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Algorithms for Wavefront Reconstruction

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(1)Algorithms for Wavefront Reconstruction Stefan Hagdahl. To cite this version: Stefan Hagdahl. Algorithms for Wavefront Reconstruction. [Research Report] RR-5696, INRIA. 2005, pp.28. �inria-00070319�. HAL Id: inria-00070319 https://hal.inria.fr/inria-00070319 Submitted on 19 May 2006. HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés..

(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. Algorithms for Wavefront Reconstruction Stefan Hagdahl. N° 5696 September 2005. N 0249-6399. ISRN INRIA/RR--5696--FR+ENG. Thème NUM. apport de recherche.

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(140) PfhjUjœ`-YTiQZ†VAyR[` 4θ = O( λ) j. j. λ. . j. (3). . (3). P(φ) (θ).  (3)  P(φ),θ (θ)   P (3) (θ + 4θ/2)  (φ),θ (3) P(φ),θ (θ + 4θ). .     =   . ¡¨thnh£.  ¯ φ(θ)  φ¯,θ (θ) , ¯ φ,θ (θ + 4θ/2)  φ¯,θ (θ + 4θ). hf Vxi4 ˜CjfhV”jœ`hBDgAWAR[C i h ¡¨tbo-£ c ,c ,c ,c , Z†V„¡"U£" Z‹RT> κ = 4 © =?>xC‡†f-PSRRT>AY[CUCjœ`hBDg6`hVACUV^RTP?`hV#RT>ACY[Z†yh>^R>xfhVxi4P[Z]iQC`b’¡¢t-nb£"Z]Pjœ`-B gxWQR[Cki ZˆR[>4¡ Uu-£$ >ACUY[C' "C'WxP[C'RT>ACC OVA` ‡†CUiAyhC9`b’ β = ∇ φ ZˆV¤RT>AC˜EI%H  #—g1`-ZˆV^R

(141) P [θ, θ + 4θ/2, θ + 4θ] © LQZˆVxjCdR[>AC¤Z†V-RTCYTg1`-‡†fbR[Z†`hV4CYTY[`-Y ’›`-Y?f jœWxsAZ†jdg6`h‡†NOVA`hBDZ]fb‡8Z†P?`h’ R[>AC¤CYTYT`hY"Cœ¬QgAYTCUPTP/Cki­ZˆV Z†‡†‡%s6Ch‘6s^Nj

(142) >A`O`-P[Z†VAy 4θ = O(√λ) ‘6`b’ R[>xCPTfbBDCP/Z†ŒCfhP’›`-YO((4θ) R[>xCEGH ) #¥ CYTYT`hYZ†V”R[>AC žcG¥ jUfhP[λC ¡¢jb© ’S©%¡¢t h£?fbV1iw¡¨t £[£š‘AZ¨© Ch© ! √   ¡¨t b£ [ Y T Y h `  Y I E H %  # T Y [ Y ` ? Y ˆ Z ^ V [ R U C [ Y 6 g h ` ] ‡  f T R ˆ Z ` V O(( λ) ) max ( , ) = max O(λ), = O(λ). λ  VR[>ACDfb‡†yh`-Y[ZˆR[>AB­PGs1CU‡ˆ` l "CŸ>xf“-CWxP[CUiqjœ`h‡†‡†`OjUfRTZˆ`-V”ZˆVR[>ACR[>AYTCC EG% H &#¥ g1`-ZˆV^R

(143) PGRT`4P[CœRdR[>AC j`OC jœZ†CV^RTP?Z†V P ‘ P fbVxi P ©'=?>AC¤Z†V-RTCYTg1`-‡†fbR[Z†`hV#CYTY[`-Y RT>ACVs6CUj`hBDCUP  Y[YT`hYEGH  # ,  YTY[`-Y?ZˆV^R[CUY[g6`h‡]fRTZˆ`-V ) = max O(λ), O √λ  = O(λ), ¡¨tbv-£ max ( >xCV4fhgAgA‡†NOZˆVAyR[>xC β¥T‘ A¥9fhVxi ρ¥¨g6`h‡†NOVA`hB©9E`hR[CdR[>1fR? ˜C¤jœ`-WA‡†i4fb‡]P/`Djœ`-BDgAWQR[C β s^NiAZ‹§8CYTCV^R[Zˆ¥ fbR[Z†VAy P ‘Q >AZ†j

(144) > "`hWA‡]i4yhZ†“hCdWxPfbV#CYTYT`hY?`b’R[>ACPTfbBDCP/Z†ŒCf-P"ZˆV„¡¢thv-£œ© <Z†Vxfb‡†‡ˆNd "C< ?fbV^R$RT`iQZ†PTjœW1P[P8RT>AC f-P/NOBDgQR[`hR[Z]j9fbVxidVOWABDCYTZ]jfb‡CUY[YT`hY

