Med Imag Sys
Med Imag Sys EECS 516
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BMEEECS 516 2004 FT Notes 1 Notes on the Fourier Transform De nition The Fourier Transform FT relates a function to its frequency domain equivalent The FT of a function gx is defined by the Fourier integral 00 427Mquot 60 mm L gxe dx for x s 6 SR There are a variety of existence criteria and the FT doesn t exist for all functions For example the function gx cos1x has an in nite number of oscillations as x gt 0 and the FT integral can t be evaluated If the FT does exist then there is an inverse FT relationship gx F 1Gs I Gsei2msds Uniqueness Given the existence of the inverse FT it follows that if the FT exists it must be unique That is for a function forms a unique pair with its FT g x lt gt GU Caveat An exception to the uniqueness property is a class of functions called massless or l 0 null functlons An example 1s the contmuous function f x O x i 0 Th1s function and others like it have the same Fourier transform as fx 0 F s 0 Thus the uniqueness exists only for a function plus or minus arbitrary null functions In practice these functions are not realizable energyless and thus for the purposes of this class we will assume that the FT is unique Alternate FT De nition In the above derivation the s is the frequency parameter There is another common FT definition that uses a radian frequency parameter a 00 7 6a mm L gxe 1 Helm2 with an inverse FT of gx F 1Gw i I Gwei dw 27239 00 Units If x has units of Q then s will have units of cyclesQ or Q39l Please note that under our definition of the FT this is not an angular frequency with units of radiansQ but just plain Q39l Please also keep in mind that x is the index of variation 7 for example we can have gx represent a velocity that varies as a function of spatial location x The function gx has units cms but x has units cm and Gs has units of cms but 3 has units of cm39l F vnmnlpc x s Time Temporal Frequency s seconds s39l Hz cycless Distance Spatial Frequency cm cm39l cyclescm BMEEECS 516 2004 FT Notes 2 Symmetry De nitions We first decompose some function gx in to even and odd components ex and 0x respectively as follows 606 gx 8030 006 H806 79036 thus 80 606 006 and ex e7x and 0x 70x A function gx is Hermitian Symmetric Conjugate Symmetric if Regx 06 and1mgx 006 thus 80 06 H006 g rx Symmetry Properties of the FT There are several related properties 1 Ifgx is real then Gs is Hermitian symmetric eg Gs G s 2 If gx is real and even Gs is real and even 3 If gx is real and odd Gs is imaginary and odd 4 If gx is real Gs can be defined strictly by nonnegative frequencies s 2 0 5 Ifgx is imaginary then Gs is AntiHermitian symmetric eg Gs G s Proof of l Gs Jgxe mdx j ex 0xcos 2m 7 isin 27rsxdx cos is even sin is odd jex cos Z xxdx jox cos 27rsxdx 7 i ex sin Z xxdx 7 i 0x sin Z xxdx je39xdx jo39xdx 7 if 0 xdx 7 if e xdx Es 0 7139 0 7139Os cos is even in s sin is odd is s I0ddx 0 E s 7 i0s E7s iO7s G7s QED Comment One interesting consequence of the symmetry properties is that if gX is real the only onehalf of the Fourier transform is necessary to specify the function 7 this follows from property 1 above More specifically gX is strictly determined by Gs for all nonnegative frequencies s BMEEECS 516 2004 FT Notes 3 Comment on negative frequencies Consider a realvalued signal 7 imagine a voltage on a wire or the sound pressure against your eardrum 7 the Fourier transform of these is completely speci ed by the nonnegative frequencies e g G s Gs We can argue that we have the concept of a frequency oscillationssecond but it doesn t really make physical sense to talk about positive or negative frequencies In this case we could argue that the having positive and negative frequencies is merely a mathematical convenience Are there cases where negative frequencies have meaning Consider the bit in a drill 7 it can turn clockwise or counter clockwise and different rotational rates Here positive and negative frequencies have physical meaning the direction of rotation As we shall see there are cases in medical imaging where this distinction is important for example the magnetic moment in MRI is a case where the sign indicates the direction of precession Convolution De nition The convolution operator is defined as 00 gx WC lg hx 615 00 The convolution operator commutes gx hx Ig hx d IgOC h d WCquot gx The delta function 5 x The Dirac