Volumetry of hypothalamic substructures by multimodal morphological image registration

Transcription

1 Volumetry of hypothalamic substructures by multimodal morphological image registration Dominik Löchel Institute for Applied and Numerical Mathematics KIT University of the State of Baden-Württemberg and National Research Center of the Helmholtz Association

2 Outline Volumetry Image Registration Distance Measure Regularisation Optimization Task 2 September 2011 Dominik Löchel Volumetry of hypothalamic substructures by multimodal morphological image registration IANM

3 Task: Volometry of hypothalamic substructures Given: MR Image Desired: Volume of each Structure Idea: Register an Atlas 3 September 2011 Dominik Löchel Volumetry of hypothalamic substructures by multimodal morphological image registration IANM

4 The Atlas of the Human Brain (J.K. Mai et al.) 4 September 2011 Dominik Löchel Volumetry of hypothalamic substructures by multimodal morphological image registration IANM

5 Our task front side above Atlas MRI matched Atlas 5 September 2011 Dominik Löchel Volumetry of hypothalamic substructures by multimodal morphological image registration IANM

6 Definition of an Image d image (foto) pixel (picture element) 3d image (MRI) voxel (volume element) gray level image one value per pixel/voxel color picture three values per pixel/voxel 6 September 2011 Dominik Löchel Volumetry of hypothalamic substructures by multimodal morphological image registration IANM

7 2d example images reference image R template image T 7 September 2011 Dominik Löchel Volumetry of hypothalamic substructures by multimodal morphological image registration IANM

8 Wanted coordinate transformation φ( u, x) := x u( x), in short φ transformed template image T φ 8 September 2011 Dominik Löchel Volumetry of hypothalamic substructures by multimodal morphological image registration IANM

9 The displacement field u coordinate transformation φ( u, x) := x u( x) arrows point to position in T, where color is picked up interpolation is required September 2011 Dominik Löchel Volumetry of hypothalamic substructures by multimodal morphological image registration IANM

10 Interpolation nearest spline linear cubic image data spline linear cubic nearest neigbour (yellow bars) + easiest values jump spline + extreme smooth expensive computation overshoots linear + easy to compute intermediate values cubic + smooth + computation time ok 10 September 2011 Dominik Löchel Volumetry of hypothalamic substructures by multimodal morphological image registration IANM

11 Control points define set of (strong or weak) control points and map these points strong control points: the transformation φ has to map these points weak control points: these points shall lie close together in the matched images manual selection of control points: long-lasting and susceptible for inaccuracy possible are rigid transformations and affin-linear transformations these transformations do fit only locally 11 September 2011 Dominik Löchel Volumetry of hypothalamic substructures by multimodal morphological image registration IANM

12 Monomodal and Multimodal Matching Monomodal Matching Multimodal Matching corresponding regions similar gray levels try to fuse gray levels corresponding regions arbitrary gray level corresponding feature: contour lines 12 September 2011 Dominik Löchel Volumetry of hypothalamic substructures by multimodal morphological image registration IANM

13 Contour lines Surface plot of an atlas slice with two different assignments of gray levels 13 September 2011 Dominik Löchel Volumetry of hypothalamic substructures by multimodal morphological image registration IANM

14 Contour lines Surface plot of an atlas slice with two different assignments of gray levels 14 September 2011 Dominik Löchel Volumetry of hypothalamic substructures by multimodal morphological image registration IANM

15 Features of Similarity in general, the gray levels are different the contour lines correspond 15 September 2011 Dominik Löchel Volumetry of hypothalamic substructures by multimodal morphological image registration IANM

16 Distance Measure Find a transformation φ, such that the contourlines coincide. physical motivation object consists of mass density m( x) R( x) = χ ( m( x) ) T ( x) = ζ ( m( x) ) R( x) = χ ( m( x) ) m( x) T ( x) = ζ ( m( x) ) m( x) consider crossing of contour lines local distance measure d(r, T ; x) = R( x) T ( x) or d(r, T ; x) = R( x) T T ( x) global distance measure by integration over the whole picture domain D R,T = d(r, T ; x) 2 d x Ω 16 September 2011 Dominik Löchel Volumetry of hypothalamic substructures by multimodal morphological image registration IANM

17 Optimization Task minimize u Ω ( d ( R, T φ( u); x )) 2 d x 17 September 2011 Dominik Löchel Volumetry of hypothalamic substructures by multimodal morphological image registration IANM

18 Result of Minimization the outer shape of T φ and R matches. the coherence is completly lost. this result is useless. 18 September 2011 Dominik Löchel Volumetry of hypothalamic substructures by multimodal morphological image registration IANM