(145) P6 ˜C'iA`ZˆVf?g6`hZ†V^R  e © =?>xCdZ]iQCkfD Z‹RT>P[CœR[R[Z†VAy­f­P/CR`b’9fhP[N^BDgQRT`bRTZ†j¤P[`h‡†WQR[Z†`hVxP u¯ P`hV4R[>xCjœZ†YTj‡ˆC γ Z]P?R[>xfbR ˜CdxRT>A∈CV>xf\“-C Σ Z†VAZˆR[Z]fb‡<jœ`hV1iQZ‹RTZˆ`-VxP?RT`­R[>ACP/NQP/R[CB `b’'=YTfhVxP/g6`hY[RCU\^WxfbR[Z†`hVxP?y-Zˆ“-CVZ†V¡ -£fbVxi„¡ £š©‰GCVxjCR[`­CkPSRTZ‹¥ B­fbR[CRT>AC CYTY[`-YTPGZˆV x ‘$ "CŸVACUCUi”RT`|Z†VO“hCUP/R[Z†y-fbR[CŸR[>AC CYTY[`-YgAYT`hgxfhy-fbR[Z†`hV”Z†VŠR[>xC­cd¦  ¥ P/`-‡ˆ“-CYRT>xfR "CWxP[Ch© =?>AZ]P Z]P P/R[Y

(146) fbZ†yh>^R9’›`-Y[ ?fbY

(147) i Z‹’$RT>ACGfhP[NOBDgQR[`hR[Z]jP/`-‡ˆWAR[Z†`hV­Z]P jœ`-V^R[Z†V^Wx`hWxP'fh‡ˆ`-VAyRT>ACGjœZ†Y

(148) jœ‡†Ch‘^Z¢© Ch© j`hVxP[Z]PSR¤`b’?fªA¬QCkiVOWABs1CUYd`h’˜gA>1fhP[CUPU‘$sAWQR ˜C s6C‡†ZˆCU“hCDZ‹RZ]Pf|Vx`hVQ¥¨R[YTZˆ“OZ]fb‡RTf-.P RT`iA`|RT>ACP[fhBDC Z†fhV#Vxfb’›WQ‡†NQR[P/WxZ]Y[P˜C’›`h "Y`hiQY DZ]PTjœR[`h`DV^‡†RT`OZˆjUVOfbWA‡†`-ZˆŒUWxChP?‘QP/ `-Zˆ‡ˆR[WQ>RTZˆ>A`-VxZ†yhPU>©<·wfhjUCjœWAjœY

(149) `-fhVxjjœNh‡†‘^WxR[iQ>AC-C‘QjU’›`hfbY?WxP/RTR[>AZ]Z]jdPg1Y[Ck`-fhZˆV^P[R

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(151) ku.   

(152)    . CS γ R u. inc. ¯ Σ. Z†yhWxY[Ct K9_ Y[`-sA‡ˆCUBP[CœR[Wxg*©  <¤   <œ  #¤  # œŸ< š  #  Ÿš =`gAYTCUP[CV^RRT>ACqfh‡ˆy-`hYTZ‹RT>AB­PDR[>1fR4 "CŠWxP[CUi¯RT`—YTCUj`hVxP/R[YTWxjšR­RT>ACqf-P/NOBDgQRT`bR[Z]jŠP[`h‡†WQR[Z†`hV– "CŠ>ABDCUCY[C P/RTfbR[C¤RT>ACŸVOWABDCYTZ]jfb‡%gAY[`-sA‡†CBBD`-Y[C¤’›`-Y[B­fb‡†‡†Nh©"·qCYTCœ’›CUYR[` Z†yhWxY[CŸt­fhVxi# ˜CŸBDf hCfhV O(λ ) fhgAgAYT`¬QZˆB­fRTZˆ`-V`b’¡ o^£?fhP"’›`-‡ˆ‡†` P© x`hY˜fyhZ†“hCV­PTjfbR/R[CUY[CUY ‘OfbV­Z†Vxjœ`-B Z†VAy ?f“hC fhVxiDs6`hWAVxixfbYTNjœ`-VxiQZˆR[Z†`hVxP˜fhPTP/WxB CR[>xfbR ˜C OVA` •‰C‡†BD>A`h‡ˆR[ŒP[`h‡†WQRTZˆ`-V Σu Z†V4R[>ACjZˆY

(153) jœWA‡]fbYP/R[YTZ†ug   e ¡¨tbp-£ x αλ αλ ≤ |x| ≤ R + , 0 ≤ arcsin :R− CS := {x ∈ ≤ π}. 2π 2π |x| =?>xCV*‘^’›`hY?f¤ªA¬QCkiY

