Signals and Systems View Point

Transcription

1 Signals and Sstms Viw Pint Inpt signal Ozt Mdical Imaging Sstm LOzt Otpt signal Izt r Iz r I A signalssstms apprach twards imaging allws s as Enginrs t Gain a bttr ndrstanding f hw th imags frm and what thir limitatins ar Dtrmin hw t bttr dsign imaging sstms t ptimiz sm particlar prfrmanc mtric.g. rsltin cntrast btwn halth tiss and disas..

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4 Riw f Linar Sstms Thr Eas qstin first: What is a linar sstm? Inpt signal f Mdical Imaging Sstm H{f} Sstm rspns fnctin Otpt signal g What prprtis f Hf and g wld mak this a linar sstm?

5 Riw f Linar Sstms Thr Eas qstin first: What is a linar sstm? Inpt signal f Mdical Imaging Sstm H{f} Sstm rspns fnctin Otpt signal g What prprtis f Hf and g wld mak this a linar sstm? Trick qstins: It is nl H sstm transfr fnctin that dtrmins is linar sstm? S what prprtis ds H ha t ha t mak this a linar sstm?

6 Riw f Linar Sstms Thr S what prprtis ds H ha t ha t mak this a linar sstm? a*f Mdical Imaging Sstm a*h{f} Otpt signal #1 a*g Prprt #1 f a Linar Sstm: If th inpt signal is scald b mltipling b a cnstant a thn th tpt signal is als simpl scald b mltipling b th sam cnstant.

7 Riw f Linar Sstms Thr S what prprtis ds H ha t ha t mak this a linar sstm? Inpt signal #1 f 1 Mdical Imaging Sstm H{f 1 } Otpt signal #1 g 1 Inpt signal # f Mdical Imaging Sstm H{f } Otpt signal # g Prprt # f a Linar Sstm: If thr ar tw inpt signals f 1 and f which prdc tw tpt signals g 1 and g thn a sstm is linar if

8 Riw f Linar Sstms Thr S what prprtis ds H ha t ha t mak this a linar sstm? Inpt signal #1+# f 1 + f Mdical Imaging Sstm H{f 1 +f } Otpt signal #1+# g 1 + g Prprt # f a Linar Sstm: If thr ar tw inpt signals f 1 and f which prdc tw tpt signals g 1 and g thn a sstm is linar if whn I cmbin th tw inpt signals tgthr f 1 + f th tpt is simpl th sm f th tw tpts g 1 + g.

9 Riw f Linar Sstms Thr S what prprtis ds H ha t ha t mak this a linar sstm? a 1 *f 1 +a *f Mdical Imaging Sstm H{a 1 *f 1 +a *f } Otpt signal #1+# a 1 *g 1 +a *g Prprt #1+# f a Linar Sstm: This is th mst imprtant prprt f linar sstms. It is calld linar sprpsitin and applis t nt st inpt signals bt t an arbitrar nmbr f inpt signals.

10 Riw f Linar Sstms Thr Linarit Eampl: Is this a linar sstm?

11 Riw f Linar Sstms Thr Linarit Eampl: Is this a linar sstm? f 1 = 9 g 1 = 3 f = 16 g =4 f 1 +f =9+16=5 g=5 g 1 +g Nt linar.

12 Riw f Linar Sstms Thr Ok thn th is nt linar. What ar sm ampls f linar pratrs Ths pratrs and all linar cmbinatins f thm ar linar pratrs: d d...

13 Riw f Linar Sstms Thr Q: Ar mst imaging sstms linar r nn-linar?

14 Riw f Linar Sstms Thr Q: Ar mst imaging sstms linar r nn-linar? A: Trick qstin! All ral sstms ar nn-linar if psh thm hard ngh. Th ral qstin shld b is th sstm mstl linar r th rang f inpts I plan t s.

15 Riw f Linar Sstms Thr Q: Ar mst imaging sstms linar r nn-linar? A: Trick qstin! All ral sstms ar nn-linar if psh thm hard ngh. Th ral qstin shld b is th sstm mstl linar r th rang f inpts I plan t s. Camra is satratd nn-linar Phtgraphr adsts th psr tim s that th camra is in a mr linar rgin.

16 Riw f Linar Sstms Thr Ok bt wh shld I car? What is th big dal abt linar sstms and th principl f linar sprpsitin?

17 Riw f Linar Sstms Thr Q: Ok bt wh shld I car? What is th big dal abt linar sstms and th principl f linar sprpsitin? A: Fr linar sstms I can tak r cmplicatd signals and brak thm p int small simpl pics. If I can prdict hw ach f th small pics rspnds t m sstm thn I can prdict hw th ntir cmplicatd signal als rspnds b simpl adding all th littl rspnss p! Cmplicatd Simpl Simpl Simpl

