optical Per frmance: Scalar Thery and Experiment", Richard E. Swing, Optical Engineering, Nvember-December 1976, Vl. 15, N. 6. A Zeis SMP05 micrdensitmeter was used t scan the images, having been set up fr image scanning with verfilled ptics using incherent illuminatin. Illuminatin was prvided by a tungsten surce with a green filter in the light path, which effectively prduced a mnchrmatic illuminatin f 500 nm wavelength. The required minimum numerical aperture fr the efflux bjective was calculated t be A 0.1NA bjective was used fr the efflux bjective. A minimum numerical aperture f was calculated fr the influx bjective and a 0.2NA bjective was used. A sampling aperture f 5ym was calculated t be the minimum usable aperture width. wever, since a Sum slit is capable f reslving infrmatin ut t 200 cycles/ 5 mm (f max= 1000/slit width, f max- 1000^ym= 200 cycles/mm) and the film was judged nt t cntain infrmatin past 100 cycles/mm, a 10um slit was used instead.

26 21 Using the sampling criteria f Ax<l/2f t and having de cided n 100 cycles/mm as the maximum frequency f interest, a maximum sampling interval f 5ym was calculated. A mini mum frequency reslutin f 1 cycle/mm fr the Furier trans frmer was chsen. Frm Af=^/x a minimum scan length f 1 mm was fund t be required. The edge was scanned ver a 1 mm interval taking measurements every 5pm, while Grup 8, Element 5 f the reslutin test target was scanned ver a 1.35 mm interval (in rder t scan the entire image) using the same sampling rate. This prduced a 200 sample repre sentatin f the edge and a 270 sample representatin f the bartarget. When an bject is scanned with a micrdensitmeter, its image is mdified r degraded by the mdulatin transfer characteristics f the instrument. Unless the micrdensi tmeter cntains a perfect imaging system (has a flat MTF), then the image will be a degraded representatin f the bject. The prblem is t recver the riginal bject frm the image knwing smething abut the respnse f the system. In this wrk the micrdensitmeter was set up t prduce a linear respnse. Because f this, the Furier transfrm f the riginal bject, 0(f), can be recvered by 21 means f inverse filtering. 0(f), can be fund by divid ing the Furier transfrm f the image, 1(f), by the fre quency respnse f the micrdensitmeter (f). That is,

27 22 since I(f) = 0(f) (f) Equatin 40 fr a linear system, then <f)= ^777 Equatin 41 h( I). Once 0(f) has been calculated, then the riginal bject can be recnstructed by cmputing the inverse Furier transfrm f 0(f). The frequency respnse f the micrdensitmeter can be apprximated by the prduct f the MTF f the efflux bjec- 22 tive and the frequency respnse f the scanning slit, if the micrdensitmeter is f high quality and has been set up prperly as in this wrk. The. efflux bjective was a dif fractin limited lens which means its MTF is defined as MTF(f)= F(V" c.0s(f)sin(<p))^ ^ert f- csffm) Equatin 43 and f is the cutff frequency f the lens, which in this case was calculated t be 400 cycles/mm. Let u=f/fc then MTF(f)= W\ s LiJi~ W' Equatin 44 Expansin f Equatin 44 int a pwer series can easily be accmplished. Using nly the first three terms f the expan sin f each term and then cllecting terms prduces

28 23 MTF(f)= /- lr * * \-~frlfj* rr[fj. Equatin 45 Equatin 45 is a very clse apprximatin t Equatin 43 and 23 much simpler t implement. The transmissin characteris tics f the scanning slit can be mathematically defined by a rectangle functin. The Furier transfrm f a rectangle functin is a sine functin and s the frequency respnse f the scanning slit is a sine functin. Fr a 10-im wide slit the frequency respnse is sinf^ulfl FCscanning aperture]= rrf Equatin 46 The prduct f Equatins 45 and 46 is a very gd apprxima tin t the frequency respnse f the micrdensitmeter, therefre (f)= (y (7c)*~7fl7cJ I Pf Equatin 47 A prblem arises when calculating the Furier transfrm f the riginal bject frm Equatin 41. (f), as shwn in Equatin 47 is the prduct f a sine functin and anther functin. The sine functin in Equatin 46 falls ff t zer at 100 cycles/mm causing 1(f) t be divided by very tiny quantities as f appraches 100 cycles/mm. This creates a very large errr in the high frequencies f 0(f). T get arund