(154) fhiQZ†WxP R ‘ α fbVxi­“fhY[Z]fbsA‡†C θ = fbVAy-‡ˆC (x) ‘Q "CIP[CUfhYTj

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(156) fbVx, iAAfhYT,i4λ)Y

(157) fNDR[Y

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(159)    =?>xCDEI%H  #¥ fh‡ˆy-`hYTZ‹RT>AB Z†‡ˆ‡9s6C­BDCV^R[Z†`hVACkiZ†V„LOCUjœR[Z†`hV ©  >ACUY[CD "CD>xf“hCZˆVO“hCkPSRTZˆy^fRTCUif#VACU ’›CkfRTWAY[C¤`h’RT>AC¤`hgQRTZˆBDZ†ŒUfRTZˆ`-V”fb‡†yh`hYTZˆR[>AB fh‡ˆYTCUf-iQN­gAYTCUP[CV^RZˆV ' *¨© inc. 1. 2. j. m(θ). m(θ). j=1. 2. j. j. ikφj (θ). j. λ. j. j. e[0,j]. j=1. j. MN J2MPO.

(160) ".   

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(167) fbBDCR[CY

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(169) fN­R[Y

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(177) fbVxj

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(184) fbV1j

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(188) CfbRTV1Y[j

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(190) fhVAiQ`bZ†RTWxCUPi `hRT’'>xjœfWARkYT‘-“Zˆf’$R[R[Wx>AY[CUC Y[CGρ =>AC(4φ) `h’RT>AC ?f“hC’›Y[`-V-Rk‘QsON|WxP[ZˆVxy­‡ˆ`Qjfh‡*VA`-VQ¥¨‡†Z†VACUfhY`hgQRTZˆBDZ†ŒUfRTZˆ`-V”`hV ρ Z†VR[>AC’›`-‡ˆ‡†` Z†VAyfbVxfh‡ˆN^RTZ†jžcG¥ P[`h‡†WQRTZˆ`-V ‘ r ¡ -th£ ρ e , u ¯ (x, ρ) = u ¯ (x ) ρ + (|Q(x)| − ρ) >xCYTC ˜Cf-P[P[WABDCUi#f­jœZ†YTjWA‡]fbY ?f“hCPSR

(191) fbY[R[Z†VAy­fR x − ρ [cos β, sin β] fbVxi "C¤WxP/Cki#R[>ACŸVA`bR

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(193) > Z

(194) jZˆY

(195) j (r, δ, ρ)

(196)

(197)

(198) dδ, ¡ -£

(199) ¯ min

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(201) ½ ¼  U¹  ¸  k¹ š½ /¼

(202) ½ bºI» ½À ­¸  ½ ¹¨¾kº  š½ ¼ %º  ¼T¹ š ½ ˜¼

(203) ½À­¹¨¾º  ¢¸ ¹?¹ ¨¼

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(205) t.   

(206)    . >xCYTC

(207) −1/2 

(208)

(209)

(210) · F` (ρ; δ, β, λ, r) =

(211) ρr [cos (`∆δ), sin (`∆δ)] − [cos (β), sin (β)]

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(213) jœ‡†Cœ¥ Y

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(218) . 5 C0

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(220) PZ†V P fhVxi P ’›`hY#fb‡†‡ m sAY

(221) fbVxj

(222) >ACkP©l·wCW1P/C¯¡ Uu-£fbV1i RT`Dj`hBDgAWQRTC¤R[>ACYTZ†yh>^R">1fbVxiP[Z†iQC¤Z†V#R[>AC‡]fhP/R?R[>AYTCCCk\-W1fR[Z†`hV1P˜Z†V„¡¨thnb£œ© ∇ φ = [cos(β), sin(β)] q. q. q. q. q. q. q. q. q. q. q. q. q. q. (4) q,j,φ. . q. q. (3) q,j,A. x. ,. > 5!I9 7 C0 9 02II 7.< 1  4.AC< 1 5 7.6 40-, #IP[P[WABDCRT>xfR" ˜CI>xf“hCfhgAgA‡†ZˆCki # ‡†yh`-Y[ZˆR[>xB t¤R[`P[Cy-BDCV^R S fbVxi­‡†CœR c fhVxi c . . 4.1-A-6879D. . . . .

