Algorithmic solutions toward applications of compressed sensing for optical imaging

(1)

HAL Id: tel-00950365

https://pastel.archives-ouvertes.fr/tel-00950365v2

Submitted on 24 Jul 2015

HAL is a multi-disciplinary open access

archive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished 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.

sensing for optical imaging

Yoann Le Montagner

To cite this version:

Yoann Le Montagner. Algorithmic solutions toward applications of compressed sensing for optical imaging. Signal and Image processing. Télécom ParisTech, 2013. English. �NNT : 2013ENST0065�. �tel-00950365v2�

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

9 : 9

:

θ

(10)

(11)

(12)

(13)

(14)

(15)

(16)

(17)

• Z • R • C • R˚ 0 N˚ Z˚ C˚ • R` 0 • Ed _{d P N} ˚ d E • Em ˆ n m, n P N˚ m n E Em _Em ˆ 1 • F pΩÑ E q Ω E Ω Rd Zd • a, b a, b P Z n aď n ď b • |Ω| Ω • ttu t P R t t • rts t P R t t • pa mod N q a P Z N P N˚ a N r q P Z a “ q ¨ N ` r • p_n “ _{p!pn ´ pq!}n ! ﬃ • z z P C z

(18)

• xk fk kt h ﬃ x P CN f • }x}_p p P N˚ Y t ` 8 u lp x P CN }x}_p“ ˜ ÿ k |xk|p ¸ 1 p p P N˚ }x}₈ “ max k |xk| • }x}₀ ﬃ l0 x P CN • xx|yy “ ř _kxk ¨ yk x,y P CN • x ˆ y P CN x,y P CN x y • 0,1 0 1 • ek P CN k P 0, N ´ 1 kt h CN 0 kt h 1 • I d P CN ˆ N CN ˆ N 0 N 1 • W˚ _W _{P C}M ˆ N _W _W _W˚ • ~W~ “ sup_x }Wx}2 }x}₂ W P C M ˆ N • Tr pWq “ ř _kwk,k W P CN ˆ N wk,l k, l P 0, N ´ 1

ﬀ

f P F `RN Ñ R˘ f x P RN ∇f pxq P RN x ∇2_{f p}_x_{q P R}N ˆ N _f ∇f pxq “ » — — — — — — — – Bf Bx0 pxq Bf Bx1 pxq .. . Bf BxN ´ 1 pxq fi ffi ffi ffi ffi ffi ffi ffi fl ∇2_{f p}_x_{q “} » — — — — — — — — – B2_f Bx2 0 pxq _BxB2f 0Bx1 pxq ¨ ¨ ¨ B2_f Bx0BxN ´ 1 pxq B2_f Bx1Bx0 pxq B2_f Bx2 1 pxq ¨ ¨ ¨ _Bx B2f 1BxN ´ 1 pxq .. . ... ... B2_f BxN ´ 1Bx0 pxq B2_f BxN ´ 1Bx1 pxq ¨ ¨ ¨ B2_f Bx2 N ´ 1 pxq fi ffi ffi ffi ffi ffi ffi ffi ffi fl B Bxk k t h x P RN

(19)

• E t X u X • V t X u X • N pµ,Σq µ P RN Σ P RN ˆ N µ Σ • P pλq λ P RN ` N kt h λk k

(20)

(21)

ﬃ

(22)

ff ffi ffi ff ff PG-URE

(23)

y “ Φx P CM x M N M ! N Φ P CM ˆ N y x x Ψ s P CL _x _“ _Ψ_s ﬃ Ψ P CN ˆ L

(24)

F pΩÑ E q Ω E ﬀ • F pΩÑ Rq Ω Ă R2 Ω Ă Z2 • F pΩÑ Rq Ω Ă R3 _{Ω Ă} _Z3 • Ω • F pΩÑ Rcq c P N˚ x P F pΩÑ Rq f Ω Ω Zd F pΩÑ E q E|Ω| ϕ : 0, |Ω| ´ 1 Ñ Ω Ω ϕ x P F pΩÑ E q x P E|Ω| x • xk k P 0, |Ω| ´ 1 x E|Ω| • x rus u PΩ x F pΩÑ E q

(25)

xk x rus u “ ϕpkq x Ω Ă Zd x Zd Ω “ 0, n1´ 1 ˆ 0, n2´ 1 ˆ ¨ ¨ ¨ ˆ 0, nd´ 1 Ă Zd x x ru1, . . . , uds “ x ru1mod n1, . . . , udmod nds u “ pu1, . . . , udq P Zd x x P CN S 0 ď S ď N S ffi S x S }x}₀ l0 1 x x }x}₀! N }x}₀ ffi x x ffi x P CN Ψ P CN ˆ L sP CL x “ Ψs s x Ψ Ψ ΨP CN ˆ L _{L “ N} L ą N L ă N CN S “ 0 S “ N S CN S N S S S x P CN ˜x x P CN _˜x _{P C}N px ´ ˜xq x 1 } ¨}₀ } ¨}₀ l0

(26)

0 10 20 30 40 50 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 u x ru s 0 10 20 30 40 50 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 u p D x q ru s x Dx pDxqrus “ x rus´ x ru ´ 1s 0 10 20 30 40 50 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 u y ru s 0 10 20 30 40 50 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 u p D y q ru s y Dy ﬃ Dy ﬃ Dx x P CN r ą 1 Cr r ˇ ˇx_pkqˇˇ_ď _C_r _{¨ pk ` 1q}´ r _{k P 0, N ´ 1} ` xp0q, xp1q, . . . , xpN ´ 1q ˘ x ˇ ˇx_p0qˇˇ_ě ˇ_ˇx p1q ˇ ˇ_ě _{¨ ¨ ¨}_ě ˇ_ˇx pN ´ 1q ˇ ˇ x P CN S 1ď S ď N S ˜xS P CN x S l1 › ›_x _´ _˜xS›› 1

(27)

› ›_x _´ _˜xS›_› 1ď Cr r ´ 1¨ 1 Sr ´ 1 ˜xS x Cr Cr x Cr Cr ˜xS S › ›_x _´ _˜xS›_› p lp 1ď pă ` 8 ˜xS _S _x ﬃ Sd Ss N Sd` Ss ă ? N 2 x P CN y “ Φx y P CM Φ P CM ˆ N x arg min xPCN }x}₀ y “ Φx P0 P0 Φ 2N y N

(28)

l0 l1 P0 P1 arg min xPCN }x}₁ y “ Φx P1 y P CM Φ “ ” φ1 φ2 ¨ ¨ ¨ φN ı P CM ˆ N _φ k Φ x P CN _y _“ _Φ_x }x}₀ă 1 2 ˆ 1 ` 1 M pΦq ˙ M pΦq “ max k‰l |xφk|φly| }φk}2¨ }φl}2 x P0 P1 y ﬃ Φ • P0 • P1 P0 ΦP CM ˆ N S P N˚ S Φ δS P R p1 ´ δSq}x}2₂ď }Φx}2₂ď p1 ` δSq}x}2₂ x P CN }x}₀ď S

(29)

δS 0 Φ S δS Φ P0 P1 x P CN Φ P CM ˆ N _y _“ _Φ_x _` _b b }b}₂ ď ˆx x 2 ˆx “ arg min xPCN }x}₁ }Φx ´ y}₂ď δ2S ă ? 2´ 1 S x ˆx }ˆx ´ x}₂ď A ¨ ` ?B S › ›_x _´ _˜xS›_› 1 A B δ2S ˜xS _S x x S ﬀ y A ¨ B ? S › ›_x _´ _˜xS›› 1 › ›_x _´ _˜xS›_› 1 x Φ Φ P CM ˆ N M U P CN ˆ N _Φ_“ _Σ_U Σ P t 0, 1uM ˆ N U ΦΦ˚ _“ _{I d} _Φ˚_Φ _“ _U˚ _p_Σ˚_Σ_q_U _Σ˚_Σ 2 b

(30)

x P CN Φ x P CN }x}₀ ď S Φ P CM ˆ N M U P CN ˆ N M ě C ¨ µ pUq2¨ S ¨ N log pN q µ pUq “ max k,l |uk,l| C µ pUq uk,l U y “ Φx P1 x y 3 ffi µ pUq U U µ pUq “ N´ 1{2 S N S log pN q 4 _{µ p}_U_{q “ 1} U “ ΦΨ Φ Ψ φ0,φ1, . . . ,φN ´ 1 ψ0,ψ1, . . . ,ψN ´ 1 Φ Ψ µ pUq “ max k,l |xφk|ψly| Φ Ψ x P CN _s _{P C}L _Ψ _P CN ˆ L x “ Ψs Ψ 3 ff ff U 4 _S ffi

(31)