29 24 the prblem f dividing 1(f) by very tiny numbers, the fre quency respnse, f.the.micrdensitmeter (f.), is set equal t a 24 cnstant aer falling t a specific amplitude. In this wrk (f) was set equal t 0.25 fr all frequencies greater than 75 cycles/mm, 0.25 being the value f (f) at that frequency. By ding this, imprtant infrmatin at lw frequencies is restred withut intrducing a large errr at the higher frequencies where little r n significant infrmatin exists. All data was stred and prcessed by the RIT Xerx Sigma 9 cmputer. The Furier transfrms f the images were cmputed with an FFT prgram capable f handling anv number f data pints, nt just 2 number f pints which mst FFT are nlv capable f handling. This allwed a cnvenient number f samples t be used. When taking the frward Furier transfrm the image data had t be trans frmed int a series f cmplex numbers. This was dne by setting the data equal t the real part and setting the imaginary part equal t zer. When cmputing the inverse Furier transfrm, the real and imaginary part f the trans frm had t be specified in the frm described previusly, and then the cmplex cnjugate cmputed befre using the FFT prgram. The mdulus f the Furier transfrm f the data was nrmalized t 1.0 by dividing by the value f the transfrm at f=0. The mdulus f the inverse Furier

30 25 transfrm (cmputed when recnstructing; the images) was nrm alized bv dividing bv the number f pints in rder t preserve the riginal magnitude f the data defining the images. The Furier transfrms f the images were first filtered using ideal lw and high pass filters. The images were then recnstructed by cmputing the inverse Furier transfrm f the filtered image transfrms. Filtering w'as accmplished bv multiplying the Furier transfrm f the image by a filter functin (.f). That is G(f)=I(f)(f), Equatin 48 where G(f) is the filtered transfrm. 1(f) the Furier trans frm f the riginal image and (f) the filter functin. The ideal lw pass filter (ILPF) is a rectangle functin riginating at f=0. The ILPF is defined as J ///* h (f) = Equatin 49 0 Ifl > h where fc is the cutff frequency. 25 (f) i. fc Figure 10

31 26 By applying the ILPF, the magnitude f all frequencies less than f are unchanged while the magnitude f thse frequen cies greater than fc becmes zer. The cutff frequency f the ILPF was decreased frm f=100 cycles/mm t f=0 cycles /.mm in increments f 5Af fr each image. The ideal high pass filter (IPF) is als a rectangle functin, hwever it riginates at the highest frequency cntained in the transfrm, in this case 100 cycles/mm, and nds at sme 26 cutff frequency. The IPF is defined as = (f) Equatin 50 (f) 1.0 fc Figure 11 When applying the IPF the magnitude f the frequencies less than f becmes zer while thse frequencies greater than fc remain unaffected. The cutff frequency f the IPF was increased at intervals f 20Af frm f=0. until the images were ttally bliterated.

32 /2~~ - 27 The image transfrms were next filtered using expnen tial lw and high pass filters. The expnential filter is a smth filter. Rather than having a sharp cutff the expnential filter causes attenuatin t be dne mre grad ually. The cutff frequency is specified as that frequency at which the filter equals a specific value. In mst cases the value f the filter at the cutff frequency is specified t be 1/ f the maximum value f the filter which is usually 1.0. The expnential lw pass filter (ELPF) causes the amplitude f the higher frequencies t be decreased relative t the lwer frequencies. The ELPF is defined as (f)= [umw/^ e Equatin 51 where n cntrls the rate at which the filter falls ff. 27 In this wrk n was set equal t 2. (f) / Figure 12

