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HT iBbrc(u, v) = (IUnf − IUnf iB)(u, v) HT iB
prt(u, v) = (IUnf • iB −IUnf)(u, v) + 8 + % % # B # i QQQ•Q 8 + + K% (
) /40 K ! K /0 + K + K ( J :(= + K I + K(
:(=* < > " 3 I IUnf INor
./ ! 012
# B % ! !% %H + ( ' # ! + U5>V U44V( % K K %
;4
! / K ? 0 % + *
IMej(u, v) =∥∥∥IDFT2
‖DFT2 INor(u, v)‖k DFT2 INor(u, v)
∥∥∥ ,
/:(60 #* (u, v) ∈ SNB,NTR
ij % % SNB ,NTRij (i, j)
INor NBBNB NTR B% DFT2 IDFT2 + % k B # ‖•‖ (
U5>V k = 0, 6 U44V 8 k = 1, 4( ) H ! % % * K 8 H (
' + # / 0 & % #% # k % + ! HB % # HB ( HB + % K % # + # + B % # ! H B ! +( J :( NB = 32% NTR = 20 k = 1, 0(
#' 0
' 8 % K ? 7 U;;V% U:>V%U:5V% U5:V% P + 7 U4;V% U56V% ! + 7 8 % U4=V% U46V% U6V( ) ! G% & % P% # H 8 UV% U>>V% # UV
;
:(* & !(2+ "
+P U=V( . % P% G 8 % G # H (
• 7 !B (u, v) I G # + # K (u, v) B K / :(>0( ' 8 % ! % % # ( ' Θ % 7 K NF ilBNCol Θ(i, j) (i, j) I% #* NFil ≤ N I
F il NCol ≤ N ICol(
• 7 P K# + (" K B
;:
% P 1( % K ! 8% P 0% ? # 8 P 1% # 8 / 0 P 0(
:(>* = % ,"
( 3% &
TH H + ( ' #H H K G ( + U6V% U:V U>=V% U;>V% U6V U;V( *
8$($ < # Gx Gy% + B (
! $$ . % ! (
T? (
;;
8$($ < #
) Gx(u, v) Gy(u, v)( B + K +* H U:V% K + H U:V ( + K H % 7 " (
! $$
J K # U6V U;V U6V U5V(
) # B B % 7 Gp(u, v)% # # G ( H% # K % ?H # % # # 180o(
Gd(u, v) = (Gx(u, v) + jGy(u, v))2
Gd(u, v) = (G2x(u, v)−G2
y(u, v)) + j(2Gx(u, v)Gy(u, v)),
/:(=0
% K Skij Sk
ij% #H Gp(i, j)*
Gp(i, j) = 1N2
k
∑(u,v)∈Sk
ij
Gd(u, v)
Gp(i, j) = (Γxx(i, j)− Γyy(i, j)) + j(2Γxy(i, j)), /:(0
*
Γxx(i, j) =1
N2k
∑(u,v)∈Sk
ij
G2x(u, v), /:(>0
Γyy(i, j) =1
N2k
∑(u,v)∈Sk
ij
G2y(u, v), /:(50
;6
Γxy(i, j) =1
N2k
∑(u,v)∈Sk
ij
Gx(u, v)Gy(u, v), /:(40
% # Φ(i, j) *
Φ(i, j) =1
2arctan
(2Γxy(i, j)
Γxx(i, j)− Γyy(i, j)
), /:(440
#* Φ(i, j) ∈ [−π/2, π/2] arctan (y/x) y/x x y(
) P H 7 *
Coh(i, j) =‖Gp(i, j)‖
1N2
k
∑(u,v)∈Sk
ij
‖Gd(u, v)‖ /:(40
/:(0 # Gd(u, v) % # + Φ(i, j) Coh(i, j)% B % Gd(u, v) G # Gd(u, v) #% # P( % 2 U6V 8 # + + H (