18 Riw f Linar Sstms Thr Q: Ok bt a fw f qstins still 1 Hw d I chs which simpl signals t s? Hw d I brak p th cmplicatd signal int a sm f simpl ns? 3 Hw d I dtrmin hw m linar sstm rspnds t th simpl signals and d I add p all th rspnss prprl t gt what I wld gt with th cmplicatd signal? Cmplicatd Simpl Simpl Simpl

19 Riw f Linar Sstms Thr Q: Ok bt a fw f qstins still 1 Hw d I chs which simpl signals t s? A: Thr ar a larg nmbr f ptntial simpl fndamntal fnctins w cld slct frm. All f ths fnctins ha th prprt that I can bild mr cmplicatd fnctins b adding thm p. Th als sall d nt inl r cmplicatd math. Cmplicatd Simpl Simpl Simpl

20 Riw f Linar Sstms Thr Q: Ok bt a fw f qstins still 1 Hw d I chs which simpl signals t s? A: Thr ar a larg nmbr f ptntial simpl fndamntal fnctins w cld slct frm. All f ths fnctins ha th prprt that I can bild mr cmplicatd fnctins b adding thm p. Th als sall d nt inl r cmplicatd math. Thr ar tw r spcial fndamntal signals that w ftn s in imaging and will s tnsil in this class 1 D Dlta fnctin r Pint fnctin Sinsidal signals r cmpl pnntials

21 Riw f Linar Sstms Thr A: Thr ar a larg nmbr f ptntial simpl fndamntal fnctins w cld slct frm. All f ths fnctins ha th prprt that I can bild mr cmplicatd fnctins b adding thm p. Th als sall d nt inl r cmplicatd math. 1st Elmntar Fnctin dlta fnctin r pint fnctin: Th tw-dimnsinal dlta fnctin δ. δ has infinitsimal width and infinit amplitd. KEY PROPERTIES: δ Sifting prprt f δdd 1 f δ 1 f 1 1 dd f 1 δ dd

22 Riw f Linar Sstms Thr A: Thr ar a larg nmbr f ptntial simpl fndamntal fnctins w cld slct frm. All f ths fnctins ha th prprt that I can bild mr cmplicatd fnctins b adding thm p. Th als sall d nt inl r cmplicatd math. nd Elmntar Fnctin cmpl pnntial: cs sin Ths fnctins frm th basis f D Frir analsis. I cannt rstimat its imprtanc twards mdical imaging!

23 Riw f Linar Sstms Thr Th D dlta fnctin and its applicatin twards th D cnltin Intgral KEY PROPERTIES: δ Sifting prprt f δdd 1 f 1 f 1 1 δ dd f 1 δ dd

24 Riw f Linar Sstms Thr Th D dlta fnctin and its applicatin twards th D cnltin Intgral Rcall th sifting prprt f f δ dd Inpt signal f Imaging Sstm L{f} Linar pratr Otpt signal g lns

25 Riw f Linar Sstms Thr Th D dlta fnctin and its applicatin twards th D cnltin Intgral Rcall th sifting prprt f f δ dd Inpt signal f Mdical Imaging Sstm L{f} Linar pratr g L[ f ] Otpt signal g g L f δ dd

26 Riw f Linar Sstms Thr Inpt signal f Mdical Imaging Sstm L{f} g L[ f ] Otpt signal g g L f δ dd Hr is th first trick! Bcas L is a linar pratr and bcas th intgrals ar linar pratrs I can changd th rdr f ths pratins witht changing th rslts g L f δ dd

27 Riw f Linar Sstms Thr Inpt signal f Mdical Imaging Sstm L{f} Otpt signal g g L f δ dd Hr is th scnd trick! Fr r al f and th fnctin f is st a cnstant. S it can b md tsid f th linar pratr L. g f L δ dd

28 Riw f Linar Sstms Thr Inpt signal f Mdical Imaging Sstm L{f} Otpt signal g g f L δ dd Qstin: What ds th trm rprsnt in an imaging sstm? L δ phsicall

29 Riw f Linar Sstms Thr Inpt signal f Mdical Imaging Sstm L{f} Otpt signal g g f L δ dd Qstin: What ds th trm phsicall rprsnt in an imaging sstm? Answr: Hw a pint in th bct plan maps int th imag plan. lns L δ

30 Riw f Linar Sstms Thr Inpt signal f Mdical Imaging Sstm L{f} Otpt signal g g f L δ dd h ; L [ δ ] In imaging applicatins h; is rfrrd t as th pint sprad fnctin r PSF. It mathmaticall dscribs hw a pint src at lcatin imags t a pint in th imag plan

31 Riw f Linar Sstms Thr Inpt signal f Mdical Imaging Sstm L{f} Otpt signal g h ; L [ δ ] Air Disk gd glass lns PSF frm hman

32 Riw f Linar Sstms Thr Inpt signal f Mdical Imaging Sstm L{f} Otpt signal g L g f L δ dd δ is gin a spcial nam calld th impls rspns r pint sprad fnctin dntd b h ; L [ δ ] g f h ; dd Sprpsitin Intgral

33 Riw f Linar Sstms Thr Inpt signal f Mdical Imaging Sstm L{f} Otpt signal g h ; L [ δ ] g f h ; dd Sprpsitin Intgral Qstin: What phsicall maning f th sprpsitin intgral in an imaging sstm?

