multilevel approaches for the total least squares method in deblurring problems malena español,...
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Multilevel approaches for the total least squares method in deblurring problems
Malena Español, Tufts UniversityMisha Kilmer, Tufts University
Dianne O’Leary, University of Maryland, College Park
ICIAM 2007 - Zurich, Switzerland
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Outline Problem Multilevel Method Algorithm Numerical Example Conclusion and Future Work
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Problem
noiseGaussian additiveunknown contain and
matrix dconditione-ill large, a is
where
,
system theand ,given , Find
bA
RA
bAx
bAx
nm
ICIAM 2007 - Zurich, Switzerland
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Total Least-Squares Method
rbxEALxrE
rbxEArE
bAx
p
p
p
xE,r
xE,r
)( s.t. min
isn formulatio TLS dRegularizeA
)( s.t. min
solve we, system thesolve To
2
2
2
F,
2
2
2
F,
Regularization by Truncated TLS by Fierro, Golub, Hansen and O’Leary (1997)
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Classical Two-Level Method
cii
ici
iiii
iTiii
iii
Tiiii
iii
iiii
iii
xxx
rxA
xAbr
ecPxx
reA
PAPA
rPr
xAbr
bxA
Solve""
Solve""
Solve""
1
111
1
1
Pre-Smoothing
Post-Smoothing
Coarse-Grid Correction
Restriction
Prolongation
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Classical Multilevel Method
cii
ici
iiii
iTiii
iii
Tiiiiiii
iiii
iii
iii
xxx
rxA
xAbr
xPxx
),bV(Ax
PAPArPr
xAbr
bxA
bxA
),bV(Ax
Solve
,
Solve
Else
Solve
gridcoarsest If
function
1
111
11
000
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Two-level TLS Method
00 , xE
c
c
TT
xxxEEE
rrxEA
ePxxPEPEE
11
1
00101
,~~
~~)~~
(
,~
rreEA ~)~~
( 00
“Post-Smoothing”
Pre-Smoothing
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Restriction and Prolongation Operators
11000000
00110000
00001100
00000011
11000000
00110000
00001100
00000011
2
2W
Haar wavelet transform V
W1V1
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Changing basis
.ˆ and ˆ ,ˆ ,ˆ
where
ˆˆˆ)ˆˆ(
have webasis of change aBy
)(
WrrWbbWEWEWAWA
rbxEA
rbxEA
TT
Optical Tomography: Zhu, Wang and Zhang (1998)
ICIAM 2007 - Zurich, Switzerland
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Properties
A
43
21
ˆˆ
ˆˆˆ
AA
AAA
.ˆin are ˆ of entrieslargest The 3)
.ˆˆ and, symmetric are ˆ ,ˆ then symmetric, is If 2)
Toeplitz. are ˆ and ˆ,ˆ,ˆ then Toeplitz, is If 1)
1
2341
4321
AA
AAAAA
AAAAAT
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Bandwidth reduction
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p
p
p
FExxLbxEAE
xEArbxEA
r
r
b
b
x
x
EE
EE
AA
AA
rbxEA
1
2
21111
2
1ˆ,ˆ
22211111
2
1
2
1
2
1
43
21
43
21
ˆˆˆ)ˆˆ(ˆmin
ˆ)ˆˆ(ˆˆˆ)ˆˆ(
ˆ
ˆˆ
ˆ
ˆ
ˆˆˆ
ˆˆ
ˆˆ
ˆˆ
ˆˆˆ)ˆˆ(
11
Pre-Smoothing
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2
11
3
11
2
1132
4
2
4
2
ˆ
ˆˆ
ˆ
ˆˆ
ˆ
ˆˆˆˆ
ˆ
ˆ
ˆ
ˆ
r
rx
A
EA
b
bxEx
E
E
A
A
Post-Smoothing
p
p
Tp
FFFEEEx x
xLWxEx
E
E
A
AEEE
2
1
2
2
132
4
2
4
22
3
2
4
2
2ˆ,ˆ,ˆ,ˆ ˆ
ˆˆˆˆ
ˆ
ˆ
ˆ
ˆˆˆˆmin
4322
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Two-Level Method
bA,
p
p
Tp
FFFEEEx x
xLWrxEx
E
E
A
AEEE
2
1
2
2
132
4
2
4
22
3
2
4
2
2ˆ,ˆ,ˆ,ˆ ˆ
ˆˆˆˆˆ
ˆ
ˆ
ˆ
ˆˆˆˆmin
4322
p
p
p
FExxLbxEAE 1
2
21111
2
1ˆ,ˆˆˆˆ)ˆˆ(ˆmin
11
V
V1 W1
xEAbr
VEVExVx TT
)(
ˆ,ˆ 11111
WEE
EEWE
xWxVx
T
TT
43
21
2111
ˆˆ
ˆˆ
ˆˆ
ICIAM 2007 - Zurich, Switzerland
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Multilevel Method
V
V1
V2
V3 W3
W2
W1
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Numerical Example
01.1 Operator, DerivativeFirst
)01.0,0( with )~
(
solution edgy :
)1.0,0(~
h matrix wit Toeplitz symmetric, :~
matrix symmetric Toeplitz, Gaussian, :
2
2
pL
bNeexEAb
x
ANEE
A
true
true
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Numerical Example
truex b
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Numerical Example
truex
recx
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Conclusions and Future work
General Cases – Non Structured Matrices
Extension to 2D, 3D Parameter Selection Additional Constraint on E