Transcript Detection of geometric Deformation

Detection of
3D Geometric Distortion in MRI
A local estimation method
F.G.C.M.v.d. Heuvel
s446087
[email protected]
Supervisor PMS:
Marcel Breeuwer
[email protected]
Supervisor TU/e:
Bart ter Haar Romeny
[email protected]
Contents
•
•
•
•
•
•
•
•
•
Geometric Distortion
State of the art
Local Estimation
Mathematics
Software
Validation
Discussion
Conclusions
Future
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Geometric Distortion 1/5
Geometric Distortion
Overview:
•
•
•
•
What is geometric distortion ?
Types
Causes
Consequences
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Geometric Distortion 2/5
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
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Geometric Distortion 3/5
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
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Geometric Distortion 4/5
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
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Geometric Distortion 5/5
Consequences
• Appearance of structure different from reality

– Size
– Shape
– Intensity
• May lead to wrong diagnosis / therapy
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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
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State of the art 1/13
State of the art
Overview:
•
•
•
•
Phantom Objects
Estimation method
Correction method
Advantages, Problems & restrictions
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State of the art 2/13
Phantom Objects
• Number of reference structures with exactly
known size and location
• MR phantom
• CT phantom
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State of the art 3/13
Phantom Objects
• MR Phantom for body coil
• For MR higher order polynomial  more complex structure 
future
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State of the art 4/13
Phantom Objects
• CT Phantom
• Only up to affine transformation  simple structure
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State of the art 5/13
Phantom Objects
Synthetically generated phantom scan
Breeuwer, Holden, Zylka, Proceedings SPIE Medical Imaging, February 2001, San Diego, USA
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State of the art 6/13
Estimation method
• Deformation expressed as nth order polynomial
transformation
• Finding transformation for entire dataset
 Estimating polynomial parameters
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State of the art 7/13
Estimation method
• Mathematically expressed as polynomial
transformation:
d  t  A  x  B  x 2  C  x3    M  x n
with:
 xm 


x 
m
 
 y

 y2 


m
 x
z


2


 
z 
2
m
m

1


x   y , x 
and x 
x y .
 xy 
 m 1 
z
 
 
 x z 
xz
 
  
 yz 
 m 2 
 
yz 
x
2
u
 
d   v ,
 w
 
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State of the art 8/13
Estimation method
•
Transformations exists of or as combinations of :
– Rigid:
• Translation
• rotation
– Affine:
• Scaling
• Shear
Atrans
1

0

0

0

0
1
0
0
0
0
1
0
dx 

dy 
dz 

1 
 cos  cos 

 sin  cos 
Arot  
 sin 


0

 sx

0
Ascaling  
0

0

 1 s xy

0 1
Ashear  
0 0

0 0

0
sy
0
0
s yz
1
0
cos  sin  cos   sin  sin 
sin  sin  cos   cos  sin 
cos  sin 
0
0

0
0

0 
1

1
1

1
0
0
sz
0
s xz
cos  sin  sin 
sin  sin  sin   cos  cos 
cos  sin 
0
1

1
1

1
– Elastic: 2th and higher order transformation matrices
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State of the art 9/13
Estimation method
• Combine all in system of equations:
 1 x1 y1

 1 x2 y 2
 


 1 x N y N



0






0




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z1 

z2 
 

z N 
0
1 x1

1 x2
 

1 x
N

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0
y1
y2

yN
0
z1 

z2 
 

z N 
0
1 x1

1 x 2
 

1 x
N

y1
y2

yN
  t x   u1 
    
  a11   u2 
 a    
  12   
  a13   u N 
 t  v 
  y   1 
  a21   v2 
    
  a22    
  a23   v N 
    
z1    t z   w1 

z 2    a31   w2 
    
    a32    

z N    a33   wN 
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State of the art 10/13
Estimation method
• Estimating parameters t and a’s using SVD [alg.
From Num Rec in C]
• Will be explained later on…
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State of the art 11/13
Estimation method
• Schematic representation of estimation and
correction procedure

