OPTICAL TRANSFORMS IN DIGITAL HOLOGRAPHY

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1 L. Yoslvsy OPICAL RASFORMS I DIGIAL HOLOGRAPHY Holo-5, Vn, My -5, 5

he poblem o mtl coesponence between optcl tnsomtons n the compte epesenttons s esse n eent compte epesenttons o bsc optcl tnsoms sch s convolton, Foe n Fesnel ntegl tnsoms e bely evewe.

2 he poblem o mtl coesponence between optcl tnsomtons n the compte epesenttons s esse n eent compte epesenttons o bsc optcl tnsoms sch s convolton, Foe n Fesnel ntegl tnsoms e bely evewe. Dect mgng: convolton ntegl ~, ξ η (, y) ( ξ, y η) h( ξ η) nsom mgng: Integl Foe nsom () F Inpt plne () Z F Otpt plne y y y λf (, y ) (, y) nsom mgng: Integl Fesnel nsom (, ) y (, y) ( ) ( y) λz y y

3 he conomty n mtl coesponence pncples between nloge n gtl sgnl tnsomtons Inpt gtl sgnl Otpt gtl sgnl Dscetzton n qntzton Dgtl sgnl tnsomtons n compte Reconstcton o the contnos sgnl Eqvlent contnos tnsomton Contnos npt sgnl (mge o hologm) Contnos otpt sgnl (otpt mge o compte genete hologm) he conomty pncple eqes tht gtl epesentton o sgnl tnsomtons shol pllel tht o sgnls. Mtl coesponence between contnos n gtl tnsomtons s s to hol both ct to tnsom entcl npt sgnls nto entcl otpt sgnls. Accong to these pncples, gtl pocessos ncopote nto optcl nomton systems shol be ege n tete long wth sgnl gtzton n sgnl econstcton evces s ntegte nlogos nts

4 Mthemtcl omlton o sgnl scetzton n econstcton Let ϕ ( ) ( ) be contnos sgnl s ncton o sptl co-ontes gven by vecto ( ) ( ) ( ) ϕ ( Δ) n ϕ ( ) ( ) ( ) ϕ ( Δ) ( ) ( ), () e scetzton n econstcton bss nctons ene n the scetzton n econstcton ( ) ( ) evces coontes { } n { }, espectvely, Δ s vecto o the smplng ntevls, s vecto o sgnl smple nces. At the sgnl scetzton, sgnl smples { } e compte s ( ) ( ( ) () ϕ Δ) ssmng cetn eltonshp between sgnl n smplng evce coonte systems {} n ( { ) }. Sgnl econstcton om the set o the smples { } s escbe s ( ) ~ ϕ Δ (3) o the sgnl econstcton om ts scete epesentton, t cn seve s It s nestoo tht lthogh eslt ~ obtne ccong to Eq. 3 s not, n genel, entcl to the ntl sgnl sbsttte o the ntl sgnl n the gven pplcton. Accong to the conomty pncple, Eqs. n 3 om the bse o eqte scete epesentton o sgnl tnsomtons.

5 DISCREE REPRESEAIO OF HE COVOLUIO IEGRAL b ( ξ ) h( ξ ) ξ h( ξ ) ( ξ ) ξ (4) b Dscetzton bss nctons [ ( ) Δ] [ ( ) Δ] ( ) ( ) ( ) ϕ ϕ ( ) ( ) ϕ ϕ Sgnl econstcton bss nctons ( ) ( ) whee n e shts, n ctons o the scetzton ntevl, o smple postons wth espect to npt n otpt sgnl coonte systems, coesponngly. n [ ] Sgnls belong to the clss: ( ) ( ) ϕ n Δ ( ) ( ) n n [ ( n ) Δ] ϕ Fo sch sgnls, ( ) ( ) Smples o b [ ] b X [ ξ ( ) Δ] h( ξ ) ξ h ξ ξ ϕ ξ ( ) ( ~ ϕ ) Δ ( h ξ ϕ ) ( ) X whee ( ) ( ) h h ξ mδ Δ ϕ ϕ ξ ξ m e: Fo gtl lteng, ths eqton s eplce by n [ ] ( ) ( ( Δ ϕ [ ξ n ) ) Δ] ξ n n h n n h b h n (5) n n [ ] Eqton (4) epesents s the cnoncl eqton o sgnl omn gtl lteng. Eqton (5) enes how scete pont spe ncton o gtl lte { h m } cn be on tht coespons to gven convolton pont spe ncton h().. (6)

