revisiting horn and schunck: interpretation as gauß-newton...
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MICCAI 2010 Tutorial
Intensity-based Deformable Registration
Similarity Measures
Christian Wachinger
Computer Aided Medical Procedures (CAMP), TU München, Germany
Quantify the similarity between images
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• Do images contain the same object?
Image retrieval
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• How are images correctly aligned?
Image registration
Quantify the similarity between images
Requirements on similarity measure
• Extremum for correctly aligned images
• Smooth, best convex
• Fast computation
• Differentiable
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Optimization
Energy ModelTransformation
Model
Transfo.
Source Image
Target Image
Difference Measure
Regularization Term
Overview
1. Standard similarity measures
2. Probabilistic framework
3. Pre-processing steps
4. Recent approaches
5. Linear vs. non-linear registration
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Standard Similarity Measures
PART I
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• SSD-SAD
• Cross Correlation
• Mutual information
• Derivatives
Difference Measures
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Volume X Volume T(Y)
i
ii yxN
SSD 2)(1
Sum of Squared Differences
i
ii yxN
SAD1
Sum of Absolute Differences:
Less sensitive on large intensity
differences than SSD
Volume X Volume T(Y)
Limitations of SSD
• Illumination change affects similarity function
• Idea: normalization of images
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SSD
rotation
SSD
rotation
SSD on Normalized Images
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Normalized Cross Correlaiton (NCC)
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Volume X Volume T(Y)
i yx
ii yyxx
NNCC
))((1
Normalized Cross Correlation:
Expresses the linear relationship between voxel
intensities in the two volumes
pixels ofNumber :
deviation Standard:
Mean:
N
x
x
NCC - example
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Source: wikipedia
Sets of (x,y) points, with the NCC of x and y for each set.
Multi-Modal Registration
• More complex intensity relationship
• Approaches:
– Simulate one modality from the other one
– Apply sophisticated similarity measures
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?
CT MR
Information Theoretic Approach
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Histogram calculation
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Image Histogram
Bins
4 3 3 6
1/4 3/16 3/16 3/8
counts
probs
Joint histogram calculation
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Image X Image Y
Y
X
1
1
1
2 1 3
Overlap
Joint Histogram
Information Theoretic Approach
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Source: W. Wells, MICCAI 2009
Registered Not Registered
Joint Histogram
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px(i)
py(i)
pxy(i,j)
px(i)
py(i)
pxy(i,j)
X and Y identical X and Y misaligned
Joint Histogram
• Histogram for images from different Modalities
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Joint HistogramTarget Image
Source ImageNot Aligned
Aligned
Source: PhD Thesis, L. Zöllei
Joint Histogram
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Intensities of Source X
Intensities of Target Y
SSD OptimumY = X
NCC OptimumY = a*X + b
Information Theoretic Approach
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• How to quantify the quality of alignment between MR and CT?
Measure the structure of the joint distribution
• How to measure the structure?
Shannon Entropy
Source: W. Wells, MICCAI 2009
Entropy
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Shannon Entropy, developed in the 1940s(communication theory)
i
ii ppH log
i
pi
i
pi
uniform distribution→ maximum entropy
any other distribution→ less entropy
Maximize or minimize entropy?
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Registered Not Registered
Mutual Information (MI)
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i j yx
xy
xyjpip
jipjip
YXHYHXHYXMI
)()(
),(log),(
),()()(),(
• Maximized if X and Y are perfectly aligned
• H(X) and H(Y) help to make the measure more robust
• Maximization of mutual information leads to minimization of joint entropy
Historical Note
• Minimum Entropy Registration– Collignon A., Vandermeulen, D., Suetens, P., and Marchal, G. 3D
multi-modality medical image registration using feature space clustering. CVRMED April 1995.
• Maximum Mutual Information Registration– Viola, P. and Wells, W.. Alignment by maximization of mutual
information. In Proceedings of the 5th International Conference of Computer Vision, June 20 – 23, 1995.
– Collignon A, Maes F, Delaere D, Vandermeulen D, Suetens P, Marchal G, Automated multi-modality image registration based on information theory. IPMI June 26, 1995.
– Viola, P. Alignment by maximization of Mutual Information. MIT PhD Thesis, June 1995.
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Normalization of MI
• Problem: changing overlap affects MI
• Entropy Correlation Coefficient (ECC)
• Normalized MI (NMI)
• Revisiting overlap invariance
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C. Studholme, D.L.G.Hill, D.J. Hawkes, An Overlap Invariant Entropy Measure of 3D Medical Image Alignment, Pattern Recognition, Vol. 32(1), Jan 1999, pp 71-86.