ﬀ • x s ˆs “ arg min sPCL }s}₁ }ΦΨs´ y}₂ď Psynt hesis ˆx ˆx “ Ψˆs • x ˆx ˆx “ arg min xPCN › ›_Ψinv_x›_› 1 }Φx ´ y}2ď Panalysis Ψinv P CL ˆ N 5 x s x “ Ψs s L “ N Ψ Ψinv Ψ Ψinv “ Ψ´ 1 Ψ L ą N Ψ L " N x Panalysis s Psynt hesis ﬀ Ψ Ψ 6 5 _Ψi nv _{Ψ Ψ} 6 Ψ “ ” ψ0 ψ1 ¨ ¨ ¨ ψL ´ 1 ı P CN ˆ L α ą 0

(32)

x P CN Ψ y “ Φx ` b Panalysis Ψinv “ Ψ˚ }ˆx ´ x}₂ ˆx l1 Panalysis T V x ˆx “ arg min xPCN }x}_{T V} }Φx ´ y}₂ď PT V }x}_{T V} x P F pΩÑ Cq Ω Ă Z2 }x}_{T V} “ ÿ pu,vqPΩ b |pDhxqru, vs|2` |pDvxqru, vs|2 Dh Dv pDhxqru, vs “ x ru ` 1, vs´ x ru, vs Dv ﬀ ﬀ T V PT V x x ˆx T V PT V

l1 Panalysis Psynt hesis

ﬃ s x l1,2 CL pω1, . . . ,ωGq 0, L ´ 1 ΨΨ˚ _“ _α_{I d} ř L ´ 1 k “ 0|xψk|sy| 2 “ α}s}2₂ s P CL α“ 1

(33)

ωg P 0, L ´ 1 Ť_G g“ 1ωg“ 0, L ´ 1 ωgXωg1“ H pg, g1q g ‰g1 sP CL l1,2 s }s}_1,2“ G ÿ g“ 1 d ÿ kPωg |sk|2 l1 l1,2 Panalysis Psynt hesis ffi s ωg ffi pω1, . . . ,ωGq ffi sk ωgXωg1“ H l2 lp ffi y ptq t y ptq “ ¡ x pr qexp ´ ´ 2iπ A k ptq ˇ ˇ ˇr E¯ d3r x pr q

(34)

y ptq k ptq k ptq k k ff ffi k k Panalysis ff ff ff E EL O E

(35)

ff I I “ |E ` EL O|2“ |E_{l ooooooomooooooon}L O|2` |E |2 Zero-order ` E ¨ El ooomooonL O Real image ` E ¨ El ooomooonL O Twin image ff E EL O ff I I y “ E ¨ EL O x y I y rξ,ηs “ ĳ x ru, vsexp ˆ iπ λd ´ pu ´ ξq2` pv ´ ηq2 ¯ ˙ du dv λ pu, vq pξ,ηq d ff

(36)

(37)

y “ Φx Φ“ » — — — — — – φ0,0 φ0,1 ¨ ¨ ¨ φ0,N ´ 1 φ1,0 φ1,1 ¨ ¨ ¨ φ1,N ´ 1 .. . ... ... φM ´ 1,0 φM ´ 1,1 ¨ ¨ ¨ φM ´ 1,N ´ 1 fi ffi ffi ffi ffi ffi fl P t 0, 1uM ˆ N ffi φk,l lt h kt h _φk,l _{“ 1} φk,l “ 0 M x M N Φ Φ “ I d M x Φ ffi x M N 40% 66% p Ap p xpP RN Ap xp

(38)

Opt ical axis B a ck li g h t so u rc e

SLM Object Telecentric syst em CMOS/ CCD sensor

Pixel p Point Ap Ap p Ap xpP RN ypP RM yp“ Φxp ΦP RM ˆ N` kt h Φ kt h _Φ p xp Ap xp M xp Φ 9{ 7 ﬃ ˘ 1 Φ M N “ 2.5% M N “ 10% M N “ 100% ﬀ

(39)

M y “ Φx P CM

(40)

x P CN

y “ Φx P CM

Panalysis Psynt hesis PT V

1 arg min xPCN f pxq }Φx ´ y}₂ď CCSR f pxq f pxq “ }x}₁ Psynt hesis CN ě 0 ﬀ ﬀ y 2 y y σ “ σ a M ` 2?2M CCSR x CN }Φx ´ y}₂ ď 1 CCSR

(41)

CCSR ﬃ arg min xPCN 1 2}Φx ´ y} 2 2` λf pxq BPDN f pxq CCSR λ ě 0 CCSR CCSR BPDN ě 0 ˆxCCSR CCSR λ‹ ě 0 λ ˆxCCSR BPDN λ ě 0 ˆxBPDN_λ BPDN ‹ _ě ₀ _ˆxBPDN λ CCSR λ‹ ‹ CCSR BPDN Φ y λ λ‹ ‹ CCSR BPDN y ˆxCCSR CCSR BPDN CCSR CCSR BPDN arg min xPCN }Φx ´ y}₂ f pxqď τ LASSO LASSO τ λ τ ﬀ y LASSO CCSR BPDN CCSR BPDN f pxq “ }x}₁ l1 LASSO τ BPDN LASSO

(42)

y,Φ

p Ð 0 Λ0Ð H x0Ð 0 r0Ð y

p Ð p ` 1

kpÐ arg maxkP 0,N ´ 1 |hk| h “ Φ˚rp Ź

ΛpÐ Λp´ 1Y t kpu

xpÐ arg minx}y ´ Φx}2 Supp pxqĂ Λp Ź

rpÐ y ´ Φxp xp x P CN y “ Φx P CM x }x}₀! N p ΛpĂ 0, N ´ 1 xp Λp rp“ y ´ Φxp kp Λp ekp kp arg max_kxΦek|rp´ 1y xp rp Supp pxpqĂ Λp }xp}₀“ S S }rp}₂ď ® Φ

(43)

ﬃ O pN log N q S S S ﬃ Psynt hesis CCSR 2 CCSR • f pxq }¨}₁ }¨}_{T V} }¨}_1,2 • CN RN N ě 106 CCSR f pxq “ }x}_{T V} }¨}_{T V} CCSR 2

(44)

3 arg min xPRN nPRN ÿ k nk # pDhxq2_k` pDvxq2_k ď n2k nk ě 0 k }Φx ´ y}2₂ď 2 CCSR f pxq “ }x}_{T V} px,nq 2N CCSR CCSR ® l1 arg min XPR2N hppX q X “ px,nq hppX q “ ÿ k nk´ 1 αp ÿ l log p´ glpx,nqq gl glpx,nqď 0 px,nq pαpq_pPN ˆ X p αp limpÑ 8 Xˆ p“ Xˆ ˆ X αp pX p,qq_qPN q X p,q` 1 hp X p,q X p,q` 1“ arg min XPR2N hppX p,qq` x∇hppX p,qq|X y ` 1 2 @ X ˇˇ∇2hppX p,qq¨X D limqÑ 8 X p,q“ Xˆ p Xp,q` 1“ ´ ` ∇2_h ppX p,qq ˘_{´ 1} ¨∇hppXp,qq 3 f pxq ﬀ f pxq }x}₁ }Ψ˚_x_} 1 }x}1, 2 f pxq “ }x}₁ “ 0 x f pxq “ }x}_{T V}

(45)

∇2_h ppX p,qq X p,q` 1 ﬃ CCSR f pxq “ }x}_{T V} f pxq “ }Ψ˚_x_} 1 Ψ f pxq f pxq 4 f pxq “ max zPQ xz|Wxy W P RL ˆ N Q RL f pxq f pxq fµpxq fµpxq “ max zPQ xz|W xy ´ µ 2}z ´ z0} 2 2 µ ą 0 z0 P Q |f pxq´ fµpxq| µ fµpxq ﬀ Lµ “ 1_µ~W~2 CCSR f pxq fµpxq 4 _{f p}_x_{q “ }}_Ψ˚_x_} 1 W “ Ψ ˚ _{D “} _u _}_u_} 8 ď 1 ( l8

(46)

4.3 Π Π pxq “ arg min zPRN }z ´ x}₂ }Φz ´ y}₂ď Φ pΦ˚_Φ_q2 “ Φ˚_Φ _Φ˚_Φ ﬃ 5 Π pxq “ ˆ I d´ λ 1 ` λΦ ˚_Φ ˙ px ` λΦ˚yq λ “ max ˆ 0,1}Φx ´ y}₂´ 1 ˙ pΦ˚_Φ_q2 “ Φ˚_Φ _Φ BPDN f pxq }x}_{T V} }Ψ˚_x_} 1 Ψ Φ 6 f pxq “ }x}_{T V} 5 ﬀ λ 1 Lµ λ λ Π 6 _Φ ®