6 ~ b DISCREE REPRESEAIO OF COVOLUIO IEGRAL: contnos PSF o gtl lte Accong to the mtl coesponence pncple, gven pont spe ncton o gtl lte h n, pont spe ncton o n eqvlent contnos lte cn be on s ollowng. b ( ) ~ b h ( ( ) ) ~ bϕ Δ hn n ϕ ( Δ) n h hn n ~ n~ Δ ξ h ~ ξ h ϕ Δ ϕ ( ) [ ξ ] Contnos npt sgnl n n Sgnl scetzton Inpt gtl sgnl Dgtl lte Eqvlent contnos lte b ~ econstcte heq, om ξ ts smples hnϕ { Δ} ~ [ ( ~ ) ] ( ~ ) ξ ϕ ξ n Δ ξ ϕ Δ whee n ~ n Otpt gtl sgnl ( ) ( h ) ~ ( b ) ( ) ~ ϕ [ ξ ( n~ ) Δ] n n Reconstcton o the contnos sgnl ϕ. { h ϕ n } (). () ~ { } ( ) Contnos otpt sgnl

7 DISCREE REPRESEAIO OF COVOLUIO IEGRAL: contnos MF o the gtl lte Feqency esponse (MF) o the gtl lte H (, p) h (, ξ ) [ ( pξ )] ξ h b h n h ϕ n eq eq ~ ( ) ~ Δ ϕ [ ξ ( n~ ) Δ] ( pξ ) [ ] n h n ξ ~ [ Δ] ( ~ ( ) ) ( pnδ ϕ ϕ ) ( ξ ) ( pξ ) ξ ( p) whee SV DFR H (, p) ( ) (, p DFR p Φ ( ) Φ ) ( p) SV ( p) eq, h h ( ~ p hn pnδ) hn [ p( n ) Δ] n n ( ) ( ( ) ϕ ) ( ) Φ ( ) ( ( p) ϕ ) ( p) Φ [ ( p) Δ] [ ( p) Δ] sn snc, sn [ ( p) Δ] Dscete eqency esponse (DFR) o the gtl lte Feqency esponse o the sgnl econstcton evce Feqency esponse o the sgnl smplng evce em esponsble o lte spce vnce

8 DISCREE REPRESEAIO OF COVOLUIO IEGRAL: contnos MF o the gtl lte ( ) ( ( ) ϕ ) ( ) Φ p ( ) ( Φ ( p) ϕ ) ( p) Bse bn DFR(p) h hn n SV(-p) snc [ p( n ) Δ] [, ( p) Δ] DFR(p) n the bse bn [-/Δ, /Δ ]

9 DISCREE REPRESEAIO OF COVOLUIO IEGRAL: contnos MF o the gtl lte (ctn) Flte spce vnce s ssocte wth nteness o the nmbe o sgnl smples: lm SV (, p) lm snc[, ( p) Δ] δ ( p) heoem. DF coecents o the gtl lte mplse esponse e smples o ts Dscete Feqency Response heoem DFR h h ( p) hn [ pnδ] whee n n η snc h n η hn n h h ; pδ h DFR o the gtl lte s scete snc-ntepolte ncton o ts smples Contnos eqency esponse o the gtl lte wth PSF tht comptes sgnl locl men n the wnow o 5/64 o sgnl sze { η } Flte bse bn /Δ

In stncton to the snc-ncton, scete snc-ncton s peocl ncton wth peo Δ o Δ epenng on whethe s o o even nmbe.

