detection of 3d geometric distortion in mri a local estimation method f.g.c.m.v.d. heuvel s446087...

Detection of 3D Geometric Distortion in MRI

A local estimation method

F.G.C.M.v.d. [email protected]

Supervisor PMS:Marcel [email protected]

Supervisor TU/e:Bart ter Haar [email protected]

Philips Medical Systems Medical IT - Advanced Development 2

Contents• Geometric Distortion• State of the art• Local Estimation• Mathematics• Software• Validation• Discussion• Conclusions• Future


Geometric Distortion Overview:

• What is geometric distortion ?• Types• Causes• Consequences

Geometric Distortion 1/5


What is Geometric Distortion?

• Change of position of anatomical structures– Shape change of entire image (global)– Shape change of parts of image (local)

• Characteristic for type of scanner– MRI– CT



Types of Geometric Distortion

• Expressed as (combinations of) polynomial transformations:

– 1th order:• Rigid translation, rotation• Affine shear, scaling (, mirror)

– 2th and higher order:– Elastic

• Both global and/or local



Causes

• Field-in-homogeneity– Especially for fast scan protocols

• Patient induced field changes– Watery environment in body plus ionic

substances Eddy current influence on the field



Consequences

• Appearance of structure different from reality – Size– Shape– Intensity

• May lead to wrong diagnosis / therapy



How to solve the problem• State of the art:

– Creating completely known phantom object– Finding transformation from un deformed data set to

deformed data set– Estimating polynomial parameters for entire dataset

Global estimation

• But :– local and sharp deformation not detected correctly

• Therefore new approach:– Estimating polynomial parameters for parts of data set

Local estimation


State of the artOverview:

• Phantom Objects• Estimation method• Correction method• Advantages, Problems & restrictions

State of the art 1/13


Phantom Objects

• Number of reference structures with exactly known size and location

• MR phantom

• CT phantom



Phantom Objects• MR Phantom for body coil

• For MR higher order polynomial more complex structure future



Phantom Objects• CT Phantom

• Only up to affine transformation simple structure



Phantom Objects

Synthetically generated phantom scan

Breeuwer, Holden, Zylka, Proceedings SPIE Medical Imaging, February 2001, San Diego, USA



Estimation method

• Deformation expressed as nth order polynomial transformation

• Finding transformation for entire dataset Estimating polynomial parameters



• Mathematically expressed as polynomial transformation:

with:

nxMxCxBxAtd 32

.,,

2

1

12

2

2

2

yzx

zx

yx

z

y

x

and

yz

xz

xy

z

y

x

z

y

x

w

v

u

m

m

m

m

m

m

m

xxxd


Estimation method


• Transformations exists of or as combinations of :– Rigid:

• Translation

• rotation

– Affine:• Scaling

• Shear

– Elastic: 2th and higher order transformation matrices

1000

100

010

001

z

y

x

trans d

d

d

A

0000

0sincossincossin

0sincoscossinsincoscossinsinsincossin

0sinsincossincossinsincoscoscos

rotA

1000

100

100

100

z

y

x

scaling s

s

s

A

1000

1100

110

11

yz

xzxy

shear

s

ss

A


Estimation method


• Combine all in system of equations:

N

N

N

z

y

x

NNN

NNN

NNN

w

w

w

v

v

v

u

u

u

a

a

a

t

a

a

a

t

a

a

a

t

zyx

zyx

zyxzyx

zyx

zyxzyx

zyx

zyx

2

1

2

1

2

1

33

32

31

23

22

21

13

12

11

222

111

222

111

222

111

1

1

11

1

11

1

1

00

00

00


Estimation method


Estimation method

• Estimating parameters t and a’s using SVD [alg. From Num Rec in C]

• Will be explained later on…



Estimation method• Schematic representation of estimation

and correction procedure



• find position d corresponding to x:

• Place intensity on d at position x by means of interpolation– Trilinear– Cubic spline– Truncated sinc

Correction methodState of the art 12/13

dxF(x)