(223). Z†VxiQCUg1CUVxiQCUV-R`b§ λ ©·qCªxVxiŽRT>AC|P[ZˆŒUC­`b’?’›`h‡†‡ˆ` Z†VAyŠjZˆY

(224) jœ‡†CP/CUyhBDCV^R Z†BDg1`^P/Z†VAy R[>xCfbgAgAYT`¬QZˆB­fbR[CdCYTY[`-Y q. max eq+1 (θ˜q+1 ) = |˜ uλ − u|L∞ ,q+1 ≈ c1 |. m X j=1. 1. Sq+1. s1CIRS "`Ÿjœ`-VxPSR

(225) fbV^RTP sON : {θ ...θ } 2. min q+1. max q+1. max max min u ¯j (θ˜q+1 ) − u(θ˜q+1 )| ≈ c2 λ min |¯ uj (θq+1 )|, j. ¡hv-£ MN J2MPO.

(226) ".   

(227)    

(228)    

(229). Z†YTjWA‡]fbY ?f“hCœ’›YT`hV^R

(230) P jœZ†YTj (x) → u¯jZˆY

(231) j (δ) u ¯ . r. r. Q. δ. u(0). EIH% #¥ jZˆY

(232) jœ‡†C. −ρ [cos (β), sin (β)]. uinc. u=0. <Z†yhWAYTC  K-*`QjUfb‡8`hgQRTZˆBDZ†ŒUfbR[Z†`hV`b’<YTf-iQZˆW1P?`b’<jWAYT“fbR[WAYTCh© fhVxi θ Z]P|R

(233) f-CV–RT`™s6CRT>ACwg1`-ZˆV^R >ACYTCqfhgAgAYT`¬OZ†B­fRTC‡†N ∂ e fR/R

(234) fbZ†V ZˆRTP#B­f¬QZˆBWAB Z†V ‘QZ¢© Ch© S˜ : {θ . . . θ˜ } ¡hp-£ e (θ ) − e (θ ) ZˆR[> , max θ =θ + `δθ ≤ θ˜ . δθ  V3¡-v-£Ÿ ˜CWxP[C|RT>ACg1`-‡ˆNOVA`-B­PZ†V #G‡ˆy-`hYTZ‹RT>AB tŠR[`„fbgAgxY[`¬QZ†BDfbR[C­RT>AC”fhP[NOB gAR[`bRTZ†j#P[`h‡†WQR[Z†`hV fbRR[>xCŸCV1i”g1`-ZˆV^Rd`b’'P/CUyhBDCV^R S˜ ¡›Z¨© C-©G "CCœ¬OR[Y

(235) fbg6`h‡]fRTC¤R[>ACg1`-‡ˆNOVA`-B­PGfhPTP[`OjZ†fbR[Cki|RT`4P[Cy-B CUu˜V^R £š©¯=?>AC "  -  2qfb‡†yh`-Y[ZˆR[>AB RT`ªxV1i Z]PDB `-Y[C4`hYD‡†CUPTPfhY[sAZˆR[Y

(236) fbYTNŽsAWQRDR[`wBDZˆVxZˆBDZ†ŒC#R[>AC S j`hBD=?gA>AWQCR

(237) fYTRTCUZˆf-`-P/Vx`-fbVŸ‡*’›j`-`-Y9P/R Wx¤P[Zˆ ˜VAy¤C¤RTgA>AYTC`hg6s6`-`hP[WACIVxR[iD>xfbZ†V”RC¡ -θ˜vhjœ£<Z†Z†CP9VxR[j>xN|fbZ]R P? ˜gAC?YTZˆ ?`-Y[fbZˆV^R[Z†R9ŒCkR[`¤i$© >xf“-CfP/B­fb‡†‡AP[CyhBDCUV-R'Zˆ’6R[>AC P fhVxi |¯u |PGj

(238) >xfhVAyhCP[WAsxP/RTfhV-RTZ†fh‡ˆ‡†N4fhVxi“OZ†jC“hCUYTPTfA©"=?>ACYTCUf-P/`-V4’›`-YWxP[ZˆVxy|jœYTZ‹RTCYTZ†f¡ hp-£Z†PR[>xfbRGφ ˜C ?fbV^R R[`iQCœRTCUjšR9RT>ACVACU ™sxYTfhVxj

(239) >ACUP<RT>xfR˜BDfNP/>A` ¯WAg­iQZ]P[j`hV^R[Z†VOWA`hW1P/‡†NŸfb‡†`hVxy γ ©9cG’*j`hWAY

(240) P/C?Zˆ’8R[>AC M œ¼T¹ ½ ½œ¹ hº ¢¼T»A¹ ¼ š½k¸[Á ½À¼ Gº  º ºS½œ¹$¸  œº ¸ k¸¨ºSÀ¹ k¹¨º ¹¢¾ºS½Iº/¼ ¾Gº š ¼  ¼T¹ š½  b½ºSºSÀ max q+1 min q+1 q+1. θ q+1. max q+1. q+1. `+1. q+1. `. min q+1. `. `. max q+1. λ. q+1. . q. max q+1. . j. j. . !( . $ ! . JKJ ½ LMNON. 8 &. -.  # 4. . /+!-.(;. u. 8+. #. . . 65. O(λ−1 )  .  *. ( !.