(47)

arg min xPRN dhPRN dvPRN 1 2}Φx ´ y} 2 2` λ ÿ k b pdhq2_k ` pdvq2_k` β 2 ´ }dh ´ Dhx}2₂` }dv ´ Dvx}2₂ ¯ λ BPDN Dh Dv β ą 0 7 _BPDN _{f p}_x_{q “ }}_x_} T V β Ñ ` 8 pdh,dvq x x pdh,dvq • pdh,dvq ppdhqk, pdvqkq O pN q • x ` Φ˚Φ` βD˚_hDh` βD_v˚Dv ˘ x “ Φ˚y` βD_h˚dh` βD_v˚dv N ˆ N `Φ˚Φ` βD_h˚Dh ` βD˚vDv ˘ Dh Dv Φ Φ˚_Φ O pN log N q CCSR f pxq “ }x}₁ Φ Psynt hesis T V PT V LASSO ﬀ τ ˆxLASSO τ LASSO › ›_Φ_ˆxLASSO τ ´ y › › 2“ 7

(48)

f pxq ˆx CCSR }Φx ´ y}₂ď }Φˆx ´ y}₂“ 0 χpτq LASSO χpτq “ ››ΦˆxLASSO τ ´ y › › 2 τ‹ _χ_p_τ_{q “} _ˆxLASSO τ‹ CCSR χ ff ˆxLASSO_τ LASSO τ‹ _χ_p_τ_{q “} _LASSO ffi Π Πpxq “ arg min zPRN }z´ x}₂ }z}₁ď τ x O pN log N q 15% ffi ffi T V l1 ffi T V l1

(49)

T V l1 ﬀ • }x}_{T V} }Ψ˚x}₁ • • pdh,dvq x LASSO O pN log N q ﬀ ® BPDN CCSR λ ˆxBPDN_λ › ›_Φ_ˆxBPDN λ ´ y › › 2“ ˘ 2% CCSR BPDN λ BPDN CCSR

(50)

ff ff • 400 ˆ 400 • Φ ® ff • ® • ®

(51)

425 ˆ 425 2230 136 14 2254 26 1 3977 73 10 3985 283 47 453 ˆ 453 1115 171 23 1169 17 1 3154 69 12 3157 80 24 716 ˆ 716 13.6 ˆ 103 86 23 13.7 ˆ 103 15 1 35.4 ˆ 103 63 21 35.4 ˆ 103 276 132 960 ˆ 960 1384 206 78 1486 23 2 6475 63 38 6400 90 119 992 ˆ 992 815 198 91 981 14 2 13.1 ˆ 103 63 46 13.0 ˆ 103 577 987 1024 ˆ 1024 3315 159 75 3504 26 3 10.2 ˆ 103 66 47 10.1 ˆ 103 3103 5823 1024 ˆ 1024 4962 179 82 5177 20 2 15.6 ˆ 103 68 47 15.7 ˆ 103 1693 3176 T V l1

(52)

(53)

ﬀ

ff ff l2 ˆx ff ff ff

(54)

ffi ff x0 yu “ Φux0 yg “ Φgx0 Φu Φg ˆxu ˆxg T V PT V ˆxu ˆxg ff ˆxg ˆxu ff

(55)

x0 ˆx ff T V PT V 15% ff • pdq • ff πpωq ffi ω πpωq “ p0¨ ˆ 1 ´ }ω}2 ωmax ˙ _α α ą 0 α ωmax ω p0ą 0 α 1 6 pbq pf q • k

(56)

peq • ffi πpωq ffi ω πpωq “ exp ˆ ´ ˆ }ω}₂ ρ ˙ _α˙ α ą 0 ρ ą 0 α “ 3.5 α “ 2 α “ 3.5 α “ 2 pcq pgq • ffi πk kt h ffi π‹ π‹ “ arg min π ˆ max k,l |xφk|ψly| ? πk ˙ $_& % 0ă πk ď 1 k ř kπk “ M φ0,φ1, . . . ,φN ´ 1 Φ ψ0,ψ1, . . . ,ψN ´ 1 Ψ M ff ffi • ff

(57)

α“ 2 α“ 2 ` α“ 5 α“ 3.5 ff 10% k π‹ _π‹ ff ff α “ 5 ff

(58)

π‹ ffi ff ff k ffi π‹ ff ffi ffi ffi M N S ffi

(59)

τ τ “ M_N ff x0 τ ˆx T V PT V RecErr “ }ˆx ´ x0}2 }x0´ µ01}₂ µ0 x0 ffi µ0 T V }x0´ µ01}2 RecErr pτq 1 τ Ñ 0 ˆx x0 τ ff ρ “ 22 256 ˆ 256 RecErr “ f pτq • τ PT V

(60)

τ “ 2 .9 ˆ 1 0 ´ 3 τ “ 6 .4 ˆ 1 0 ´ 3 τ “ 2 .2 ˆ 1 0 ´ 2 τ “ 5.9 ˆ 10´ 3 τ “ 7.4 ˆ 10´ 3 10´ 3 ₁₀´ 2 ₁₀´ 1 0 1 2 1 τ ρ“ 22 τ “10´ 3_{, 10}´ 1‰ ff τ τ “ 2.2 ˆ 10´ 3 _x 0 • τ ffi ffi • τ‹ ff ρ RecErr “ f pτq τ‹ τ‹ RecErr “ f pτq ffi S

(61)

ρ“ 5 ρ“ 13 ρ“ 36 ρ“ 99 10´ 3 10´ 2 10´ 1 0 1 2 1 τ τ‹5 “ 2. 4ˆ 10 ´3 τ‹13 “ 4. 2ˆ 10 ´3 τ ‹ 36 “ 9 .6 ˆ 1 0 ´ 3 τ‹99 “ 2. 6ˆ 10 ´2 ρ τ‹ 0 10 20 30 40 50 60 70 80 90 100 0 0.005 0.01 0.015 0.02 0.025 ρ τ ‹ τ‹ _ρ τ‹ ρ

(62)

τ‹ _{RecErr “} f pτq τ‹ RecErr pτ‹q “ 1_{₂ 1_{₂ τ‹ ffi τ‹ RecErr pτ‹q “ 1_{₂ PT V ffi τ‹ τ‹ ρ τ‹ ρ τ‹ ρ τ‹ τ‹ S ffi τ‹ γ ff T V τ‹ “ 10´ 2, 1.7 ˆ 10´ 2‰ ff

(63)

γ “ 1 γ “ 0 .3 7 γ “ 0 .1 4 γ “ 0 .0 5 γ“ 1 γ“ 0.37 γ“ 0.14 γ“ 0.05 10´ 3 ₁₀´ 2 ₁₀´ 1 0 1 2 1 τ τ‹ P “ 250 γ τ‹ x0 ffi τ‹ τ‹ N ffi S T V ffi S

(64)

0 200 400 600 800 1000 1200 0 0.01 0.02 0.03 0.04 0.05 P τ ‹ pP ,τ‹_q x0 τ‹ P • P pP ,τ‹_q pP ,τ‹_q τ‹

(65)

• P P P ff RecErr “ f pτq ff ρ RecErr “ f pτq ρ “ 3 ρ “ 5 τ ă 10´ 3 τ ą 10´ 1 ff ff

(66)

τ “ 4.0 ˆ 10´ 4 τ “ 1.0 ˆ 10´ 3 τ “ 7.3 ˆ 10´ 3 τ “ 2.3 ˆ 10´ 1 ρ“ 3 ρ“ 5 ρ“ 13 ρ“ 36 10´ 3 ₁₀´ 2 ₁₀´ 1 0 1 2 1 τ ρ ρ“ 5 τ “ 2.3 ˆ 10´ 1

(67)

paq pbq pcq pdq 10´ 2 10´ 1 1 0 1 2 1 τ ‹ pa q “ 5 .2 ˆ 1 0 ´ 2 τ ‹ pb q “ 1 .5 ˆ 1 0 ´ 1 τ paq pbq pcq pdq ﬃ S

(68)

ﬃ • • 256 ˆ 256 • • τ • 10´ 1 10´ 2 • ﬀ S τ “ 100%

(69)

ﬀ

(70)

(71)

(72)

X P CN T _T xt P CN 0ď tď T ´ 1 Φ Y P RM 1 » — — — — — – y0 y1 .. . yT ´ 1 fi ffi ffi ffi ffi ffi fl l ooomooon Y “ » — — — — — – φ0 φ1 . .. φT ´ 1 fi ffi ffi ffi ffi ffi fl l oooooooooooooomoooooooooooooon Φ ¨ » — — — — — – x0 x1 .. . xT ´ 1 fi ffi ffi ffi ffi ffi fl l ooomooon X

Panalysis Psynt hesis

X Y

Y T yt P Cmt

yt xt

(73)

φt ffi ffi • • ffi Φ Φ δS φt }xt}2₂ mt φt φt X X Y Ψ

(74)

