General theorems of Optical Imaging systems

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Σίλας Ελευθεριάδης
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1 Gnral thorms of Optcal Imagng sstms

2 Tratonal Optcal Imagng Topcs Imagng qualt harp: mags a pont sourc to a pont Dstorton fr: mags a shap to a smlar shap tgmatc Imagng Imags a pont sourc to a nfntl sharp pont Prfct Imagng Imag s (mathmatcall) smlar to th obct Contmporar optcal magng Optcal magng optcal sstm sgnal procssng Computatonal magng magng pr- or postprocssng

3 Gnral Optcal Imagng Thorms Rlats mostl wth th tratonal optcal magng topcs Mawll s thorm Prfct magng s onl possbl for thr s no magnfcaton Abb sn conton Rla conton for transvrsal obcts Hrschl cosn conton Rla conton for longtunal obcts

4 tgmatc Imagng tgmatc magng P P 0 H(rr ) P 0 P H s th transfr functon btwn 0 to 0 plans. For fr spac for ampl H E P 0 : (00) at 0 plan ( ) ( ) δ ( 0 0) 0 mag to 0 P 0 : (00) at 0 plan

5 Transfr functon of stgmatc magng P 0 : (00) at 0 plan so E mag to ( ) E ( ) ( 0) HE( ) ( ) H const. δ const. P 0 : (00) at 0 plan ( 00 0) ( ) thrfor H ( ) ( ) ( ) ( )

6 Transfr functon of stgmatc magng H ( ) ( ) ( ) ( ) H s th transfr functon btwn 0 (n th obct spac) to 0 plan (n th mag spac) If on pont sourc n th obct spac s mag nto an nfntl sharp pont n th mag spac H has to assum th abov form. What wll happn to othr pont sourc at 0 plan? Can H also mag othr pont sourc to prfct ponts?

7 Othr pont sourcs at 0 plan H Fourr transform for spctrum P : ( ) at 0 plan E δ It s stll a constant but wth ffrnt phas.

8 Tracng n th spctrum oman E nstrumnt transfr from 0 plan to 0 plan P : ( ) at 0 plan has th spatal spctrum: H H

9 Output mag at 0 A A A E 0 0 0

10 tgmatc Imagng conton for transvrsal plan (D Imagng) E ( ) A 0 If γ γ ar constants ( ) A δ ( γ γ ) E 0 Pont sourcs on 0 plan ar all mag nto prfct ponts f / an / ar constants.

11 Abb sn conton for Prfct Imagng n transvrsal plans Prfct Imagng prsrvs mathmatcal smlart thrfor for prfct magng γ γ an E A0 δ γ γ Abb sn conton: for ρ snθ ρ snθ sn θ or snθ snθ snθ snθ sn θ f th mag spac has sam n of rfracton as th obct spac. ρ P H(rr ) P ρ

12 Othr pont sourcs on th optcal as H Fourr transform for spctrum P : (00) at plan E δ It s stll a constant but wth ffrnt phas.

13 Othr pont sourcs on th optcal as E Fr spac transfr to 0 plan E nstrumnt transfr from 0 plan to 0 plan P : (00) at plan has th spatal spctrum: H Fr spac transfr from 0 plan to plan H

14 Output mag at A A E 0 0

15 tgmatc Imagng conton for longtunal obcts Pont sourcs at orgn of 0 an plans ar mag nto prfct ponts at orgn of 0 an plans. It s also prfct magng. (Wh?) If 00 0δ A E A A E 0 0 thn For all ( ) an

16 Hrschl cosn conton for Prfct Imagng of longtunal obcts ( ) ( ) 0 0 Hrschl cosn conton: cosθ cosθ H(rr )

17 Prfct magng n 3D Abb sn conton an Hrschl cosn conton hav to b smultanousl hol wth th sam scalng factor ρ snθ ρ snθ or ρ ρ snθ snθ an an cosθ cosθ cosθ cosθ For prfct magng: ρ ρ or sn θ cosθ snθ cosθ

18 Prfct magng n 3D: Mawll s Thorm sn θ cosθ snθ cosθ tanθ tanθ For nonro cosθ ρ P H(rr ) P ρ Thrfor for -/ < θ < / θ θ an M ρ ρ n n

19 Prfct magng n 3D: Mawll s Thorm M ρ ρ n n Mawll s Thorm Prfct magng s onl possbl for th magnfcaton of n/n or for sam n an n magnfcaton of.

20 Contmporar Optcal Imagng Contmporar optcal magng Optcal magng optcal sstm sgnal procssng 3D Imagng 3D prfct magng from (gtal) sgnal procssng Computatonal magng pr- or postprocssng

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