(241) .   

(242)    . VxC –sAY

(243) fbVxj

(244) >xCUP'>1f“hCIfP/B­fh‡ˆ‡8fbBDgA‡†Z‹RTWxiQCGR[>ACUVR[>xCN­ Z†‡ˆ‡6VA`bR?s6CdiQCR[CkjšR[CkisON R[>ACdjœYTZ‹RTCYTZ†fx©9cIV|R[>AC `hR[>ACUY>xfhVxi”Zˆ’9R[>ACUN#fhY[C "CUf |RT>ACVRT>ACN”fbYTCgAYT`hsxfhsA‡†N|VA`hRdfhPZˆBDg6`hY[RTfbV^RG >xCV”RT>AC ?f“hCœ’›YT`hV^R

(245) P fhY[CI·–R[`­>As6CUCV gAYT`hgxfhy-Z†P¤fbR[P[CUCœi*R© ˜CD ?fbV^Rjfb‡†‡'EGH  # ZˆVqR[>xZ†P¤g6`hZ†V^RdRT`s1CfhsA‡ˆCR[`YTWAV #‡†yh`hYTZˆR[>ABKt’›`hY ©<‰G` ˜θCU“hCYk‘kR[`iQCkjœYTCUfhP[C'RT>AC?Y[Z]P ¤R[>xfbR< "Cjfb‡†‡^R[>ACEI%H  #¯Z†V fIg1`-ZˆV^R R[>1fR'fbYTC Z†VŸRT>AC?“OZ†jZˆVAZˆRSN S `h’<fDjfhWxP/R[Z]jjœWAYT“hCd ˜CiQ`DRT>AC¤’›`h‡†‡†` ZˆVAy1© >ACkj ­’›`hYj

(246) >1fbVAy-CdZ†V m(θ) sONZˆV1jœYTCUfhP[Z†VAy θ fhjUjœ`hY

(247) iQZ†VAyR[` ¡ u-£ 2r θ →θ + , >xCYTC r Z†P%yhZ†“hCVsOND¡ -oh£š‘bfhVxi¤Y[WxV¤R[>AC"EG%H &#¥ fb‡†yh`hYTZˆR[>ABR©  ’QRT>ACYTC Z]P*VA`Ij

(248) >xfbVAy-C9ZˆVR[>AC˜V^WxBŸs6CY%`h’ sxYTfhVxj

(249) >ACUP9j`hBDgxfhY[CkiŸRT`R[>xCGEG%H &#–Z†V RT>ACV "Cjfh‡ˆ‡ #‡†yh`-Y[ZˆR[>AB fbV1isxWAZˆ‡]i­VACU ®g1`-‡ˆNOVA`-BDP Z†V S Z‹RT> #‡†yh`-Y[ZˆR[>xB tA©  ’9RT>ACYTC Z†θPdfbVŠZ†VxjY[CkfhP[CZˆVV^WxBŸs6CYd`b’ sAY

(250) fbV1j

(251) >ACUPIfhPTP/WABDCRT>xfRdR[>xCN >xCYTC˜Z†V^R[YT`OiAWxjœCkiŸC¬Qf-jšRT‡ˆNŸfbR fbVxisAWAZ†‡†iDfIªxY

(252) PSR9`-YTiQCUY9fbgAgAYT`¬QZˆB­fbR[Z†`hV`h’xR[>xCjœ`hYTYTCUP[g1`-VxiQZ†VAy g6`h‡†NOVA`hB­PsON#WxP[ZˆVAyR[>ACŸEI%H  θ#¥ ixfRTfZˆV θ © fb‡†&‡ #‡†yh`-Y[ZˆR[>xB fhVxisAWAZ†‡†iVxC g1`-‡ˆNOVA`-B­PZ†V ZˆR[> #‡†yh`-Y[ZˆR[>xB tQ© S  ’RT>ACYTCZ]PfŠiACUjœYTCUf-P/C­Z†V„VOWABŸs6CY`b’sAY

(253) fbV1j

(254) >ACUPf-P[P[WABDC­RT>xfRRT>ACN„iQZ]PTfbgAg6CUfhY[CkiqC¬AfhjšRT‡ˆNŽfR fhVxi4P[CœR"R[>xCZ†Y?j`hYTY[CkP/g6`hV1iQZˆVxyfbBDgA‡†ZˆR[WxiQCkP˜R[`­ufbR θ © fb‡†‡ #‡†yh`-Y[ZˆR[>AB fbV1isAWAZ†‡]i|VACU θg6`h‡†NOVA`hB­P?Z†V ZˆR[> #G‡ˆy-`hYTZ‹RT>AB tQ© EG`bR[CGR[>xfbR SZˆR˜Z]P'VA` ¯VxfRTWAY