33 28 The ELPF was used fr varius cutff frequencies, starting frm f= 5Af and ranging t f=40af. The expnential high pass filter (EPF) is very similar t the ELPF, except it is reflected abut f=0 and shied t the right. This causes the amplitude f the lw frequencies t be reduced t a much larger extent than that f the^ higher frequencies. The EPF is defined by (f)=, Equatin 52 where fg is the desired starting frequency f the filter (the maximum frequency reprted by the transfrm). In this case f0=100 cycles/mm and again n=2. The value f the cutff fre quency f the EPF was ranged frm f=80af t f=20af. (f) l-oi Figure 13 Anther filter similar t the EPF was used, this ne being a high frequency emphasis expnential high pass filter. This filter des nt attenuate the amplitude f the lwer frequencies but rather bsts r magnifies the amplitude f the higher frequencies. This is dne by adding a cnstant 28 t the EPF. The high frequency filter used, is defined as

34 29 (f )= e Equatin 53 where again n=2, and fq=100 cycles/mm. f This filter was used n the Furier transfrms f the degrad ed images f the edge and bartarget fr varius frequencies. The idea being t enhance the degraded images by increasing the amplitude f the higher frequencies. The effect f aliasing was investigated by sampling the image data at varius increments ther than ne, and then cmputing the Furier transfrm. The resulting transfrm was then cmpared t the riginal transfrm f the image at the same frequencies in rder t detect any changes due t aliasing. The image data was sampled by taking every ther pint, every third pint and every furth pint. Since the ttal width ver which the functin was sampled remained cnstant the interval between the frequencies reprted stayed the same s that valid cmparisn culd be made be tween the different transfrms. Only the number f frequen cies reprted was mdified by the carser sampling rates emplyed.

35 30 Results The Furier transfrms f the edge and the bartarget were crrected fr the frequency respnse f the micrdensi tmeter. The result was t cause the value f the transfrms t increase slightly at lw frequencies (5% at 10 cycles/mm) and t increase t a very large extent at high frequencies (up t 300% at 99 cycles/mm). The micrdensitmeter has a limiting respnse and will cause attenuatin at all frequen cies. Attenuatin will be greater at high frequencies since the respnse f the instrument drps ff as the frequency f the input signal increases. Because f this, the greatest change in the crrected Furier transfrms f the images will ccur at higher frequencies. It shuld be nted that the largest increases in the crrected transfrms tk place at high frequencies, the magnitude f thse frequencies was s small cmpared t that f the lw frequencies that the relative change in magnitude f the high frequencies was fr the mst part insignificant. The images prduced frm the crrected Furier transfrms demnstrated n significant change frm the riginal images ther than a slight increase in nise which may well be as much f a result f errr due t calculatin as frm the increased magnitude f the higher frequencies. Crrectin fr the frequency respnse f the micrdensi tmeter did prduce a difference in the recnstructed image,

36 31 that being in the end pints f the images f the edge. This errr des nt ccur in the crrected image f the bartarget. A test was cnducted wherein the rignal data frm scanning the edge and the bartarget was Furier transfrmed and then the resulting functin inverse Furier transfrmed. The resulting images shuld be identical t the riginals. The image f the bartarget shwed n significant change. The edge hwever, demnstrated the same errr at, the end pints (an scillatin f increasing magnitude) encuntered befre, but t a lesser degree. An errr is being intrduced int the image by the calculatin f the discrete Furier trans frm. The discrete Furier transfrm treats the sampled functin as a peridic functin. The bartarget used is a peridic type functin, terminating at the same level frm which it riginated aer several identical cycles. The edge hwever, cntains a large discntinuity. That is, aer starting at a lw level f transmissin it increases in value and then levels ff at sme higher level f transmissin where it terminates. The functin never returns t its riginal starting values and s cntains a large discntinu ity. Figure 15 demnstrates hw the discrete Furier trans frm views an edge r step functin. Figure 15

