Lecture 22 Exemplary Inverse Problems including Filter Design

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Lecture 22 Exemplary Inverse Problemsincluding
Filter Design
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Syllabus
Lecture 01 Describing Inverse ProblemsLecture
02 Probability and Measurement Error, Part
1Lecture 03 Probability and Measurement Error,
Part 2 Lecture 04 The L2 Norm and Simple Least
SquaresLecture 05 A Priori Information and
Weighted Least SquaredLecture 06 Resolution and
Generalized Inverses Lecture 07 Backus-Gilbert
Inverse and the Trade Off of Resolution and
VarianceLecture 08 The Principle of Maximum
LikelihoodLecture 09 Inexact TheoriesLecture
10 Nonuniqueness and Localized AveragesLecture
11 Vector Spaces and Singular Value
Decomposition Lecture 12 Equality and Inequality
ConstraintsLecture 13 L1 , L8 Norm Problems and
Linear ProgrammingLecture 14 Nonlinear
Problems Grid and Monte Carlo Searches Lecture
15 Nonlinear Problems Newtons Method Lecture
16 Nonlinear Problems Simulated Annealing and
Bootstrap Confidence Intervals Lecture
17 Factor AnalysisLecture 18 Varimax Factors,
Empircal Orthogonal FunctionsLecture
19 Backus-Gilbert Theory for Continuous
Problems Radons ProblemLecture 20 Linear
Operators and Their AdjointsLecture 21 Fréchet
DerivativesLecture 22 Exemplary Inverse
Problems, incl. Filter DesignLecture 23
Exemplary Inverse Problems, incl. Earthquake
LocationLecture 24 Exemplary Inverse Problems,
incl. Vibrational Problems
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Purpose of the Lecture
solve a few exemplary inverse problems image
deblurring deconvolution filters minimization of
cross-over errors
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Part 1
image deblurring
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three point blur(applied to each row of pixels)
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null vectors are highly oscillatory
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solve with minimum length
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note that GGT can deduced analytically
and is Toeplitz might lead to a computational
advantage
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Solution Possibilities

• Use sparse matrix for G
• together with mestG((GG)\d)
• (maybe damp a little, too)
• 2. Use analytic version of GGT
• together with mestG(GGT\d)
• (maybe damp a little, too)
• 3. Use sparse matrix for G
• together with bicg() to solve GGT?d
• (maybe with a little damping, too)
• and then use mestGT?

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Solution Possibilities

• Use sparse matrix for G
• together with mestG((GG)\d)
• (maybe damp a little, too)
• 2. Use analytic version of GGT
• together with mestG(GGT\d)
• (maybe damp a little, too)
• 3. Use sparse matrix for G
• together with bicg() to solve GGT?d
• (maybe with a little damping, too)
• and then use mestGT?

we used the simplest, which worked fine
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image blurred due to camera motion (100 point
blur)
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Part 2
deconvolution filter
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Convolution
general relationship for linear systems with
translational invariance
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Convolution
general relationship for linear systems with
translational invariance
model m(t) and data d(t) related by linear
operator
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Convolution
general relationship for linear systems with
translational invariance
only relative time matters
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underlying principlelinear superposition
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causes the output of d(t)g(t)
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Then the general input m(t)
spike of amplitude, m(t0)
m(t)
time, t
t0
causes the general output d(t)m(t)g(t)
m(t0)g(t-t0)
d(t)
time, t
t0
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convolution dmg
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discrete convolution dmg
standard matrix from dGm
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seismic reflection sounding
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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)
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actual airgun pulse is ringy
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so construct a deconvolution filter m(t) so that

• g(t) m(t) d(t)

and apply it to the data
p(t) g(t) r(t)
p(t)m(t) g(t)m(t)r(t) r(t)
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so construct a deconvolution filter m(t) so that

• g(t) m(t) d(t)

this is the equation we need to solve
and apply it to the data
p(t) g(t) r(t)
p(t)m(t) g(t)m(t)r(t) r(t)
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use discrete approximation of convolution

• g(t) m(t) d(t)

discrete approximation of delta function
Gm d
1 0

0
m
...
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solve with damped least squares

• mest GTG e2I-1 GTd

with d 1, 0, 0, ..., 0T (or something
similar) matrices GTG and GTd can be calculated
analytically
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approximately Toeplitz with elements
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approximately Toeplitz with elements
autocorrelation of g
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cross-correlation of g and d
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Solution Possibilities

• Use sparse matrix for G
• together with mest(GG)\(Gd)
• (maybe damping a little, too)
• 2. Use analytic versions of GTG and GTd
• together with mestGTG\GTd
• (maybe damp a little, too)
• 3. Never form G, just work with its columns, g
• use bicg() to solve GTG m GTd
• but use conv() to compute GT(Gv)
• 4. Same as 3 but add a priori information of
• smoothness

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Solution Possibilities

• Use sparse matrix for G
• together with mest(GG)\(Gd)
• (maybe damping a little, too)
• 2. Use analytic versions of GTG and GTd
• together with mestGTG\GTd
• (maybe damp a little, too)
• 3. Never form G, just work with its columns, g
• use bicg() to solve GTG m GTd
• but use conv() to compute GT(Gv)
• 4. Same as 3 but add a priori information of
• smoothness

we used this complicated but very fast method
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(A) Original
d(t)
(B) After deconvolution
d(t)m(t)
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Part 3
minimization of cross-over errors
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true
latitude
longitude
gravity anomaly, mgal
estimated
note streaks
latitude
longitude
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general idea

• data s is measured along tracks
• data along each track is off by an additive
  constant
• theory
• sjobs (track i) sjtrue (track i) m(track i)
• goal is to estimate the constants by minimizing
  the error at track intersections

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• ith intersection has
• ascending track Ai and descending track Di
• sAiobs sAitrue mAi
• sDiobs sDitrue mDi
• subtract
• sAiobs-sDiobs mAi- mDi
• has form
• dGm

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the matrix G is very sparse
every row is all zeros, except for a single 1
and a single -1
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note that this problem has an inherent
non-uniqueness m is determined only to an
overall additive constantone possibility is to
use damped least squares, to choose the smallest
m(you can always add a constant later)
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the matrices GTG and GTd can be calculated
semi-analytically
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recipestarting with zeroed GTG and GTd
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Solution Possibilities

• Use sparse matrix for G
• together with damped least squares
  mest(GGe2speye(M,M))\(Gd)
• 2. Use analytic versions of GTG and GTd
• add damping directly to the diagonal of GTG
• then use mestGTGpe2I\GTd
• 3. Use sparse matrix for G
• together with bicg() version of damped least
  squares
• 4. Methods 1 or 2, but use hard constraint
  instead of damping to implement Si mi 0

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Solution Possibilities

• Use sparse matrix for G
• together with damped least squares
  mest(GGe2speye(M,M))\(Gd)
• 2. Use analytic versions of GTG and GTd
• add damping directly to the diagonal of GTG
• then use mestGTG\GTd
• 3. Use sparse matrix for G
• together with bicg() version of damped least
  squares
• 4. Methods 1 or 2, but use hard constraint
  instead of damping

our choice
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