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State of the art 12/13
Correction method
• find position d corresponding to x:
F(x)
x d
• Place intensity on d at position x by means of
interpolation
– Trilinear
– Cubic spline
– Truncated sinc
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State of the art 13/13
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
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Local Estimation 1/5
Local Estimation
Overview:
• Not entire 3D dataset but sub volume
• Estimating transformation for every sub
volume
• Expected Advantages
• Expected Disadvantages
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Local Estimation 2/5
Not entire 3D dataset but sub volume
• Use of 3D data subsets
 overlapping sub
volumes

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Local Estimation 3/5
Estimating transformation for every sub
volume
• n sub volumes  n sets of polynomial parameters
 t  A  x  B  x 2  C  x3    M  x n 
1


2
3
n
t  A  x  B  x  C  x    M  x 2 





2
3
n
t  A  x  B  x  C  x    M  x 
n

• So a system of equation for every subvolume
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Local Estimation 4/5
Expected Advantages
• Better estimation of very local or higher-order
global deformations
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Local Estimation 5/5
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
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Mathematics 1/12
Mathematics
Overview:
•
•
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
–
–


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as maximum likelihood
fit
2
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Mathematics 2/12
Polynomial Transformation
t  A  x  B  x 2  C  x3    M  x n
Number of coordinate combinations and transformation
parameters to be estimated as function of order for every
volume or sub volume:
Order
1
2
3
4
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Coords
4
10
20
35
Pars
12
30
60
105
28
Mathematics 3/12
Used solution method for Least Squares
Problem
• Rewriting transformation as system of
equations:
Ax  b
• A: design matrix, containing the coordinate combinations
• b: vector with deformed point coordinates
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Mathematics 4/12
Used solution method for Least Squares
Problem
• Singular Value Decomposition
A  UWV
T
• U, V: orthogonal,left and right singular vectors resp.
• W: diagonal matrix with singular values
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Mathematics 5/12
Methods used in SVD computation
• Computing singular values by using:
– Householder reduction
– Givens Rotations
• Left and right singular vectors
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Mathematics 6/12
Singular values 1/4
Householder reduction
• Matrix
A  :
nn
• Householder matrix:
P I
2uu Twith:
u
and
x  ai
, ith column of
2
u  x  x e1
A
Using this matrix 2 times n-2 times to bi-diagonalize A.
Full diagonalization by Givens Rotations:
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Mathematics 7/12
Singular values 2/4
Givens Rotations
• Plane rotation:
1


0


0
G i, k ,    

0


0


0


cos 

0

 sin  


  sin    cos 


0


0


0



0
i
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
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k
 0
 
 0 i


 0 k

 
 1

 
 0 M
 N
33
Mathematics 8/12
Singular values 3/4
Givens Rotations 2/
and
x  N
then:

cos   xi

yi  sin    xi  cos   xk

xj

Zero yi
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y  G(i, k , )T  x
by:
cos  
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j i 

jk 
j  i, k 
xi
x x
2
i
2
k
and sin   
 xk
xi2  xk2
34
Mathematics 9/12
Singular values 4/4
Givens Rotations 2/
Now construction of  :


~
~ ~ ~
  G j G1  PN 1 P1  A  Pj P1  G N 1 G1
With elements
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
wi.
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Mathematics 10/12
Left and right singular vectors
• Left singular vectors:
U  G N G1  PN 1 P1 
• Right singular vector:
~
~
~ ~
V  P1  PN 1  G1 G j
• Solution:
 Ui  b 
Vi
x   
i 1  wi




M
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Mathematics 11/12
Goodness of Fit estimation
•

2
as maximum likelihood estimate
• Goodness-of-Fit by means of incomplete
 - function
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Mathematics 12/12
Goodness of Fit estimation
Solution vector using SVD tot minimize:
  Ax b
2
2
Goodness of Fit:



2
 1


 
1
2
t
2
Q    Q
,  
e
t
dt

 2 2      2
2
 2
Chi-square exceedance by chance
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Software 1/3
• Overview
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Software 2/3
• Estimation loop
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Software 3/3
• Application loop
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Validation 1/17
Validation
• Performance of the local estimator:
d = deformed point
rt = retransformed point3D visualization
– Maxima, minima, st. dev. of Euclidian distance in the patch
xd  xrt  yd  yrt  zd  zrt
– Goodness of Fit measurement
•
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
2
estimate 
- fit