Nathan D. Cahill, Julia A. Schnabel, J. Alison Noble, David J. Hawkes, "Revisiting overlap invariance in medical image alignment," MMBIA, 2008
Maes F, Collignon A,Vandermeulen D, Marchal G, Suetens P. Multimodality image registration by maximization of mutual information, TMI, 1997
NMI – change of overlap
• Two images
• T2 has higher information content
• Overlap changes for deformable registration?
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m n
C. Studholme, D.L.G.Hill, D.J. Hawkes, An Overlap Invariant Entropy Measure of 3D Medical Image Alignment, Pattern Recognition, Vol. 32(1), Jan 1999, pp 71-86.
Improvements to MI
• Density estimation– Parzen window– Partial volume distribution– Uniform volume histogram– NP windows
• Spatial information• Alternative entropy measures
• Tutorial at MICCAI 2009: Information theoretic similarity measures for image registration and segmentation: Maes, Wells, Pluimhttp://ubimon.doc.ic.ac.uk/MICCAI09/a1882.html
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Derivatives of Similarity Measures
• General form of derivative of similarity metrics
– SSD:
– MI
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Hermosillo, G., Chefd'Hotel, C., Faugeras, O.: Variational Methods for Multimodal Image Matching, International Journal of Computer Vision 50(3) (2002)
Derivatives of Similarity Measures
• General form of derivative of similarity metrics
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dxI3 dyI3
dxI2 dyI2
dxI1 dyI1
Derivatives of Similarity Measures
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Images
X
Y
•
Update
=
=
dxI3 dyI3
dxI2 dyI2
dxI1 dyI1
Derivatives of Similarity Measures
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Images
X
Y
•
Update
=
=
dxI3 dyI3
dxI2 dyI2
dxI1 dyI1
Derivatives of Similarity Measures
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Images
X
Y
•
Update
=
=
dxI3 dyI3
dxI2 dyI2
dxI1 dyI1
Overview
1. Standard similarity measures
2. Probabilistic framework
3. Pre-processing steps
4. Recent approaches
5. Linear vs. non-linear registration
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Probabilistic Framework for Image Registration
PART II
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• Viola, PhD thesis, 1995
• Roche et al., 2000
• Zöllei, PhD thesis, 2006
Probabilistic Framework for registration
• Maximum Likelihood Estimation (MLE)– Probability for the model m having the observations a
– Log-likelihood function
• Formulate registration as likelihood maximization
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MLE framework for registration
• Model
• Probability function
• Log-likelihood function
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• Stationary white Gaussian noise
• Log-likelihood with Gaussian noise
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Assuming f to be the identity
• Stationary white Gaussian noise
• Log-likelihood with Gaussian noise
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SSD
Intensity relationships
• Identity : SSD
• Affine : NCC
• Functional : Correlation Ratio
• Statistical : Mutual Information
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Viola, P. Alignment by maximization of Mutual Information. MIT PhD Thesis, June 1995.Roche et al., Unifying maximum likelihood approaches in medical image registration, 2000
Pre-Processing Steps
PART III
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Similarity Measure
Optimization
Registration FrameworkImages
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Similarity Measure
Optimization
Registration FrameworkImages
?
Pre-processing
1. Image gradients
2. Entropy images
3. Phase
4. Multi-resolution
5. Attribute vectors
Image Gradients
• Normalized gradient fields
• Maximize square of cosine
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Haber, E. and Modersitzki, J., Intensity gradient based registration and fusion of multi-modal images, MICCAI 2006
“two images are considered similar, if intensity changes occur at the same locations“
Image Gradients
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Synthetic images Gradient images
Image Gradients
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Gradient of smoothed images Gradient images
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Similarity Measure
Optimization
Registration FrameworkImages
?
Pre-processing
1. Image gradients
2. Entropy images
3. Phase
4. Multi-resolution
5. Attribute vectors
Entropy Images
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0 2 4 6 8 10 12 14 16 180
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0 2 4 6 8 10 12 14 16 180
0.02
0.04
0.06
0.08
0.1
0.12
0.14
H
H
Input Image Patches PDF Entropy Image
Wachinger, Navab, Structural Images for Image Registration, MMBIA, 2010.
Entropy Images
• Examples from the RIRE dataset http://www.insight-journal.org/rire/
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CT T1 T2 PD
Original
Entropy
Wachinger, Navab, Structural Images for Image Registration, MMBIA, 2010.
