Download Computational Aspects of MRI Geometrical Transforms 1

Computational Aspects of MRI
Geometrical Transforms
References
Geometrical Transformations
David Atkinson
• Foley, van Dam, Feiner, Hughes.
Computer Graphics: Principles and
Practice. Chapter 5.
• Wolfram MathWorld
http://mathworld.wolfram.com/
[email protected]
•
•
•
•
•
•
Computational Aspects of MRI
Computational Aspects of MRI
Outline
Geometrical Transforms in Therapy
Motivation
Vectors and coordinates
Coordinate systems
Homogeneous coordinates
Geometrical transforms
DICOM
Computational Aspects of MRI
and display
[VisionRT]
Computational Aspects of MRI
and Image Registration
Non-Rigid Registration between baseline and follow-up
Jacobian map within brain area
Modat et al., “Fast FFD using the NMI gradient and GPU”, 10th September, HP-MICCAI 08 workshop
Computational Aspects of MRI
Computational Aspects of MRI
1
Computational Aspects of MRI
Geometrical Transforms
Registration Matrix Calculation
Overall registration transform is composed of a series of stages
Calibration + tracking during intervention
M1
Scanner Space
3D Image Space
The XMR suite at Guy’s Hospital
(KCL)
M2
Catheter in human heart
Computational Aspects of MRI
Geometry: Coordinates and Vectors
Dicom image header M1
Calibration M2, T, P
Tracking M3
Patient imaging λ, R
M3
[Kawal Rhode, KCL]
2D Image Space
Scalar Product (dot product, inner product)
ĵ
(4,2,1)
r
X-ray C-arm
Space
R*P
y
r = 4xˆ + 2yˆ + zˆ
r = 4ˆi + 2ˆj + kˆ
λT
X-ray Table Space
r.ˆj = r ˆj cos(θ )
r
θ
x
r length of vector r
ˆj = 1 (unit vector with length 1)
z
Norm (length) r or r 2 = 4 2 + 2 2 + 12
Scalar product is length of projection of r onto unit vector ĵ
unit vector ˆj = 0 2 + 12 + 0 2 = 1
2
Computational Aspects of MRI
Scalar product in component form
u = u1ˆi + u2 ˆj + u3kˆ
[MathWorld]
Computational Aspects of MRI
Vector (Cross) product
• Cross product of two vectors u and v is a
third vector perpendicular to u and v.
• E.g. z axis is cross product of x and y.
v = v1ˆi + v2 ˆj + v3kˆ
3
u ⋅ v = ∑ ui vi = u1v1 + u2 v2 + u3v3
i =1
u× v
u
θ
v
Computational Aspects of MRI
Computational Aspects of MRI
2
Computational Aspects of MRI
Geometrical Transforms
Vector product
u× v
Right Handed Coordinate System
u
x
u = u1ˆi + u2 ˆj + u3kˆ
y
θ
v = v1ˆi + v2 ˆj + v3kˆ
thumb
v
1st finger
u × v = ˆi (u y v z − u z v y ) + ˆj(u z v x − u x v z ) + kˆ (u x v y − u y v x )
u × v = ( u v sin θ )nˆ
z
2nd finger
Computational Aspects of MRI
Computational Aspects of MRI
Example Scanner and Patient
Coordinate Systems
Radiographic Presentations
Anterior
L Left
P Posterior
H Head
Z/H
X/L
Head: “superior” (S)
Foot: “inferior” (I)
Y/P
Left
Right
Head
For a supine
(face-up) patient
Axial / Transverse
Posterior
Head
Sagittal
Coronal
Right
Left
Posterior
Anterior
Foot
Foot
Computational
Aspectsfoot
of MRI
View patient
from Left
Computational Aspects of MRI
Rotation Transformation
• Rotation matrix
transforms
coordinates of a
point by a rotation.
b
yb
ya
xb = xa cosθ − ya sin θ
yb = xa sin θ + ya cosθ
⎡ xb ⎤ ⎡cosθ
⎢ y ⎥ = ⎢ sin θ
⎣ b⎦ ⎣
− sin θ ⎤ ⎡ xa ⎤
cosθ ⎥⎦ ⎢⎣ ya ⎥⎦
Computational Aspects of MRI
a
θ
xb xa
• Rotations and scales can be expressed
using 2x2 matrices for 2D coordinates
(3x3 for 3D).
• However, translations cannot so use
Homogeneous Coordinates.
– Add another dimension with value 1 to the
coordinates.
– Add another row and column to the transform
matrix.
Computational Aspects of MRI
3
Computational Aspects of MRI
Geometrical Transforms
Homogeneous Coordinates
• A convenient way of representing coordinates
that allows for geometric transformations
including translations.
