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A New Method of Probability Density Estimation for Mutual Information Based Image Registration

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Title: A New Method of Probability Density Estimation for Mutual Information Based Image Registration


1
A New Method of Probability Density Estimation
for Mutual Information Based Image Registration
  • Ajit Rajwade,
  • Arunava Banerjee,
  • Anand Rangarajan.
  • Dept. of Computer and Information Sciences
    Engineering,
  • University of Florida.

2
Image Registration problem definition
  • Given two images of an object, to find the
    geometric transformation that best aligns one
    with the other, w.r.t. some image similarity
    measure.

3
Mutual Information for Image Registration
  • Mutual Information (MI) is a well known image
    similarity measure (Viola95, Maes97).
  • Insensitive to illumination changes useful in
    multimodality image registration.

4
Mathematical Definition for MI

Marginal entropy
Joint entropy
Conditional entropy
5
Calculation of MI
  • Entropies calculated as follows

Marginal Probabilities
Joint Probability
6
Joint Probability
Functions of Geometric Transformation
7
Estimating probability distributions
Histograms
How do we select bin width?
Too large bin width Over-smooth distribution
Too small bin width Sparse, noisy distribution
8
Estimating probability distributions
Parzen Windows
Choice of kernel
Choice of kernel width
Too small Noisy, spiky
Too large Over-smoothing
9
Estimating probability distributions
Mixture Models Leventon98
How many components?
Local optima
Difficult optimization in every step of
registration.
10
Direct (Renyi) entropy estimation
Minimal Spanning Trees, Entropic kNN
Graphs Ma00, Costa03
Requires creation of MST from complete graph of
all samples
11
Cumulative Distributions
Entropy defined on cumulatives Wang03
Extremely Robust, Differentiable
12
A New Method
Whats common to all previous approaches?
Obtain approximation to the density
Take samples
More accurate approximation
More samples
13
A New Method
Uncountable infinity of samples taken
Assume uniform distribution on location
Transformation Location Intensity
Each point in the continuum contributes to
intensity distribution
Distribution on intensity
Image-Based
14
Other Previous Work
  • A similar approach presented in Kadir05.
  • Does not detail the case of joint density of
    multiple images.
  • Does not detail the case of singularities in
    density estimates.
  • Applied to segmentation and not registration.

15
A New Method
Continuous image representation (use some
interpolation scheme) No pixels!
Trace out iso-intensity level curves of the
image at several intensity values.
16
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17
Analytical Formulation Marginal Density
  • Marginal density expression for image I(x,y) of
    area A
  • Relation between density and local image gradient
    (u is the direction tangent to the level curve)

18
Joint Probability
19
Joint Probability
20
Analytical Formulation Joint Density
  • The joint density of images and with area
    of overlap A is related to the area of
    intersection of the regions between level curves
    at and of , and at and
    of as
    .
  • Relation to local image gradients and the angle
  • between them ( and are the level curve
    tangent vectors in the two images)

21
Practical Issues
  • Marginal density diverges to infinity, in areas
    of zero gradient (level curve does not exist!).
  • Joint density diverges
  • in areas of zero gradient of either or both
  • image(s).
  • in areas where gradient vectors of the
  • two images are parallel.

22
Work-around
  • Switch from densities (infinitesimal bin width)
    to distributions (finite bin width).
  • That is, switch from an analytical to a
    computational procedure.

23
Binning without the binning problem!
More bins more (and closer) level
curves. Choose as many bins as desired.
24
32 bins
64 bins
128 bins
256 bins
512 bins
1024 bins
Standard histograms
Our Method
25
Pathological Case regions in 2D space where both
images have constant intensity
26
Pathological Case regions in 2D space where only
one image has constant intensity
27
Pathological Case regions in 2D space where
gradients from the two images run locally parallel
28
Registration Experiments Single Rotation
  • Registration between a face image and its 15
    degree rotated version with noise of variance 0.1
    (on a scale of 0 to 1).
  • Optimal transformation obtained by a brute-force
    search for the maximum of MI.
  • Tried on a varied number of histogram bins.

29
MI Trajectory versus rotation noise variance 0.1
Our Method
Standard Histograms
16 bins
32 bins
64 bins
128 bins
30
MI Trajectory versus rotation noise variance 0.8
Our Method
Standard Histograms
16 bins
32 bins
64 bins
128 bins
31
Affine Image Registration
BRAINWEB
PD slice
T2 slice
Warped T2 slice
Warped and Noisy T2 slice
Brute force search for the maximum of MI
32
Affine Image Registration
MI with standard histograms
MI with our method
33
Directions for Future Work
  • Our distribution estimates are not differentiable
    as we use a computational (not analytical)
    procedure.
  • Differentiability required for non-rigid
    registration of images.

34
Directions for Future Work
  • Simultaneous registration of multiple images
    efficient high dimensional density estimation and
    entropy calculation.
  • 3D Datasets.

35
References
  • Viola95 Alignment by maximization of mutual
    information, P. Viola and W. M. Wells III, IJCV
    1997.
  • Maes97 Multimodality image registration by
    maximization of mutual information, F. Maes, A.
    Collignon et al, IEEE TMI, 1997.
  • Wang03 A new robust information theoretic
    measure and its application to image alignment,
  • F. Wang, B. Vemuri, M. Rao Y. Chen, IPMI 2003.
  • BRAINWEB http//www.bic.mni.mcgill.ca/brainweb/

36
References
  • Ma00 Image registration with minimum spanning
    tree algorithm, B. Ma, A. Hero et al, ICIP 2000.
  • Costa03 Entropic graphs for manifold
    learning, J. Costa A. Hero, IEEE Asilomar
    Conference on Signals, Systems and Computers
    2003.
  • Leventon98 Multi-modal volume registration
    using joint intensity distributions, M. Leventon
    E. Grimson, MICCAI 98.
  • Kadir05 Estimating statistics in arbitrary
    regions of interest, T. Kadir M. Brady, BMVC
    2005.

37
Acknowledgements
  • NSF IIS 0307712
  • NIH 2 R01 NS046812-04A2.

38
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