SAR FOCUSING: SCALED INVERSE FOURIER TRANSFORMATION

Transcription

1 Univesität Siegen Poject Secto - Optiml Signl Pocessing - Senso Dt Fusion, Remote Sensing - SAR ZESS SAR FOCUSING: SCALED INVERSE FOURIER TRANSFORMATION AND CHIRP SCALING O. Loeld, A. Hein Univesity o Siegen Institut ü Nchichtenvebeitung Pul-Bontz-Stße 9-11, 5768 Siegen, Gemny Emil: Hein@wiene.zess.uni-siegen.de, Pge 1

2 Univesität Siegen Poject Secto - Optiml Signl Pocessing - Senso Dt Fusion, Remote Sensing - SAR ZESS OVERVIEW Intoduction into the Scled Invese Fouie Tnsomtion SIFT) Pocessing Show eltionship to Chip Scling lgoithms Pesent some esults nd implementtion spects Conclusions, Pge

3 Univesität Siegen Poject Secto - Optiml Signl Pocessing - Senso Dt Fusion, Remote Sensing - SAR ZESS INTRODUCTION SAR Pocessing cn be cied out by ilteing the w dt in the twodimensionl equency domin. All ecent developments, such s the poposed Scled Invese Fouie Tnsomtion ppoch nd the well known Chip Scling techniques somehow exploit the time o equency scling popeties o chips in sequence o multiplictions nd convolutions. The ppe shotly pesents the bsic ides nd points out the common popeties o the outcoming lgoithms., Pge 3

4 Univesität Siegen Poject Secto - Optiml Signl Pocessing - Senso Dt Fusion, Remote Sensing - SAR ZESS RAW DATA SPECTRUM The tnsmitted signl: The Point Tget Response in the time domin: s T T t) = t,, ect,r t T ) = σ s exp { } j π k t R, ) T t t,,r) ) { jπ t,,r )} exp ect c Rnge migtion: t,, R ) R c Phse histoy: φ,, R ) = π t,, R ), Pge 4

5 Univesität Siegen Poject Secto - Optiml Signl Pocessing - Senso Dt Fusion, Remote Sensing - SAR ZESS The POINT TARGET RESPONSE in the equency domin is evluted by two-dimensionl Fouie Tnsomtion: ) { } R,,, R ) = I I t,,, R T t T R T,,,R ) = S T exp ) σ R, ) ect Bz cr + ) { j π } exp j4π + ) t c R c 1 v + ) c 4 v e c 4v jπ /4 3/ 4, Pge 5

6 Univesität Siegen Poject Secto - Optiml Signl Pocessing - Senso Dt Fusion, Remote Sensing - SAR ZESS The COMPLETE SCENE SPECTRUM is obtined s the two-dimensionl integl ove ll point tget spect:, ),,, ) RT = RT R d dr Solving the integl nd substituting R = Rm + Rm we get: with: S T ) R T, ) = S T σ ) ect c B z c + ) c + ), 4v c 1 v + ) e c 4 v jπ / 4 3/ 4 : equivlent lowpss spectum o the tnsmitted signl R m, Pge 6

7 c Univesität Siegen Poject Secto - Optiml Signl Pocessing - Senso Dt Fusion, Remote Sensing - SAR ZESS : nge equency : zimuth equency : cie equency : Dopple centoid equency σr, ) : bightness o point tget t R, ) c : velocity o light v : eective senso velocity B z : zimuth bndwidth R m : midswth nge : nge vition ound midswth, Pge 7

8 Univesität Siegen Poject Secto - Optiml Signl Pocessing - Senso Dt Fusion, Remote Sensing - SAR ZESS The DEFORMED SCENE SPECTRUM contins the complete inomtion depending on the nge- nd the zimuth-equency nd descibes the nge migtion by the coupling between the equencies. σ c c + ), = σ, ), ) 4v with:, ) = + ) c 4v c Ate mplitude compenstion nd nge compession, the bsic ocusing poblem is to tnsom this nonlinely equency scled two-dimensionl, spectum into the nge- nd zimuth - time domin by escling the deomtion., Pge 8

9 Univesität Siegen Poject Secto - Optiml Signl Pocessing - Senso Dt Fusion, Remote Sensing - SAR ZESS TWO BASIC PROBLEMS: Descibe the spectum in escleble om ppoximtion). σ c, ), ) = σ + ), = σ ) + b ), ) c 4v! Ate ppoximting the spectum we dditionly hve to tnsom this spectum bck into the time domin by escling it scle tnsomtion). σ! 1 1, ) = F F { σ ) + b ), )} t sclet, Pge 9

