Transcript Image Registration

Image Registration
박성진
Surface-based Registration
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The 3D boundary of an anatomical object is an
intuitive and easily characterized geometrical
feature that can be used for registration
Surface-based methods
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Point-based registration
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Determine corresponding surfaces in different images
Find the transformation that best aligns these surfaces
Aligns generally small number of corresponding points
Surface-based registration
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Aligns larger number of points for which correspondence
is unavailable
Surfaces
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Skin surface (air-skin interface)
Bone surface (tissue-bone interface)
Representations
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Point set (collection of points on the surface)
Faceted surface, e.g., triangle set
approximating surface
Implicit surface
Parametric surface, e.g., B-spline surface
Surface-based Registration
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Disparity function
N
d (T ( X ), Y ) 
w d
x
j 1
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2
j
2
(T ( x j ), Y ) 
Nx
w
j 1
2
j
|| T ( x j )  y j ||2
Given a set of surface points and a surface, find the rigid
transformation that minimizes the mean squared distance
between the points and the surface
Head and Hat Method
Iterative Closest Point
Method
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Initialization: k  1, xi(0)  xi , xi(1)  T (0) ( xi(0) )
Iteratively apply the following steps,
incrementing k after each loop, until
convergence within a tolerance is achieved:
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Compute the closest points
(k )
(k )
y

C
(
x
i ,Y )
Compute the transformation i between
the
)
initial point set
and current
T (k set
)
{ yi(kregistered
}
{ xi( 0 ) }
Apply the transformation
to produce
points
xi( k 1) the
T ( k ) ( iterative
xi( 0 ) )
Terminate
loop when
d (T ( k ) )  d (T ( k 1) )  
Intensity-based registration
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Registration based on similarity measures
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Uses some measure derived from the intensity
of the image directly
Assumes that there is a relationship between
the image intensities of both images if the
images are registered
Does not require any feature extraction, thus
the registration error is not by any errors
Generic Intensity-based
Registration Procedure
Initial transformation
Calculate cost function
For transformation T
Optimize T by maximizing
cost function C
Final transformation
Is new transformation
an improvement?
Update transformation
Intensity-based registration
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Registration-based on geometric features is
independent of the modalities from which the
features have been derived
Registration-based on voxel similarity measures
features we must make a distinction between
monomodality registration and multimodality
registration
Monomodality image
registration
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Sums of Squared Differences (SSD)
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Assumes an identity relationship between
image intensities in both images
Optimal measure if the difference between both
images is Gaussian noise
Sensitive to outliers
Monomodality image
registration
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Robust statistics can be used to reduce
the influence of outliers on the registration
Sum of Absolute Differences (SAD)
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Assumes an identity relationship between
image intensities
Less sensitive to outliers
Monomodality image
registration
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Correlation
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Assumes a linear relationship between
image intensities
Sensitive to large intensity values
Monomodality image
registration
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Normalized Cross Correlation (CC)
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Assumes a linear relationship between
image intensities
Monomodality image
registration
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Ratio of Image Uniformity (RIU)
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Normalized standard deviation
Registration Basis :
Image Intensity
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Monomodality registration
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Image intensities are related by simple function
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Identity : SSD, SAD
Linear : CC, RIU
Multimodality registration
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Image intensities are related by some unknown
function or statistical relationship
Relationship between intensities is not known a
priori
Relationship between intensities can be viewed
by inspecting a 2D histogram or co-occurrence
matrix
Multimodality image
registration
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Partitioned image uniformity (PIU)
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Used for MR-PET registration
PIU : Measure the sum of the normalized
standard deviation of voxel values in
image B for each intensity a in image A
Images as Probability
Distribution
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Images can be viewed as probability distributions
p(a)
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Probability distribution of an image can be
estimated using
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Marginal probability p(a) of a pixel having intensity a
Joint probability p(a,b) of a pixel having intensity a in
one image and intensity b in another image
Parzen windowing
Histograms
Histograms require “binning”
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Usually use 32 to 256 bins per image
Images as Probability
Distribution
Intensity-based on IT
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Entropy
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Describes the amount of
information in image A
The information content
of an image is maximal
if all intensities have
equal probability
The information content
of an image is minimal if
one intensity a has a
probability of one
Intensity-based on IT
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Joint Entropy
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Describes the amount of information in the
combined images A and B
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If A and B are totally unrelated, the joint entropy will
be the sum of the entropies of A and B
If A and B are related, the joint entropy will be similar
Registration can be achieved by minimizing
the joint entropy between both images
Intensity-based on IT
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Joint Entropy is highly sensitive to the
overlap of the two images
Mutual information
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Describes how well one image can be
explained by another images
Expressed in terms of marginal and joint
probability distributions
Intensity-based on IT
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Mutual information is still sensitive to the overlap
of the two images
Normalized mutual information can be shown to
be independent of the amount of overlap between
images
Registration can be achieved by maximizing
(normalized) Mutual Information between both
images
Registration using Similarity
Measures
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Some similarity measures assume a functional
relationship between intensities
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Other similarity measures only assume a statistical
relationship
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Identity : SSD, SAD
Linear : CC, RIU
Nonlinear : PIU, CR
Joint entropy
(Normalized) Mutual Information
All similarity measures can be calculated from a
2D histogram of the images