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including. Filter Design. Syllabus. Lecture 01 Describing Inverse Problems. Lecture 02 Probability and Measurement Error, Part 1. Lecture 03 Probability and Measurement Error, Part 2 . Lecture 04 The L. ID: 714854
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Presentations text content in Lecture 22 Exemplary Inverse Problems
Lecture 22
Exemplary Inverse Problems
including
Filter Design
Slide2Syllabus
Lecture 01 Describing Inverse Problems
Lecture 02 Probability and Measurement Error, Part 1
Lecture 03 Probability and Measurement Error, Part 2
Lecture 04 The L
2
Norm and Simple Least Squares
Lecture 05 A Priori Information and Weighted Least Squared
Lecture 06 Resolution and Generalized Inverses
Lecture 07 BackusGilbert Inverse and the Trade Off of Resolution and Variance
Lecture 08 The Principle of Maximum Likelihood
Lecture 09 Inexact Theories
Lecture 10
Nonuniqueness
and Localized Averages
Lecture 11 Vector Spaces and Singular Value Decomposition
Lecture 12 Equality and Inequality Constraints
Lecture 13 L
1
, L
∞
Norm Problems and Linear Programming
Lecture 14 Nonlinear Problems: Grid and Monte Carlo Searches
Lecture 15 Nonlinear Problems: Newton’s Method
Lecture 16 Nonlinear Problems: Simulated Annealing and Bootstrap Confidence Intervals
Lecture 17 Factor Analysis
Lecture 18
Varimax
Factors,
Empircal
Orthogonal Functions
Lecture 19 BackusGilbert Theory for Continuous Problems; Radon’s Problem
Lecture 20 Linear Operators and Their
Adjoints
Lecture 21
Fr
é
chet
Derivatives
Lecture 22 Exemplary Inverse Problems, incl. Filter Design
Lecture 23 Exemplary Inverse Problems, incl. Earthquake Location
Lecture 24 Exemplary Inverse Problems, incl.
Vibrational
Problems
Slide3Purpose of the Lecture
solve a few exemplary inverse problems
image
deblurring
deconvolution
filters
minimization of crossover errors
Slide4Part 1
image
deblurring
Slide5three point blur
(applied to each row of pixels)
Slide6null vectors are highly oscillatory
Slide7solve with minimum length
Slide8note that
GG
T
can deduced analytically
and is
Toeplitz
might lead to a computational advantage
Slide9Solution Possibilities
Use sparse matrix for
G
together with
mest
=G’*((G*G’)\d)
(maybe damp a little, too)
2. Use analytic version of
GG
T
together with
mest
=G’*(GGT\d)
(maybe damp a little, too)
3. Use sparse matrix for
G
together with
bicg
()
to solve
GG
T
λ
=d
(maybe with a little damping, too)
and then use
m
est
=
G
T
λ
Solution Possibilities
Use sparse matrix for
G
together with
mest
=G’*((G*G’)\d)
(maybe damp a little, too)
2. Use analytic version of
GG
T
together with
mest
=G’*(GGT\d)
(maybe damp a little, too)
3. Use sparse matrix for
G
together with bicg() to solve GGTλ=d (maybe with a little damping, too) and then use mest=GTλ
we used the simplest, which worked fine
Slide11image blurred due to camera motion
(100 point blur)
Slide12(A)
(B)
(C)
(D)
(E)
(F)
pixel number
pixel number
pixel number
pixel number
pixel number
Slide13pixel number
[
G
g
]
728
R
728
row number
row number
(A)
(B)
sidelobes
Slide14Part 2
deconvolution
filter
Slide15Convolution
general relationship for
linear systems
with translational invariance
Slide16Convolution
general relationship for
linear systems
with translational invariance
model
m(t)
and
data
d(t)
related
by linear operator
Slide17Convolution
general relationship for
linear systems
with translational invariance
only relative time matters
Slide18underlying principle
linear superposition
Slide19time ,
t
, after impulse
d(t)= g
(t)
0
time ,
t
, after impulse
m(t)=
δ
(t)
0
If the input of a spike
m(t)=
δ
(t)
spike
causes the output of
d(t)=
g
(t)
Slide20m(t
0
)g(t

t
0
)
m(t)
time,
t
t
0
d(t)
time,
t
t
0
spike of amplitude,
m(t
0
)
Then the general input
m(t)
causes the general output
d(t)=m(t)*g(t)
Slide21convolution
d=m*g
Slide22discrete convolution
d=m*g
standard matrix from
d
=
Gm
Slide23seismic reflection sounding
Slide24Slide25
want
airgun
pulse to be as spiky as possible
p(t) = g(t) * r(t)
pressure =
airgun
pulse * sea floor response
so as to be able to detect
pulses
in sea floor response
p(t)
≈
r(t)
Slide26time,
t
g(t)
actual
airgun
pulse is
ringy
Slide27so construct a
deconvolution
filter
m(t)
so that
g(t) *m(t) =
δ
(t)
and apply
it to the data
p(t)*m(t) =
g(t)*m(t)*r
(t
) = r(t)
p(t) =g(t) * r(t)
Slide28g(t) *m(t) =
δ
(t)
and apply
it to the data
p(t)*m(t) =
g(t)*m(t)*r
(t
) = r(t)
p(t) =g(t) r(t)
this is the equation we need to solve
so construct a
deconvolution
filter
m(t)
so that
Slide291
0
0
g(t) *m(t) =
δ
(t)
Gm
=
d
discrete approximation of delta
function
m
=
use discrete approximation of convolution
...