% Skij
Λr% #* Skij = Λ1 ∪ Λ2 ∪ . . . ∪ Λr ∪ . . . ∪ ΛM % Λr H
+ NΛ B(3 % Λr #H
P*
Gp(r) =1
NΛ
∑(m,n)∈Λr
Gd(m, n), /:(4:0
Coh(r) =‖Gp(r)‖
1NΛ
∑(m,n)∈Λr
‖Gd(m, n)‖ . /:(4;0
# % Sk
ij *
Gpn(i, j) =1
M
M∑r=1
w(r)Gp(r), /:(460
;=
7 *
w(r) =Coh(r)
‖Gp(r)‖ , /:(4=0
" /:(4=0 /:(460 *
Gpn(i, j) =1
M
M∑r=1
Coh(r)Gp(r)
‖Gp(r)‖ , /:(40
/:(4;0% /:(0 /:(=0 /:(40 *
Gpn(i, j) =1
M
M∑r=1
(Γxx(r)− Γyy(r)) + j(2Γxy(r))
Γxx(r) + Γyy(r). /:(4>0
% Φ(i, j) Coh(i, j)% 8*
Φ(i, j) =1
2arctan
(Im Gpn(i, j)Re Gpn(i, j)
)/:(450
Coh(i, j) =‖Gpn(i, j)‖1M
M∑r=1
Coh(r)
≈ ‖Gpn(i, j)‖ /:(0
) % % U5V% % # H Φ(i, j)( + Sk
ij 7 *
D1 = Ski−1,j−1 ∪ Sk
i−1,j ∪ Ski,j−1 ∪ Sk
i,j
D2 = Ski−1,j ∪ Sk
i−1,j+1 ∪ Ski,j ∪ Sk
i,j+1
D3 = Ski,j−1 ∪ Sk
i,j ∪ Ski+1,j−1 ∪ Sk
i+1,j
D4 = Ski,j ∪ Sk
i,j+1 ∪ Ski+1,j ∪ Sk
i+1,j+1
/:(40
#* Ski,j = D1 ∩ D2 ∩ D3 ∩ D4% Dr
Φr Cohr% 8 /:(440 /:(40%7 Φ(i, j) Sk
ij Φr # ? HB Cohr(' Φ% Θ(
K K Φ( K U6=V% % Φ / ? [cos(2Φ), sin(2Φ)]0 7 H wsv*
;
Υx(i, j) =∑
(m,n)∈Ssv
wsv(m, n) cos(2Φ(i + m, j + n)) /:(0
Υy(i, j) =∑
(m,n)∈Ssv
wsv(m, n) sin(2Φ(i + m, j + n)) /:(:0
Θ 8*
Φsv(i, j) =1
2arctg(
Υy(i, j)
Υx(i, j)) /:(;0
Θ(i, j) =
Φsv(i, j) Φsv(i, j) ≥ 0
Φsv(i, j) + π Φsv(i, j) < 0/:(60
J :(5 :(4 P % # 8 H #% B U6V # + 8 H #% #% # (
:(5* Θ ( % % 6?@9 % 68A9
;>
:(4* Θ B ( % % 6?@9 % 68A9
#+ /
J 8 # # 8% K B !% % P 7 ( J % K +(
# 4 52 6
"" &
+P B ! & ( 7 +P # # HB / :(440(
B % Θ% ? *
• U6>V ! ( % 7% ! ! ! Θ /K
;5
:(44* !+
K Θ 8 % + 0( ' % +P 8 # HB (
• U64V Θ%# K +P H % # + 8 7 K (i, j) Θ( ! % 87 ! & ( ' % + +P 8 (
• U>4V # Θ # +P % ! ! & # (
• U=V H P / P # & P0 ! ( & P ! ! (
J K +P% % # +P
6
% H I ! (
6 / "
+P( ) I + ( ) /,'0% /330 /N30 P % # /M0 P % /'0 ( % H % ,'% 33% N3% M '(
! % U6=V% U4=V% U>V% U5=V( H U>;V% U>V( ) ! Θ % # 2 U6V K Θ ( + G U6V( ) *
6 / . 6 ! ( " % +P ! ( " +P(
6 / =! <> " 7 ! + K G K Θ( K K +P K K(
' ? (
6 / . 6
" C + Θ% ! PΘ,C 7 XB Θ C /T :(40% ?% 8 Θ C( ) ! PΘ,C (i, j) Θ *
64