(255) fb‡ART`Z†V-RTY[`QiQW1jœCRT>ACVA`hRTfRTZˆ`-V m P/Z†VxjœCG ˜C>1f“hC?R[>xCIPTfbBDCVOWABŸs6CY `h’sAYTfhVxj

(256) >ACkPZˆVR[>AC4CV^R[Z†Y[C4P/CUyhBDCV^R © #‡]P/`VA`bRTCh‘<R[>xfbRsONŽfhPTP/WABDgQRTZˆ`-V ¡¢n-£ZˆRZ]PYTCUfhP[`hV1fbsA‡†C RT`s6C‡†ZˆCU“hCRT>xfRIR[>ACGaSWABDg`h’ O(λ) ZˆV—S ¡ u-£ Z†‡ˆ‡ f“h`-Z†iZ†VŠBD`-P/RIjf-P/CkPg6CY[’›`hYTBDZˆVAy4EG%H &#fb‡†`hVxy fjZˆY

(257) jœ‡†C >xCYTCRT>AC­VOWABŸs6CY¤`h’"sAY

(258) fbV1j

(259) >ACUP¤fbYTC VA`hVQ¥ WAVAZ]\^WACh©D‰` "C“-CYk‘8R[>AZ]P¤g6CY

(260) jœWxPTP/Z†`hVqB­fNŠVA`bR fh‡ˆ ?fNQPs6CDCVA`-WAyh>wP/Z†VxjœCDf#VAC jfbW1PSRTZ†j jœWAYT“hC B­fNŠP/>x` lWAgqZ†VŽf4iQZ]P/RTfbV1jœC ‡ˆCkP[PIR[>1fbV 2r ©=?>AC PTfbBDCjœ`-Vxjœ‡†WxP[Zˆ`-V4fhgAgA‡†ZˆCkPZ‹’¡ -vh£?fbV1iq¡ hph£˜’ fhZˆ‡]PZ†V#Z]iQCUV-RTZ‹’›NOZ†VAy­f­jfhWxPSRTZ†j¤jWAY[“-Ch© ’ Z†P|R[>ACªxY

(261) PSRP/CUyhBDCV^RRT>ACV `-VACqjUfbV jY[CkfR[Cqf„ªxY

(262) PSR4`hY

(263) iQCY4g1`-‡ˆNOVA`-B sON®WxP[ZˆVAy—R[>AC EI%H  #S¥ ixfRTf’›YT`hB `-VAC¤g1`-ZˆV^Rk© max q+1. q+1 . max q+1. max q+1. max q+1. min q+1. q+1. max q+1. max q+1. . q+1. max q+1. . max q+1. q+1. q. q. q+1. ,. . 4.1-A-6879D. . . A-6 9 7 < 1. . 4.1-A-6879D ,. "C’›`hYTC4`-VAC”jfbV®jfh‡ˆ‡ #‡†yh`hYTZˆR[>AB tq`-VACBŸWxP/RP/`-Y/RDR[>xCŠiAfR

(264) fqZ†V¯CUfhj

(265) >®EGH  #¥ g6`hZ†V-Rk© š© C-©®R[>AC EIH% # ¥ fh‡ˆy-`hYTZ‹RT>AB `hVA‡†NDgAY[`QiQW1jœCGfP/CR˜`b’ β P"fhVxi u¯ P˜Z†VCUf-j

(266) >g1`-ZˆV^R?fbVxi­ "CIiA`ŸVA`hR$OVA` ® >AZ]j

(267) >. g1fbZ†Y (β, u¯ ) ZˆV#RT>ACP/CRTP. λ. λ j.   Tmin : (β, u ¯λ )1 (θmin ), . . . (β, u ¯λ )m(θmin )(θmin ). hf Vxi   )(θ ) T : (β, u ¯ ) (θ ), . . . (β, u ¯ ) RT>xfR#s1CU‡ˆ`-VAy-P|R[`—R[>ACŽPTfbBDCsAY

(268) fbV1j

(269) >*© EG`bR[CR[>xfbR m(θ ) = m(θ ) iA`^CkP|Vx`bRVACUjCUPTP[fhY[Z†‡ˆN >1f“hC#R[`„>A`-‡†i*© ·qCqP/WAy-yhCkPSRDR[>xC’›`h‡†‡ˆ` Z†VAyP[`hY[R[Z†VAy„fb‡†yh`hYTZˆR[>AB© #GPTP[WABDCR[>xfbR| ˜C>xf“-C m sxYTfhVxj

(270) >ACUP4Z†V ‘ sAYTfhVxj

(271) >ACkP|Z†V fhVxi–RT>xfR ‘IR[>ACUV RT>AC„iQZˆ§6CUY[CUV-R g6`-PTP[ZˆsA‡†C  

(272) θA   m

(273) dR[>1fR "C¤fhY[C¤s6`hWAθV1i|RT` R[CUP/R’›`hYfhY[CmP[CœRsON­≤R[>xmC¤’›`h‡†‡ˆ` Z†VAy8K max. λ 1. max. λ m(θ max. min. max. max. min. . min. max. max. max. min. MN J2MPO.