Psynt hesis Panalysis X ffi Psynt hesis Ψ K K ě 2 ψ P CN ˆ N N K BK “ » — — — — — — — – ψ ψ ψ .. . . .. ψ ψ ψ ψ fi ffi ffi ffi ffi ffi ffi ffi fl CK “ » — — — — — — — – ψ ψ ψ .. . ... . .. ψ ψ ¨ ¨ ¨ ψ ψ ψ ¨ ¨ ¨ ψ ψ fi ffi ffi ffi ffi ffi ffi ffi fl BK CK l1 Psynt hesis ff CK l1 ffi ψ´ 1¨ pxt ´ xt ´ 1q ffi ff pxt ´ xt ´ 1q xt ψ BK ffi ψ´ 1_{¨ p}_x t ´ xt0q xt0 K ff CK

(75)

ff ff ffi ff xt xt ` 1 ffi ff pxt ` 1´ xtq xt xt ` 1

(76)

xt xt ff | xt ` 1 ´ xt | t “ 30 t “ 40 t “ 55 t “ 67 ff ff ff

(77)

xt ff pxt ` 1´ xtq ffi xt xt ffi pxt ` 1´ xtq T V-3D }X}_{T V -3D} “ T ´ 2_ÿ t “ 0 ÿ pu,vqPΩ b

|pDhxtqru, vs|2` |pDvxtqru, vs|2` |pxt ` 1´ xtqru, vs|2

T V-3D Dh Dv Ω Ă Z2 xt Xˆ X Y ˆ X “ arg min XPCN T }X }_{T V -3D} }ΦX ´ Y}₂ď Φ ě 0 Y }X }_{T V -3D} T V-3D max ˜ ÿ t }xt}T V, ÿ t }xt ` 1´ xt}1 ¸ ď }X }_{T V -3D} ď ˜ ÿ t }xt}T V ¸ ` ˜ ÿ t }xt ` 1´ xt}1 ¸ }¨}_{T V} }X }_{T V -3D} T V ř t}xt}T V l1 ﬀ ř t }xt ` 1´ xt}1 ˜ ÿ t }xt}_{T V} ¸ ` ˜ ÿ t }xt ` 1´ xt}₁ ¸ ď ?2 ¨ }X }_{T V -3D} t P 0, T ´ 2 pu, vq PΩ

(78)

b

|pDhxtqru, vs|2` |pDvxtqru, vs|2“ |pxt ` 1´ xtqru, vs| ř t}xt}T V ř t}xt ` 1´ xt}1 pu, vq t b |pDhxtqru, vs|2` |pDvxtqru, vs|2 ff |pxt ` 1´ xtqru, vs| }X }_{T V -3D} }X }_{T V -3D} l1,2 X ff pxt` 1´ xtq xt ` 1 xt T V-3D arg min XPCN T }X ´ A}_{T V -3D} }ΦX ´ Y}₂ď A at 0ď t ď T ´ 1 at “ µt1 µt tt h A µt Y µt “ _N1 x1|xty φt 1 µt Y φt ffi φt 1 µt µt ˆµt “ _N1 xφt1|yty φt ˆµt “ 1 N xφt1|yty “ 1 N xφt1|φtxty « 1 N x1|xty “ µt φt

(79)

T V-3D X 1 _“ _X _´ _A T V-3D T V T V-3D • l1 • l1 BK CK K “ 4 K “ 20 ψ ff T V • 256 ˆ 256 ˆ 80 ˆ ˆ • 288ˆ 352ˆ 80 • 256 ˆ 256 ˆ 80 5 25 1 3 ff Y ffi ff

(80)

R econ st r u ct i on m et h od P SN R ( d B )

3D t ot al var i at i on

3D t ot al var i at i on w i t h b ack gr ou n d cor r ect i on

B4 C4 B20 C20 PSNR 10% 20% Y PSNR T V-3D C20 1 4 T V-3D T V-3D T V-3D BK CK • BK K ﬀ • CK t

(81)

B20 T V C20 t “ 50 B20 T V C20 t “ 23

(82)

t “ 14 t “ 21 t “ 24 t “ 27 T V-3D t ` 1 t ´ 1 t ` 2 t ´ 2 T V-3D PSNR T V-3D ﬀ T V-3D BK CK T V-3D T V-3D

(83)

14.8 ˆ 10´ 2 3.9 ˆ 10´ 2 10´ 2 10´ 1 1 40 42 44 46 48 50 ﬀ T V-3D PSNR T V-3D T V PSNR T V-3D T V-3D T V-3D

(84)

(85)

xt xt ´ 1 ff x0 ff t x xt a xt ´ 1 xt y yt φ φt x P RN _y RN • Dy, x ff Dy, “ x P RN }y ´ |φx|}₂ď ( • R R ˆx Dy, R ˆx “ `ΠR ˝ ΠDy, ˝ ΠR ˝ ¨ ¨ ¨ ˝ ΠDy, ˘ pxinitq

(86)

˝ xinit ΠDy, ΠR Dy, R ΠDy, pxq “ arg min z }z ´ x}2 z P Dy, ΠR Dy, R • y “ |φx| RN φ φ˚ 0,φ˚1, . . . ,φ˚N ´ 1 t yk “ |xφk|xy|uk“ 0,...,N ´ 1 x y y_k2“ Tr pφkφ˚kx x˚q k Tr p¨q 2 x P RN X “ x x˚ _{P R}N ˆ N N2 _N N “ 6.5ˆ 104 N2 “ 4.3 ˆ 109 • y “ |φx| x u P CN uk u y ˆ u “ φx 3 2 3 xP CN x P RN

(87)

arg min x,uPCN }φx ´ y ˆ u}2₂ @k |uk| “ 1 x x “ φ: ¨ py ˆ uq φ: φ arg min uPCN xu|H uy @k |uk| “ 1 H P CN ˆ N φ u P CN _U _“ _{u u}˚ _P CN ˆ N ff • ff ff ff ff ff ff R Dy, ffi Dy, ΠDy, ffi

(88)

x1“ Fx z1“ Fz x z F P CN ˆ N ΠDy, pxq “ F ˚ _ˆz1 _ˆz1_{“ arg min} z1 › ›_z1_´ _x1›_›2 2 › ›_y_´ ˇˇ_Σ_z1ˇ_ˇ›_› 2ď Σ P t 0, 1umtˆ N φ z1 zs P Cmt _zd_{P C}pN ´ mtq _zs _“ _Σ_z1 ffi z1 Σ x1 ˆz1“ « ˆzs ˆzd ff ´ ˆzs,ˆzd ¯ “ arg min zs_,zd }zs´ xs}2₂` › › ›zd´ xd › › › 2 2 }y ´ |z s_|} 2ď ˆzd“ xd zd zs “ r ëx ppi ¨θq 4 xs “ ρëx ppi ¨ϕq ˆzs “ ˆr ëx p ´ i ¨θˆ ¯ ´ ˆr,θˆ ¯ “ arg min r,θ ÿ k ` r2_k` ρ2_k´ 2 ¨ rk ¨ρk ¨ cospθk´ ϕkq ˘ }y ´ r}₂ď rk ě 0 k rk ρk θˆ “ ϕ ˆzs xs r ˆr “ 1 1 ` λ pρ` λyq λ “ max ˆ 0,1}ρ´ y}₂´ 1 ˙ ΠDy, O pN log N q 4 _zs _“ _r _¨_{ex p}_{pi ¨}_θ_q _zs k “ rk ¨ exp pi ¨θkq k

(89)

}x}_{hT V ,w,a}“ ÿ

pu,vqPΩ

w ru, vs¨ b

pDhxqru, vs2` pDvxqru, vs2` px ´ aqru, vs2 hT V

x a ΩP Z2 _D h Dv T V-3D w Ω 0ă w ru, vsă ` 8 pu, vq PΩ w w “ 1 w hT V T V-3D X T xt 0ď t ď T ´ 1 }X }_{T V -3D} “ T ´ 2_ÿ t “ 0 }xt}hT V ,1,xt ` 1 xt a xt ` 1 R hT V Rw,a,τ “ ! x P RN }x}_{hT V ,w}_,a ď τ ) τ ą 0 Rw,a,τ ﬃ ΠRw,a,τ ΠRw,a,τ pxq “ arg min zPRN }z´ x}₂ }z}_{hT V ,w}_,a ď τ

(90)

l8 Fd d P N˚ d RN d “ 3 px,x1,x2q P F3 x,x1,x2 P RN f P Fd f ppq P RN p P 0, d ´ 1 pt h _f px,x1,x2q_p1q“ x1 f pp,qqP F2 ` f ppq,f pqq ˘ f P Fd Fd l2 xf |gy “ d´ 1_ÿ p“ 0 @ f _ppqˇˇg_ppqD f ,gP Fd }f }₂“ a xf |f y f P Fd F3 l1 l8 5 w P RN 0ă wk ă ` 8 f P F3 }f }_1,w “ ÿ k wk ¨ b ` f p0q ˘₂ k ` ` f p1q ˘₂ k` ` f p2q ˘₂ k }f }_{8 ,w} “ max k wk¨ b ` fp0q ˘₂ k` ` f p1q ˘₂ k` ` f p2q ˘₂ k ` f _ppq˘ k k t h _pt h f P F3 }¨}1,w }¨}_{8 ,w} w F3 R3N }¨}_1,w }¨}_{8 ,w} ﬀ l1 l8 R3N _w _“ ₁ hT V l1 5 _F d lp

