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  • 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-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

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    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/

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    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/

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    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/

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    Intensity-Based

    ImageRegistration

    42-431 Intro.

    BiomedicalImaging and

    Image Analysis

    Overview

    Least squares

    Cross-

    correlation

    MutualInformation

    Three main components

    http://find/

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    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/

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    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/

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    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/

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    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/

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    Intensity-Based

    ImageRegistration

    42-431 Intro.

    BiomedicalImaging and

    Image Analysis

    Overview

    Least squares

    Cross-

    correlation

    MutualInformation

    Example

    http://find/

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    Intensity-Based

    ImageRegistration

    42-431 Intro.

    BiomedicalImaging and

    Image Analysis

    Overview

    Least squares

    Cross-

    correlation

    MutualInformation

    Example

    http://find/

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    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/

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    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

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    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

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    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/

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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/

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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.

    As a consequence, sum of squared differences, cross

    correlation, not likely to succeed.

    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 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/

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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)

    Optimization

    minf

    I(s(f), t)

    Intensity-BasedE l

    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

    Mutual

    Information

    Example

    http://find/

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