intensity based registration
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7/30/2019 Intensity Based Registration
1/22
Intensity-Based
ImageRegistration
42-431 Intro.
BiomedicalImaging and
Image Analysis
Overview
Least squares
Cross-
correlation
MutualInformation
Intensity-Based Image Registration
Gustavo K. Rohde
42-431 Intro. Biomedical Imaging and Image Analysis
November 7, 2008
http://find/ -
7/30/2019 Intensity Based Registration
2/22
Intensity-Based
ImageRegistration
42-431 Intro.
BiomedicalImaging and
Image Analysis
Overview
Least squares
Cross-
correlation
MutualInformation
Overview
Intensity-based registrationGiven digital images s(m,n) and t(m,n), find spatialtransformation f such that s(fx(m,n), fy(m,n)) and
t(m,n) are similar.
http://find/ -
7/30/2019 Intensity Based Registration
3/22
Intensity-Based
ImageRegistration
42-431 Intro.
BiomedicalImaging and
Image Analysis
Overview
Least squares
Cross-
correlation
MutualInformation
Overview
Intensity-based registrationGiven digital images s(m,n) and t(m,n), find spatialtransformation f such that s(fx(m,n), fy(m,n)) and
t(m,n) are similar.
Optimization
minfC
(s(f), t ,f )
f : Rd Rd: spatial transformation model C spatial transformation class
(, , ): difference measure
http://find/http://goback/ -
7/30/2019 Intensity Based Registration
4/22
Intensity-Based
ImageRegistration
42-431 Intro.
BiomedicalImaging and
Image Analysis
Overview
Least squares
Cross-
correlation
MutualInformation
Continuous image representation
s(x, y) =pZ
qZ
c[p, q](x p)(y q)
where (x) is the basis function (sinc function, B-splines),and c are the the coefficients.
Spatial transformation
s(fx(m,n), fy(m,n)) =pZ
qZ
c[p, q](fx(m,n)p)(fy(m,n)q)
http://find/ -
7/30/2019 Intensity Based Registration
5/22
Intensity-Based
ImageRegistration
42-431 Intro.
BiomedicalImaging and
Image Analysis
Overview
Least squares
Cross-
correlation
MutualInformation
Three main components
http://find/ -
7/30/2019 Intensity Based Registration
6/22
Intensity-Based
ImageRegistration
42-431 Intro.
BiomedicalImaging and
Image Analysis
Overview
Least squares
Cross-
correlation
MutualInformation
Least squares
(s(f), t ,f ) = s(f) t2
http://find/ -
7/30/2019 Intensity Based Registration
7/22
Intensity-Based
ImageRegistration
42-431 Intro.
BiomedicalImaging and
Image Analysis
Overview
Least squares
Cross-
correlation
MutualInformation
Least squares
(s(f), t ,f ) = s(f) t2
Example
Affine transformations f(x) = Ax + a in 2D.
(s(f), t ,f ) =m
n
(s (fx(m,n), fy(m,n)) t(m,n))2
with fx(m,n) = A11m + A12n + a1 andfy(m,n) = A21m + A22n + a2.
http://find/ -
7/30/2019 Intensity Based Registration
8/22
Intensity-Based
ImageRegistration
42-431 Intro.
BiomedicalImaging and
Image Analysis
Overview
Least squares
Cross-
correlation
MutualInformation
Least squares
(s(f), t ,f ) = s(f) t2
Example
Affine transformations f(x) = Ax + a in 2D.
(s(f), t ,f ) =m
n
(s (fx(m,n), fy(m,n)) t(m,n))2
with fx(m,n) = A11m + A12n + a1 andfy(m,n) = A21m + A22n + a2.
Optimize using
pk+1 = pk (s(f), t ,f ).
http://find/ -
7/30/2019 Intensity Based Registration
9/22
Intensity-Based
ImageRegistration
42-431 Intro.
BiomedicalImaging and
Image Analysis
Overview
Least squares
Cross-
correlation
MutualInformation
Cross-correlation
Let s[n] and t[n] = s[n a] be two nice signals to bealigned. The goal is then to find a. This can be done viacross correlation
a = arg maxn
s tv[n]
For larger signals, the computation above can be donemore efficiently using the FFT.
http://find/http://goback/ -
7/30/2019 Intensity Based Registration
10/22
Intensity-Based
ImageRegistration
42-431 Intro.
