lecture8 registration

8/14/2019 Lecture8 Registration

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Image Processing

Image Registration

What is image registration?

• Image Registration is the process of estimating an

optimal transformation between two images.

• Sometimes also known as “Spatial Normalization”.

• Applications in medical imaging

– Matching PET (metabolic) to MR (anatomical) Images

2

– Atlas-based segmentation/brain stripping

– fMRI Specific

Motion Correction

Correcting for Geometric Distortion in EPI

Alignment of images obtained at different times or with different

imaging parameters

Formation of Composite Functional Maps

Anatom ical

Reference

(SCA)

Functional

Reference

(FCA)

ReferenceSignatures

MR Image

(New Subject)

PET Image

(New Subject)

MR-PET

Registration

3

Analysi s

A schematic diagram of multi-modality MR-PET

image analysis using computerized atlases.

Image Registration Through Transform

A B f

4

g

Image registration provides transformation of a source

image space to the target image space.

The target image may be of different modalities from

the source one.

Multi-modality brain image registration

• External Markers and Stereotactic Frames Based

Landmark Registration.

• Rigid-Body Transformation Based Global

Registration.

•

5

.

– Boundary and Surface Matching Based

Registration

– Image Landmarks and Features Based

Registration

Rigid-Body Transformation

Translation of z

Rotation by φ ' = + x Rx t

Rotation Translation

6

Translation of yTranslation of x

Rotation by θ

Rotation by ω




'

'

'

x x p

y y

z z

= +

=

=

(a) Translation along x-axis by p (c) Translation along z-axis by r

'

'

'

x x

y y

z z r

=

== +

7

'

'

'

x x

y y q

z z

=

= +

=

(b) Translation along y-axis by q


'

'

'

c o s s in

s in c o s

x x p

y y z

z y z

θ θ

θ θ

= +

= +

= − +

(a) Rotation about x-axis by θ (c) Rotation about z-axis by φ

'

'

'

c o s s in

s in c o s

x x y

y x y

z z

ϕ ϕ

ϕ ϕ

= +

= − +

=

8

'

'

'

c o s s in

s in c o s

x x z

y y

z x z

ω ω

ω ω

= −

=

= +

(b) Rotation about y-axis by ω


' = + x Rx t

cos sin 0 cos 0 sin 1 0 0

sin cos 0 0 1 0 0 cos sin

R R Rθ ω ϕ

θ θ ω ω

θ θ

= =

−⎡ ⎤ ⎡ ⎤ ⎡ ⎤

⎢ ⎥ ⎢ ⎥ ⎢ ⎥−

R

9

0 0 1 sin 0 cos 0 sin cosω ω ϕ ϕ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥−⎣ ⎦ ⎣ ⎦ ⎣ ⎦

p

q

r

⎡ ⎤⎢ ⎥= ⎢ ⎥⎢ ⎥⎣ ⎦

t


' = + x Rx t

'

10

1 11=⎢ ⎥ ⎢ ⎥ ⎢ ⎥

⎣ ⎦ ⎣ ⎦⎣ ⎦

R t x x

0

Rigid-Body Transformation: Affine Transform

Affine transform: translation + rotation + scaling

Scaling: '

'

'

x a x

y b y

z c z

=

=

=

11

a, b and c are scaling parameters in the three directions.

Affine transform can be expressed as

'

11

⎡ ⎤ ⎡ ⎤=⎢ ⎥ ⎢ ⎥

⎣ ⎦⎣ ⎦

x x A


A: The affine transform matrix that integrates translation,

rotation and scaling effects.

'

11

⎡ ⎤ ⎡ ⎤=⎢ ⎥ ⎢ ⎥

⎣ ⎦⎣ ⎦

x x A

12

0

1 1

⎡ ⎤ ⎡ ⎤= ⎢ ⎥ ⎢ ⎥

⎣ ⎦ ⎣ ⎦

R t S A

0 0

a

b

c

⎡ ⎤⎢ ⎥= ⎢ ⎥⎢ ⎥⎣ ⎦

S where




The overall mapping can be expressed as

'

'

'

1 0 0 cos sin 0 0 cos 0 sin 0

0 1 0 sin cos 0 0 0 1 0 0

0 0 1 0 0 1 0 sin 0 cos 0

p x

q y

r z

θ θ ω ω

θ θ

ω ω

−⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤

⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥−⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥=⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥

13

0 0 0 1 0 0 0 1 0 0 0 11

1 0 0 0 0 0 0

0 cos sin 0 0 0 0

0 sin cos 0 0 0 0

0 0 0 1 0 0 0 1

a x

b

c

ϕ ϕ

ϕ ϕ

⎣ ⎦ ⎣ ⎦ ⎣ ⎦⎣ ⎦

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥−⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ 1

y

z

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

Rigid-Body Transformation: Principal Axes Registration

Principal axes registration (PAR) is used for global matching

of binary volumes from CT, MR or PET images.

1 ( , , )( , , )

0 ( , , )

x y z o b je c t B x y z

x y z o b je c t

∈⎧= ⎨

∉⎩

T

14

, ,g g g x y z centroid of ( , , ) B x y z

, ,

, ,

( , , )

( , , )

x y z

g

x y z

x B x y z

x B x y z

=∑

∑, ,

, ,

( , , )

( , , )

x y z

g

x y z

y B x y z

y B x y z

=∑

∑, ,

, ,

( , , )

( , , )

x y z

g

x y z

z B x y z

z B x y z

=∑

∑


The principal axes of B( x,y,z) are the eigenvector of inertia matrix I :

xx xy xz

yx yy yz

zx zy zz

I I I

I I I I

I I I

⎡ ⎤− −⎢ ⎥

= − −⎢ ⎥⎢ ⎥− −⎣ ⎦

2 2

2 2

, ,

( ) ( ) ( , , ) xx g g

x y z

I y y z z B x y z⎡ ⎤= − + −⎣ ⎦∑

15

, ,

, , yy g g

x y z

x x z z x y z= − + −⎣ ⎦

2 2

, ,

( ) ( ) ( , , ) zz g g

x y z

I x x y y B x y z⎡ ⎤= − + −⎣ ⎦∑

, ,

( ) ( ) ( , , ) xy yx g g

x y z

I I x x y y B x y z⎡ ⎤= = − −⎣ ⎦∑

, ,

( ) ( ) ( , , ) xz zx g g

x y z

I I x x z z B x y z⎡ ⎤= = − −⎣ ⎦∑

, ,

( ) ( ) ( , , ) yz zy g g

x y z

I I y y z z B x y z⎡ ⎤= = − −⎣ ⎦∑


The inertia matrix I is diagonal when computed with

respect to the principal axes.

The centroid and principal axes can describe the

orientation of a volume.

16

The principal axes registration can resolve six degrees

of freedom of an object (three rotation and three

translation).

It can compare the orientations of two binary volumes

through rotation, translation and scaling.


Normalize the principal axes (the eigenvectors of I ) and define:

1 1 1 2 1 3

2 1 2 2 2 3

3 1 3 2 3 3

e e e

E e e e

e e e

⎡ ⎤⎢ ⎥= ⎢ ⎥⎢ ⎥⎣ ⎦

17

cos sin 0 cos 0 sin 1 0 0

sin cos 0 0 1 0 0 cos sin

0 0 1 sin 0 cos 0 sin cos

R R Rθ ω ϕ

θ θ ω ω

θ θ ϕ ϕ

ω ω ϕ ϕ

= =

−⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥−⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥−⎣ ⎦ ⎣ ⎦ ⎣ ⎦

R

The rotation matrix is


Let

E = R

18

31

21

32

arcsin( )

arcsin( / cos )

arcsin( / cos )

e

e

e

ω

θ ω

ϕ ω

=

= −

= −




The PAR algorithm to register V1 and V2:

1. Compute the centroid of V1 and translate it to the origin.

2. Compute the principal axes (by using E=R) of V1 and rotate

V1 to coincide with the x, y and z axes.

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3. Compute the principal axes of V2 and rotate the x, y and z

axes to coincide with it.

4. Translate the origin to the centroid of V2.

5. Scaling V2 to match V1 by using factor .3 1/ 2sF V V =


20

A 3-D model of brain

ventricles obtained

from registering 22 MR

brain images using the

PAR method. Rotated views of the 3-D brain

ventricle model in the left image.

Iterative Principal Axes Registration (IPAR)

Advantages over PAR : IPAR can be used with partial

volumes.