delta or impluse function is a mathematical construct that is infinitely high in amplitude infinitely short in duration and has unity area 0 0 and I5xdxl 00 x 60C 2 0 x at Most properties of 5 x can eXist only in a limiting case e g as a sequence of functions gquot x gt 5x or under an integral Some important properties of 5 x 15xgxdx g0 with gx continuous at x 0 5x agxdx ga with gx continuous at x a 5axgxdx l1 g0 with gx continuous at x 0 a F6x 1 Delta function properties First two are technically only defined under the integral but we ll still talk about them Similarity stretching 60m L 60C l a l ProductSifting 800506 a g 1506 11 Siftng jgxax7awc go BMEEECS 516 2004 FT Notes Convolution 80 505 50 gx 80 800506 11 506 agx 806 a Fourier Transform Theorems There are many Fourier transform properties and theorems This is a partial list Assume that Fgx Gs Fhx H s and that a and b are constants F 8010 Convolution 11730061275 GS Fgxcos27rs0x 7s0Gs Fgxsin2nxox 7s07Gsso Rayleigh s Power dx ds Cross Power gxh xdx GsH sds F 806 gx GSG S i27rsGs Moments Fxgx L Gs DC V gxdx G0 Some common FT pairs 0052713035 7 so 6s Sinamox So 5sso 2 sincs sin7rs Its t 7 l lt recx70 2 l trianglex 0 BMEEECS 516 2004 FT Notes 5 1 my 671 for x gt 00 otherwise 11392713 1 2752 The comb function c0mbx The sampling or comb function is a train of delta lnctions combx Z 5x n n7oo The Fourier transform of combx is Fcombx combs Proof Fcombx F 2 5x yo z em Fs n7oo 11700 The RHS of the above expression can be Viewed as the exponential Fourier series representation of a periodic function F s with period 1 and an l for all n Recall the Fourier series expressions are 00 Fs Z anelz m where an IFse 2mds n 72 Now let Gs rectsFs one period ofFs and thus Fs Z Gs m Now observe that m7oo V2 0 an jFsequot2 mds jGsequot2 ds FGsxn 1 7 700 One function that satis es this relationship is Gs 53 Thus one possible Fourier transform of combx is Fs 2 53 m combs m7oo By uniqueness of the Fourier transform this is the unique Fourier transform of combx BMEEECS 516 2004 FT Notes 6 Sampling and replication by combx The comb function can be used to sample or extract values of a continuous function gx Sampling with periodX can be done as gxcomb gx6 nX gx6x nXX gnX6x nX By the stretching and sifting properties of the delta function A function gx can be replicated with periodX by convolving with a comb function gm comm 2 mm 61 n X 2 gm 6x nX X z gx nX 11700 11700 11700 By the stretching and convolution properties of the delta function Sampling Theory When manipulating real objects in a computer we must rst sample the continuous domain object into a discretized version that the computer can handle As described above we can sample a function gx at frequency lX using the comb function mm gxcomb X ingwoc nX The Fourier transform is GS s Gs XcombXs Gs EX Xs m 6W Em mf gas mm Thus sampling in one domain leads to replication of the spectrum in the other domain The spectrum is periodic with period Typically only frequencies less than fx 2 can be represented in the discrete domain signal Any components that lie outside of this spectral region if 2 g s g f 2 results in aliasing 7 the misassignment of spectral information Re licated Gs Orlgmal Gs P Spectrum Spectra Aliasing BMEEECS 516 2004 FT Notes 7 The WhittakerShannon sampling theorem states that a band limited function with maximum frequency smax can be fully represented by a discrete time equivalent provided the sampling frequency satisfies the Nyquist sampling criterion 1 fs 223max If this is the case then the original spectrum can be extracted by filtering and by uniqueness of the FT the original signal can be reconstructed To reconstruct the original signal we apply a reconstruction filter Hs rects f5 rectXs 30 GssHs GssrectXs Gs if there is no aliasing In the x domain this results in sinc interpolation 06 gs x Sind gnX5x nX sinc ZSinC quotXgnX If the Nyquist criterion is met then x gx A Info from all gX g511 gX samples contribute to this pomt 1D Linear Systems Consider a system S39 with an input function x and an output or response function gx S fx This system is linear if and only if Swim mm aSlf1xl Sf2xl ag1x 200 for all a f1 and f2 More generally the superposition of an arbitrary set of input functions will yield a net response that is the superposition of responses to each input function alone BMEEECS 516 2004 FT Notes 8 Additionally if any input is scaled