Entropy Images
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Synthetic images Entropy images
Wachinger, Navab, Structural Images for Image Registration, MMBIA, 2010.
Entropy Images
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Entropy on smoothed images Entropy images
Wachinger, Navab, Structural Images for Image Registration, MMBIA, 2010.
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Similarity Measure
Optimization
Registration FrameworkImages
?
Pre-processing
1. Image gradients
2. Entropy images
3. Phase
4. Multi-resolution
5. Attribute vectors
Phase-base registration
• Fourier representation
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M. Mellor and M. Brady. Phase mutual information as a similarity measure for registration. Medical Image Analysis, 2005.
Dashed: MISolid: Phase MI
Optical Flow with Phase
• Idea: replace the assumption of brightness constancy with phase constancy
• Error for chaning noise and illumination
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L. Wietzke, G. Sommer, O. Fleischmann, The Geometry of 2D Image Signals, CVPR, 2009
blue: gradientred: phase
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Similarity Measure
Optimization
Registration FrameworkImages
?
Pre-processing
1. Image gradients
2. Entropy images
3. Phase
4. Multi-resolution
5. Attribute vectors
Multi-Resolution Registration
• Perform registration on multiple resolutions
1. Smooth
2. Downsample
• Advantages:
– Speed: down-sampled images
– Convergence: smoother cost func
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Similarity Measure
Optimization
Registration FrameworkImages
?
Pre-processing
1. Image gradients
2. Entropy images
3. Phase
4. Multi-resolution
5. Attribute vectors
Attribute Vectors
• Registration with attribute vectorsa = [ gradient, intensity, GMI ]
• GMI = geometric moment invariants
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Shen, D. and Davatzikos, C., HAMMER: hierarchical attribute matching mechanism for elastic registration, IEEE Transactions on Medical Imaging, 2002M. Wacker and F. Deinzer. Automatic robust medical image registration using a new democratic vector optimization approach with multiple Measures, MICCAI, 2009
Brain Image GMI self-similarity
Recent Approaches
PART IV
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Recent examples...
• mainly compare to MI as state of the art
• address the following problems of MI
– intensity does not represent tissue (e.g. US-CT)
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CT US US-CT
Recent examples...
• mainly compare to MI as state of the art
• address the following problems of MI
– intensity does not represent tissue (e.g. US-CT)
– intensity non-uniformity (bias)
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Minimizing Residual Complexity
• Motivation: MI fails for these images
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Myronenko A., Song X.: "Intensity-based Image Registration by Minimizing Residual Complexity", IEEE Trans. on Medical Imaging, 2010
Model
Minimizing Residual Complexity
• Compress the difference image
• Similarity measure
• Related to entropy of difference image
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Myronenko A., Song X.: "Intensity-based Image Registration by Minimizing Residual Complexity", IEEE Trans. on Medical Imaging, 2010
Buzug et al.: Image registration: convex weighting functions, CVRMed, 1997
Local and Global Statistics
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Z. Yi and S. Soatto. Nonrigid Registration Combining Global and Local Statistics, CVPR, 2009.
global local
Local and Global Statistics
• Combining local and global densities
• : manually, emphasize on local or global influence
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Zhuang, X. and Hawkes, D. and Ourselin, Unifying Encoding of Spatial Information in Mutual Information for Nonrigid Registration, IPMI, 2009Loeckx, Slagmolen, Maes, Vandermeulen, Suetens, Nonrigid image registration using conditional mutual information, IPMI, 2007
Linear vs. Non-Linear Registration
PART V
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Linear vs. Non-Linear Registration
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dxI3 dyI3
dxI2 dyI2
dxI1 dyI1
Variational
Affine
• Deformable registration is more sensitive to errors from poor modeling
• Averaged out in linear registration
x1
xn
y1
yn
x1
xn
y1
yn
1
1
1
1
...
...
......
...
...
11
11
11
2 * 2562
2 *
25
62
6
2 *
25
62
•
2 * 2562
Linear vs. Non-Linear Registration
• Choice of similarity measure has influence on the optimization strategy
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fy
fz
fx
hy
hz
hx
=
EXAMPLE: Gauß-Newton Optimization with SSD
=
Linear vs. Non-Linear Registration
• Overlap invariance only for linear ?
– MI vs. NMI
– Joint Entropy vs. MI
• Transition between linear and non-linear
– Parameterization
– Regularization
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Summary
• A large variety of concepts for measuring the similarity is available in the literature
• Choose similarity measure that best fits the application
• Appropriate choice is more important for deformable registration
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