Advantages, Problems & restrictions• Advantages

– Simple continuous description one polynomial transform

• Problems & restrictions– Unable to describe local deformations– Does not work well for “exotic” global

deformation fields



Local EstimationOverview:

• Not entire 3D dataset but sub volume• Estimating transformation for every

sub volume• Expected Advantages• Expected Disadvantages

Local Estimation 1/5


Not entire 3D dataset but sub volume

• Use of 3D data subsets overlapping sub volumes



Estimating transformation for every sub volume

• n sub volumes n sets of polynomial parameters

• So a system of equation for every subvolume

n

n

n

n

xMxCxBxAt

xMxCxBxAt

xMxCxBxAt

32

2

321

32



Expected Advantages

• Better estimation of very local or higher-order global deformations



Expected Disadvantages

• For every n sets of sub-volumes n polynomial estimations needed more calculation time

• High order needs more memory

• Risk of edge effects needs large amount of patch-overlap 3Dspace



MathematicsOverview:

• Polynomial transformation• Used solution method for

Least Squares Problem:– Singular Value Decomposition

• Methods used in SVD computation:– Singular values σi:

Householder and Givens– Left and right eigenvectors

using the σi

• Error calculation and testing – as maximum likelihood– fit

2

Mathematics 1/12


Polynomial TransformationnxMxCxBxAt 32

Number of coordinate combinations and transformation parameters to be estimated as function of order for every volume or sub volume:

Order Coords

Pars

1 4 122 10 303 20 604 35 105

Mathematics 2/12


Used solution method for Least Squares Problem

• Rewriting transformation as system of equations:

• A: design matrix, containing the coordinate combinations

• b: vector with deformed point coordinates

Mathematics 3/12

bAx


• Singular Value Decomposition

•U, V: orthogonal,left and right singular vectors resp.

•W: diagonal matrix with singular values

Used solution method for Least Squares Problem

Mathematics 4/12

TUWVA


Methods used in SVD computation• Computing singular values by using:

– Householder reduction– Givens Rotations

• Left and right singular vectors

Mathematics 5/12


Singular values 1/4Householder reduction

• Matrix :

• Householder matrix: with: and , ith column of

Using this matrix 2 times n-2 times to bi-diagonalize A.

Full diagonalization by Givens Rotations:

nnA

2

2

u

uuIP

T

1exxu

iax A

Mathematics 6/12


Mathematics 7/12

Singular values 2/4Givens Rotations

• Plane rotation:

NkiM

k

i

kiG

0000

1000

0cossin0

0sincos0

0001

,,


Singular values 3/4Givens Rotations 2/

xkiy T ),,( GNx

kijx

kjxx

ijx

y

j

ki

i

i

,

cossin

cos

and

then:

Zero by:iy 2222

sincoski

k

ki

i

xx

xand

xx

x

Mathematics 8/12


Singular values 4/4Givens Rotations 2/

Now construction of :