10 Dscete snc-ncton (, Δ) snc sn sn ( Δ) ( Δ) Dscete snc-ncton s scete nlog o the contnos smplng snc-ncton, whch s pont spe ncton o the el low-pss lte. In stncton to the snc-ncton, scete snc-ncton s peocl ncton wth peo Δ o Δ epenng on whethe s o o even nmbe. Its Foe spectm s smple veson o the eqency esponse o the el low pss lte s n o nmbe Δ s n even nmbe Δ Contnos (e ots) n scete (ble lne) snc-nctons o o n even nmbe o smples Feqency esponse o the el low pss lte (e) n Foe tnsom o the scete snc-ncton (ble)

11 Dscete Repesentton o Integl Foe nsom: ( ) Contnos smple sgnl ϕ [ ( ) Δ] Sgnl spectm smples () ( ) v Δ Δ [ ( )( v) ΔΔ ] Contnos sgnl spectm ϕ [ ( ) Δ] ( ) ( ) ϕ ( ) Φ ( ) φ [ ( v) Δ ] [ ( ) Δ] Φ ( ) s eqency esponse o sgnl econstcton evce [ ( ) Δ] Φ ( ) φ [ ( v) Δ ] [ ( v) Δ ] φ ( ) [ ( ) Δ] Φ Secon tem [ ( )( v) ] s sege Δ Δ (7)

12 Dscete Repesentton o Integl Foe nsom: DF, Shte DF, DC n DcS Cnl smplng: Δ Δ, no smplng g shts (, v ) Dscete Foe nsom (DF) Cnl smplng: Δ Δ, smplng g shts, v Shte Dscete Foe nsom (SDF(,v)) Res. : DF plys nmentl ole n gtl hologphy thns to the vlblty o Fst Foe nsom (FF) lgothm., v v Usng SDFs, one cn cy ot contnos spectm nlyss wth sb-pel esolton n bty sgnl e-smplng wth el scete-snc-ntepolton Res. Impotnt specl cses o Shte DFs e Dscete Cosne (DC) n Dscete cosne-sne (DcS) nsoms cos DC DcS sn () DC n DcS e SDF(/,) o sgnls tht ehbt even n, coesponngly, o symmety ({ ± ). hey hve st compttonl lgothms tht belong to the mly o st Foe } nsom lgothms. Usng st DC n DCs lgothms, one cn ecently mplement st bony eect ee gtl convolton.. L.P. Yoslvsy, Shte Dscete Foe nsoms, In: Dgtl Sgnl Pocessng, E. by V. Cppelln, n A. G. Constntnes, Avemc Pess, Lonon, 98, p (8) (9)

13 Dscete Repesentton o Integl Foe nsom: Shte n Scle DF Smplng n -scle coontes: Δ Δ, no smplng g shts (, v ): Scle Dscete Foe nsom (ScDF; t s lso nown ne nmes chp-tnsom n Fctonl Foe nsom Re.-4 ):, ; v Fo compttonl pposes, t s convenent to ess ScDF v cnoncl DF tht cn be compte sng FF lgothms ( enotes element-wse, o Hm poct o vectos). () IDF DF DF hs lgothm enbles sgnl e-smplng, n bty scle (sb-smplng n p-smplng), wth el scete snc-ntepolton Smplng n -scle coontes: Δ Δ, smplng g shts (, v ) () Shte Scle Dscete Foe nsom (ShScDF,):, v v (3). Rbne L.R., Sche R.W., Re C.M., he chp z-tnsom lgothm n ts pplctons, Bell System ech. J., 969, v. 48, Rbne L. R., Gol B., heoy n pplctons o gtl sgnl pocessng, Pentce Hll, Englewoo Cls,.J., Bley D. H., Swtztbe P.., he ctonl Foe nsom n pplctons, SIAM Rev., 99, v. 33, 97-3

14 Pont spe ncton PSF (,) o the scete Foe nlyss v v,, [ ] v Δ ϕ [ ] v Δ ϕ [ ] v Δ ϕ [ ],,, v DFA v PSF Δ ϕ PSF v DFA v,,,,, PSF (,) lns sgnl spectm n ts smples obtne by sgnl DFs o ts smples: { } v, It cn be on s: (4) (5)