(91)

}x}_{hT V ,w,a} “ }px ´ a,Dhx,Dvxq}_1,w ∇ : RN _{Ñ F} 2 ∇x “ pDhx,Dvxq p´ ∇q D i v D i vf “ ´ D_h˚f p0q´ D˚vf p1q f P F2 f ,gP F3 xf |gyď }f }_1,w ¨ }g}_{8 ,}1 w 1 w P R N 1 wk f P F3 gP F3 g‰0 gP F3 f P F3 }¨}_1,w gP F3 λ ě 0 λ}g}_{8 ,}1 w “ sup_f_PF 3 xf |gy }f }_1,w ď λ }¨}_{8 ,}1 w f P F3 sup gPF3 ´ xf |gy ´ τ }g}_{8 ,}1 w ¯ “ # 0 }f }_1,w ď τ ` 8 l1,w τ 1 ! }f }_1,w ď τ ) f ,g P F3 xf |gy “ ÿ k ` f p0q ˘ k ¨ ` gp0q ˘ k ` ` f p1q ˘ k¨ ` gp1q ˘ k` ` f p2q ˘ k¨ ` gp2q ˘ k ď ÿ k wk¨ b ` f p0q ˘₂ k` ` f p1q ˘₂ k` ` f p2q ˘₂ k ¨ 1 wk ¨b `gp0q ˘₂ k` ` gp1q ˘₂ k` ` gp2q ˘₂ k l ooooooooooooooooooooomooooooooooooooooooooon ď}g}_{8 ,}1 w xf |gyď }f }_1,w¨ }g}_{8 ,}1 w f P F3 g

(92)

» — – ` g_p0q˘ k ` g_p1q˘_k ` gp2q ˘ k fi ffi fl “ $ ’’’ ’’’ ’’’ & ’’’ ’’’ ’’’ % wk b pfp0qq 2 k` pfp1qq 2 k` pfp2qq 2 k » — – ` f _p0q˘ k ` f _p1q˘_k ` f p2q ˘ k fi ffi fl » — – ` f _p0q˘ k ` f _p1q˘_k ` f p2q ˘ k fi ffi fl ‰ » — – 0 0 0 fi ffi fl » — – 0 0 0 fi ffi fl gP F3 f » — – ` f p0q ˘ k ` f _p1q˘ k ` f _p2q˘_k fi ffi fl “ $ ’’’ ’’’ ’’’ & ’’’ ’’’ ’’’ % 1 wk¨ b pgp0qq 2 k` pgp1qq 2 k` pgp2qq 2 k » — – ` g_p0q˘_k ` gp1q ˘ k ` g_p2q˘_k fi ffi fl k “ k0 » — – 0 0 0 fi ffi fl k0 k0“ arg maxk w1k b ` gp0q ˘₂ k` ` gp1q ˘₂ k` ` gp2q ˘₂ k g f }g}_{8 ,}1 w “ 1 }f }_1,w “ 1 Rw,a,τ z‹ inf zPRN 1 2}z ´ x} 2 2` 1 ! }pz ´ a,Dhz ,Dvz q}_1,w ď τ ) inf zPRN _fsup_PF 3 1 2}z ´ x} 2 2` @ f _p0qˇˇz´ aD` @f _p1,2qˇˇ∇zD´ τ }f }_{8 ,}1 w z f z sup fPF3 ˆ ´ τ }f }_{8 ,}1 w ´ @ f p0q ˇ ˇ_aD_` _inf zPRN ˆ 1 2}z´ x} 2 2` @ f p0q´ D i vf p1,2q ˇ ˇ_zD ˙ ˙ z f 1 2}x} 2 2´ 12 › ›_x _´ _f _p0q_` _{D i v}_f _p1,2q››2 2 z “ x ´ f _p0q` D i vf _p1,2q

(93)

ΠRw,a,τ pxq “ z ‹ _“ _x _´ _f ‹ p0q` D i vf ‹ p1,2q f ‹ “ arg min fPF3 τ }f}_{8 ,}1 w ` @ f _p0qˇˇaD` 1 2 › ›_x _´ _f p0q` D i vf p1,2q › ›2 2 F3 RN l8 l8 F3 f ÞÑ }f }_{8 ,}1 w λ ě 0 6 _f _Þ_Ñ _λ_}_f _} 8 ,_w1 ﬃ f P F3 P r oxλ} ¨}_{8 ,}1 w pf q “ arg min gPF3 λ}g}_{8 ,}1 w ` 1 2}g´ f } 2 2 inf gPF3 sup hPF3 }h}1,wďλ xh|gy ` 1 2}g´ f } 2 2 sup hPF3 }h}_1,_wďλ inf gPF3 xh|gy ` 1 2}g´ f } 2 2 g 1 2}f } 2 2´ 12}h´ f } 2 2 g“ f ´ h 7 6 7 ϕ ψ H ψpyq “ sup_x_PH xx|yy´ ϕpxq yP H ψ ϕ x“ P r oxϕ pxq` P r oxψpxq x P H ϕ ψ

(94)

P r oxλ} ¨} 8 ,_w1 pf q “ g ‹ _“ _f _´ _h‹ h‹ “ arg min hPF3 }h´ f }₂ }h}_1,w ď λ f l1,w l1 f l1 w h‹ 8 » — — — – ´ h‹_p0q ¯ k ´ h‹_p1q ¯ k ´ h‹_p2q ¯ k fi ffi ffi ffi fl “ max ¨ ˝ 0, 1 ´ ν¨ wk b ` f _p0q˘2_k` `f _p1q˘2_k` `f _p2q˘2_k ˛ ‚ » — – ` f _p0q˘ k ` f _p1q˘_k ` f p2q ˘ k fi ffi fl k ν ě 0 k • }f }_1,w ď λ ν “ 0 • ν }h‹_} 1,w “ λ ν }f }_1,w ą λ rk “ _w1_k b ` f _p0q˘2_k` `f _p1q˘2_k ` `f _p2q˘2_k k ϕ 0, N ´ 1 ffi rk rϕp0qď rϕp1qď ¨ ¨ ¨ď rϕpN ´ 1q l1 h‹ }h‹}_1,w “ N ´ 1_ÿ l “ 0 w_ϕ2_{pl q}¨ max`0, r_ϕ_{pl q}´ ν˘ sk k P 0, N ´ 1 l1 h‹ ν rϕpkq sk 8 ν h‹ h‹ _{“ arg min} hPF3 1 2}h´ f} 2 2` ν}h}1,w ν

(95)

sk “ N ´ 1_ÿ l “ k` 1 w_ϕ2_{pl q}¨`r_ϕ_{pl q}´ r_ϕ_pkq˘ k P 0, N ´ 1 pskqk“ 0,...,N ´ 1 k0 k0“ arg min kP 0,N ´ 1 k sk ď λ k0 sN ´ 1“ 0ď λ k0ě 1 ´ }h‹}_1,w ν“ r_ϕ_pk₀_q ¯ “ sk0 ď λ ă sk0´ 1“ ´ }h‹}_1,w ν“ r_ϕ_pk₀_{´ 1q} ¯ ν }h‹}_1,w “ λ ν “ r_ϕ_pk₀_{´ 1q}, r_ϕ_pk₀_q‰ }h‹}_1,w “ λ ν ν “ $ ’’’ & ’’’ % r_ϕ_p0q¨ }f }1,w ´ λ }f }_1,w ´ s0 k0“ 0 r_ϕ_pk₀_{´ 1q}¨ λ´ sk0 sk0´ 1´ sk0 ` r_ϕ_pk₀_q¨ sk0´ 1´ λ sk0´ 1´ sk0 k0ě 1 O pN log N q prkqk“ 0,...,N ´ 1 ν ΠRw,a,τ • f ÞÑ @f _p0qˇˇaD` ₂1››x ´ f _p0q` D i vf _p1,2q››2 2 ﬀ ´ 1 ` ~D i v~2 ¯ 9 ~D i v~2 • f ÞÑ τ}f }_{8 ,}1 w Jp

(96)

p J‹ pJp´ J‹q 1 p2 1 p ΠRw,a,τ O pN log N q ΠDy, Dy, Rw,a,τ τ Dy, ﬀ τ Rw,a,τ X Dy, τ τ‹ _a _R w,a,τ τ “ 0 τ ă τ‹ τ τ0 xcandidat e τ ΠRw,a,τ xin fin xout fout • xin xout • fin • fout f ‹