111111

~~~~GGPPAPPGG NjNj

With elements .

iw

Mathematics 9/12


Left and right singular vectors

• Left singular vectors:

• Right singular vector:

• Solution:

Mathematics 10/12

M

ii

i

i

w1

VbU

x

111 PPGGU NN

jN GGPPV~~~~

111


Goodness of Fit estimation

• as maximum likelihood estimate

• Goodness-of-Fit by means of incomplete

- function

Mathematics 11/12

2


Goodness of Fit estimation

Solution vector using SVD tot minimize:

Goodness of Fit:

Chi-square exceedance by chance

22 bxA

2

122

2

2

2

1

2,

2

dtteQQ t

Mathematics 12/12


Software 1/3• Overview


• Estimation loop

Software 2/3


• Application loop

Software 3/3


Validation• Performance of the local estimator:

d = deformed pointrt = retransformed point3D visualization

– Maxima, minima, st. dev. of Euclidian distance in the patch

– Goodness of Fit measurement

• estimate - fit

rtdrtdrtd zzyyxx

Validation 1/17

2


Validation

• Global up to 4th order deformation– Origin in center – Origin in corner

• Local deformation– Divide space into four parts

Validation 2/17


Global Deformation

Applied to both the cornered as centered data set

888888888888888

888888888888888

888888888888888

5555555555

5555555555

5555555555

444444

444444

444444

101010101010101010101010101010

101010101010101010101010101010

101010101010101010101010101010

,

10101010101010101010

10101010101010101010

10101010101010101010

,

101010101010

101010101010

101010101010

,

100

010

001

D

C

BA

Validation 3/17


Origin in center 1• 3D display:

Validation 4/17


Origin in center 2• Histogram of introduced error by initial manual deformation

Validation 5/17


Origin in center 3• Error histogram between initial deformed and globally re-

transformed data set

order1 order2

0 0

5.9733 0.9816

order3 order4

0 1

0.0469 0

Validation 6/17

2

2


Origin in center 4• Error histogram between initial deformed and locally re-

transformed data setAver. 1

σ 0

max 1

min 1

Aver. 0

0

max. 0

min. 0

Validation 7/17


Origin in corner 1• 3D display:

Validation 8/17


Origin in corner 2• Histogram of introduced error by initial manual deformation

Validation 9/17


Origin in corner 3• Error histogram between initial deformed and globally re-


Order1 Order2

0 0

0.5453 0.5443

Order3 Order4

0 1

0.0469 0

Validation 10/17

2

2


Origin in corner 4• Error histogram between initial deformed and locally re-

transformed data setAver. 1

σ 0

max 1

min 1

Aver. 0

0

max. 0

min. 0

Validation 11/17


– Space divided in four parts– Each part another deformation up to

third order

Validation 12/17

Local Deformation


Local Deformation• Part1 Not deformed

• Part 2 only 2th order

• Part 3 only 3th order:

• Part 4 2th and 3th order:

000000

000101010

000101010333

333

B

0000000000

0000000101010

0000000101010555

555

C

0000000000

0000000101010

0000000101010

000000

000101010

000101010

555

555

333

333

C

B

Validation 13/17


Division in four parts• 3D display

Validation 14/17


Division in four parts• Histogram of introduced error by initial manual deformation

Validation 15/17


Division in four parts• Error histogram between initial deformed and globally re-


4blocks order1 order2

0.0000 0.0000

2.4536 1.4357

order3 order4

0.0000 0.0000

1.1996 1.0735

Validation 16/17

2

2


Division in four parts• Error histogram between initial deformed and locally re-

transformed data set 4blocks order3, ps3

Aver. 0.6518

σ 0.4674

max 1.0000

min 0.0000

Aver. 0.2277

0.3357

max. 1.8826

min. 0.0000

Validation 17/17


Discussion• Not tested on read MRI data• Only a limited amount of tests

performed• Only one type of patch• Only tests used with patch shifts of

only 1


Conclusions

• For global deformation not much difference between global and local estimation

• For local deformation, local estimation gives better description of deformation

• Discontinuous deformation– Global estimation results in very large

errors– Local estimation also not perfect


Future Plans 1/2• Better detection of reference

structure in real hardware phantom

• Adaptation of order and patch size to type and amount of local distortion

• Spherical subvolumes (patches) instead of cubic shaped


• More dependence on type of deformations– First global estimator, then after error

analysis, local estimation where necessary– Adapted for type of scan protocol (order &

patch size)?

• Perhaps more complex structured phantom for higher order estimation

Future Plans 2/2


Learning value

• More business-like environment• More performance driven• First time use of real programming

language C• Learning to use work of other people

– Useful to see others ideas– Very difficult to understand

undocumented code, especially encoded mathematics


Questions ??

detection of 3d geometric distortion in mri a local estimation method f.g.c.m.v.d. heuvel s446087...

Documents

local estimation slide

field geometric distortion

advanced development9

advanced development8

advanced development4

phantom scan breeuwer

art overview

advanced development14