15 [ ], v PSF Δ ϕ v v Δ Φ Δ ; snc Pont spe ncton o scete Foe nlyss (ctn) PSF Φ Δ Δ ; snc, v Δ Δ Sgnl scetzton Inpt gtl sgnl Contnos npt sgnl Dscete Foe nsoms (ScShDFs) Dscete Foe nsome Inpt sgnl spectm smples { } v,, ϕ Φ

16 Resolvng powe o scete spectm nlyss 5-tmes zoome spectm o snsol sgnl wth eqency 9 5-tmes zoome spectm o snsol sgnl wth eqency tmes zoome spectm o snsol sgnl wth eqency tmes zoome spectm o snsol sgnl wth eqency tmes zoome spectm o two bove snsol sgnls tmes zoome spectm o two bove snsol sgnls Resolvng spect o two snsol sgnls wth close eqences (9 n 3, (let) n 9 n 3.5 (ght) nts)

17 PSF OF UMERICAL RECOSRUCIO OF HOLOGRAMS RECORDED I FAR DIFFRACIO ZOE Fo hologm smplng evce wth eqency esponse Φ(.), pont spe ncton o nmecl econstcton o Foe hologms s obtne s: PSF FZ λz (, ) Φ snc[, ( Δ) Δ] whee λ - wve length, Z - object-to-hologm stnce; - nmbe o hologm smples, Δ λz SH λz Δ, Δ - hologm smplng ntevl (6) he pont spe ncton s peocl ncton o : PSF FZ g ( ) FZ g PSF ; (7) (g s ntege). It genetes smples o object wveont mse by the eqency esponse o the hologm econg n smplng evce, the smples beng ten wth scetzton ntevl wthn the object sze S o λz/ Δ. Δ/ λz/ S H λz/ Δ he cse coespons to cnl econstcte object wveont smple wth scetzton ntevl Δ λz/ S H λz/ Δ. When >, econstcte scete wveont s -tmes ove-smple, o -tmes zoome-n. One cn show tht n ths cse the econstcte object wveont s scete snc-ntepolte veson o the cnl one.

18 Dscete Repesentton o -D Integl Foe nsom: -D Sepble, Rotte n Ane DFs Sepble cnl smplng: Δ,, wth no shts n coonte systems tht Δ Δ Δ conse wth those o sgnl n ts -D spectm ls Sepble -D Dscete Foe nsom, s, l (8) (-D DF) l A B~ Smplng n coonte system ne tnsome wth espect to tht o the sgnl y C D ~ y Ane Dscete Foe nsom (ADF), s whee A / AΔ~ Δ ; B / BΔ~ yδ ; C / CΔ~ Δ ; y D / DΔ~ yδ ; y Δ ~ ; Δ ~ y - sgnl smplng ntevls; Δ, Δ - sgnl spectm smplng ntevls y s, l l A C B D l sl (9) Smplng, wth eql smplng ntevls n coonte system otte wth espect to tht o sgnl thogh ngle θ cosθ y snθ snθ ~ cosθ ~ y Rotte Dscete Foe nsom (RotDF) sl s l, s, l cosθ snθ l ()

Ognl mge Avlblty o sht n scle n otton ngle

mge sclng, otton n genel e-smplng wth el scete

5 75 o -otte mge wth bcbc ntepolton (let),

Compson o mge otton sng bcbc splne (top) n

19 Ognl mge Avlblty o sht n scle n otton ngle pmetes n SDF, ScDF n RotDF enbles st lgothms o mge sclng, otton n genel e-smplng wth el scete snc-ntepolton Re o -otte mge wth bcbc ntepolton (let), otton eo(mle) otton eo spectm ght) Bse bn Compson o mge otton sng bcbc splne (top) n scete snc-ntepolton 75 o -otte mge wth sncntepolton (let), otton eo(mle) otton eo spectm ght) Bse bn 5. L. Yoslvsy, Dgtl Hologphy n Dgtl Imge Pocessng, Klwe Acemc Pbl., Boston, 4