4( "? Cr + # 7 #OP Θ% ! r W Cr H # NCrBNCr % #* NCr = 2r + 1 r > 0% r = 0% Cr H # 2B2( , H% :(4(
:(4* 1, ! & %
) Cr ( " # % # I ( % % (
( PΘ,C(i, j) + 8 ? Cr( " εx εy
x y Cr% # # (0, 0) % ! H *
PΘ,C(i, j) =
N2Cr
−1∑k=0
∆ij(k), /:(=0
*
6
∆ij(k) =
δij(k) |δij(k)| < π/2π + δij(k) δij(k) ≤ −π/2π − δij(k) outro caso
, /:(0
#*
δij(k) = Θ(i+ εx(k +1), j + εy(k +1))−Θ(i+ εx(k), j + εy(k)). /:(>0
PΘ,C(i, j) B 0o,±180o,±360o % (i, j) H * B 0o% B 180o B −180o% :(4:(
:(4:* 1B & % r = 1
:( ' PΘ,C Θ K # + ! ( # # ( ' % H /T :(4;0(
;( " % +P ! ( " +P% % # + ' H +PH + G(
6 / =! <>
' + G U6V '( ) H + G
6:
:(4;* ,% & % & % ,B - '
K G K (i, j) Θ( ) K B 0 K + ( ) H *
4( (i, j) Θ ! % K 8 qBq /q = 30 (i, j) Θi,j( , KΘi,j Π(
( K G Θi,j( K *
Σij =1
q − 1
q∑k=1
(Θkij − Uij)(Θ
kij − Uij)
T , /:(50
Θkij k K Θij Ui,j
K Θij (
6;
:( HB Σij ( ) + G (i, j) H HB & K G Σij % λij(
;( ) /40 /:0 # (i, j) Π% + G% K Ψ / 7 :(46(0(
6( Ψ K max(Ψ)/6 / max(Ψ) HB Ψ0% #% 6 + U44V( 3 % KK( . # +P H /7 :(46(0(
:(46* !+ : - ! , ' - B B ! , ' !+ :
66
6 / ! !?$"$ .< " !#!
' 7 # ?7 K% # ! 7# ( ' % H ! B % B H +P 8 '% # +P 8 H ! # H ! # 7 +P '( % # ? + # +P ## % # ( ' B # # % # QI ! Q(
B + 8 XB 8 + +P( % :(4=% # XB +P # % 0o% XB + ( % # +P P #K 0o( U>4V +P # K ! # +P H K Θ ? #K ?0o( . + B +P / +P H #K 0o0 7 +P '( % 8 H Θ I ! #K % # #K 8 K 0o ? K % H (
[ Θ #K Q / ?% ! Q 8 π/Q 0 K Θ0% Θ1% · · · % ΘQ−1*
6=
:(4=* ," !+
Θ(i, j) =
Q−1∑k=0
kπ
QΘk(i, j), /:(:0
#*
Θk(i, j) =
1 k =
⌊Θ(i, j)Q
π
⌋0 caso contrario
, /:(:40
* • B (. Θk H 1
# Θ # kπ/Q% K # Θk k / #K 0 ( H I ! H K Θk / ! H0 # ! Θ ? ( # ! Θ +P 7 *
ΘBd (i, j) = Θm(i, j) + Θn(i, j), /:(:0
#* m = mod (−d, Q) n = mod (Q−1−d, Q)% % d G # 7 ! H ( #ΘB
d H / :(40% 8 mπ/Q nπ/Q 1% # 8 7 8 + ΘB
d 7 /:(:0( 8 ,'% N3% 33% M ,3 +P
H # ? H K
6
:(4* I Θ ( " 32 &, ΘB0 Θ
B1 Θ
B2 !Θ
B3
ΘBd / :(4>( :(4>(+0 '
+P H 7 ? H ΘB
d / :(4>(0 % & # ΘB
d % ! +P% +P 7 ( K U4V 7 *
ΘCd (i, j) =
Q−1∑r=0
Q−1∑k=0