(97)

xinit,τinit,∆t olvar,α p Ð 0 τ Ð τinit px0,f 0q Ð pxinit,0q p Ð p ` 1 Ź xp´ 1_{₂Ð ΠDy, pxp´ 1q Ź pxp,f pq Ð ΠRw,a,τ ` xp´ 1{2,f p´ 1 ˘ Ź δpÐ }xp´ xp´ 1}₂{ }xp´ 1}₂ Ź x δpă ∆t olvar xcandidat eÐ xp Ź x τ Ð α¨τ Ź τ α 0ă α ă 1 xcandidat e Ź x xinit τinit ∆t olvar α ∆t olvar τ α 0ă α ă 1 1 ΠRw,a,τ fin ΠRw,a,τ τ pδpq p‹ plogδpq p‹ ´ ∆p ` 1ď pď p‹ _∆_p _s τ s smax

(98)

plogδpq ∆p smax ∆p “ 100 smax “ ´ 10´ 4 25 • 256ˆ 256ˆ 80 ˆ ˆ 5 25 1 3 ﬀ • 256ˆ 256ˆ 80 w hT V RMSE t

(99)

t “ 3 t “ 15 t “ 30 t “ 70 15% t “ 0 w “ 1 t “ 3 t “ 15 t “ 30 t “ 70 25% t “ 0 w “ 1

(100)

0 16 32 48 64 80 0 0.04 0.08 0.12 0.16 0.2 0 0.003 0.006 0.009 0.012 t ? M S E ? M S E t t ă 10 ﬀ • RMSE t ﬀ • RMSE t “ 22 t ą 30 t “ 22 t ă 22 w hT V w “ 1 w ru, vs

(101)

w“ 1 w κ“ 1 w κ“ 10 w κ“ 100 w κ“ 1000 t “ 70 ﬀ w pu, vq PΩ xt xt ´ 1 w ru, vs “ exp ˆ ´ κ b pDhxt ´ 1qru, vs2` pDvxt ´ 1qru, vs2 ˙ κ ą 0 κ “ 100 ﬀ κ

(102)

w “ 1 w w ru, vs “ 1 η` b pDhxt ´ 1qru, vs2` pDvxt ´ 1qru, vs2 η ą 0 w ff T V-3D ff ffi

(103)

ﬃ MSE PG-URE MSE PG-URE PG-URE ﬀ

(104)

9 : 9

:

θ

(105)

ffi ff ffi ffi ˆx “ arg min x 1 2}x ´ y} 2 2` λ}x}T V y ˆx }¨}_{T V} λ ą 0 λ λ Ñ 0 ffi ff ff k ˆxk “ 1 Z pkq ÿ l w pk, l qyl

(106)

y l w pk, l qě 0 k l Z pkq “ ř _lw pk, l q w pk, l q w pk, l q “ exp ˜ ´ }πpkq´ πpl q} 2 2 h2 ¸ πpkq y k h ą 0 λ ﬀ y f θ θ ˆx “ f θpyq θ ˆx x MSE MSE “ 1 N }f θpyq´ x} 2 2 N

(107)

y _ˆx ˆx1“ ˆx2“ ˆx3“ y1“ y2“ y3“ FT FT FT F u si o n CS CS CS y yr y ffi yr ˆxr PT V T V ˆxr ˆx ff ffi yr r “ 1, . . . , R ˆxr PT V ˆxr ˆx x yr y ˆxr ˆxr g ˆx “ gpˆx1, . . . ,ˆxRq R

(108)

PT V ffi yr ˆxr x MSE SURE SURE SURE y x b y “ x ` b b„ N `0,σ2I d˘ σ 1_f _SURE SURE “ 1 N }f pyq´ y} 2 2´ σ2` 2σ2 N Divf pyq Divf pyq “ ř _k Bfk Byk pyq f 2 _MSE _SURE b E t MSEu “ E t SUREu SURE MSE SURE « MSE ff MSE SURE x x SURE MSE θ1, . . . ,θK K f θ 1 _θ fθ 2 _f ff E !_ř k ˇ ˇ ˇBfk Byk pyq ˇ ˇ ˇ ) ă ` 8 f f ff

(109)

θk‹ MSE SURE pθkq y SURE SURE SURE SURE Bfk Byk pyq f ff ff y SURE ff ff SURE

(110)

yk xk yl y „ P pxq ﬀ PURE MSE PURE “ 1 N ´ }f pyq}2₂` }y}2₂´ 2 A y ˇ ˇ ˇf r´ 1spyq E ´ x1|yy ¯ f r´ 1spyq ´ f r´ 1s ¯ kpyq “ fkpy ´ ekq k f f r´ 1spyq f r´ 1spyq « f pyq´ Bf pyq Bf pyq pBf q_kpyq “ Bfk Byk pyq k PURE “ 1 N ´ }f pyq´ y}2₂` 2 xy|Bf pyqy ´ x1|yy ¯ A y ˇ ˇ ˇf r´ 1spyq E xy|Bf pyqy SURE y f ﬃ SURE

(111)

σ“ 0 σ“ 10´ 1.5 _σ_{“ 10}´ 1 _σ_{“ 10}´ 0.5 ζ “ 0 ζ “ 1 0 ´ 2 ζ “ 1 0 ´ 1 ζ “ 1

Increasing Gaussian behavior

In c re a si n g P o is so n b e h a v io r y ﬀ σ ζ σ “ 0 ζ “ 0 x

(112)

3 y “ ζz` b $ ’& ’% z „ P ˆ x ζ ˙ b„ N `0,σ2I d˘ z b • σ ě 0 b • ζ ě 0 4 ζ “ 0 σ ą 0 ζ “ 1 σ“ 0 ﬀ ζ σ σ ζ y ζ ě 3 PG-URE MSE PG-URE “ 1 N ´ }f pyq}2₂` }y}2₂´ 2 A y ˇ ˇ ˇf r´ ζspyq E ` 2σ2Divf r´ ζspyq´ ζx1|yy ¯ ´ σ2 f r´ ζspyq 3 4 ζ“ 0 y“ x` b ζz z„ P ´ x ζ ¯ x ζÑ 0

(113)

´ f r´ ζs ¯ kpyq “ fkpy´ ζekq E t PG-UREu “ E t MSEu f f r´ ζspyq f r´ ζspyq « f pyq´ ζBf pyq PG-URE PG-URE “ 1 N ´ }f pyq´ y}2₂` 2@ζy ` σ21ˇˇBf pyqD´ 2σ2ζ@1ˇˇB2_f _p_y_qD_´ ζx1|yy ¯ ´ σ2 Bf pyq B2_f _p_y_q pBf q_kpyq “ Bfk Byk pyq `B2_f ˘ kpyq “ B2fk By2 k pyq k PG-URE ζ f r´ ζspyq « f pyq´ ζBf pyq PG-URE SURE PURE σ ζ ζ f r´ ζs Bf B2_f ﬃ MSE x f r´ ζs Bf B2_f f f

(114)

f y f pyq f r´ ζspyq f N y py´ ζekq k “ 0 N ´ 1 N Bf B2_f _ff ff 1`fkpy ` ekq´ fkpyq ˘ kt h _Bf _p_y_q f py ` ekq k “ 0 N ´ 1 PG-URE f N f @ ζy` σ21ˇˇBf pyqD f y y ą 0 δ xu|Bf pyqy u “ ζy` σ21 u u “ 1 xu|Bf pyqy “ Divf pyq f ff f py ` δq “ f pyq` ÿ l δl Bf Byl pyq` r p q r p q 0 Ñ 0 lim Ñ 0 B δˆ u ˇ ˇ ˇ ˇ f py ` δq´ f pyqF “ ÿ k,l ukδkδl Bfk Byl pyq k l

(115)

δ • δk δ • δk 0 1 5 _δ Eδ " lim Ñ 0 B δˆ u ˇ ˇ ˇ ˇ f py ` δq´ f pyqF * “ ÿ k uk Bfk Byk pyq lim Ñ 0Eδ " B δˆ u ˇ ˇ ˇ ˇ f py ` δq´ f pyqF * “ xu|Bf pyqy ffi xu|Bf pyqy 1_x_δ_{ˆ u}_|_f _p_y _` _δ_q´ _f _p_y_qy ffi xu|Bf pyqy xu|Bf pyqy “ 1@δˆ uˇˇf py ` δq´ f pyqD f @vˇˇB2_f _p_y_qD _v _“ ₁ v “ 1 v f ff f py ` δq “ f pyq` ÿ l δl Bf Byl pyq` 2 2 ÿ l ,m δlδm B2f BylBym pyq` 2r p q f py ´ δq 5 y y Eδ

