Introduction to Medical Imaging. Lecture 4: Fourier Theory = = ( ) 2sin(2 ) Introduction

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Brian Melton
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Introduction Introduction to Mdical aging Lctur 4: Fourir Thory Thory dvlopd by

spctrum S(k) Klaus Mullr s() forward transform DC (avrag) trm πik Sk ( ) Fs { ( )}

s ( ) F { Sk ( )} Sk ( ) dk = = Etnsion to Highr Dimnsions Th Fourir transform

{ } sy ddy invrs transform = = πi( k+ ly) sy F { S( k, l)} S( k, l) dkdl = =

= π kl πki πk = AL sinc( π kl) iπkl iπkl ( ) sin( ) Fourir pair W s that a finit

1 Introduction Introduction to Mdical aging Lctur 4: Fourir Thory Thory dvlopd by Josph Fourir (768-83) Th Fourir transform of a signal s() yilds its frquncy spctrum S(k) Klaus Mullr s() forward transform DC (avrag) trm πik Sk ( ) Fs { ( )} s ( ) d = = Computr Scinc Dpartmnt Stony Brook Univrsity S(k) invrs transform πik s ( ) F { Sk ( )} Sk ( ) dk = = Etnsion to Highr Dimnsions Th Fourir transform gnralizs to highr dimnsions Considr th D cas: forward transform πi( k+ ly) Skl Fsy { } sy ddy invrs transform = = πi( k+ ly) sy F { S( k, l)} S( k, l) dkdl = = Calculation: ct Function + L iπk iπk S( k) = F{ AΠ ( )} = A ( ) d A d Π = L A A = = π kl πki πk = AL sinc( π kl) iπkl iπkl ( ) sin( ) Fourir pair W s that a finit signal in th -domain crats an infinit signal in th k-domain (th frquncy domain) th sam is tru vic vrsa

Proprtis Scaling: considr th rct (bo): th gratr L th highr th spctrum (factor AL) th narrowr th spctrum (factor L) th scaling rul is thrfor: k Fsa {( )} = S( ) a a Symmtry: F{ S( )} = s( k) Linarity:

2 Proprtis Scaling: considr th rct (bo): th gratr L th highr th spctrum (factor AL) th narrowr th spctrum (factor L) th scaling rul is thrfor: k Fsa {( )} = S( ) a a Symmtry: F{ S( )} = s( k) Linarity: F{ as( ) + bs( )} = F{ as( )} + F{ bs( )} Translation: Fs {( )} = Sk ( ) πi k phas shift Convolution: Fs { ( ) s( )} = S( k) S( k) Fs { ( ) s( )} = S( k) S( k) Sk ( ) = ALsinc( π kl) a> shrinks s a< strtchs s Scaling Proprty Th rct function provids good insight into th rlationship of fin dtail and frquncy bandwidth a thin rct can rprsnt/rsolv fin dtail (think of a signal bing rprsntd as an array of thin rcts a thin rct givs ris to a wid frquncy lob this illustrats that signals with mor dtail will hav broadr frquncy spctra or, in turn, signals with thin frquncy spctra will hav low spatial rsolution Influnc of Transfr Function H W know (from th last lctur) that: ( ) ( ) πik ( ) = i s S k H k dk s ( ) = s ( ) h( ) S ( k) H( k) = S ( k) o i i o Calculation: Dirac puls For s()=δ(): iπk iπk ( ) = { ( )} = ( ) = = S k F δ δ d Lt s look at a concrt ampl: H is a lowpass (blurring) filtr: it rducs th highr frquncis of S mor than th lowr ons aftr application of H k call that th Dirac is an trmly thin rct function th frquncy spctrum is thrfor trmly broad ( vrywhr) This illustrats a ky fatur of th Fourir Transform: th narrowr th s(), th widr th S(k) sharp objcts nd highr frquncis to rprsnt that sharpnss

3 portant Fourir Pairs: Sinusoids Sinusoids of frquncy k giv ris to two spiks in th frquncy domain at ±k cos( πk ) ( δ( k+ k) + δ( k k)) / sin( πk) i( δ( k+ k) δ( k k)) / call th pointr analogon in th compl plan for th cos(): th ral signal is givn by th addition of th two vctors (dividd by ), projctd onto th ral ais portant Fourir Pairs: Sinusoids Sinusoids of frquncy k giv ris to two spiks in th frquncy domain at ±k cos( πk ) ( δ( k+ k) + δ( k k)) / sin( πk) i( δ( k+ k) δ( k k)) / call th pointr analogon in th compl plan for th sin(): th ral signal is givn by th addition of th two vctors (dividd by ), projctd onto th imaginary ais (not th i in th quation) -k k = -k k = Mor portant Fourir Pairs δ ( ) δ ( k) cos( πk) ( δ( k+ k) + δ( k k)) / σ sin( πk) i( δ( k+ k) δ( k k)) / Π( ) sinc( π Lk) Λ( ) Lsinc ( π Lk) k σ Som ots In th D transform, if f(,y) is sparabl, that is, f(,y)=f()f(y), on may writ: π { } ( ) πik Skl = Fsy = s y ( s( ) d) dy π ily πik sy F { S( k, l)} S() l ( s( k) dk) dl = = this coms in handy somtims th Gaussian width is invrsly rlatd

4 Som ots Somtims th factor πk is usd as ω: ( ) ( ) iω s ( ) = Si ω H ω dω So far, w hav only discussd th continuous spac with (potntially) infinit spctra and signals that is whr it maks sns to us ω but in rality w dal with finit, discrt signals (hr k mattrs) w shall discuss this nt Fourir Transform of Discrt Signals: DTFT Discrt-Tim Fourir Transform (DTFT) assums that th signal is discrt, but infinit S( ω) = s( n) n= + π sn ( ) = S( ω) π iωn th frquncy spctrum is continuous, but is priodic (has aliass) s(n) iωn S(ω) aliass cntrd at π Fourir Transform of Discrt Signals: DFT Discrt Fourir Transform (DFT) assums that th signal is discrt and finit Sk ( ) = sn ( ) sn ( ) = Sk ( ) iπkn iπkn n= n= now w hav only sampls, and w can calculat frquncis th frquncy spctrum is now discrt, and it is priodic in Fourir Transform in Highr Dimnsions Th D transform: Sparability: M Skl = snm m= n= m= n= i π ( kn+ lm) M M snm = Skl M i π ( kn+ lm) M s(n) S(ω) M iπlm M Skl = Pkm whr Pkm = snm M m= n= M iπlm M snm = pnl whr p( nl, ) = Snm M l= k= if M=, complity is O( 3 ) iπ kn iπ kn

5 Fast Fourir Transform () cursivly braks up th FT sum into odd and vn trms: Fast Fourir Transform () Givs ris to th wll-known buttrfly architctur: iπkn / iπkn / i πk(n+ ) Sk ( ) = sn ( ) = s( n ) + s(n+ ) n= n= n= / iπkn iπk / iπkn / / svn() n sodd () n n= n= = + sults in an O(n log(n)) algorithm (in D) O(n log(n)) for D (and so on)

Fourier Transforms and the Wave Equation. Key Mathematics: More Fourier transform theory, especially as applied to solving the wave equation.

Fourier Transforms and the Wave Equation. Key Mathematics: More Fourier transform theory, especially as applied to solving the wave equation. Lur 7 Fourir Transforms and th Wav Euation Ovrviw and Motivation: W first discuss a fw faturs of th Fourir transform (FT), and thn w solv th initial-valu problm for th wav uation using th Fourir transform