Lecture 22 Exemplary Inverse Problems - PowerPoint Presentation

Slide1

Lecture 22

Exemplary Inverse Problems

including

Filter DesignSlide2

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

2

Norm and Simple Least Squares

Lecture 05 A Priori Information and Weighted Least Squared

Lecture 06 Resolution and Generalized Inverses

Lecture 07 Backus-Gilbert 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 Backus-Gilbert 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

ProblemsSlide3

Purpose of the Lecture

solve a few exemplary inverse problems

image

deblurring

deconvolution

filters

minimization of cross-over errorsSlide4

Part 1

image

deblurringSlide5

three point blur

(applied to each row of pixels)Slide6

null vectors are highly oscillatorySlide7

solve with minimum lengthSlide8

note that

GG

T

can deduced analytically

and is

Toeplitz

might lead to a computational advantageSlide9

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

GG

T

λ

=d

(maybe with a little damping, too)

and then use

m

est

=

G

T

λ

Slide10

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 fineSlide11

image blurred due to camera motion

(100 point blur)Slide12

(A)

(B)

(C)

(D)

(E)

(F)

pixel number

pixel number

pixel number

pixel number

pixel numberSlide13

pixel number

[

G

-g

]

728

R

728

row number

row number

(A)

(B)

sidelobesSlide14

Part 2

deconvolution

filterSlide15

Convolution

general relationship for

linear systems

with translational invariance Slide16

Convolution

general relationship for

linear systems

with translational invariance

model

m(t)

and

data

d(t)

related

by linear operatorSlide17

Convolution

general relationship for

linear systems

with translational invariance

only relative time mattersSlide18

underlying principle

linear superpositionSlide19

time ,

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)Slide20

m(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)Slide21

convolution

d=m*gSlide22

discrete convolution

d=m*g

standard matrix from

d

=

GmSlide23

seismic reflection soundingSlide24
Slide25

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) Slide26

time,

t

g(t)

actual

airgun

pulse is

ringySlide27

so 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) Slide28

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)

this is the equation we need to solve

so construct a

deconvolution

filter

m(t)

so that Slide29

1

0

0

g(t) *m(t) =

δ

(t)

Gm

=

d

discrete approximation of delta

function

m

=

use discrete approximation of convolution

...Slide30

solve 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 analyticallySlide31
Slide32

approximately

Toeplitz

with elementsSlide33

approximately

Toeplitz

with elements

autocorrelation

of

gSlide34
Slide35

cross-

correlation

of

g

and

dSlide36

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

Slide37

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 methodSlide38

time,

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)Slide40

Part 3

minimization of cross-over errorsSlide41

longitude

latitude

latitude

true

estimated

gravity anomaly,

mgal

longitude

note streaksSlide42

general 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 intersectionsSlide43

5

6

7

8

1

2

3

4

cross-over pointsSlide44

i

th

intersection has

ascending track

A

i

and descending track

D

i

s

Ai

obs

=

s

Ai

true

+

mAisDiobs= sDitrue + mDisubtractsAiobs-sDiobs= mAi- mDi

has form

d

=

GmSlide45

the matrix

G

is very sparse

every row is all zeros, except for a single

+1

and a single

-1Slide46

note that this problem has an inherent non-uniqueness

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)Slide47

the matrices

G

T

G

and

G

T

d

can be calculated semi-analyticallySlide48
Slide49

recipe

starting with zeroed

G

T

G

and

G

T

d

Slide50

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 = 0Slide51

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=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 choiceSlide52

longitude

longitude

latitude

latitude

latitude

latitude

(A)

(B)

(D)

(C)

gravity anomaly,

mgal

longitude

longitude