(116)

lim Ñ 0 B δˆ v ˇ ˇ ˇ ˇ f py ` δq´ 2f pyq` f py´ δq 2 F “ ÿ k,l ,m vkδkδlδm B2fk BylBym pyq δ κ δk δk Eδtδkδlδmu k “ l “ m Eδ δ3k ( “ κ ‰0 lim Ñ 0Eδ " B δˆ v ˇ ˇ ˇ ˇ f py ` δq´ 2f pyq` f py´ δq 2 F * “ κ@vˇˇB2_f _p_y_qD δ ﬃ @ vˇˇB2_f _p_y_qD @ vˇˇB2_f _p_y_qD_“ 1 2_κ @ δˆ vˇˇf py ` δq´ 2f pyq` f py ´ δqD PG-URE f PG-URE “ 1 N }f pyq´ y} 2 2´ ζ N x1|yy ´ σ 2_` 2 N 9 A 9 δˆ `ζy ` σ21˘ ˇˇˇf ´ y ` 9δ9 ¯ ´ f pyq E ´ 2σ 2_ζ N :2_κ A : δ ˇ ˇ ˇf ´ y ` :δ: ¯ ´ 2f pyq` f ´ y ´ :δ: ¯ E PG-URE • δ9 f 9 δk 0 1 9 δk ´ 1 1 1_{₂ PG-URE 9 δ • δ: : δ 6 _E ! : δk ) “ 0 E ! : δ2 k ) “ 1 6 :_δ k

(117)

E ! : δ_k3 ) “ κ ‰ 0 PG-URE : δ π π ˆ : δk “ ´ c q p ˙ “ p π ˆ : δk “ c p q ˙ “ q $ & % p “ 1 2` κ 2 ` κ2` 4˘´ 1{2 q “ 1 ´ p κ π κ 1 • 9 : f 9 : PG-URE PG-URE 4 Cf ` O pN q Cf f PG-URE PG-URE 9 δ δ: V _δ,9_δ: t PG-UREu 1_{2 PG-URE δ9 δ: 9 δ δ: PG-URE V _δ,9_δ: t PG-UREu “ 1 N2V δ9 # ÿ k,l ak,lδ9kδ9l + l ooooooooooooomooooooooooooon Vδ9 ` 1 N2_κ2V δ: # ÿ k,l ,m bk,l ,mδ:kδ:lδ:m + l ooooooooooooooooooomooooooooooooooooooon Vδ: ak,l bk,l ,m

(118)

ak,l “ 2 ` ζyk ` σ2 ˘ Bf_k Byl pyq bk,l ,m “ 2σ2ζ B2fk BylBym pyq V_δ9 9 δ pt h 9 δ m9p“ E ! 9 δ_kp ) V_δ9 N2V_δ9“ ÿ k,l ,m ,n ak,lam ,nE_δ9 ! 9 δkδ9lδ9mδn9 ) ´ ÿ k,l ak,kal ,l 9 δk m9₁“ 0 E_δ9 ! 9 δkδ9lδm9 δn9 ) 0 k l m n ﬀ m92“ 1 N2V_δ9“ m94 ÿ k a2_k,k` ÿ k,l k‰l ` ak,kal ,l ` a2k,l` ak,lal ,k ˘ ´ ÿ k,l ak,kal ,l V_δ9 V_δ9“ 9 m₄´ 1 N2 ÿ k a2_k,k` 1 2N2 ÿ k‰l pak,l` al ,kq2 V_δ9 PG-URE 9 δk 9 m4 m91 “ 0 m92 “ 1 9 m4 ě 1 9 m₄ “ 1 ´ 1 1 1_{₂ k l N pN ´ 1q k, l “ 1 N k “ l pak,l` al ,kq2 0 ak,l Bf_Byk l pyq k l f yl kt h fkpyq pak,l ` al ,kq2 N V_δ9 _N1 N V_δ9 PG-URE δ9

(119)

256 ˆ 256 r0, 1s ffi f PG-URE V_δ: δ: ffi bk,l ,m m:p“ E ! : δ_kp ) V_δ9 V:_δ _N1 f V_δ: δ: κ‹ κ κ‹ ffi bk,l ,m f κ “ 1 PG-URE MSE 9 δ δ: PG-URE 9 : PG-URE “ MSE 256 ˆ 256

(120)

256 ˆ 256 0 1 x y • σ“ 10´ 1.5 ζ “ 10´ 2 • σ“ 10´ 1 _ζ _{“ 10}´ 2 • σ“ 10´ 1.5 ζ “ 10´ 1 • σ“ 10´ 1 _ζ _{“ 10}´ 1 θ • f W So_θ pyq “ W ´1_¨_T θpW ¨yq W Tθ ﬃ w sign pwq ¨ max p0, |w| ´ θq • T V f T V_θ pyq “ arg min x }x}T V }x ´ y}2ď θ • f NL M_θ pyq 5 ˆ 5 • yk xk f S-W So_θ pyq “ S´ 1˝ f W So_θ ˝ Spyq f S-T V_θ f S-NL M_θ Spyq Skpyq “ 2 ζ sign ptq a |t| t “ ζyk ` 3 8ζ 2_` _σ2 Skpyq 1 xk

(121)

θ f θpyq MSE PG-URE ff 9 : ® 9 : PG-URE 9 : 9 : ff PG-URE PG-URE “ T0` T1p9q` T2p: q T0“ 1 N }f pyq´ y} 2 2´ ζ N x1|yy ´ σ 2 T1p9q “ 2 N 9 A 9 δˆ `ζy ` σ21˘ ˇˇˇf ´ y ` 9δ9 ¯ ´ f pyq E T2p: q “ ´ 2σ2ζ N :2_κ A : δ ˇ ˇ ˇf ´ y ` :δ: ¯ ´ 2f pyq` f ´ y´ :δ: ¯ E T0 PG-URE 9 : T1p9q T2p: q f T0 T1p9q T2p: q θ ff 9 : 9 T1p9“ 0.1q T1p9“ 1q T1p9q 9 9ď 10´ 3 T1p9q 9 9Ñ 0 T1p9q 9 10´ 7 1 100.02 9“ 10´ 7, 10´ 6.98, 10´ 6.96, . . . θ MSE MSE‹ T1p9q ∆T1 ∆T1“ 1 MSE‹ 9Pr 9St dDevm i n, 9m axs T1p9q St dDev_{9Pr 9}_{m i n}_{, 9}_{m ax}_sT1p9q T1p9q 9 r 9min, 9maxs “ 10´ 7_{, 1}‰

(122)

70 80 90 ₁₀0 ₂₀0 ₃₀0 ₄₀0 ₅₀0 ´2 ˆ 10 ´3 0 2ˆ 10 ´3 4ˆ 10 ´3 6ˆ 10 ´3 8ˆ 10 ´3 θ M S E P G -U R E T0 T1 p 9q T2 p: q 9“ 10´ 4 9“ 10´ 3 9“ 10´ 2 9“ 10´ 1 9“ 1 : “ 10´ 4 : “ 10´ 3 : “ 10´ 2 : “ 10´ 1 : “ 1 9“ 10´ 4 : “ 10´ 4 9“ 10´ 3 _{: “ 10}´ 3 9“ 10´ 2 : “ 10´ 2 9“ 10´ 1 : “ 10´ 1 9“ 1 : “ 1 T1p9q T2p: q T0 M SE PG-URE ` f S-T V_θ T V MSE PG-URE θ PG-URE T0 T1p9q T2p: q 9 : PG-URE 9“ 10´ 4 _{: “ 10}´ 2

(123)

0.4 0.5 0.6 0.7 0.8 0.9 1 2 ´2 ˆ 10 ´3 0 2ˆ 10 ´3 4ˆ 10 ´3 6ˆ 10 ´3 8ˆ 10 ´3 10 ´2 θ M S E P G -U R E T0 T1 p 9q T2 p: q 9“ 10´ 4 9“ 10´ 3 9“ 10´ 2 ₉ “ 1₀ ´ 1 : “ 1 0 ´ 4 : “ 10´ 3 : “ 10´ 2 : “ 10: “ 1´ 1 ` f S-NL M_θ

(124)

L o w n o is e M o st ly G a u ss ia n M o st ly P o is so n H ig h n o is e L o w n o is e M o st ly G a u ss ia n M o st ly P o is so n H ig h n o is e fW So_θ fT V θ fN L M θ f S-W So_θ fS-T V_θ fS-N L M_θ ∆T1 9“ 10´ 6, 10´ 5.98, . . . , 10´ 3.02, 10´ 3 1% ∆T1 r 9min, 9maxs “ “ 10´ 6, 10´ 3‰ T1p9q 9 MSE 9 “ 10´ 6, 10´ 3‰ T1p9q PG-URE 9“ 10´ 4 0 1 ﬀ ﬀ 0 255 9 “ 255 ˆ 10´ 4 T1p9q : : T2p: q : “ 10´ 4, 10´ 3.99, . . . , 10´ 0.02, 10´ 0.01, 1 θ MSE T2p: q : T2p: q : Ñ 0 T2p: q : ď 10´ 3 T2p: q