eg by a then the output will also be scaled by the same amount Based on the sifting properties of the delta function we know that 00 fx j f 5x w fx 60c 00 which is the superposition of an in nite number of weighted and shifted delta functions Based on linearity the output of this system gx S fx is gx S j f 5x 615 j f S5x 161 The system operates on functions of x and g is a constant scaling factor S 5x is a special function know as the impulse response and can is defined as hoe 5 S6x 5 is the response to an impulse located at X 5 and gx j f hx w 00 is known as the superposition integIal This representation for the output is valid for any linear system Now consider a system that is shift invariant or time invariant for functions of time We define a system as being shift invariant if and only if gOCa Sl xa for all g and a For linear shift invariant systems the impulse response can be written as hx S6x 5 m 0 hoe 5 The superposition integral then becomes 00 gx j f hx w fx hoe 00 the convolution of the input function with the impulse response hx For linear shift invariant systems or linear timeinvariant systems only we can then consider the Fourier domain equivalent GU F SHS Where H s is known as the transfer function or system spectral response BMEEECS 516 2004 FT Notes 9 Notes on the 2D Fourier Transform De nition The 2D Fourier Transform FT relates a function to its frequency domain equivalent The FT of a function gxy is defined by the 2D Fourier integral Guv Fgx y j j gxyequot dxdy There is also an inverse FT relationship gx y F391Gu v J IGuve 2 WVydudv Uniqueness Given the existence of the inverse FT it follows that if the FT exists it must be unique That is for a function forms a unique pair with its FT gx y lt gt Gu v 2D FT in Polar Coordinates We consider a special case where the functional form of gxy is separable in polar coordinates that is grt9 gRrgdH Since gag is periodic in 9 it has a Fourier series representation gem i ate It can be shown that quotHo FDgre z39quote quot I 2ngRrJ2mprdr where the part under the integral in known as the Hankel transform of order n and J n is the n3911 order Bessel function of the rst kind 1 7r J a exasmwn d I 2 L go Derivation of the Hankel transform relationship relies on e m wlyv e 2 quot sg Thus the 2D FT in polar form is Gm Fgrg 6 Z a z39quote quot 0 2ngRrJ2mprdr For the special case of circular symmetry of g that is grt9 gRr then Gm Gltpgt 240 gRrJo2mprdr which is also a circularly symmetric function The inverse relationship is the same gm 240 GpJo2mppdp Symmetry Properties of the FT If gxy is real then Guv is Hermitian Symmetric that is Guv G u v Ifgxy is real and even that is gxy g x y then Guv is also real and even Finally as described above if we have a real and circularly symmetric function grt9 gRr the Gp Gp a real and circularly symmetric function BMEEECS 516 2004 FT Notes 10 The delta function 5 x y The delta function in two is equal the to product of two lD delta functions 5 x y 5 x5 y In a manner similar to the 1D delta function the 2D delta function has the following definition 6 zoox0andy0 dJJEOC dxd 1 x y 0 otherwise an a y y Most properties of 5 x y can be derived from the 1D delta function There is also a polar coordinate version of the 2D delta function 5 x y 5r 727 Fourier Transform Theorems Let a and b are nonzero constants and F gxy Guv and F hxy H MN Linearity F 1806 y 106 0 GOA V 51101quot Magni cation F gwc by Ial bIG Shift Fgx 7 a y 7 12 Guvequot2 a b Complex Modulation Fgx ye 2 myb Gu 7 a v 7 b ConvolutionMultiplication gx y hx y g 77x 7 g y 7 77d d77 Fgx y hx y GuvHuv Fgx who y Gm Huv Separability 800 8X 3082 y F gx y F1DxgX xF1Dy gy y G X GY V Power x y2 d Cdy HIG gtVIZ dudv goc yh x ydxdy ijuvH uvdudv Axis Reversal Fg xgt y G A Some common 2D FT pairs 7ay ie 6 6 e e cos27 x cos27zx l 1 5u circr 0 Vi 1 J12 p jincP BMEEECS 516 2004 FT Notes 11 combxy cnmh cnmh I combuv cnmhh cnmh I The comb function in 2D combx y The 2D sampling or comb function is de ned as combxycombxcomby and has the 2D FT F combx y combuv Formally the 2D comb function is de ned as combx y 25x n y m In a manner similar to the 1D case we can prove that Fourier transform of the 2D comb function is also a 2D comb function as given in the above table Sampling Theory in 2D In a manner similar to sampling in 1D sampling in 2D can be modeled as multiplying a function times the 2D comb function With sample