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Summary Medical Image Analysis (8DC00)
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Samenvatting van de colleges van Medical Image Analysis 8DC00 van studiejaar in het Engels
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Summary Medical Image Analysis 8DC00 DYWRegistration
Image registration: determination of a geometrical transformation that aligns one view of an
object with another view of that object or another object.
- It aligns the moving axial scan with the fixed one.
Geometrical transformations: can be combined by multiplying the transformation matrices,
not commutative.
- Translation: adding a 2D translation vector to every vertex of x.
- Rotation: rotation angle 𝜃𝜃 defines the number of degrees of rotation and is typically
done about the image origin.
cos 𝜃𝜃 − sin 𝜃𝜃
• 𝑅𝑅 = � �.
sin 𝜃𝜃 cos 𝜃𝜃
- Scaling: the straightness of lines and the angles between them are preserved.
𝑠𝑠𝑥𝑥 0
• 𝑆𝑆 = � 0 𝑠𝑠 �.
𝑦𝑦
- Reflection: mirror transformation of an image along a given axis.
• Horizontal: 𝑥𝑥 ′ = −𝑥𝑥 and 𝑦𝑦 ′ = 𝑦𝑦.
• Vertical: 𝑥𝑥 ′ = 𝑥𝑥 and 𝑦𝑦 ′ = −𝑦𝑦.
- Shearing: each point is displayed by a distance proportional to the perpendicular
distance from the point’s parallel line.
1 ℎ𝑥𝑥
• 𝐻𝐻 = � �.
ℎ𝑦𝑦 1
Rigid transformation: consists of only rotation and translation.
- In 2D it had 3 degrees of freedom: rotation angle and 2D translation vector.
- Orthogonal Procrustes problem: the minimalization of the error of rigid registration.
Affine transformation: no restriction on the transformation parameters. It is a composition of
rotation, translation, scaling, reflection and shearing, which results in 9 parameters. However,
7 are dependent.
- [𝜙𝜙 𝑠𝑠𝑥𝑥 𝑠𝑠𝑦𝑦 ℎ𝑥𝑥 ℎ𝑦𝑦 𝑟𝑟𝑥𝑥 𝑟𝑟𝑦𝑦 𝑡𝑡𝑥𝑥 𝑡𝑡𝑦𝑦 ].
1. However, reflections do not usually occur in medical image registration so this
should be excluded from affine registration.
2. The inverse of a affine transformation can be computed by taking the inverse
of the transformation matrix.
3. If the determinant of the transformation matrix is positive we are talking about
a rotation, if negative a reflection.
- In 2D it uses 6 degrees of freedom of the transformation matrix. So, there are 6
unknowns, thus there are at least 3 corresponding points pairs needed. This, because
every corresponding point pair produces 2 equations (one for the x and one for the y
coordinate).
- In 3D it has 12 degrees of freedom and since every point produces 3 equations (one
for the x, one for the y and one for the z coordinate) there are at least 4 corresponding
point pairs needed to solve all.
- The optimal transformation can be found by solving 𝑋𝑋 ′ 𝑋𝑋 𝑇𝑇 (𝑋𝑋𝑋𝑋 𝑇𝑇 )−1 .
Forward mapping: specifies where every pixel after transformation will go. However, gaps
and / or overlaps may occur.
- Inverse mapping and interpolation can avoid these things.
1
, Summary Medical Image Analysis 8DC00 DYW
1. Define grid of the output image.
2. Map the points on the grid to the input image with 𝑇𝑇 −1 .
3. Determine the intensity value of those location with image interpolation.
Inter-modal image registration: comparing two images which are created by different imaging
techniques.
Intra-model image registration: comparing two images taken at different times, but with the
same equipment.
Uni-modal involves aligning images obtained using the same imaging technique.
Multi-modal: aligns images obtained via differing imaging techniques.
Point-based registration: image registration driven by a set of reference points on both fixed
and moving views. The corresponding point pairs can be used to calculate the parameters of
the transformation matrix, since it is assumed that 𝑥𝑥𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 = 𝑇𝑇𝑥𝑥𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 . This matrix will be used
to translate the moving image with −𝑇𝑇.
- Fiducial points: selected points in the fixed image.
1. Intrinsic features: anatomical landmarks.
2. Extrinsic features: implanted markers.
- Optimalization: finding the best parameter according to an objective function, which
is either minimised or maximised.
1. This can be done via a full search of the parameter space, however this is not
the most efficient optimization approach.
2. Start with a certain initial set of transformation parameters, hence point
coordinates, and (iteratively) update the parameters such that the distance
between the fixed and moving points become smaller.
3. Perfect fiducial alignment is impossible, therefore the best alignment will be
found using the transformation with the lowest error.
4. So, to find the minimum of the error 𝐸𝐸(𝑇𝑇) with respect to the parameters. The
partial derivative / gradient can be set to zero.
2
𝐸𝐸 (𝑇𝑇) = ∑𝑛𝑛𝑖𝑖=1 ��𝑇𝑇𝑥𝑥′𝑖𝑖 − 𝑥𝑥𝑖𝑖 �� .
2
- Target registration error (TRE): can be used to evaluate the image registration.
1. Target points: should be in locations that are relevant and are used for
optimalization.
They should not be the same as for the computation, so the fiducial
points. Since, their distances are by definition minimized and might
even be close to zero after the transformation and will thus not say
anything about how well the rest of the image is registered.
2. A low error means a high accuracy.
Image intensity-based registration: it uses an iterative optimalization method, where the
similarity slowly increases. This can be done using different similarity measures:
2
- Sum of square differences: 𝑆𝑆𝑆𝑆𝑆𝑆 (𝐼𝐼, 𝐽𝐽, 𝑇𝑇) = ∑𝑛𝑛𝑖𝑖=1�𝐼𝐼 (𝑖𝑖 ) − 𝐽𝐽𝑇𝑇 (𝑖𝑖 )� , with 𝐼𝐼 is the fixed image
and 𝐽𝐽 the transformed moving image.
1. The SSD will be lowest when the image is perfectly aligned (SSD=0). It will
increase with misalignments.
2. Optimal if two images only differ by Gaussian noise.
3. Not true for inter-modality registration.
4. Rarely true for intra-modularity registration, but can still be used with success.
2
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