(274) n.   

(275)    

(276)    

(277). L^R[`-Y[C¤fb‡†`hVxyYT` P Z†V4f B­fRTY[Zˆ¬ C fh‡ˆ‡$g6`-PTP/Z†sA‡†Cdj`hBsAZˆV1fR[Z†`hV1P'RT` gxZ†j  P/CR T © C Z]Pf . min. CU‡ˆCUBDCV^RTP˜’›YT`hB. mmax. mmin ! × mmax − mmax )!. ­B fR[YTZˆ¬8© A`-Y˜Ckfhj

(278) >­YT` Z†V jY[CkfR[CIfb‡†‡xg6CYTBŸWQR

(279) fR[Z†`hV1P `h’6RT>ACdC‡†CBDCV^RTP fhVxiP/R[`hYTCRT>ACB«Z†VY[` P ZˆV”f B­fRTY[Zˆ¬ PωC © P CC Z]Pf mmax !(mmin. . C r. mmin ! × mmax (mmin − mmax )!. B­fR[YTZˆ¬8© Y[CkfRTCfjœ`hVAªxyhWAY

(280) fRTZˆ`-V sONŸgAZ]j OZˆVAyYT` ˆZ V ‘- ZˆR[>­‡†CVxybR[> ‘OfbV1i f-P[P[`QjœZ]fRTC C‡†CBDCV^R j ZˆV ω R[`”CU‡ˆcCUBDCV^R j Z†V„P/CR T ω ©|?= >APC Cm gxfbZ†YTP¤`h’ m(β, u¯ ) VA` iACœªxVAC jœ`-VQªxyhWxYTfbR[Z†`hV c fbVxi4>ACV1jœCdR[>ACUY[CfhY[C . PC r. r. PC r. max. max. max. λ j. r. mmin ! (mmin − mmax )!. œj `-VQªxyhWxYTfbR[Z†`hVxP˜R[>1fRGfbYTCIR[`­s6C¤R[CUP/R[Cki$© x`hYICUfhj

(281) >qjœ`hVAªxyhWAY

(282) fRTZˆ`-V "CŸjœ`-BDgAWQR[C ZˆR[> #‡†yh`-Y[ZˆR[>AB t ’›`-YGRT>ACŸB­fbRTj

(283) >ACki g1fbZ†YTP¤fbV1iq ZˆR[>„f4ªxY

(284) PSR`-cYTiACYfbgAgxY[`¬QZ†BDfb(PR[Z†`hV’›`hY¤, R[P>xC­WAVAB­) fR

(285) j

(286) >ACUi$©­·wC­gAZ]j ŠRT>ACjœ`-VQªxyhWxYTfbR[Z†`hV RT>xfRBDZ†VAZ†B Z†ŒC |˜u (θ) − u(θ)| fbgAgxY[`¬QZ†BDfbR[CU‡ˆN-‘^Z¨© Ch© r. (φ,r,j). (r) λ. ZˆR[>. (A,r,j). L1. min r. max m q

(287) `X X.

(288)

(289)

(290) (r) uj (θ` ) − u(θ` )

(291) ,

(292) ¯. ›’ `-YP/`-B C δθ © =θ + `δθ ≤ θ ’θ ˜ ¤ C A g T Y Q ` œ j  C k C | i † Z V”fDP/Z†BDZˆ‡]fbYB­fbVxVACYf-P"fhs1`“-Ch© "Cm‡†` ">CŠWxmP[CŠR[>xC’›`-‡ˆ‡†` ZˆVxyVA`hBDCUVxjœ‡]fRTWAY[C-© # "f“-Cœ’›YT`hV^R >AZ]j

(293) > >xf-P’›`-‡†iQCki–`“hCUYZˆRTP[C‡ˆ’ gx Y[>x`QCiQVwWxjZ‹RCUP’›`-f­‡†iAP[PUCœ‘%RIf#`b’9iQVA`hCB­ fhZˆsAVwY