wd(k, r)Θ mod (k−d,Q)(i, j), /:(::0
wd(k, r) = exp
(− 1
2σ2(k − mod(r − d, Q))2
). /:(:;0
)% wd + Θk% # # 8 mπ/Q nπ/Q ! % K K + # 8 ΘC
d % ! G # # 8mπ/Q nπ/Q / :(450(
+ 8 I ! ΘBd ΘC
d % # +P
65
0, π/Q, · · · , (Q− 1)π/Q(( ! #
K P Coh 8 Π( % I Π% # NPri% I Coh # + tQ% # NCoh( 7 [Qc(tQ) = NCoh/NPri% % H # Qc(tQ) H 1% #% tQ 7 Qc(tQ) /& # tQ B 1 8 7 #0(
:( d = 0, 1, · · · , Q/8− 1% K*
• " Qc(tQ) # 0.8% ΘBd % H
ΘCd ( J # ΘC
d ? % + % K 8 mπ/Q nπ/Q% # H H ΘB
d (
• $ ΘBd Π
Θ (
• K Bm ΘBd (
ΘBd ? 1
# ( , #* Bm(i, j) = 1 (i, j) ?H ( / :(0
• 7K Pe K Bm(' Bm 7 N K I K U;V% # 8 7K H 7/ Bm 70% 7K # ? H ΘB
d ( , 8 K 7K Pe = (ip1, jp1), (ip2, jp2), · · · (• NK 7 Pe # 8 # H ΘB
d ( % # ? W % # ? ? 1 # 2W %
=
:(* " B % Bm D 1 ΘB
d "
# # + ΘBd /
ip ≥ Nfil/1.6 0W # % B K 3B3 ΘB
d (ip, jp) ?H + + 0% 2% (
;( 7 H BT % # ΘBd %
?% BT = ΘB0 ∪ ΘB
1 ∪ · · · ∪ΘBQ/8−1% BT
% # K * K H BTp = (BT ∩ Π) ∪ (BT ∩ Π) % • % H BTp BT ? H ? # BTp(
6( H Pe% B BT % ? 1 # % /6 0 Pe( ,% B ? 1
=4
/ # # BT
?H 0(
=( " I H Pe # 0 I BT 1% % H Pe # BT + / # +PH # 0(3 % +P H / # +P + 0(
( " I H Pe 0 I BT 1% 7 +P Π(
>( " I H Pe 0 I BT 1% # + '% +P I BT % H BT # ( % % BT ? 1 # ? % ? 1 # BT % ? 1W % BT /;0W % I BT W # % # 7 +P # 1(
5( " +P /=0 />0 H % +P Π% H % +P /=0 K K P% Coh% B H ( B! K Coh +P% 11B11 7 +P K # P(
J :(4 #OP % #K Θ 32 / :(4(0% H
=
ΘBd #
? H ΘBd
K /7 :(4(0W :(4(+ H BT # % ! +P + W :(4( H +P K Coh :(4( +P 7 K(
:(4* Θ ( " 32 &, ΘB0 Θ
B1 Θ
B2 Θ
B3 !BT
B !+ " " Coh !+
#. /
J ! + B K 8 ! G # ? 7 / :(0% 8 # # H + % # + +P # K 8 + (
J 7 % # K + % + (
%
'
. .'"<" +P% K 7 *
• 1A + %
• kK B
• / !0(
' 7 + + .'"<" B 7 U4V( ' 7 1A + kK B + + B ! % # 7 + H + / ?% % 8 +P ?H 7 K +0( ' % .'"<" + 7 % B + # 7 (
) . ! 8% 7 ! % % B ! G ( " % K%
! ,
=;
=6
? ! ! 7 % 7% 7 ! K ( J I ! (
' (*
K ! 7 ! /J % K B 0( K ! 7 % # 7 # ( . ?H % # .'"<"% # K 7 *
5 . ' Θ # % 9(
( !$ . 9 # H ( K H /.'0% # % 7 ! 4(
$ ) 7 K +8 Di% # ( ) ! 4 7 HB + U6V(
' ? (
( 7 !""