34 Riw f Linar Sstms Thr Inpt signal f Mdical Imaging Sstm L{f} Otpt signal g h ; L [ δ ] g f h ; dd Sprpsitin Intgral Qstin: What phsicall maning f th sprpsitin intgral in an imaging sstm? Answr: Th tpt frm a linar imaging sstm can b cmptd b taking a wightd sm f th rspns frm a bnch f pints.

35 Riw f Linar Sstms Thr Inpt signal f Mdical Imaging Sstm L{f} g f h ; dd f PSF g Otpt signal g

36 Riw f Linar Sstms Thr Inpt signal f Mdical Imaging Sstm L{f} Otpt signal g h ; L [ δ ] g f h ; dd Qstin: What phsicall maning f th sprpsitin intgral in an imaging sstm? Answr: Th tpt frm a linar imaging sstm can b cmptd b taking a wightd sm f th rspns frm a bnch f pints. Qstin: Hw ds that mak r lif asir?

37 Tim inarianc A sstm is tim inariant if its tpt dpnds nl n rlati tim f th inpt nt abslt tim. T tst if this qalit ists fr a sstm dla th inpt b t 0. If th tpt shifts b th sam amnt th sstm is tim inariant i.. ft gt ft - t gt - t inpt dla tpt dla Is ft fat gt an adi cmprssr tim inariant?

38 Tim inarianc A sstm is tim inariant if its tpt dpnds nl n rlati tim f th inpt nt abslt tim. T tst if this qalit ists fr a sstm dla th inpt b t 0. If th tpt shifts b th sam amnt th sstm is tim inariant i.. ft gt ft - t gt - t inpt dla tpt dla Is ft fat gt an adi cmprssr tim inariant? ft - t fat fat t - tpt f adi cmprssr fat t - shiftd rsin f tpt this wld b a tim inariant sstm. S ft fat = gt is nt tim inariant.

39 Spac r shift inarianc A sstm is spac r shift inariant if its tpt dpnds nl n rlati psitin f th inpt nt abslt psitin. If shift inpt Th rspns shifts bt in th plan th shap f th rspns stas th sam. If th sstm is shift inariant h ; = h and th sprpsitin intgral bcms th D cnltin fnctin: g f h dd Ntatin: g = f ** h ** smtims implis tw-dimnsinal cnltin as ppsd t g = f * h fr n dimnsin. Oftn w will s * with D and 3D fnctins and impl D r 3D cnltin.

40 Riw f Linar Sstms Thr Minimal Shift Varianc Lts f Shift Varianc h ; h h ; h

41 If a sstm is bth linar and shift-inariant it is calld LSI g f h dd In a LSI imaging sstm th shap f th pint sprad fnctin h is indpndnt f th lcatin f th pint src. W mdl mst mdical imaging sstms as LSI

42 D CONVOLUTION INTEGRAL g f h g f h dd

43 D CONVOLUTION INTEGRAL APPROXIMATED DIGITALLY g d d h f g 1 1 n m i N n N m n m i h f

44 MATLAB DEMO

45 Cmpl Epnntials sin cs An imprtant cntins signal is th cmpl pnntial s c 1 1 sin 1 1 cs and ar calld spatial frqncis

46 Cmpl Epnntials sin cs s c 1 1 sin 1 1 cs

47 Cmpl Epnntials sin cs s c 1 1 sin 1 1 cs 0 0

48 Cmpl Epnntials sin cs s c 1 1 sin 1 1 cs If I start adding mltipl cmpl pnntials tgthr

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54 Sstm transfr fnctin H h h H dd dd Impls rspns r PSF

55 Frqnc Rspns: Eampl Assm th PSF f a LSI imaging sstm can b dscribd as: b a b a b a rct h Dtrmin th transfr fnctin

56 Frqnc Rspns: Eampl Assm th PSF f a LSI imaging sstm can b dscribd as: b a b a b a rct h Dtrmin th transfr fnctin b b a a dd H dd b a rct H dd h H

57 Frqnc Rspns: Eampl Assm th PSF f a LSI imaging sstm can b dscribd as: b a b a b a rct h Dtrmin th transfr fnctin 1 1 a a a b b H d d H dd H b b a a b b a a

58 Frqnc Rspns: Eampl Assm th PSF f a LSI imaging sstm can b dscribd as: b a b a b a rct h Dtrmin th transfr fnctin a a b b ab H H a a a b b H a a b b sin sin

59 Frqnc Rspns: Eampl Assm th PSF f a LSI imaging sstm can b dscribd as: h rct a b a 0.5a 0.5b 0.5b Dtrmin th transfr fnctin H ab H ab H absinc b sinc a sin b sin a b a sinc b sinc a