10 Univesität Siegen Poject Secto - Optiml Signl Pocessing - Senso Dt Fusion, Remote Sensing - SAR ZESS APPROXIMATION The deomed scene spectum σ,), ) tem: ) ) contins nonline equency c = + c 4v, This tem cn be modiied to the expected om esulting in:,) ) + b ) with the scling ctos: ) = c 1 1 ) 4v λ b = 1 c 4v λ, Pge 1

11 Univesität Siegen Poject Secto - Optiml Signl Pocessing - Senso Dt Fusion, Remote Sensing - SAR ZESS SCALED INVERSE FOURIER TRANSFORMATION The scled invese Fouie tnsomtion is given by: st) = S ) o jπ t ) e d S ISFT The ollowing sequence o chip multipliction, chip convolution nd chip multipliction chieves the desied scled invese Fouie tnsom: S ) exp exp ) j π ) jπ exp 1 s ) j π ) Figue 1: Scled Invese Fouie Tnsom elized with chips, Pge 11

12 Univesität Siegen Poject Secto - Optiml Signl Pocessing - Senso Dt Fusion, Remote Sensing - SAR ZESS SCALED INVERSE FOURIER TRANSFORMATION IDEA 1 jπ s t) = S ) e t s d with: e jπt = e jπt 1 1 jπt jπ jπ ) ) t t = e S e e ) s t ) jπ jπ e d ) jπt jπ ) jπ t = e S e e e S ) exp j π ) 1 s t) exp ) j π exp ) j π t, Pge 1

13 Univesität Siegen Poject Secto - Optiml Signl Pocessing - Senso Dt Fusion, Remote Sensing - SAR ZESS IMPLEMENTATION, ) S m e m jπ N Rw Dt T t, ) D- FFT m ) Hk1 T,, R IFFT H k, ) FFT s IFFT m, ) e e m jπ N m jπ N Figue : Block digm o ISFFT H ISFFT k 3, ) Focused Scene 1D- IFFT σ, ) + Figue 3: Complete pocessing digm o the SIFT Pocesso H kt, ) nd H k3t, ) e line phso unctions to ocus the scene into the slnt nge, zimuth dopple domin nd e evluted in the ppe., Pge 13

14 Univesität Siegen Poject Secto - Optiml Signl Pocessing - Senso Dt Fusion, Remote Sensing - SAR ZESS RELATIONSHIP TO CHIRP SCALING ALGORITHM Stting with the deomed scene spectum, skipping the nge compession nd mplitude coection we get: σ c c + ), = σ, ), ) = σ ) + b ), ) 4v! We simpliy: σ! σ [ ) + b ) ), ] = S + b ) the second coodinte hs been neglected o convenience, Pge 14

15 Univesität Siegen Poject Secto - Optiml Signl Pocessing - Senso Dt Fusion, Remote Sensing - SAR ZESS Tnsoming the SIMPLIFIED spectum bck into the RANGE DOMAIN F 1 1 { Sσ + b ) } = sσ exp jπb = n 1 Now the gol is to ind scling technique tht tnsoms: Time scling n n) 1 scling Fequency scling N scling ) N ) Hving ound such scling device we cn esily obtin the desied imge by: [ jπ b ] sσ) = n) exp, Pge 15

16 Univesität Siegen Poject Secto - Optiml Signl Pocessing - Senso Dt Fusion, Remote Sensing - SAR ZESS Applying FREQUENCY SCALING PROCEDURE with chips we cn emove this scling. N ) exp j α ) exp j γ ) N ) exp ) jβ ) exp j ε α π Figue 4: Fequency scling with chips due to Ppoulis) The ollowing undmentl eltionships between the chip tes must be obeyed 3 equtions o 4 unknowns): γ α = 1 γ + α = β ε = γ β α, Pge 16

17 Univesität Siegen Poject Secto - Optiml Signl Pocessing - Senso Dt Fusion, Remote Sensing - SAR ZESS We cn lso shit the lst multipliction om the output to the input. α [ ] Unshited Chip: N ) = N ) exp jε α [ ] Modiied Shited Chip: N ) = N ) exp jε π π N ) α ) exp j exp j γ ) ) N α π ε ) exp j j β ) exp Figue 5: Modiied equency scling with chips, Pge 17

