A New Method of Probability Density Estimation for Mutual Information Based Image Registration Ajit Rajwade, Arunava Banerjee, Anand Rangarajan. Dept.
Download ReportTranscript A New Method of Probability Density Estimation for Mutual Information Based Image Registration Ajit Rajwade, Arunava Banerjee, Anand Rangarajan. Dept.
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
• 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.
Mathematical Definition for MI
MI
(
I
1 ,
I
2 )
H
(
I
1 )
H
(
I
1 |
I
2 )
MI
(
I
1 ,
I
2 )
H
(
I
1 )
H
(
I
2 )
H
(
I
1 ,
I
2 )
H
(
I
1 |
I
2 )
H(I
1
,I
2
) H
(
I
2 |
I
1 )
H
(
I
1 )
H
(
I
1 ) MI (
I
1 ,
I
2 )
H
(
I
1 ,
I
2 ) Marginal entropy Joint entropy
H
(
I
2 )
H
(
I
1 |
I
2 ) Conditional entropy
Calculation of MI
• Entropies calculated as follows:
H H H
(
I
1 ) (
I
2 (
I
1 , )
I
2 ) 1
p
1
p
2 ( ( 2 1 2 1 2 ) ) log
p
12 log ( 1
p
1 ,
p
2 ( 2 1 ( ) ) 2 ) log
p
12 ( 1 , 2 )
p
12
(
1 , 2
)
Joint Probability
p
1
p
2 ( 1 ( 2 ) ) Marginal Probabilities
I
1
Joint Probability
I
2 Functions of Geometric Transformation
p
12
(i
,
j) H
(
I
1 ,
I
2 ) MI (
I
1 ,
I
2 )
Estimating probability distributions
Histograms How do we select bin 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
How many components?
Mixture Models [Leventon98] Local optima Difficult optimization in every step of registration.
Direct (Renyi) entropy estimation
Minimal Spanning Trees, Entropic
kNN
Graphs [Ma00, Costa03] Requires creation of MST from complete graph of all samples
Cumulative Distributions
Entropy defined on cumulatives [Wang03] Extremely Robust, Differentiable
A New Method
What’s common to all previous approaches?
Take samples More samples Obtain approximation to the density More accurate approximation
A New Method
Assume uniform distribution on location Uncountable infinity of samples taken
Transformation
Location Intensity Each point in the continuum contributes to intensity distribution Distribution on intensity 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 interpolation scheme) No pixels!
Trace out iso-intensity level curves of the image at several intensity values.
P
( Curves at at and
Analytical Formulation: Marginal Density
• Marginal density expression for image
I(x,y)
of area
A
:
p
( ) 1
A
lim[ 0 ]
I
dxdy
• Relation between density and local image gradient (
u
is the direction tangent to the level curve):
p
( ) 1
A I
du
|
I
(
x
,
y
) |
Joint Probability
Joint Probability
P
( 1 Level 1 1 at 1 area Level and ( 2 , 2 Intensity 2 ) in I 2
I
2 1 , 1 in I 1 2 1 and ) 2 in in ) I 1 I 2
Analytical Formulation: Joint Density
• The joint density of images
I
1 (
x
,
y
) and
I
2 (
x
,
y
) with area of overlap
A
is related to the area of intersection of the 2 2 2
I
2 1 1 1 0 , 2 1 of 0
I
1 , • Relation to local image gradients and the angle between them (
u
1 and
u
2 vectors in the two images): are the level curve tangent
p
( 1 , 2 ) 1
A I
1 1 ,
I
2 2 |
I
1 (
x
,
du
1
du
2
y
)
I
2 (
x
,
y
) sin |
Practical Issues
p
( 1 , 2 )
p
( 1
A
I
1 ) 1 ,
I
1 2
A
2
I
|
du du
1
du
2 |
I
1
x I
, (
y x
) ,
y I
) 2 | (
x
,
y
) sin | • Marginal density diverges to infinity, in areas of zero gradient (level
curve
does 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 Method
Pathological Case: regions in 2D space where both images have
constant intensity
Pathological Case: regions in 2D space where
only 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
MI Trajectory versus rotation: noise variance 0.8
Standard Histograms Our Method
Affine Image Registration
BRAINWEB
PD slice Brute force search for the maximum of MI
Affine Image Registration MI 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.