spacing of X and Y in the X and y directions the sampled function is gx y gxycomba gov y 2 5 7 m XY i6x nXymYgxJ nmroo XY 2 6x nX y mYgnX mY The discrete domain equivalent is gdnm gnX mY g5nX mY In the Fourier domain the result is GS u v Gu v XYcombXu Yv Guv Ef e ave ZGuv Thus sampling in one domain leads to replication of the spectrum in the other domain Spacing of the replicated spectra is lX 1 Y The WhittakerShannon sampling theorem in 2D states that a band limited function with maximum frequencies 3mm and smugV can be fully represented by a discrete time equivalent provided the sampling frequency satis es the Nyquist sampling criterion i 2 2s and1 2 2erX X Y 39y Under these circumstances there is no spectral overlap or aliasing the original spectrum and by uniqueness of the FT the original signal can be reconstructed BMEEECS 516 2004 FT Notes 12 To reconstruct the original signal we apply a reconstruction lter H u v rectXurectYv Ru v GS u vH u v GS u vrectXurectYv Gu v if there is no aliasing In the Xy domain this corresponds to sinc interpolation in 2D sincX M mc 96 y gs x y Sincsinc x nXy mYgnX mY sinc sinc nmroo Y ZsincU X sincy39quot ygnX mY Zsincmwkinca yogdmm The last line demonstrates how the original continuous signal can be retrieved from the discrete sampled version of gxy BMIE IEECS 516 2004 FT Notes Examples of Fourier Transforms Low spatial freq data image domain image domain Low spatial freq Fourier domain data Fourier domain High spatial freq data image domain High spatial freq data Fourier domain BMEEECS 516 2004 FT Notes 14 rectx2 2sinc2S 1 05 0 05 5 0 5 circ jinc BMEEECS 516 2004 FT Notes rectx2recty2 circ 5 5 5 5 4sinc2usinc2v jinc 4sinc2usinc2v jinc BMEEECS 516 2004 gxy rectxrecy Guv smusmcv sa1mgmagm canon properly sa1mgmagm canon properly shx zng property modulation FT Notes AbsGouner RealFouner ImagGouner BMEEECS 516 2004 FT Notes gXy sincXsincy GuV recturectV sampling pattern with AX Ay In the Fourier transform we have the replication pattern with spacing lAX lAy sampling pattern with AX lt Ay In the Fourier transform we have the replication pattern with spacing lAX gt lAy sampling pattern with AX ltlt Ay This has aliasing in the y V direction Image Data Fourier Data BlVTEEECS 516 2004 FT Notes 18 2D Linear Systems Consider an imaging system with an input image 11xy that goes through some system S39 and produces an output image 12xy e g 12xy S11xy 1 01 T S A 12 quot33 Several properties that we are interested in are 0 Linearity which as two parts superposition and scaling Thus a system is linear if and only if Sagltx y hx w aSgltx ygt 58 w For a linear system we can de ne an impulse response as the output of a system for an impulse located at position 5 17 hCayman S5x y 77 and in general we can de ne the output given some input image 11xy using the superposition integral 120w I I11ltnhxy nd dn OO 00 0 Space Invariance A system is space invariant if and only if 12xayb S11xayb for all a b and 11 Alternately a system is space invariant if and only if the impulse response can be expressed in terms of the shifts of the delta function hx y 77 S5x y 77 Thus hx y S 5 x y is all that is needed to specify the system The superposition integral becomes 12xy I I11 77hx y77d d77 OO 00 11x y hx y where indicates 2D convolution BMEEECS 516 2004 FT Notes 19 Example of a 2D Imaging System Here we consider a pinhole imaging system Izlxz Y2 I1 11 VI EMAb4 FIG 22 Pinhole imaging system with magni cation htxz V2 i It Is this system linear Assuming that the aperture is open or close and that light always travels in straight lines through the hole then yes the system is linear We should then be able to determine the impulse response Let s rst consider two different magni cations factors one for the object and one for the pinhole aperture For object magni cation we imagine the pinhole in in nitely small For a shift of 77 in the input we will get a shift of 277 in the output plane Thus we de ne an a 3 input source magni cation term as M 01 Consider then a system for a delta function at position rf 77 hoe ma 77 S6ltx 5 y 77 C6x M y M77 where the delta function appears scaled by C and at location M M77 Is this system space invariant No the above expression cannot be written as a function of 965 and y 77 Now given an input image 11xy we can still determine the output image using the superposition integral remember the system is still linear