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43 32 Nte that Figure 15 is nt a cntinuus functin. The dis cntinuity in the edge results in the errr prduced in the end pints f the recnstructed images. As the amplitude f the higher frequencies increases the greater the errr prduced at the end pints as is evidenced by the image prduced frm the crrected Furier transfrm f the edge. The ILPF has sme interesting effects n the images f the bartarget and the edge. The filtered irffages f the bartarget demnstrate n significant changes until a cutff frequency f abut 80 cycles/mm was reached. At fc ~ 30 cycles/mm the image is becming much smther and the edges are lsing sharpness. Ringing is increasing as the cutff frequency is decreased. Finally at fc * 15 cycles/mm the image is almst unacceptable and at fc 5 cycles/mm is cm pletely destryed. The edge ges thrugh a similar transi tin as the cutff frequency f the ILPF is decreased. N significant change in the image ccurs until fc is apprxi mately 50 cycles/mm. "Ringing" becmes very evident at fc ~ 40 cycles/mm. Smthing and "ringing" increases until the image f the edge is all but bliterated at a cutff frequen cy equal t 5 cycles/mm. In the images f the edge, the errr at the end pints increases as fc decreases. As the cutff frequency f the ILPF is decreased, the nise in the images is reduced and the images becme smther. As suspected, n significant infrmatin was fund t exist

44 33 past 75 cycles/mm. This is demnstrated by the fact that the amplitude f the transfrms is very small at frequencies greater than 75 cycles/mm and that reducing it t zer in that regin had n significant effect n the images. Of interest, is the effect that takes place in the images as the cutff frequency is reduced. By the cnvlutin threm, the Furier transfrm f the prduct f tw functins is the cnvlutin f their Furier transfrms. Aa is well knwn the cnvlutins f tw funcitns results in a third much smther functin. When the cutff frequency is decreased the amunt f smthing f the final image increases. This is a result f the fact that as a functin becmes narrwer in ne dmain, its crrespnding functin in the ther d main increases in width. Therefre as fc decreases, the width f the Furier transfrm f the filter increases caus ing mre smthing t ccur when it (the Furier transfrm f the filter) is cnvlved with the image. "Ringing" is a result f the fact that the ILPF is a rectangle functin whse Furier transfrm is a sine functin. The sine func tin is nt a smthly decreasing functin but rather, cntains side-lbes which create "ringing" when the functin is cnvlved with the image. As the width f the rectangle functin which defines the ILPF decreases with decreasing values f fc, its Furier transfrm increases in width caus ing the side-lbes f the transfrm t increase in width als. This in turn causes the amplitude and perid f the "ringing t increase.

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48 34 The results f the applicatin f the IPF are harder t analyze. When the amplitude f the initial frequency at f= 0 is reduced t zer bth images preserved much f their rigi nal infrmatin except that the level f the signal in each is reduced, and thse regins which riginally had lw trans missin, becme much higher in. transmissin, even greater than thse regins which riginally pssessed high transmis sins. This effect ccurs thrughut all cases where the IPF is used and the image is nt ttally bliterated. As the cutff frequency is increased, the average level f the sig nal is reduced and nise becmes mre dminant in the images. Als as the cutff frequency is increased nly sharp trans itins are preserved while the general shape f the image is destryed. Edges remain sharp. Infrmatin abut sharp transitins must therefre be cntained at the higher fre quencies f the transfrm since the amplitude f the lw frequencies have been reduced t zer. The level f the signal in the filtered images is reduced because the ampli tude f the frequencies unaffected by the filter is small. The level f the signal is therefre dependent upn the abslute magnitude f the frequencies f which it is cm prised. As the cutff frequency is increased, the magnitude f the nise stays practically the same. wever, the mag nitude f the nise relative t the signal increases since the level f the signal is reduced as fq increases. The signal t nise rati decreases when the IPF is used. This

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53 35 fllws frm what was determined earlier, that the nise is cntained at the high frequencies f the transfrm. Since the magnitude f the high frequencies is unaffected by the IPF, the abslute magnitude f the nise will remain cn stant, while the level f the signal is reduced. The reasn fr the increase in transmissins f the riginally lw transmissin areas is nt fully understd, but may be a result f the fact that the IPF prduces a large discnti nuity in the Furier transfrm f the riginal image. The filtered transfrm is a step functin which tails ff with the values f the riginal transfrm aer the cutff fre quency. The recnstructed images are therefre the result f the Furier transfrm f a step and the remaining part f the riginal transfrm. It is this discntinuity in the filtered transfrm which may result in the strange effect n the areas f lw transmissin. The effects f the ELPF parallel thse achieved using the ILPF except with ne majr difference. At a cutff frequency f 30 cycles/mm the ELPF destryed r remved all nise frm the images. Sharp crners became runded and the edges lst sme sharpness, but nt much. Since the level f the signal remains the same and the nise is all but destryed, the signal t nise rati is greatly increased by the ELPF. As the cutff frequency f the filter is reduced, the images becme smther and the less sharp the edges becme. At a cutff frequency f abut 5-10 cycles/mm the images were