' U4V% U4:V% U4V U5V K Θ% ?*
v(i, j) = β(i, j)[
cos(Θreg(i, j)) sin(Θreg(i, j))], /;(40
% Θreg(i, j) β 7 7( 7 # K
==
/180o ↔ 0o0 ? ( 7K *
β(i, j) = 1, /;(0
#% Θ G( ) 9 7 v( J ;(4 #OP 9% Θ +P% Θreg / ;(4(0% % Θreg % v / ;(4(0 7 v% 9 / ;(4(0(
;(4* < E Θ= Θ Θreg v
( 8 "
" Θ 64B64 K 7 Θ 8192 /Y64B64B20% # I H% 7 7 ( % # H K ( K .'"<" .'(
=
8$ !" "$ @ A
" # ? 8 + Θ* 91, · · · , 9p m H% ? K # 4 n H% #% n < m( ) 7 K + B # ? 9 4 K # B
4 = BT9,E[9] = 0. /;(:0
)% E[•] (' .' U6:V + K
K K G ? K ( ' K G ? 8 ? 8 K *
Σz =1
p− 1
p∑j=1
9j9Tj , /;(;0
Σx =1
p− 1
p∑j=1
4j4Tj , /;(60
% p I 8( " # E[9j ] = 0 E[4j ] = 0( Σz #7 8 ! H ? % H G # H B G # ( . ? K % H K # ! G & H( . H + B% #% K Σx( .' K # Σz K Σx% B B (
Σz< = λ<. /;(=0
) λ ≥ 0 < ∈ Rm − 0 Σz( ' /;(=0% 7 Σz % *
> ,
=>
= CT ΣzC, /;(0
C < Σz λ Σz( ' /;(:0 /;(60*
Σx =1
p− 1BT ΣzB. /;(>0
. /;(0 /;(>0% # Σx H KH B K C(
' .' G(
4( . 8 91, · · · , 9p K 8 D ∈ RmpW D = [91, 92, · · · , 9p] / ;((0(
( . D% K ( ∈ Rm1W ( =1pD p1% p1 ∈ Rp1 1 /
;((0(
:( "! ( D% K E ∈ RmpWE = D− ( 1p% 1p ∈ R1p 1(
;( K G Σz ∈ RmmW Σz =1
p−1EET W 1
p−1(EET )C =
C( ' K C ∈ Rmm% <1,<2, · · ·<m Σz K ∈ Rmm% m λ1, λ2, · · ·λm Σz(
6( N C + Σz(
=( " n C 7 K + B * B = [<1,<2, · · · ,<n] / ;((0(
( ? 8 K 7 K + BW 4 =BT (9− () / ;((0(
" m > p /I H # I 80 K /;0% J % EET ∈ Rmm% K ETE ∈ Rpp% ?*
=5
1
p− 1ETEC = C, /;(50
/;(50 E% *
1
p− 1(EET )(EC) = (EC). /;(40
% EC p EET % m−p K 7 ( 'K EC % % H k EC
√pλk.
;(* " , , % = , D- " !