Slide30solve with damped least squares
m
est
= [
G
T
G
+
ε
2
I
]
1
G
T
d
with
d
= [1, 0, 0, ..., 0]
T(or something similar)matrices GTG and GTd can be calculated analytically
Slide31Slide32
approximately
Toeplitz
with elements
Slide33approximately
Toeplitz
with elements
autocorrelation
of
g
Slide34Slide35
cross
correlation
of
g
and
d
Slide36Solution Possibilities
Use sparse matrix for
G
together with
mest
=(G’*G)\(G’*d)
(maybe damping a little, too)
2. Use analytic versions of
G
T
G
and
G
T
d
together with
mest
=GTG\GTd (maybe damp a little, too)3. Never form G, just work with its columns, g use bicg() to solve GTG m = G
Td
but use
conv
()
to compute
G
T
(
Gv
)
4. Same as 3 but add a priori information of
smoothness
Solution Possibilities
Use sparse matrix for
G
together with
mest
=(G’*G)\(G’*d)
(maybe damping a little, too)
2. Use analytic versions of
G
T
G
and
G
T
d
together with
mest
=GTG\GTd (maybe damp a little, too)3. Never form G, just work with its columns, g use bicg() to solve GTG m = G
Td
but use
conv
()
to compute
G
T
(
Gv
)
4. Same as 3 but add a priori information of
smoothness
we used this complicated but very fast method
Slide38time,
t
time,
t
time,
t
g(t)
m(t)
m(t)*g(t)
(A)
(B)
(C)
Slide39(A) Original
(B) After
deconvolution
d(t)
d(t)*m(t)
Slide40Part 3
minimization of crossover errors
Slide41longitude
latitude
latitude
true
estimated
gravity anomaly,
mgal
longitude
note streaks
Slide42general idea
data
s
is measured along tracks
data along each track is off by an additive constant
theory
s
j
obs
(track
i
)
=
s
j
true
(track
i
)
+ m(track i)goal is to estimate the constants by minimizing the error at track intersections
Slide435
6
7
8
1
2
3
4
crossover points
Slide44i
th
intersection has
ascending track
A
i
and descending track
D
i
s
Ai
obs
=
s
Ai
true
+
mAisDiobs= sDitrue + mDisubtractsAiobssDiobs= mAi mDi
has form
d
=
Gm
Slide45the matrix
G
is very sparse
every row is all zeros, except for a single
+1
and a single
1
Slide46note that this problem has an inherent nonuniqueness
m
is determined only to an overall additive constant
one possibility is to use damped least squares, to choose the smallest
m
(you can always add a constant later)
Slide47the matrices
G
T
G
and
G
T
d
can be calculated semianalytically
Slide48Slide49
recipe
starting with zeroed
G
T
G
and
G
T
d
Solution Possibilities
Use sparse matrix for
G
together with damped least squares
mest
=(G’*G+e2*
speye
(M,M))\(G’*d)
2. Use analytic versions of
G
T
G
and
G
T
d
add damping directly to the diagonal of
GTG then use mest=GTGpe2I\GTd3. Use sparse matrix for G together with bicg() version of damped least squares4. Methods 1 or 2, but use hard constraint instead of damping to implement Σi mi = 0
Slide51Solution Possibilities
Use sparse matrix for
G
together with damped least squares
mest
=(G’*G*e2*
speye
(M,M))\(G’*d)
2. Use analytic versions of
G
T
G
and
G
T
d
add damping directly to the diagonal of
GTG then use mest=GTG\GTd3. Use sparse matrix for G together with bicg() version of damped least squares4. Methods 1 or 2, but use hard constraint instead of damping
our choice
Slide52longitude
longitude
latitude
latitude
latitude
latitude
(A)
(B)
(D)
(C)
gravity anomaly,
mgal
longitude
longitude