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58 36 very degraded and represented sinusids mre than the rig inal images. One effect that the ELPF had n the edge that it did nt crrespndingly have n the bartarget, was t prduce large errr at the end pints f the image. As fc decreased, the errr in the endpints f the edge increased. The mst imprtant difference between the effects f the ILPF and the ELPF is that results frm the use f the ELPF d nt cntain "ringing". The frm f the ELPF use.d is Gaussian and the Furier transfrm f a Gaussian functin is anther Gaussian functin. The Gaussian functin is a very smth cntinuus functin and s its cnvlutin with the image upn filtering prduces a very smth image. As the cutff frequency is decreased the width f the Furier transfrm f the filter increases, prducing a wider Gaussian functin in the spatial dmain, which is why the images becme much smther as fc is decreased. The EPF did nt have the destrying effect that the IPF had n the images. At a cutff frequency f 80 cycles/ mm bth images are cmpsed primarily f nise. At fc= 60 cycles/mm the images are very nisy but bth the edge and the bartarget are easily distinguished. As with the IPF, the EPF reduced the level f the signal in the images. The lwered level f the signal prduced by the EPF causes the signal t nise rati t decrease substantially frm the riginal images. As the cutff frequency used is reduced,

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62 37 the level f the signal is als reduced thereby prducing a lwer signal t nise rati. An interesting effect encun tered when applying the EPF is that f versht at the edges. Oversht is a cnditin where the transmissin f the image at a pint n r near a majr transitin is much higher than the level f the signal shuld be cmpared t the surrunding regin. This is knwn as the Gibb ' s phen menn. This cnditin f versht is in ppsitin t the cnditins previusly mentined, that is, the reductin f the signal t nise rati. The cnditin f versht causes the image t appear t be sharper because the change in the level f the signal ccurs ver a larger range f transmissin. wever the reduced signal t nise rati causes the detectin f a signal frm the backgrund t becme mre difficult. The high frequency emphasis expnential high pass filter had very little effect n the images until a cutff frequency f abut 20 cycles/mm was reached. At lw cutff frequencies the high frequency emphasis filter caused the level f the signal t increase, but did nt affect the image in any ther way. The quality f the edges and the magnitude f the nise were unaffected by the filter. Little r n effect n the sharpness f the edges and the magnitude f the nise was prduced because f the fact that the magnitude f the Furier transfrms at high frequencies is s small. Even

63 38 where the amplitude f the higher frequencies was increased by a factr f tw, the change relative t the amplitude f the lwer frequencies is negligible. Only when the filter starts t affect the relative amplitude f the lwer fre quencies did a change ccur in the images. The transmissin range f the image is increased as wuld be expected since the amplitude f thse frequencies frm which the structure f the image is based is being increased. The effect f aliasing culd nt be graphically demn strated because the changes prduced were small in cmpar isn t the maximum magnitude f the transfrms. Bth transfrms fell t very small values at relatively lw frequencies and s even large changes were very small in cmparisn t the magnitude f the transfrm at very lw frequencies. wever when the amplitude f the transfrms f the differently sampled images were cmpared frequency fr frequency, differences became mre apparent. A signifi cant increase in the amplitude f the Furier transfrm f the image sampled at loym intervals was detected at abut 30 cycles/mm. The increase being in the range f 10-20%. Fr sampling dne at 15ym intervals a significant difference was detected at 20 cycles/mm. Increases again fell in the 10-20% range. Aliasing did nt becme significant until abut 15 cycles/mm fr a sampling interval f 20ym. Actual lcatins where aliasing becmes significant were hard t

64 39 pinpint because f nise in the tranfrm. N change in the transfrm wuld appear at a relative maximum but then an ad jacent sample wuld demnstrate a significant change in value due t aliasing and then the sample adjacent t that pint wuld be unchanged. In any case, the effect f aliasing in this wrk was small due t the fact that s little signifi cant infrmatin was cntained in the transfrms past 30 cy cles/mm. Any increase in the amplitude f the higher frequencies was insignificant relative t the amplitude f the lwer frequencies.