BiomedicalImaging and
Image Analysis
Overview
Least squares
Cross-
correlation
MutualInformation
Example
http://find/ -
7/30/2019 Intensity Based Registration
11/22
Intensity-Based
ImageRegistration
42-431 Intro.
BiomedicalImaging and
Image Analysis
Overview
Least squares
Cross-
correlation
MutualInformation
Example
http://find/ -
7/30/2019 Intensity Based Registration
12/22
Intensity-Based
ImageRegistration
42-431 Intro.
BiomedicalImaging and
Image Analysis
Overview
Least squares
Cross-
correlation
MutualInformation
Normalized Cross-correlation
What if we have s[n] and t[n] = As[n a], with unknown
A?
Use normalized cross correlation:
(s,t,f) =s[n], t[f(n)]
s[n]t(f[n])
http://find/ -
7/30/2019 Intensity Based Registration
13/22
Intensity-Based
ImageRegistration
42-431 Intro.
BiomedicalImaging and
Image Analysis
Overview
Least squares
Cross-
correlation
MutualInformation
Example
http://find/ -
7/30/2019 Intensity Based Registration
14/22
Intensity-Based
ImageRegistration
42-431 Intro.
BiomedicalImaging and
Image Analysis
Overview
Least squares
Cross-
correlation
MutualInformation
Example
http://find/ -
7/30/2019 Intensity Based Registration
15/22
Intensity-Based
ImageRegistration
42-431 Intro.
BiomedicalImaging and
Image Analysis
Overview
Least squares
Cross-
correlation
MutualInformation
Mutual Information
Key problem
Matching images of different modalities.
Example
Matching CT and MRI.
http://find/ -
7/30/2019 Intensity Based Registration
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Intensity-Based
ImageRegistration
42-431 Intro.
BiomedicalImaging and
Image Analysis
Overview
Least squares
Cross-
correlation
MutualInformation
Intensity values not linearly related
Joint histogram: no linear relationship between intensityvalue, even when the images are optimally aligned. This
is generlly the case when matching images of differentmodalities.
http://find/ -
7/30/2019 Intensity Based Registration
17/22
Intensity-Based
ImageRegistration
42-431 Intro.
BiomedicalImaging and
Image Analysis
Overview
Least squares
Cross-
correlation
MutualInformation
Intensity values not linearly related
Joint histogram: no linear relationship between intensityvalue, even when the images are optimally aligned. This
is generlly the case when matching images of differentmodalities.
As a consequence, sum of squared differences, cross
correlation, not likely to succeed.
d
http://find/ -
7/30/2019 Intensity Based Registration
18/22
Intensity-Based
ImageRegistration
42-431 Intro.
BiomedicalImaging and
Image Analysis
Overview
Least squares
Cross-
correlation
MutualInformation
I t it B d
http://find/ -
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Intensity-Based
ImageRegistration
42-431 Intro.
BiomedicalImaging and
Image Analysis
Overview
Least squares
Cross-
correlation
MutualInformation
I t sit B s dM t l I f ti
http://find/ -
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Intensity-Based
ImageRegistration
42-431 Intro.
BiomedicalImaging and
Image Analysis
Overview
Least squares
Cross-
correlation
Mutual
Information
Mutual Information
Let Ps(f),t(s(f), t) denote the joint PDF of images s(f)and t. The mutual information is given by:
I(s(f), t) =s(f),p
Ps(f),t(s(f), t)log Ps
(f),t(s(f), t)
Ps(f)(s(f))Pt(t)
Intensity BasedM t l I f ti
http://find/ -
7/30/2019 Intensity Based Registration
21/22
Intensity-Based
ImageRegistration
42-431 Intro.
BiomedicalImaging and
Image Analysis
Overview
Least squares
Cross-
correlation
Mutual
Information
Mutual Information
Let Ps(f),t(s(f), t) denote the joint PDF of images s(f)and t. The mutual information is given by:
I(s(f), t) =s(f),p
Ps(f),t(s(f), t)log Ps(f),t(s(f), t)Ps(f)(s(f))Pt(t)
Optimization
minf
I(s(f), t)
Intensity-BasedE l
http://find/ -
7/30/2019 Intensity Based Registration
22/22
Intensity-Based
ImageRegistration
42-431 Intro.
BiomedicalImaging and
Image Analysis
Overview
Least squares
Cross-
correlation
Mutual
Information
Example
http://find/
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