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Validation 2/17
Validation
• Global up to 4th order deformation
– Origin in center
– Origin in corner
• Local deformation
– Divide space into four parts
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Validation 3/17
Global Deformation
10 4 10 4 10 4 10 4 10 4 10 4 
1 0 0


A  0 1 0, B  10  4 10  4 10  4 10  4 10  4 10  4 ,
10  4 10  4 10  4 10  4 10  4 10  4 
0 0 1


10 5 10 5 10 5 10 5 10 5 10 5 10 5 10 5 10 5 10 5 


C  10 5 10 5 10 5 10 5 10 5 10 5 10 5 10 5 10 5 10 5 ,
10 5 10 5 10 5 10 5 10 5 10 5 10 5 10 5 10 5 10 5 


10 8 10 8 10 8 10 8 10 8 10 8 10 8 10 8 10 8 10 8 10 8 10 8 10 8 10 8 10 8 


D  10 8 10 8 10 8 10 8 10 8 10 8 10 8 10 8 10 8 10 8 10 8 10 8 10 8 10 8 10 8 
10 8 10 8 10 8 10 8 10 8 10 8 10 8 10 8 10 8 10 8 10 8 10 8 10 8 10 8 10 8 


Applied to both the cornered as centered data set
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Validation 4/17
Origin in center 1
• 3D display:
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Validation 5/17
Origin in center 2
•
Histogram of introduced error by initial manual deformation
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Validation 6/17
Origin in center 3
• Error histogram between initial deformed and globally re-transformed data set

2

2
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order1
order2
0
0
5.9733
0.9816
order3
order4
0
1
0.0469
0
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Validation 7/17
Origin in center 4
• Error histogram between initial deformed and locally re-transformed data set
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Aver. 
1
σ
0
max 
1
min 
1
Aver.
0
s C2
0
max. C2
0
min. C2
0
48
Validation 8/17
Origin in corner 1
• 3D display:
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Validation 9/17
Origin in corner 2
•
Histogram of introduced error by initial manual deformation
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Validation 10/17
Origin in corner 3
• Error histogram between initial deformed and globally re-transformed data set
Order1
Order2

0
0
2
0.5453
0.5443
Order3
Order4
0
1
0.0469
0

2
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Validation 11/17
Origin in corner 4
• Error histogram between initial deformed and locally re-transformed data set
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Aver. 
1
σ
0
max 
1
min 
1
Aver. C2
0
ss C2
0
max. C2
0
min. C2
0
52
Validation 12/17
Local Deformation
– Space divided in four parts
– Each part another deformation up to third order
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Validation 13/17
Local Deformation
• Part1  Not deformed
• Part 2  only 2th order
• Part 3  only
3th
order:
• Part 4  2th and 3th order:
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10 3 10 3 10 3

B  10 3 10 3 10 3
 0
0
0

0 0 0

0 0 0
0 0 0
10 5 10 5 10 5

C  10 5 10 5 10 5
 0
0
0

0 0 0 0 0 0 0

0 0 0 0 0 0 0
0 0 0 0 0 0 0
10 3 10 3 10 3

B  10 3 10 3 10 3
 0
0
0

0 0 0

0 0 0
0 0 0
10 5 10 5 10 5

C  10 5 10 5 10 5
 0
0
0

0 0 0 0 0 0 0

0 0 0 0 0 0 0
0 0 0 0 0 0 0
54
Validation 14/17
Division in four parts
• 3D display
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Validation 15/17
Division in four parts
• Histogram of introduced error by initial manual deformation
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Validation 16/17
Division in four parts
• Error histogram between initial deformed and globally re-transformed data set
4blocks
order1
order2
0.0000
0.0000
2.4536
1.4357
order3
order4

0.0000
0.0000
2
1.1996
1.0735

2
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Validation 17/17
Division in four parts
• Error histogram between initial deformed and locally re-transformed data set
4blocks order3, ps3
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Medical IT - Advanced Development
Aver. 
0.6518
σ
0.4674
max 
1.0000
min 
0.0000
Aver. C2
0.2277
s C2
0.3357
max. C2
1.8826
min. C2
0.0000
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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
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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
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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
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Future Plans 2/2
• 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
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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
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Questions ??
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