65 40 CONCLUSIONS The relative magnitude f the Furier transfrm f an image at varius frequencies determines their imprtance in the cnstructin f the image. Varius regins f the Furier transfrm cntrl r affect different prperties f the image. The amplitude f the lw frequencies cntrls the level f the signal and the shape r frm f the image. The amplitude f the high frequencies cntrls the sharp transitins in the image but nt their relative size. Mst nise is cntained in the high frequency regin f the Furier transfrm. Remval r attenuatin f the lwer frequencies in the Furier transfrm f the image causes the average level f the signal t decrease, affects the general shape f the image, and lwers the signal t nise rati. Remval r attenuatin f the high frequencies causes the image t becme smther and the signal t nise rati t increase. Filters must be designed t fit the specific needs f a given prblem. There is n ne ideal filter fr every situatin. There are always tradeffs which must be cn sidered befre a filter can be applied t a prblem. A lw pass filter will reduce the nise in an image but at

66 41 CONCLUSIONS, cntinued the expense f fine detail and sharp edges. A high pass filter will cause the image t appear sharper but at the expense f a lwer signal t nise rati. Therefre the perfect filter des nt exist, but filters can be designed s that the desired results are ptimized befre ther factrs begin t enter int the prblem. A knwledge f the image and its Furier transfrm is vital in the prper design f filters s that unwanted results are nt btained and that time and energy are nt wasted n hit r miss attempts at the slutin f the prblem.

67 42 ACKNOWLEDGMENTS I wuld like t express my gratitude twards Prfessr Mhamed Abuelata and Mr. Jim J. Jakubwski whse guidance made this wrk pssible.

68 1? Brigham, The Fast Furier Transfrm, p. S5. 43 REFERENCES Rnald N. Bracewell, The Furier Transfrm and Its Applicatins (New Yrk: McGraw-ill, 1978), p Ibid. 3 Oran E. Brigham, The Fast Furier Transfrm (Englewd Cliffs, New Jersey: Prentice-Kail, Inc., 1974), p Rafael C. Gnzalez and Paul Wintz, Digital Image Prcessing (Reading, Massachusetts: Addisn-Wesley, 1977), p Brigham, The Fast Furier Transfrm, p Ibid., 7Ibid., 8Ibid., p. 96. p. 93. p Jack D. Gaskill, Linear Systems, Furier Transfrms, and Optics (New Yrk: Jhn Wiley and Sns, 1978), pp. " Brigham, The Fast Furier Transfrm, p :LIbid., p Gnzalez and Wintz, Digital Image Prcessing, p Gnzalez and Wintz, Digital Image Prcessing, p Brigham, The Fast Furier Transfrm, p Ibid., p Ibid. 18Ibid., 19Ibid., p p. 18.

69 44 REFERENCES, cntinued 20Gnzalez and Wintz, Digital Image Prcessing, p Ibid., p ^"Jim Jakubwski, Xerx Crpratin, persnal cmmunicatin. 23Prfessr Jhn F. Carsn, Rchester Institute f Technlgy, persnal cmmunicatin. "Jim Jakubwski, persnal cmmunicatin. 2DGnzalez and Wintz, Digital Image Prcessing, p GIbid., 27Ibid., p p Ibid., p Gaskill, Linear Systems, Furier Transfrms and Optics, 110.

70 45 BIBLIOGRAPY Bracewell, Rnald N. Applicatins. New Yrk: McGraw-ill, The Furier Transfrm and Its Brigham, E. Oran. The Cliffs, New Jersey: Prentice-all, Inc., Fast Furier Transfrm. Englewd Gaskill, Jack D. Linear Systems, Furier Transfrms, and Optics. New Yrk: Jhn Wiley and Sns, Gnzalez, Rafael C. and Paul Wintz. Digital Image Prcessing Reading, Massachusetts: Addisn-Wesley, 1977.