20 Dscete Repesentton o Integl Fesnel ~ ~ nsom ( ~ ) ~ ( ) [ ( ) ] λz Smplng sgnl tnsom wth smplng g sht vδ ( ) φ [ ( v) Δ ] Inpt sgnl smple epesentton wth smplng g sht ϕ [ ( ) Δ] Δ [ ( Δ Δ Δ vδ ) ] he lst two tems escbe contbton o sgnl n tnsom smplng evces. In the ssmpton tht PSFs o smplng n econstcton evces e elt-nctons they cn be gnoe Fo scete epesentton o Fesnel [ ( ) ϕ ] ( Δ Δ Δ vδ ) [ ] [ ( ) φ ( )] ( Δ Δ Δ vδ ) [ ] [ ] ( ) ntegl, only ths tem s se: Δ Δ Δ vδ

21 Dscete Repesentton o Integl Fesnel nsom: Dscete Fesnel nsoms Cnl smplng: Δ λz Δ, wth no shts, n coonte systems collne wth those o sgnl n ts tnsom Cnoncl Dscete Fesnel μ / μ () nsom (DF): μ λz Δ DF cn be esse v DF n compte sng FF: μ μ () Smplng n -scle coontes : Δ λz Δ, wth shts Δ, vδ n coonte systems collne wth those o sgnl n ts tnsom Shte Scle Dscete Fesnel nsom (ShScDF): μ λz Δ ( μ / μ w) (3) w μ v / μ Smplng n -scle coontes : Δ λz Δ, wth shts Δ, vδ n coonte systems collne wth those o sgnl n ts tnsom; chp-ncton n the tnsom s gnoe Shte Scle Ptl Dscete Fesnel nsom (ShScPDF): μ ( wμ) (4)

Dscete Fesnel nsoms, ctn: Cnl smplng: Δ λz Δ, wth shts w μ v

] Focl plne nvnt Dscete μ / μ Fesnel nsom (FPIDF): (4) μ λz

DF mecl econstcton o mges on eent stnces om hologm sng Focl

22 Dscete Fesnel nsoms, ctn: Cnl smplng: Δ λz Δ, wth shts w μ v / μ μ, n coonte systems collne wth those o sgnl n ts tnsom [ ] Focl plne nvnt Dscete μ / μ Fesnel nsom (FPIDF): (4) μ λz Δ mecl econstcton o mges on eent stnces om hologm sng cnonc DF mecl econstcton o mges on eent stnces om hologm sng Focl plne nvnt DF Imges e estoe om hologm cope om PDF le o the ppe: E.Cche, P. Mqet, Ch. Depesnge, Sptl lteng o zeo oe n twn-mge elmnton n gtl o-s hologphy, Appl. Opt., v. 3, o. 3, Ag.

23 Invetbltyo Dscete Fesnel nsoms n nc-ncton { } I one comptes, o smple sgnl,,,,...,, ect Shte DF wth epth n sht pmetes ( μ, w ) n then nvets t wth nvese Shte DF wth epth n sht pmetes ( μ, ), one obtns w ± ± ( ) μ, w μ w ( nμ w ) Whee q μ μ, w± w μ w μ n n nc ± n ( ; q; n w q ) nc q ( ; q; ) (5) s nc-ncton, n nlog o snc-ncton o the DF, entcl to t when. In nmecl econstcton o hologms, nc-ncton s convolton enel tht lns object n ts ot o ocs econstcton.

24 Dscete nc-ncton n ts ocl plne nvnt veson: epenence on ocsng pmete q nc ; q; ( q ) bs(nc) q nc ( ; q; ) q ( ) q.5.4 q..5.4 q..3 q.8.3 q... q.8 q.3 q.5.. q.4 q.8 q

Fnc-ncton: ppomtons nc As t ws shown n Re. YoChn, ( ; q; ) ect q q q Mgnte Fnc(48;.5;s-Vq):Mgnte ( ) Fnc(56,q,) o q:.:.56.8 5.6.4. q - - 4 6 8 4 6 8 z 3 Phse Fnc(48;.

