Towards a better comprehension of Similarity Measures used in Medical Image Registration Alexis Roche, Grégoire Malandain, Nicholas Ayache, Sylvain Prima INRIA Sophia Antipolis - EPIDAURE Project 2004, route des Lucioles B.P. 93 06902 Sophia Antipolis Cedex (France) [email protected]

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Motivations









Methodology for selecting an appropriate measure Model-based registration Define optimal measures A general formulation of intensity-based registration

2

Classification of existing measures Assumed relationship Identical

Adapted measures Sum of Squared Differences Sum of Absolute Differences Measures based on image difference (Buzug et al, 1997)

3

Classification of existing measures Assumed relationship Affine

Adapted measures Correlation Coefficient

4

Classification of existing measures Assumed relationship Functional

Adapted measures Woods criterion (Woods et al, 1993) Robust Woods criterion (Nikou et al, 1997) Correlation Ratio (Roche et al, 1998)

5

Classification of existing measures Assumed relationship Statistical

Adapted measures Joint Entropy Mutual Information (Viola & Wells, 1997; Maes et al, 1997) Normalised Mutual Information (Studholme, 1998)

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Maximum Likelihood Registration Formulations based on explicit dependence models (Costa et al, 1993) (Viola, 1995) (Leventon & Grimson, 1998) (Bansal et al, 1998) We propose to generalize these approaches 



7

Maximum Likelihood Formulation 

General dependence model θS

Random field

Scene S

T

θI

Spatial transfo.

Memoryless channel

θJ Memoryless channel



Image I

Image J

Inference principle max P ( I , J | T ,θ I ,θ J ,θ S ) T ,θ I ,θ J ,θ S

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Maximum Likelihood Formulation First example 

Image dependence model T Image J

Spatial transfo.

σ Gaussian channel

Image I

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Maximum Likelihood Formulation First example 

Image dependence model I ( xk ) = J (T ( xk )) + ε k , ε k ≡ N (0, σ )



Maximum likelihood parameters Noise variance σˆ 2 =

1 ( I (xk ) − J (Tˆ(xk )) )2 ∑ n k

Transformation Tˆ = argmin ∑( I (xk ) − J (T (xk )) )2 T

k

Sum of Squared Differences

10

Maximum Likelihood Formulation Second example 

Image dependence model Image J

T

f

σ

Spatial transfo.

Intensity mapping

Gaussian channel

Image I

11

Maximum Likelihood Formulation Second example 

Image dependence model I ( xk ) = f [ J (T ( xk ))] + ε k , ε k ≡ N (0, σ )



Maximum likelihood parameters

Intensity mapping fˆ = argmin ∑( I (xk ) − f (J (Tˆ (xk ))) )2 f k

f affine f arbitrary

Noise variance 1 σˆ 2 = ∑( I (xk ) − fˆ ( J (Tˆ (xk ))) )2 n k

Transformation Tˆ = argmin ∑( I (xk ) − fˆ (J (T (xk ))) )2 T k

Correlation Coefficient Correlation Ratio 12

Maximum Likelihood Formulation Third example 

stationarity

Image dependence model Spatial transfo.

Random field

Memoryless channel

Image I

Scene S

stationarity spatial independance

Memoryless channel

Image J

stationarity

13

Maximum Likelihood Formulation Third example 

Image dependence model Image J



T

P(i | j)

Spatial transfo.

Memoryless channel

Image I

Maximum likelihood parameters Transition probabilities Pˆ (i | j ) = h(i, j )

∑ h(i′, j ) i′

Transformation Tˆ = argmax ∑log Pˆ ( I ( xk ) | J (T ( xk )) ) T

k

h(i,j): image joint histogram

Mutual Information 14

Experimental Comparison Correlation Ratio

vs.

(-) simplistic model (+) few parameters 

Mutual information (+) more realistic model (-) many parameters

Accuracy study CT / MR rigid registration

PET / MR rigid registration 

Robustness study Ultrasound / MR rigid registration

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Ultrasound / MR registration Data from the Roboscope EC Project Courtesy of ISM Salzburg (Austria) and Max Planck Institute (Germany)

MR image

Manual registration

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Ultrasound / MR registration Data from the Roboscope EC Project 



200 registrations with random initial positions Results Similarity measure

MR filtered by anisotropic diffusion

Failures

Correlation ratio

14%

Mutual information

51%

Correlation ratio

12.5%

Mutual information

28%

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Ultrasound / MR registration Data from the Roboscope EC Project 



200 registrations with random initial positions Results ∆t = 20 mm ∆r = 15 deg

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Conclusion 

 



 

At the theoretical level Maximum likelihood: a general framework Systematic derivation of optimal measures At the experimental level Different assumptions yield different performances Need for realistic and low-dimensional models

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Thank you for your attention

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Image overlap

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Towards a better comprehension of Similarity Measures ...

Medical Image Registration. Alexis Roche, Grégoire Malandain, Nicholas Ayache, Sylvain Prima. INRIA Sophia Antipolis - EPIDAURE Project. 2004, route des ...

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