18 Univesität Siegen Poject Secto - Optiml Signl Pocessing - Senso Dt Fusion, Remote Sensing - SAR ZESS INVERSE TRANSFOMATION Modiied equency scling N ) α ) exp j exp j γ ) ) N IFT α π ε ) exp j j β ) exp Modiied time scling n exp j ε ) j β exp n ) exp j α exp j γ, Pge 18

19 Univesität Siegen Poject Secto - Optiml Signl Pocessing - Senso Dt Fusion, Remote Sensing - SAR ZESS CHIRP SCALING As beoe mentioned since we hve 3 equtions o 4 unknowns, one chip te cn be eely choosen so, the ist chip te is set equl with the nge chip te π k! = ε The emining chip tes e esily identiied by using α = π k [ 1 ] 1, β = π k, γ π = k 1 [ 1 ] n exp jπk Rw dt in Rnge-Dopple Domin exp Figue 6: Chip Scling ) exp ) jπk ) ) [ ] ) k [ ] jπk 1 exp jπk 1 exp jπk b ) s σ, Pge 19

20 Univesität Siegen Poject Secto - Optiml Signl Pocessing - Senso Dt Fusion, Remote Sensing - SAR ZESS SUMMARIZING Stting with the deomed scene spectum, we hve the ollowing options to exploit the scling popeties o chips to SAR ocusing: 1. Scled Invese Fouie Tnsomtion ou ppoch to SAR pocessing). Fequency Scling 3. Time scling known s Chip Scling Chip Scling is the most eicient one, since one FFT cn be sved in compison with the SIFT ppoch. This computtionl dvntge, howeve, is pid o by the equiement o line equency modulted nge signls., Pge

21 Univesität Siegen Poject Secto - Optiml Signl Pocessing - Senso Dt Fusion, Remote Sensing - SAR ZESS RESULTS Pt o Flevolnd) On DEC Pesonl Woksttion 5 equipped with 768 MBytes o coe memoy, Windows-NT) the esults wee identicl. The pocessing o 496 zimuth) x 56 nge) scene took: 13 min o the Chip Scling- nd 18 min o the SIFT-ppoch, Pge 1

22 Univesität Siegen Poject Secto - Optiml Signl Pocessing - Senso Dt Fusion, Remote Sensing - SAR ZESS CONCLUSIONS Two ppoches to eiciently compess SAR dt hve been pesented nd comped. The lgoithms exploit the scling popeties o chips in dieent wys but with the sme esults. It hs been shotly shown tht the SIFT-pocesso nd chip scling ppoches hve common bckgound. As opposed to chip scling, the slightly moe demnding SIFT pocesso does neithe equie line FM chip o the tnsmitted signl no does the sequence o nge nd zimuth compession hve to be modiied., Pge

23 Univesität Siegen Poject Secto - Optiml Signl Pocessing - Senso Dt Fusion, Remote Sensing - SAR ZESS APPROXIMATION ERROR The investigtions hee wee cied out with ERS-1 pmetes B MHz, B z khz) Rel. ppox. eo o DC = 3 Hz. Mx.: 4.17E-11 Rel. ppox. eo o DC = 1 khz. Mx.: 3.55E-7 APPROXIMATION ERROR SECONDARY RANGE COMPRESSION, Pge 3

24 Univesität Siegen Poject Secto - Optiml Signl Pocessing - Senso Dt Fusion, Remote Sensing - SAR ZESS A compison between the ppoximtion nd the exct tnse unction: c, ) = + ) ) + b ) shows tht the SIFT lgoithm intoduces two-dimensionl phse betion o like the nge dopple nd the chip scling pocesso): Φ e,r ) = 4v c πr c 3 c v 1 4v which hs stong dependence on dopple, but only wek dependence on nge R. 3/, Pge 4

25 Univesität Siegen Poject Secto - Optiml Signl Pocessing - Senso Dt Fusion, Remote Sensing - SAR ZESS APPROXIMATION ERROR SECONDARY RANGE COMPRESSION It cn be shown esulting eective FM chip te: k e,r ) = 1 c R k 1 wich chieves when mtched in the pocesso the so clled secondy nge compession SRC. The SIFT pocesso evlutes the SRC duing the nge compession. k 4v 4v λ λ 3/, Pge 5