(294) fb V1Z‹j

(295) RT>A>wCUR[PU>A$© YT#CC jœR[sAWxY

(296) fhfb‡ˆVx‡†Nhj

(297) ‘6>AfbCkVxPfh‘$‡ˆVxNQCœP[¬OZ†PRP[R[>AZ†BD` C P?ZˆRRT>x’›f`hR‡]iA’›P¤YT`hN-B `hW`hVAy-CœCŸR¤ "ªxf“-“-CCœ’›sAYTY

(298) `hfbV^V1R"j

(299) N->A`hCUWP¤yhfbCV1RUi ‘ P[` `hV*©9‰` "C“-CYk‘h >ACUV#P[C“-CY

(300) fb‡6 "f“-Cœ’›YT`hV^RTP˜fbYTCGgAYTCUP[CV^R?Z‹RZ]P"ZˆBDg6`-PTP/Z†sA‡†CGRT`DiAZ†P/R[Z†VAyhWxZ†P[>|s6CœRS "CCUV RT>xfR "C¤>xf“hC¤P[C“hCUYTfh‡8 "f“-Cœ’›YT`hV^RTP˜`hYP[C“-CY

(301) fb‡6sAYTfhVxj

(302) >ACkP"’›Y[`-B `hVAC¤ ?f“hC’›Y[`-V-Rk© #IP[P[WABDCR[>1fRRT>ACYTCfbYTC`hVA‡†N|`-VAC ?f“hCd’›Y[`-V^RgAY[CkP/CUV^RR[>ACUV” ˜CB­f -C¤R[>AC’›`h‡†‡ˆ` Z†VAyDY[CUB­fbY 6©  RD "`hWA‡]i„>xf“-C|s6CCV™“hCUY[N„fR[R[Y

(303) fhjœR[Z†“hCRT`qW1P/C|R[>AB C OVA` ‡†CUiQy-C­R[>xfbR Vx`wjUfbWxP/R[Z]j|jWAY[“-C4jY[`^P[PŸR[>AC P[Cy-BDCV^R˜P[ZˆVxjCRT>ACV4 ˜C OVA` ®RT>xfR˜R[>ACIY[CU‡†fbR[Z†“h$C /`hY

(304) iQCUY `b’%P[Z†ŒC `h’$R[>ACdsAY

(305) fbVxj

(306) >ACkP φ (θ), . . . φ (θ) fhY[C hCgAR¤j`hVxP/RTfhV-Rk© š© C-©dZˆ’ >A`h‡]iAPIR[>ACUVqfh‡†P[` BWxPSR >x`h‡]i$©9=?>ACP/`-Y/RTZˆVAy fb‡†yh`-Y[ZˆR[>AφB«(θ "`hWA‡]i) >1<f“hφCd(θs1CUCV4R[)YTZˆ“OZ]fb‡6R[` ZˆBDgA‡†CBDCUφV-R(θZ†V#P[Wxj

(307) )>#<fφjUfhP[(θCh©9‰`) "C“-CYk‘ fhgxfbY[R’›YT`hB R[>xfbRI "CB­fN4>xf“-CP/CU“hCUYTfh‡* "f“-Cœ’›YT`hV^RTP? "C`-VA‡ˆN OVA` WAg”R[`4fbV”Z†V-RTCy-CY ω ¡ jb© ’S© C¬QgAY[CkP[P[Z†`hVq¡¨t £[£" >AZ]j

(308) ># Z†‡†‡*B­f hCIR[>xZ†P?RSNOg6C`b’ P/`-Y/RTZˆVAyDWAVA¥¢’›CkfhP[ZˆsA‡†Chφ© M ½ ¼T¹  ¼ š½º /¼

(309) ½dÀ M ½ ¼T¹  ¼ š½º /¼

(310) ½dÀ      6¹  šºS¹%¸ 8¾¤¹ ¼ šº ¹%¼T¹ ¸ -¨¾Á ¼ ¼T¹ hÁ min q+1 min. q+1,` max. `=1 j=1. max q+1. 3. min. 1. min. 3. j.    . JKJ ½ LMNON. 8$ 8$. . . 7+ 7+.  

(311) 

(312)     mmin  mmax ,*    mmax .* + /-. + " 1. /-.  1. max. 1. 1. m. max. j.

(313) ko.   

(314)    . ,. . . 4.1-A-6879D. 5 C0

(315) 86 AC<$9 ? 6 ACI5!1-59 7AC< 5=<> /(5 :. . . . 6 5 7 < 1 ,. x`hY?Z†VA>A`hBD`-yhCVxC`hW1P B­fRTCYTZ†fh‡xRT>ACP/`Djfh‡ˆ‡†CUi4 ?f“hCœ’›YT`hV^R˜j`hVxP/R[YTWxjœR[Z†`hVRTCUj

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