• Add an extra dimension.
w
P
1
x
[Foley]
xb = xa cosθ − ya sin θ
yb = xa sin θ + ya cosθ
⎡ xb ⎤ ⎡cosθ
⎢ y ⎥ = ⎢ sin θ
⎣ b⎦ ⎣
− sin θ ⎤ ⎡ xa ⎤
cosθ ⎥⎦ ⎢⎣ ya ⎥⎦
⎡ xb ⎤ ⎡cosθ
⎢ y ⎥ = ⎢ sin θ
⎢ b⎥ ⎢
⎣⎢ 1 ⎦⎥ ⎣⎢ 0
− sin θ
cosθ
0
0⎤ ⎡ xa ⎤
0 ⎥ ⎢ ya ⎥
⎥⎢ ⎥
1⎦⎥ ⎣⎢ 1 ⎦⎥
⎡ xb ⎤ ⎡1 0 t x ⎤ ⎡ xa ⎤
⎢ y ⎥ = ⎢0 1 t ⎥ ⎢ y ⎥
y ⎥⎢ a ⎥
⎢ b⎥ ⎢
⎣⎢ 1 ⎦⎥ ⎣⎢0 0 1 ⎦⎥ ⎢⎣ 1 ⎦⎥
xb = xa + t x
y b = ya + t y
y
Computational Aspects of MRI
Computational Aspects of MRI
Alternative row-vector x matrix
representation
[xb
y b ] = [ xa
⎡ cosθ
ya ] ⎢
⎣− sin θ
[xb
y b 1] = [xa
ya
[xb
y b 1] = [xa
⎡⋅
⎢
ya 1] ⎢ ⋅
⎣⎢t x
Note transpose!
sin θ ⎤
cosθ ⎥⎦
⎡ cosθ
1] ⎢− sin θ
⎢
⎣⎢ 0
⋅
⋅
ty
(MP) T = P T M T
Coords as columns of P post multiplying M
sin θ
cosθ
0
0⎤
0⎥
⎥
1⎦⎥
0⎤
⎥
0⎥
1⎥⎦
MATLAB maketform
but makehgtform uses
matrix vector
convention!
Row vectors pre-multiplying MT
Computational Aspects of MRI
Computational Aspects of MRI
Comments
Affine Transforms
• Homogeneous coordinates allow
representation of affine transformations.
• Transforms can be concatenated
(composed) into a series of matrix
multiplications - in the correct order.
• Take care assembling the matrix – is it for
row-vector*matrix, or, matrix*columnvector?
Computational Aspects of MRI
• An affine transform keeps parallel lines
parallel.
• Includes:
– translations,
– rotations,
– scale changes,
– skews.
Computational Aspects of MRI
4
Computational Aspects of MRI
Geometrical Transforms
Affine Transformation Matrices
(3D, matrix * column-vector format)
⎡s x
⎢0
scale ⎢
⎢0
⎢0
⎣
0
0
sy
0
0
0
sz
0
… 3D rotations
0
⎡1
⎢0 cosθ
⎢
Rx =
⎢0 sin θ
⎢
0
⎣0
⎡ cosθ 0
⎢ 0
1
Ry = ⎢
⎢− sin θ 0
⎢
0
⎣ 0
0⎤
0⎥
⎥
0⎥
1⎥⎦
⎡1
⎢λ
2D skew or shear ⎢ y
⎢0
⎢0
⎣
λx
1
0
0
0 0⎤
0 0⎥
⎥
1 0⎥
0 1⎥⎦
⎡1
⎢0
translation ⎢
⎢0
⎢0
⎣
0 0 tx ⎤
1 0 ty ⎥
⎥
0 1 tz ⎥
0 0 1 ⎥⎦
⎡cosθ
⎢ sin θ
Rz = ⎢
⎢ 0
⎢
⎣ 0
Computational Aspects of MRI
Applying transforms to coordinates
• transform applied to
multiple coordinates to
transform an object.
• in homogeneous
coordinates, matrices
can be multiplied to
compose transforms
0
− sin θ
cosθ
0
sin θ
0
cosθ
− sin θ
cosθ
0
0
0
0⎤
0⎥
⎥
0⎥
⎥
1⎦
0⎤
0⎥
⎥
0⎥
⎥
1⎦
0 0⎤
0 0⎥
⎥
1 0⎥
0 1⎥⎦
Computational Aspects of MRI
• Compose transforms into one matrix.
• Arrange multiple coordinates into one
Transform
matrix
⎡ xb ⎤ ⎡ ⋅ ⋅ ⋅ ⎤ ⎡ xa ⎤
⎢ y ⎥ = ⎢ ⋅ ⋅ ⋅⎥⎢ y ⎥
⎥⎢ a ⎥
⎢ b⎥ ⎢
⎣⎢ 1 ⎥⎦ ⎣⎢0 0 1⎥⎦ ⎢⎣ 1 ⎥⎦
⎡ xb
⎢y
⎢ b
⎣⎢ 1
ub L
vb L
1
1
One coordinate
⎤ ⎡ ⋅ ⋅ ⋅ ⎤ ⎡ xa
⎥ = ⎢ ⋅ ⋅ ⋅⎥⎢ y
⎥⎢ a
⎥ ⎢
⎦⎥ ⎣⎢0 0 1⎦⎥ ⎣⎢ 1
ua L
va
L
1
1
⎤
⎥
⎥
⎦⎥
Multiple coordinates
Computational Aspects of MRI
Computational Aspects of MRI
Coordinate and Rotation Origins
Summary
• Coordinate origin usually does not
coincide with image centre.
• Affine R matrix assumes rotation about
coordinate origin.
• Recipe: Translate image such that origin is
at centre, apply rotation, translate back.
These can be composed, note order is
important.
Computational Aspects of MRI
• Work in a right-handed coordinate system.
• To compose rotations, translations, or any
affine transform, use homogeneous
coordinates.
• The order of transformations is important.
Computational Aspects of MRI
5