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CVPR2010: Advanced ITinCVPR in a Nutshell: part 4: Isocontours, Registration
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Isocontours and Image Registration
Anand Rangarajan
2. Image Registration
The need for information-theoretic measures
When there is no clearly established analytic relationship between
two or more images, it is often more convenient to minimize an
information-theoretic distance measure such as the negative of the
mutual information (MI).
Figure: Left: MR-PD slice. Right: Warped, noisy MR-T2 slice.
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4. Density and Entropy estimation
Density estimation
Histogramming
Parzen windows
Mixture models, wavelet densities (and other parametrizations)
Entropy estimation
Entropy estimation from the joint density (or distribution)
Direct entropy estimation (kNN, MST, Voronoi etc.)
Entropy estimation from the cumulative distribution (cdf)
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5. Moving away from samples
The underlying commonality in all of the previous approaches
All previous approaches are sample-based. Our new approach does
not begin with the idea of individual samples.
Obtain approx. to
Take samples density and entropy
Obtain improved
Take more samples approximation
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6. Image-based density estimation
Uncountable infinity
Assume uniform distribution of samples taken
on location
Transformation Each point in the
Location continuum contributes
to intensity
Intensity distribution
Distribution on intensity Image-Based
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8. Isocontour area-based density
Isocontour density estimation
Area trapped between level sets α and α + ∆α is proportional to the
probability Pr(α ≤ I ≤ α + ∆α). The density function is
ˆ
1 1
p(α) = du
A I (x,y )=α | I (x, y )|
Level sets at I (x, y ) = α
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9. Isocontour area-based density
Isocontour density estimation
Area trapped between level sets α and α + ∆α is proportional to the
probability Pr(α ≤ I ≤ α + ∆α). The density function is
ˆ
1 1
p(α) = du
A I (x,y )=α | I (x, y )|
Level sets at I (x, y ) = α and I (x, y ) = α + ∆α
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10. Isocontour area-based density
Isocontour density estimation
Area trapped between level sets α and α + ∆α is proportional to the
probability Pr(α ≤ I ≤ α + ∆α). The density function is
ˆ
1 1
p(α) = du
A I (x,y )=α | I (x, y )|
Area in between I (x, y ) = α and I (x, y ) = α + ∆α
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14. Joint Probability
Level sets at I1 (x, y ) = α1 and I2 (x, y ) = α2
The cumulative area of the black regions is proportional to
Pr(α1 ≤ I1 ≤ α1 + ∆α1 , α2 ≤ I2 ≤ α2 + ∆α2 ).
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15. Joint Probability
Level sets at I1 = α1 , α1 + ∆α1 and I2 = α2 and α2 + ∆α2
The cumulative area of the black regions is proportional to
Pr(α1 ≤ I1 ≤ α1 + ∆α1 , α2 ≤ I2 ≤ α2 + ∆α2 ).
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16. Joint Probability
Areas: α1 ≤ I1 ≤ α1 + ∆α1 and α2 ≤ I2 ≤ α2 + ∆α2
The cumulative area of the black regions is proportional to
Pr(α1 ≤ I1 ≤ α1 + ∆α1 , α2 ≤ I2 ≤ α2 + ∆α2 ).
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17. Joint Probability Expression
The joint density of images I1 (x, y ) and I2 (x, y ) with area of
overlap A is related to the area of intersection of regions
between level curves at α1 and α1 + ∆α1 of I1 and at α2 and
α2 + ∆α2 of I2 as ∆α1 → 0, ∆α2 → 0.
The joint density
ˆ ˆ
1 du1 du2
p(α1 , α2 ) =
A I1 (x,y )=α1 ,I2 (x,y )=α2 | I1 (x, y ) I2 (x, y ) sin(θ)|
where u1 and u2 are the level curve tangent vectors in I1 and I2
respectively and θ the angle between the image gradients.
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18. When there’s no joint density
Pathological cases
1
Examine | I1 (x,y ) I2 (x,y ) sin(θ)| :
Level curves of Image 2
at intensities α2 and
α2+∆α Level curves of Image 1
Region in Image 2 at intensities α1 and
of constant intensity α1+∆α
α2
Region in Image 1
Region in Image 1 with constant intensity
of constant intensity
α1
α1 Level curves of Image 2
at intensities α2 and
Area of intersection Area of intersection α2+∆α
Area where level curves
of the two regions (contribution to from images 1 and 2
[contribution to P(α1,α2)] P(α1,α2) are parallel
Figure: Left: Both images flat. Middle: One image flat. Right: Gradients
run locally parallel.
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23. Information-theoretic formulation
Mutual Information-based registration
Given two images I1 and I2 , a now standard approach to image
registration minimizes
E (T ) = −MI (I1 , I2 (T )) = H(I1 , I2 (T )) − H(I1 ) − H(I2 (T ))
where the mutual information (MI) is unpacked as the sum of the
marginal entropies minus the joint entropy. The entropies (Shannon)
can be easily estimated from the iscontour density estimators (as well
as other estimators such as histogramming and Parzen windows).
The transformation T (usually rigid or affine) is applied to only I2 in
this formulation.
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24. Comparison with std. histograms
32 bins
Left: Standard histogramming. Right: Isocontours
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25. Comparison with std. histograms
64 bins
Left: Standard histogramming. Right: Isocontours
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26. Comparison with std. histograms
128 bins
Left: Standard histogramming. Right: Isocontours
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27. Comparison with std. histograms
256 bins
Left: Standard histogramming. Right: Isocontours
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28. Comparison with std. histograms
512 bins
Left: Standard histogramming. Right: Isocontours
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29. Comparison with std. histograms
1024 bins
Left: Standard histogramming. Right: Isocontours
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37. Discussion
With piecewise linear interpolation, much faster than upsampled
histogramming
Extended to multiple image registration and 3D
Statistical significance (Kolmogorov-Smirnov) tests run
Other groups (Oxford etc.) involved - analytic studies
Applied to mean shift filtering and unit vector density estimation
Drawbacks: Non differentiable, no clean extension to higher
dimensions
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