a new method of probability density estimation for mutual information based image registration
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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. Image Registration: problem definition. - PowerPoint PPT PresentationTRANSCRIPT
A New Method of Probability Density Estimation for Mutual Information Based
Image Registration
Ajit Rajwade, Ajit Rajwade,
Arunava Banerjee,Arunava Banerjee,
Anand Rangarajan.Anand Rangarajan.
Dept. of Computer and Information Sciences & Engineering,
University of Florida.
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.
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.
)|()(),( 21121 IIHIHIIMI
)|( 21 IIH
),()()(),( 212121 IIHIHIHIIMI
)|( 12 IIH),IH(I 21
),(MI 21 II)( 1IH )( 2IH
)( 1IH ),( 21 IIH
Marginal entropy Joint entropy
Mathematical Definition for MI
)|( 21 IIH
Conditional entropy
Calculation of MI
• Entropies calculated as follows:
)(p 2112 ,)(
)(
22
11
p
pJoint Probability
Marginal Probabilities
),(log),(),(
)(log)()(
)(log)()(
2112211221
22222
11111
1 2
2
1
ppIIH
ppIH
ppIH
Joint Probability
j)(ip ,12
1I 2I
),( 21 IIH ),(MI 21 II
Functions of Geometric Transformation
Estimating probability distributions
Histograms
How do we selectbin width?
Too large bin width:Over-smooth distribution
Too small bin width:Sparse, noisy distribution
Estimating probability distributions
Parzen Windows
Choice of kernel
Choice of kernel width
Too large:Over-smoothing
Too small:Noisy, spiky
Estimating probability distributions
Mixture Models[Leventon98]
How many components?
Difficult optimization in every step of registration.
Local optima
Direct (Renyi) entropy estimation
Minimal SpanningTrees, Entropic kNN Graphs
[Ma00, Costa03]
Requires creation of MSTfrom complete graph of all samples
Cumulative Distributions
Entropy definedon cumulatives
[Wang03]
Extremely Robust,Differentiable
A New Method
What’s common to allprevious approaches?
Take samplesObtain approximation
to the density
More samples More accurateapproximation
A New Method
Assume uniform distribution on location
TransformationLocation
Intensity
Distribution on intensity
Uncountable infinityof samples taken
Each point in thecontinuum contributes
to intensitydistribution
Image-Based
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.
A New Method
Continuous image representation (use some interpolationscheme) No pixels!
Trace out iso-intensity level curves of the imageat several intensity values.
Intensity at Curves Level andIntensity at Curves Level regionbrown of area )( IP
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):
I
dxdy
Ap ] 0lim[
1)(
I yxI
du
Ap
|),(|
1)(
Joint Probability
2211 Iin and Iin Intensity at Curves Level 2222
1111
Iin ),( and
Iin ) ,(Intensity at Curves Level
regionblack of area
),( 22221111 IIP
Joint Probability
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):
1 11
2 22 0,0 21
2211 , 21
2121 |sin),(),(|
1),(
IIyxIyxI
dudu
Ap
),(1 yxI ),(2 yxI
1I
2I
1u 2u
Practical Issues
• Marginal density diverges to infinity, in areas of zero gradient (level curve does not exist!).
I yxI
du
Ap
|),(|
1)(
2211 , 21
2121 |sin),(),(|
1),(
IIyxIyxI
dudu
Ap
• 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.
Work-around
• Switch from densities (infinitesimal bin width) to distributions (finite bin width).
• That is, switch from an analytical to a computational procedure.
Binning without the binning problem!More bins = more (and closer) level curves.
Choose as many bins as desired.
Standard histograms Our Method32 bins64 bins128 bins256 bins512 bins1024 bins
Pathological Case: regions in 2D space whereboth images have constant intensity
Pathological Case: regions in 2D space whereonly one image has constant intensity
Pathological Case: regions in 2D space where gradients from the
two images run locally parallel
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.
MI Trajectory versus rotation: noise variance 0.1
Standard Histograms Our Method
16 bins32 bins64 bins128 bins
MI Trajectory versus rotation: noise variance 0.8
Standard Histograms Our Method
16 bins32 bins64 bins128 bins
PD slice T2 slice
Affine Image Registration
BRAINWEB
Warped T2 sliceWarped and Noisy T2 slice
Brute force search for themaximum of MI
Affine Image RegistrationMI with standard
histograms
MI with our method
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.
Directions for Future Work
• Simultaneous registration of multiple images: efficient high dimensional density estimation and entropy calculation.
• 3D Datasets.
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/
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.
Acknowledgements
• NSF IIS 0307712
• NIH 2 R01 NS046812-04A2.
Questions??