25 Fnc-ncton: ppomtons nc As t ws shown n Re. YoChn, ( ; q; ) ect q q q Mgnte Fnc(48;.5;s-Vq):Mgnte ( ) Fnc(56,q,) o q:.: q z 3 Phse Fnc(48;.5;s-Vq):Phse z Rto o the let n ght pts o the eqton Fo ntege, nc ( ;; ) *q 5 5 he vle o the ocsng pmete q s the theshol te whch lsng begns In nmecl econstcton o hologms, qλz/δ 5 5 5

26 Dscete epesentton o ntegl Fesnel nsom: Convoltonl Dscete Fesnel nsom Integl Fesnel nsom cn lso be ege s convolton n epesente thogh two Foe nsoms: Δy p p p μ s [ ] Assmng tht smplng ntevls o sgnl Δ n ts tnsom Δ e entcl: w μ s s nc ( ; μ ; w ) Smlly to the bove scete Foe n scete Fesnel tnsoms ConvDF s n othogonl tnsom wth nvese ConvDF ene s s w μ s s nc p ( ; μ ; w ) When μ, ConvDF egenetes nto n entcl tnsom. When μ, t s entcl to the cnoncl DF. Althogh ConvDF cn be nvete o ny μ, n nmecl econstcton o hologms t cn be pple only o μ. I μ >, lsng my ppe n om o ovelppng peocl copes o the econstcton eslt. (6) (7)

27 PSF o econstcton o hologms ecoe n ne cton zone: Foe econstcton lgothm In gtl hologphy, DF s se, ne nme o Foe econstcton lgothm, o nmecl econstcton o optcl hologms ecoe n ne cton zone. hs pocess cn be chcteze by ts pont spe ncton (PSF) tht lns object wve ont n object smples obtne om smples o ts hologm n the nmecl econstcton. Smlly to the bove scsse cse o DF, PSF o nmecl econstcton o hologms by DF epens on pmetes o the lgothm n o PSF o the hologm smplng evce. Genel omls e pesente n Re. 5. Hee we, s n llstton, pove PSF o the econstcton pocess o the cse when pont spe ncton o the hologm smplng evce cn be ege s elt-nctons. In ths cse o n ocs econstcton ( / Δ) Z PSF, n ocs econstcton PSFFo (, ; μ) μo snc[ ; ( Δ) Δ] (8) PSF, ot o ocs econstcton: [ ] [ ] Z μ o / Δ / μ / PSF, ; μ nc; ;( Δ) Δ μ μ whee μ s ocsng pmete o the econstcton lgothm n μ s ts vle tht coespons to the n ocs econstcton. (9) As one cn see lsng ee object sze s eql to the peo S o Δ. Gven sze o the hologm S h Δ, the peo s S o μ S H. heeoe lsng ee object econstcton sng Foe econstcton lgothm s possble μ λz Δ Othewse the lgothm wos s mgnyng glss cpble o econstctng o smll -th cton o the o object om μ -th cton o the hologm pove the est o the hologm s zeoe. μ 5. L. Yoslvsy, F. Zhng, I. Ymgch, Pont spe nctons o gtl econstcton o gtlly ecoe hologms, In: Poceengs o SPIE Vol. 564, Inomton Optcs n Photoncs echnology, Gogng M, Fncs. Y, Sgn Jtml, Etos Jn 5;

Hologm econstcton: Foe lgothm vs Convolton

28 Hologm econstcton: Foe lgothm vs Convolton lgothm Foe econstcton Foe econstcton o the centl pt o the hologm ee o lsng Convolton econstcton Imge s estoye e to the lzng Z33mm; μ.439 Alsng tcts Z83mm; μ.668 All estotons e entcl Z36mm; μ Hologm cotesy D. J. Cmpos, UAB, Bcelon, Spn

29 L. Yoslvsy, Ph.D., D. Sc. Phys&Mth, Poesso Dept. o Intescplny Stes, Fclty o Engneeng, el Avv Unvesty, el Avv, Isel

Advanced image processing Lab

Advanced image processing Lab L. Yoslvsy Advnced mge pocessng Lb A Ttol Eopen Sgnl Pocessng Confeence EUSIPCO, Tmpee, Fnlnd Sept. 4, Leond Yoslvsy, Pofesso, Dept. of Intedscplny Stdes, Fclty of Engneeng, Tel Avv Unvesty, Tel Avv 69978