Assumption made in IPAR : the field of view (FOV)

of a functional image (such as PET)) is less than the

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MR image) covers the entire brain.

Algorithm: refer to pages 259 ~ 264 of the textbook.

Iterative Principal Axes Registration (IPAR)

Sequential slices of MR (middle rows) and PET

(bottom rows) and the registered MR-PET brain

images (top row) using the IPAR method.

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Landmarks and Features Based Registration

With the corresponding landmarks/features identified

in the source and target images, a transformation can be

computed to register the images.

Non-rigid transformations have been used in

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an mar s ea ures ase reg s ra on y exp o ng e

relationship of corresponding points/features in source

and target images.

Two point-based algorithms will be introduced.

Similarity Transformation

X: source image

Y: target image

x : landmark points in X

y: landmark points in Y

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T ( x ): non-rigid transformation

' ( )T s= = ⋅ ⋅ + x x r x t

scaling rotation translation



Similarity Transformation

The registration error:

( ) ( ) E T s= − = + − x x y rx t yThe optimal transform is to find s, r and t to minimize:

25

22

1

N

i i i

i

w s=

+ −∑ rx t y

wi: the weighting factor for landmark x i

N : the number of landmark points

Similarity Transformation: Algorithm

1. Set s=1.

2. Find r through the following steps:

2 N

2a) Compute the weighted centroids of x and y.

2 N

26

1

2

1

i i

i

N

i

i

w

=

=

=

∑ x

1

2

1

i i

i

N

i

i

w

=

=

=

∑ y

i i= −

x x x

2b) Compute the distance of each landmark from

the centroid.

i i= −

y y y

Similarity Transformation: Algorithm (continued)

2c) Compute the weighted co-variance matrix.

2

1

N t

i i i

i

w=

= ∑ Z x y

Singular value decomposition

t = Λ Z U V

27

t t = =UU VV Ι

1 2 3 1 2 3( , , ) 0diag λ λ λ λ λ λ Λ = ≥ ≥ ≥

where

2d) Compute

(1,1, det( )) t diag= ⋅ ⋅r V VU U

Similarity Transformation: Algorithm (continued)

3. Compute the scaling factor:

2

1

N

i i i

i

w

s ==∑ rx y

28

2

1

i i i

i

w=

∑ rx x

4. Compute

s= −t y rx

Iterative Features Based Registration (WFBR)

X i : a data set representing a shape in source image

x ij : points in X i

T ( x ): transform operator

'

Y i : the corresponding data set in target image

yij : points in Y i

29

= = ⋅ ⋅Disparity function to be minimized:

22

1 1

( ) ( )s i N N

ij ij ij

i j

d T w T = =

= −∑∑ x y

N s: number of shapes

N i

: number of points in shape X i

WFBR: Iterative Algorithm

1. Determine T by the similarity transformation method.

2. Let T (0)=T and initialize the optimization loop for k=1 as

(0)

ij ij= x x

(1) (0) (0)=

30

ij ij

3. For the points in shape X i, find the closest points in Y i as

( ) ( )( , ) 1,2,3,...,k k

ij i ij i iC Y j N = = y x

where C i is the operator to find the closest point.



WFBR: Iterative Algorithm (continued)

4.Compute the transformation T (k ) between { x ij(0)} and

{ yij(k )} with the weights {wij} using the similarity

transformation method.

5. Compute

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( 1) ( ) (0)( )k k

ij ijT + = x x

6. Compute d (T (k )) – d (T (k +1)). If the convergence criterion

is met, stop; otherwise go to step 3 for the next iteration.

Elastic Deformation Based Registration

One of the two volume is considered to be made of elastic

material while the other serves as a rigid reference. Elastic matching

is to map the elastic volume to the reference volume.

The matching starts in a coarse mode followed by fine

adjustments.

32

e cons ra n s n e op m za on nc u e smoo ness an

incompressibility.

The smoothness ensures that there is continuity in the deformed

volume while the incompressibility guarantee that there is no

change in the total volume.

For detail algorithm, refer to pp. 269 ~ 272 in the textbook.

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Results of the elastic deformation based registration of 3-D MR brain images:

The left column shows three reference images, the mi ddle column shows the

images to be registered and the right column shows the registered brain images.

End of Lecture

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lecture8 registration

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