(125)

10´ 4 ₁₀´ 3 ₁₀´ 2 ₁₀´ 1 ₁ -0.5 0 0.5 1 1.5 : 1 M S E T2 p: q f W So θ fS-N L Mθ fS-T V θ fS-W Soθ fS-N L M_θ fS-N L M_θ T2p: q : θ MSE T2p: q MSE ﬀ f ´ y ` :δ: ¯ ´ 2f pyq` f ´ y ´ :δ: ¯ f ´ y ˘ :δ: ¯ f pyq : T2p: q : Ñ 0 ﬀ : : : 9 : “ 10´ 2 T2p: q ∆T2 ∆T2“ 1 MSE‹ : Pr:St dDevm i n,:m axs T2p: q :

(126)

L o w n o is e M o st ly G a u ss ia n M o st ly P o is so n H ig h n o is e L o w n o is e M o st ly G a u ss ia n M o st ly P o is so n H ig h n o is e fW So_θ fT V θ fN L M θ f S-W So_θ fS-T V_θ fS-N L M_θ ∆T2 5ˆ 10´ 3ď : ď 2ˆ 10´ 2 100.01 ∆T2 r:min, :maxs “ “ 5 ˆ 10´ 3, 2 ˆ 10´ 2‰ T2p: q PG-URE : 1% MSE T2p: q f NL M_θ : f S-T V_θ r:min, :maxs ∆T2 : “ 10´ 2 PG-URE θ : θ PG-URE θ f θ θ MSE PG-URE 7 θ‹ M SE θ‹PG-URE

MSE PG-URE ˆxPG-URE “

f θ‹

P G -U R E pyq

(127)

L o w n o is e M o st ly G a u ss ia n M o st ly P o is so n H ig h n o is e L o w n o is e M o st ly G a u ss ia n M o st ly P o is so n H ig h n o is e f W So_θ 31 dB 27 dB 25 dB 24 dB 33 dB 30 dB 27 dB 27 dB f T V_θ 36 dB 32 dB 28 dB 27 dB 38 dB 35 dB 31 dB 30 dB f NL M θ _{35 dB} _{32 dB} _{27 dB} _{26 dB} _{39 dB} _{36 dB} _{29 dB} _{29 dB} f S-W So_θ 30 dB 27 dB 23 dB 22 dB 33 dB 30 dB 26 dB 24 dB f S-T V θ _{35 dB} _{31 dB} _{27 dB} _{24 dB} _{37 dB} _{34 dB} _{28 dB} _{27 dB} f S-NL M_θ 36 dB 32 dB 27 dB 25 dB 39 dB 36 dB 29 dB 28 dB ∆ PSNR ˆxM SE PG-URE ˆxM SE “ f θ‹ M SE pyq MSE ﬀ ∆ “ }ˆxPG-URE ´ ˆxM SE} 2 2 }x ´ ˆxM SE}22 ∆ l2 l2 x ∆ ˆxM SE PSNR “ ´ 10 log₁₀ ´ 1 N }x ´ ˆxM SE} 2 2 ¯ ˆxM SE ˆxPG-URE θ_{M SE}‹ θ‹_PG-URE MSE PG-URE ∆ 5% ﬀ

(128)

ˆxM S E ˆxP G -U R E fW So θ fT Vθ fS-N L Mθ f S-T Vθ ˆxM SE ˆxPG-URE ﬀ PSNR ˆxM SE ˆxPG-URE ∆ ď 20% ˆxM SE ˆxPG-URE PG-URE MSE f S-T V_θ θ MSE PG-URE : ∆T2 9 δ δ: PG-URE 9 δ δ: T1p9q T2p: q

(129)

PG-URE PG-URE PG-URE f PG-URE θ PG-URE PG-URE MSE PG-URE E t PG-UREu “ E t MSEu y “ x ` b x P RN b „ N `0,σ2I d˘ φ : RN Ñ RN ﬀ E ! ˇ ˇ ˇBφk Byk pyq ˇ ˇ ˇ ) ă ` 8 k E xb|φpyqy( “ σ2E Divφpyq(

(130)

z P RN z „ P pxq zk xk ψ : RN Ñ RN E t |ψkpzq|uă ` 8 k E xx|ψpzqy( “ E ! A z ˇ ˇ ˇψr´ 1spzq E) y “ ζz ` b b „ N `0,σ2I d˘ z „ P ´ x ζ ¯ b z φ : RN Ñ RN ﬀ E t |φkpyq|uă ` 8 E ! ˇ ˇ ˇBφk Byk py´ ζekq ˇ ˇ ˇ ) ă ` 8 k E xx|φpyqy( “ E ! A y ˇ ˇ ˇφr´ ζspyq E ´ σ2Divφr´ ζspyq ) ψb : RN Ñ RN ψbpzq “ φpζz ` bq E xx|φpyqy( “ E " ζEz " B x ζ ˇ ˇ ˇ ˇψbpzq F * * “ E ! ζEz ! A z ˇ ˇ ˇψbr´ 1spzq E) ) “ E ! A y ´ b ˇ ˇ ˇφr´ ζspζz ` bq E) “ E ! A y ˇ ˇ ˇφr´ ζspyq E) ´ E ! Eb ! A b ˇ ˇ ˇφr´ ζspyq E) ) “ E ! A y ˇ ˇ ˇφr´ ζspyq E) ´ E ! σ2Eb ! Divφr´ ζspyq ) ) “ E ! A y ˇ ˇ ˇφr´ ζspyq E ´ σ2Divφr´ ζspyq ) MSE E t MSEu “ 1 NE }f pyq} 2 2´ 2 xx|f pyqy ` xx|yy ( φ “ f φ “ I d PG-URE f V_δ: : δ PG-URE

(131)

ﬃ bk,l ,m m: p“ E ! : δp_k ) : δ : mp V_δ: V_δ: • ck “ bk,k,k k • dk,l “ bk,k,l` bk,l ,k ` bl ,k,k k ‰l • Dl “ ř k,k‰ldk,l l • ek,l ,m “ bk,l ,m ` bk,m ,l ` bl ,k,m ` bm ,k,l ` bl ,m ,k ` bm ,l ,k pk, l , mq k ‰l k ‰m l ‰m : m1“ 0 :m2“ 1 :m3“ κ V_δ: N2κ2V_δ: “ V _δ: # ÿ k,l ,m bk,l ,m:δkδ:l:δm + “ ÿ i ,j ,k,l ,m ,n bi ,j ,kbl ,m ,nE_δ: ! : δi:δj:δkδl: δm: δn: ) ´ ˜ ÿ k,l ,m bk,l ,mE_δ: ! : δkδl: δ:m ) ¸ 2 E_δ: ! : δkδ:lδ:m ) “ 0 k “ l “ m : δ m:1 “ 0 N2κ2V_δ: “ ÿ i ,j ,k,l ,m ,n bi ,j ,kbl ,m ,nE_δ: ! : δiδ:jδ:kδ:l:δmδ:n ) l ooooooooooooooooooooooomooooooooooooooooooooooon S6 ´ κ2ÿ k,l ckcl S6 E_δ: ! : δi:δjδ:kδ:lδm: :δn ) “ 0 ﬀ S6 E_δ: ! : δiδ:j:δk:δl:δmδ:n ) S6“ :m6T6` :m4T4,2` κ2T3,3` T2,2,2 • T6 S6 T6“ ř kc2k • T4,2 i “ j “ l “ n ‰k “ m

(132)

• T3,3 • T2,2,2 8 T4,2“ ÿ k‰l d2_k,l ` 2ÿ k ckDk T3,3 “ ÿ k‰l ckcl ` ÿ k‰l dk,ldl ,k T2,2,2“ ÿ k D2_k´ ÿ k‰l d2_k,l` 1 6 ÿ k‰l ‰m e2_{k,l ,m} N2V_δ: “ : m6´ :m24´ κ2 κ2 ÿ k c2_k ` m:4´ κ 2_{´ 1} κ2 ÿ k‰l d2_k,l ` 1 κ2 ÿ k p:m4ck` Dkq2 ` 1 2 ÿ k‰l pdk,l` dl ,kq2` 1 6κ2 ÿ k‰l ‰m e2_{k,l ,m} mp Hp Hp“ » — — — — — — — — — – 1 m1 m2 ¨ ¨ ¨ mp m1 m2 . . . _.. . m2 . . . _.. . .. . . .. ... mp ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ m2p fi ffi ffi ffi ffi ffi ffi ffi ffi ffi fl : m6´ :m24´ κ2ě 0 m:4´ κ2´ 1ě 0 V_δ9 V_δ: PG-URE f B2fk BylBym pyq k l m bk,l ,m ffi dk,l ek,l ,m N V_δ: _N1 8 ř k ‰l ‰m pk, l , mq k ‰l l ‰m m ‰k ‰

(133)