( ""+
J K # 7 % 7 1A + % kK B /kJJ0 7
/$3 ) "0 ! /JJ " !&0% 7 ! .'% ? ! 4( . 7 +8 # 7 H HB ( ' K B 7 K % *
• L* I W L = 4, 5 / 4 /L = 40 8 7 N3% M% 33% ,''% 5 /L = 50 8 7 N3% M% 33% ,' '0%
• Ωi* i %
• P (Ωi)* Ωi%
• P (Ωi)* P (Ωi)%
• n* ! / K 0%
• 4W ! /! .'0%
• 4(i)j * i ! j %
• (i* Ωi%
• (i* (i%
• Σi* K G Ωi%
• Σi* Σi%
• d (4,F) = (4−F)T (4−F)* G 4 FW 4% F ∈ Rn%
• Di(4)* i + W 1 ≤ i ≤ L% 4 ∈ Rn(
$ EF ! =" !$ @$ !$A
7 H 1A + H Ωi( % HI 7 % # *
• p(4)* + 4 ∈ S ⊆ Rn% %∫S
p(4)d4 = P (4 ∈ S)%
4
• p(4|Ωi)* + 4 ∈ S ⊆ Rn # Ωi% %
∫S
p(4|Ωi)d4 = P (4 ∈ S|Ωi)%
• P (Ωi|4)* 4 % P (Ωi|4) =P (4 Ωi)(
) 7 1A # # K % 7 # 7 K 7 K 1( J % 7 1A 7 4 & Ωi # P (Ωi|4) ( 1A*
P (Ωi|4) =P (Ωi)p(4|Ωi)
p(4). /;(440
) 7 7 4 & Ωi P (Ωi|4) HB( 7 J Ωi # + *
p(4|Ωi) =1
(2π)n2 |Σi|
12
exp
(−1
2(4− (i)
T Σ−1i (4− (i)
), /;(40
% |Σi| Σi( J H (i Σi % ?* (i Σi( " /;(40 /;(440 8 (i Σi P (Ωi|4)( 7 + 7 % Di(4)% P (Ωi|4)*
Di(4) = ln(P (Ωi))− 1
2ln(
∣∣∣Σi
∣∣∣)− 1
2(4− (i)
T Σ−1i (4− (i), /;(4:0
/40 # + Ωi # BK P (Ωi|4) (2π)−
n2 (
& 4 HB + U6V(
c = arg maxi=1,2,··· ,L
Di(4) , /;(4;0
% c ! ! 4 arg max ! HB + (
$ kG<92 ! "H4!
) H K B E U:;V( 7 B ( J % +8 % P K +8 ( % B E + % 4% k B 4( # ? (. E U=V + U:;V% 1K B k = 1 / 1JJ% BF0% kK B /kJJ0( J ? B 1A( % B kJJ & B 1A% I 7 % 7 *
P ∗ ≤ P ≤ P ∗(
2− c
c− 1P ∗
), /;(460
% P ∗ B 1A P B kJJ% 7 c / ;(:0(
;(:* 0 k->> c = F 6GH9
:
' 7 1K B B %# ? 7 4% # ? % K B 4 ? ( "? 41NN K B 4% # 41NN Ωc% 4 Ωc(
' 7 + 1JJ *
Di(4) = − minj=1,2,··· ,Li
d2
(4,4(i)
j
), /;(4=0
1JJ 7 /;(4;0( R # K B 4 +K 7 + ? ( B 1JJ 7 kK B % kJJ% # kK B 4 ? P 4 & k K ( ' 7 + kJJ *
Di(4) = card(i)
x
, /;(40
% (i) ? #
& i # kK B 4 card I ? ( ' kJJ 7 /;(4;0(
" ($G!
R +% K % # # # B # K# ? G H % ?% ( $3 /) "0( J XB + % # & / ! 4 0% !( B % 7 % $3 P / !0*
• N (i)* I F i /i = 0, 1, 20% N (0) = n N (2) = L(
• f(s)* + /s ∈ R0(
;
• b(k)i * i F k /k = 1, 20(
• w(k)ij * i F k j F
(k − 1) /k = 1, 2W 1 ≤ i ≤ N (k)W 1 ≤ j ≤ N (k−1))%
• n(k)i (4)* ! i F k /k = 1, 2W 1 ≤
i ≤ N (k)0(
;(;* +
) + B *
• ! 4 (
• . F % + f % ! *
n(1)j (4) = f
b
(1)j +
N(0)∑k=1
w(1)jk xk
, /;(4>0
6
• F !% ! !*
n(2)i (4) = f
b
(2)i +
N(1)∑j=1
w(2)ij n
(1)j (4)
. /;(450
' 7 + $3 * Di(4) =
n(2)i (4)% 7 /;(4;0(' + /;(450 G K
H H ( '# + 7 # H(
% K H K K #H ? % K + ? % #OP 1 0 7 B (
) + + ( B H % ' K # + % ? / 0( , # * /K 0 / ? 80( J % + K % % U;V( / 0 0 N J $',3'1(
($ //$.
, JJ /" !&0% + " U5V( R + # B ? ( ' JJ F K P % B *
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