19 Regularization preserve the coherence/topology the displacement field u of one voxel and the neighbouring voxel must be similar, i.e. u( x) u( x + ɛ h) for small ɛ > 0 and any direction h. at least the slope u i ( x) 2 must be small. maybe the curvature u i ( x) must be small, too. Introduce a penalty term (α > 0) ( ) 1 d minimize u D R,T φ( u) + α u i ( x) 2 2 d x 2 Ω i=1 }{{} =:S 1 ( u) ( ) 1 d ( minimize u D R,T φ( u) + α ui ( x) ) 2 d x 2 Ω i=1 }{{} =:S 2 ( u) 19 September 2011 Dominik Löchel Volumetry of hypothalamic substructures by multimodal morphological image registration IANM

20 Advanced Minimization Problem minimize u D R,T ( φ( u) ) + α S( u) }{{} I α( u) The number of parameters in u is the number of voxels times the dimension. For example > 50 millionen unknowns. Ordinary minimization procedures like steepest descent or the simplexmethod of Nelder and Mead take to much time In the minima of the functional I α ( u) holds u I α( u) h = 0 for each direction h H 2 (Ω) n We get a non-linear boundary value problem. 20 September 2011 Dominik Löchel Volumetry of hypothalamic substructures by multimodal morphological image registration IANM

21 Boundary Value Problem harmonic BVP with S 1 and Dirichlet b.c. α u i = ( f Ω ) i u i = 0 auf Ω auf Ω. biharmonic BVP with S 2 and Dirichlet b.c. α 2 u i = ( f Ω ) i u i = 0 u i = 0 auf Ω auf Ω auf Ω fω = ( ( d) t R ) ( T ) φ 21 September 2011 Dominik Löchel Volumetry of hypothalamic substructures by multimodal morphological image registration IANM

22 Advanced Minimization Problem Going down the hill in case of two unknowns. α becomes step size. 22 September 2011 Dominik Löchel Volumetry of hypothalamic substructures by multimodal morphological image registration IANM

23 Local Minima 0 Distance of rotated image D September 2011 Dominik Löchel Volumetry of hypothalamic substructures by multimodal morphological image registration IANM

24 Local Minima 0 Distance of rotated image 1 D September 2011 Dominik Löchel Volumetry of hypothalamic substructures by multimodal morphological image registration IANM

25 Local Minima 0 Distance of rotated image 1 D September 2011 Dominik Löchel Volumetry of hypothalamic substructures by multimodal morphological image registration IANM

26 Local Minima 0 3 Distance of rotated image 1 D September 2011 Dominik Löchel Volumetry of hypothalamic substructures by multimodal morphological image registration IANM

27 Local Minima 0 3 Distance of rotated image 1 4 D September 2011 Dominik Löchel Volumetry of hypothalamic substructures by multimodal morphological image registration IANM

28 Local Minima 0 3 Distance of rotated image D September 2011 Dominik Löchel Volumetry of hypothalamic substructures by multimodal morphological image registration IANM

29 Basic Algorithm from smooth to sharp while images get more similar do compute direction of deformation apply this direction with optimal step size end while end from 29 September 2011 Dominik Löchel Volumetry of hypothalamic substructures by multimodal morphological image registration IANM

30 Given images Reference R Template T 30 September 2011 Dominik Löchel Volumetry of hypothalamic substructures by multimodal morphological image registration IANM

31 Weak stiffness (S 1 ) in the elastic transformation coordinate transformation Templatebild T y weak Artificial deformations occur locally. The stiffness is to weak. 31 September 2011 Dominik Löchel Volumetry of hypothalamic substructures by multimodal morphological image registration IANM

32 Medium stiffness (S 3/2 ) in the elastic transformation coordinate transformation Template T y medium A few local artificial deformations occur. The quality is better but not satisfactory. 32 September 2011 Dominik Löchel Volumetry of hypothalamic substructures by multimodal morphological image registration IANM

33 Strong stiffness (S 2 ) in the elastic transformation coordinate transformation Template T y strong No artificial deformations. The strong stiffness is good, but it inhibits strong local deformations. 33 September 2011 Dominik Löchel Volumetry of hypothalamic substructures by multimodal morphological image registration IANM

34 Elastic Image Registration Reconstruction of the Surface Atlas brain Patient brain matched Atlas 34 September 2011 Dominik Löchel Volumetry of hypothalamic substructures by multimodal morphological image registration IANM

35 Volumetry Example Sheet1 Volumetry Label BI6Tog.nii BI6Trv.nii BK1Trv.nii BSDTrv.nii BSGTrv.nii GC3Tog.nii GC3Trv.nii HD6Tog.nii September Dominik 1093 Löchel 944 Volumetry 933 of hypothalamic 874 substructures by multimodal 903 morphological image registration IANM

36 Summary Multimodal image matching relies on contour lines. The non-fitting of contour lines defines the distance of both images. The transformation minimizes the distance. The elastic model of transformation preserves the coherence. Smoothing prevents to be stuck in a local minima. Volumes of each structure can be obtained easily from the registered atlas. 36 September 2011 Dominik Löchel Volumetry of hypothalamic substructures by multimodal morphological image registration IANM

37 For further information, see 37 September 2011 Dominik Löchel Volumetry of hypothalamic substructures by multimodal morphological image registration IANM