Registration by continuous optimisation. Stefan Klein Erasmus MC, the Netherlands Biomedical Imaging Group Rotterdam (BIGR)

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1 Registration by continuous optimisation Stefan Klein Erasmus MC, the Netherlands Biomedical Imaging Group Rotterdam (BIGR)

2 Registration = optimisation C t x t y 1

3 Registration = optimisation C t x t y 1

4 Registration = optimisation C t x t y 1

5 Registration = optimisation C t x t y 1

6 Example 2

7 Example fixed image moving image 2

8 Example fixed image moving image 2

9 Example fixed image moving image 2

10 Math F(x) = fixed image, M(x) = moving image x = voxel coordinate Transformation function: T(x ; p) p = vector of transformation parameters Cost function: C( p ) measures similarity of fixed image F(x) and deformed moving image M( T(x; p) ) Find p that minimises C 3

11 Iterative optimisation p k+1 = p k + a k. d k d k = search direction a k = step size gradient descent: d C p k ( pk) g k 4

12 Gradient descent p k+1 = p k - a k. g k P 1 p 1 g 1 p 2 p 2 g 2 p 3 = p 3 - a k. g 3 : : : : : : k+1 k k 5

13 Gradient descent p k+1 = p k - a k. g k p 1 p 1 g 1 p 2 p 2 g 2 = C p 1 k p 3 = p 3 - a k. g 3 : : : : : : k+1 k k 5

14 6 Cost function derivative Example for mean of squared differences: x x x x p T p x T x p p x T x p p x T x p M )) ; ( M( ) F( N 2 M )) ; ( M( ) F( N 2 C )) ; ( M( ) F( N 1 ) C( t 2

15 Choice of d k p k+1 = p k + a k. d k 7

16 Choice of d k gradient descent C p 1 p 2 8

17 Choice of d k smarter steps C p 1 p 2 8

18 Choice of d k cheaper steps C p 1 p 2 8

19 Choice of d k p k+1 = p k + a k. d k gradient descent: Newton: quasi-newton: d k = - g k d k = - [H k ] -1 g k d k = - B k g k smarter steps conjugate gradient: d k = - g k + β k d k-1 stochastic gradient: d k - g k cheaper steps 9

20 Experimental comparison Cardiac CT, 97x97x97 voxels, artifically deformed 10

21 Experimental comparison Cardiac CT, 97x97x97 voxels, artifically deformed 11

22 Experimental comparison Error measure: e 1 N x T ( x) Tˆ( x) 12

23 Experimental comparison 3 gradient descent quasi-newton conjugate gradient stochastic gradient e [mm] computation time

24 Choice of a k p k+1 = p k + a k. d k 14

25 Choice of a k Too small steps C p 1 p 2 15

26 Choice of a k Too large steps C p 1 p 2 15

27 Choice of a k p k+1 = p k + a k. d k constant: a k = a slowly decaying: a k = f ( k ) = a / ( A + k ) a exact line search: a k = argmin a C ( p k + a d k ) inexact line search: a k argmin a C ( p k + a d k ) [Wolfe conditions] adaptive: a k = F ( progress in previous iterations ) 16

28 Stochastic gradient descent with adaptive strategy for a k p k1 p k f(t k ) g k 20 f(t k ) a/(a t k ) a t k1 t k sigmoid( g T k g k1 )

29 Stochastic gradient descent with adaptive strategy for a k p k1 p k f(t k ) g k 20 f(t k ) a/(a t k ) a t k1 t k sigmoid( g Choose a such that: T k g k1 max. voxel displacement per iteration < (with 95% probability) ) 1 0 [mm]

30 Experimental comparison 6 prostate MR image pairs: nonrigid registration evaluation measure: overlap of manual segmentations after registration 18

31 Experimental comparison [mm] A non-adaptive [mm] A adaptive non-adaptive adaptive 19

32 Experimental comparison Experiments with: brain, lung, prostate CT, MRI sum of squared differences, mutual information, normalized mutual information rigid, nonrigid A 20, voxelsize good results in all experiments! 20

33 Local similarity measures MI = mutual information assumes grey-value distribution does not vary over image domain LMI = localised mutual information = 1 MI ( x ) (aka: regional MI, conditional MI, spatial information encoded MI) x N 21

34 Local similarity measures MI = mutual information assumes grey-value distribution does not vary over image domain LMI = localised mutual information = 1 MI ( x ) (aka: regional MI, conditional MI, spatial information encoded MI) x N can be efficiently implemented with stochastic gradient descent! 21

35 Summary Parametric formulation can be solved by continuous optimisation Derivative-based methods: require Extensive literature C p Basic method: gradient descent Popular choice: quasi-newton or conjugate gradient icm inexact line search Recommended : stochastic gradient descent with adaptive step sizes 22

36 Literature Nocedal & Wright: Numerical Optimization IEEE Trans. Image Processing Klein, Staring, Pluim Evaluation of optimization methods for nonrigid medical image registration using mutual information and B-splines Int. J. Computer Vision Klein, Pluim, Staring, Viergever Adaptive stochastic gradient descent optimisation for image registration IEEE Trans. Image Processing Thevenaz, Unser Optimization of mutual information for multiresolution image registration 23

37 Rigid and nonrigid registration Various cost functions, transformation models, multiresolution strategies etc. Many optimisation algorithms implemented Free: Based on Insight ToolKit (ITK): IEEE Trans. Medical Imaging Klein, Staring, Murphy, Viergever, Pluim elastix: a toolbox for intensity based medical image registration 24