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SIAM Annual Meeting 2005 1
An Iterative, Projection-Based Algorithm for General Form Tikhonov Regularization
Misha Kilmer, Tufts University Per Christian Hansen, Technical University of
Denmark Malena Español, Tufts University
SIAM Annual Meeting 2005 2
Outline
Problem Background Algorithm Numerical Examples Conclusion and future work
SIAM Annual Meeting 2005 3
Discrete Ill-Posed problem
holds condition Picard Discrete
noise (white)unknown is
gap without aluessingular v Decaying
:Properties
e
matrix dconditione-ill large, a is where
,
model theand ,given , Find
nm
truetrue
true
RA
ebbAx
bAx
0 10 20 30 4010
-15
10-10
10-5
100
105
i
i|uiTbtrue|
SIAM Annual Meeting 2005 4
Need for regularization
n
ii
eui
butrue
n
i
Tiii
T
i
Ti
i
trueTix
uVUA
1
1
bygiven issolution exact The
.SVD thebe Let
0 10 20 30 4010
-15
10-10
10-5
100
105
i| uiTbtrue|
| uiTe|
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Tikhonov Regularization
. 0
min
or
min
:Problem dRegularize Tikhonov The
2
2
2
2
22
2x
bx
L
A
LxbAx
x
.0parameter tion regularizaon depends ngConditioni
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Tikhonov Method: choosing
L-curve (Lawson-Hansen)
2 log bAx
2 log Lx
101.84
101.86
101.88
101.9
101.92
101.94
102.2
102.3
102.4
102.5
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Tikhonov Regularization
employed. LSQR) (CGLS, solvers Iterative
operator. derivative discrete (scaled) aoften is L
inverse.-pseudo weighted-A theis where
, min
: If
. Tikhonov, form-standardIn
)()(2
2
22
2y
A
AA
L
yLxybyAL
IL
IL
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Bidiagonalization
'82 Saunders and Paige '96, Zha'65,Kahan -Golub
explicitlyion factorizat QR forming of need No
. of form bidiagonalupper an
and of form bidiagonallower a Obtaining
usly.simultaneo computed becan and ofization Bidiagonal
. ,
ionfactorizat QR heConsider t
L
A
LA
LTLA
TA
L
A
Q
Q
IQQQQRQ
Q
L
A
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Relating A,L
.ˆ ,
such that matrix
invertible and , bidiagonalupper an
, bidiagonallower a ,ˆmatrix
unitary ,matrix unitary exist There
. andLet :Theorem
11
ZBULUBZA
Z
nnRB
RBUpp
Umm
RL RA
np
nm
npnm
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Projected Problem
.on depend and Only
.0
min
problem projected theofsolution theis
,solution its with so )(But
.,,span with ,0
min
withproblem Tikhonov theReplace
2
11
111
1
2
kk
k
k
y
k
kkkk
kkZx
yx
ey
B
B
y
yZxbeU
zzZb
xL
A
k
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Choosing
2
2
2
2
2
211
2
2 and
.each for defined as ,for :Theorem
curve-L
kkkkkk
kk
yBLxeyBbAx
yx
101.84
101.86
101.88
101.9
101.92
101.94
102.2
102.3
102.4
102.5
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Iterative Method
e.convergencfor check 4.
norms, constraint and residual update 3.
,1 stepat computed quantities from
each for desired) if ,( compute 2.
ization,bidiagonaljoint of stepth thecompute 1.
,2 ,1For
k
xy
k
k
kk
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Regularizing algorithm
},{ of GSVD theofion approximat
an get we},{ ofion decomposit CS with the
behavior. decayingsimilar have aluessingular v the
: from properties ngconditioni inherites
. with , min
problem the toapplied LSQR
similar to worksalgorithm that theshowcan We
12
2
LA
BB
AQ
yRxbyQ
kk
A
Ay
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TGSVD of the Projected Problem
klk
kreg
l
ii
i
TiBk
l
kkB
B
k
k
kkk
k
k
y
yZxheu
y
BBHM
N
U
U
B
B
yZxe
yB
B
k
k
k
then and )()(
is problem projected theosolution t TGSVD The
}. ,{ of GSVD thebe 0
0 Let
. with ,0
min
:Problem Projected theRecall
1
11
1
2
11
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Numerical Examples
1 that so scaled ,
.or 1 dim of op. derivative where,
).01.0,0( with and rice""
)10()( blur(),
Hansen.by Toolstion Regulariza Matlab, ,deblurring Image 2D
22
11
1
9
LALA
LLIL
LIL
bNeeAxbx
OAkA
truetrue
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Original and blurred images
Original Blurred + Error
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Restoration L =derivative operator
Original Restored
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Restoration L= Laplacian
Original Restored
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Comparing with L=I and L=Laplacian
Original Blurred + Error
L=Derivative Op. L=Laplacian Op. L=Identity
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Conclusion and Future work
oningPreconditi
accuracyiteration Inner
: workFuture
priori aknown not
orecompute/st todifficult is
: whenusefulmation transfor
form-standard avoids that algorithm Iterative
AL