RIGID IMAGE REGISTRATION

Transcription

1 RIGID IMAGE REGISTRATION Duygu Tosun-Turgut, Ph.D. Center for Imaging of Neurodegenerative Diseases Department of Radiology and Biomedical Imaging

2 What is registration? Image registration is the process of estimating an optimal geometric transformation to bring homologous points of two s as close as possible

3 Why registration? Longitudinal analysis Motion correction Distortion correction Multi-modal integration Atlas generation Kinematic analysis Morphometry

4 Image registration categories Feature-based Intensity-based

5 How is intensity-based registration done?

6 The moving is the that will be transformed and resampled in to the fixed coordinate system Registered moving

7 How do we represent an? (1,1,c) columns (1,1,1) (1,n,1) p The intensity at a given voxel p of a 3D I is usually expressed as Rows l I(r,c,z) (m,1,1) Coordinates are in units of voxels

8 How do we represent an? (1,1,c) columns (1,1,1) (1,n,1) p Always ask for the spatial resolution Rows (m,1,1) Row voxel size in mm Medical registration is done in physical coordinates, not in voxel coordinates Column voxel size in mm

9 How do we represent an? y Coordinates are in units of mm (0,0,0) z x The center of the physical coordinate system is usually given by the scanner Be aware that still many s are not isotropic In the physical coordinate ystem we usually express the position of a point p as [x y z]

10 How do we go from one coordinate system to another? We need a geometric transformation A geometric transformation maps coordinates from one coordinate system to another The transformation model is often described by its Degrees of Freedom (DoF), which is the number of independent ways that the transformation can be changed In general, increasing the number of DoF allows the transformations greater scope to make one match the other

11 Rigid transformation In 3D it has 6 DoF: 3 translations! " x' y' z' 1 $! = % " T t x t y t z $! %" x y z 1 $ %

12 Rigid transformation In 3D it has 6 DoF: 3 translations 3 rotations! " x' y' z' 1 $! = % " M z cos(θ) sin(θ) 0 0 sin(θ) cos(θ) $! %" x y z 1 $ %

13 Rigid transformation In 3D it has 6 DoF: 3 translations 3 rotations! " x' y' z' 1 $! = % " M y cos(ϕ) 0 sin(ϕ) sin(ϕ) 0 cos(ϕ) $! % " x y z 1 $ %

14 Rigid transformation In 3D it has 6 DoF: 3 translations 3 rotations! " x' y' z' 1 $! = % " M x cos(ω) sin(ω) 0 0 sin(ω) cos(ω) $! %" x y z 1 $ %

15 Affine transformation In 3D it has 9 DoF: 3 translations 3 rotations 3 scalings! " x' y' z' 1 $! = % " S s x s y s z $! %" x y z 1 $ %

16 Affine transformation! " In 3D it has 12 DoF: 3 translations 3 rotations 3 scalings 3 skew x' y' z' 1 $! = % " K z 1 0 a b $! %" x y z 1 $ % Shear along any pair of axes is proportional to the third axis e.g., to shear along z: x and y are altered by an amount proportional to z, leaving z unchanged. x' = x + az y' = y + bz z' = z

17 Affine transformation Maps lines to lines Maps parallel lines to parallel lines Preserves ratios of distance along a line Does NOT preserve absolute distances and angles

18 ations

19 ations In medical imaging, you usually need some combination of these basic transformations This is accomplished by concatenating the transformations: y y M z y T x = Composition x x x

20 ations Matrix multiplication is not commutative! y M z x Composition y T x x M z M T M T M z y T x y M z Composition x x

21 Nonlinear transformation The non-linear transformation model includes all transformations that do not fit into the affine transformation model In 3D, it can go from those that are nearly linear with fed DoF to the most general transformations which have a separate displacement (3 DoF) for each voxel, giving well over a million DoF for typical s

22 Rigid transformation example Intra-subject longitudinal registration Baseline scan Follow-up scan Follow-up scan After rigid registration

23 Rigid transformation example Intra-subject longitudinal registration Baseline scan Follow-up scan Follow-up scan After rigid registration

24 Affine transformation example Inter-subject registration Model Young females (n=94) No registration Rigid registration

25 Affine transformation example Inter-subject registration Model Young females (n=94) No registration Affine registration

26 Affine transformation example Inter-subject registration Model Young females (n=94) No registration Nonlinear registration

27 Affine transformation example Inter-subject registration Rigid Affine Nonlinear

28 How do we apply the transformation to get an aligned moving? How to change from voxel coordinates to physical coordinates Voxel coordinates Voxel coordinates Physical coordinates (r,c) (r,c) (x,y) (x,y)

29 How do we apply the transformation to get an aligned moving? To go from voxel coordinates to the physical coordinates system we need a transformation matrix that includes Voxel coordinates scalings and translations so that: Voxel coordinates Physical coordinates (r,c) x = c x Spacing x + O x y = r x Spacing y + O y After the geometric transformation has been completed, change from the physical coordinate system to the fixed voxel space (r,c) (x,y) (x,y)

30 How do we apply the transformation to get an aligned moving? Once in the physical space we can apply the geometric transformation to align the moving to the fixed Voxel coordinates Physical coordinates Voxel coordinates However, remember that the final goal is to have the moving aligned to the fixed in the space of the fixed, so we have to take into account the mapping of the voxel intensities (r,c) (r,c) (x,y) (x,y)

31 Forward mapping Forward mapping voxels are mapped onto the fixed Multiple moving voxels could hit the same fixed voxel: The transformation has to keep track and accumulate this overlapping The registered moving could have holes voxels with no hits

32 Backward mapping Backward mapping coordinates are mapped back onto the moving Output voxel values must be interpolated from a neighborhood in the moving All fixed coordinates are scanned sequentially thus avoiding holes and overlaps in the output This can sometimes become a problem if the mapping function is non-invertible

33 Image registration in medical imaging is commonly done with backward mapping requiring intensity interpolation The parameters of the transformation are thus those that map each voxel position in the fixed to the moving

34 Intensity interpolation Nearest neighbor Linear Spline Sinc

35 Linear interpolation Intensity at a given point is computed based on a weighted average of the surrounding voxels (e.g., 4 in 2D and 8 in 3D) Weights are calculated based on the distances between the coordinates of the point of interest and those of the surrounding voxels

36 Linear interpolation I M (x, y) Δx I M (x +1, y) Δy p I M (x, y +1) I M (x +1, y +1) I(p) = (1 Δx)(1 Δy)I M (x, y) +Δx(1 Δy)I M (x +1, y) +(1 Δx)ΔyI M (x, y +1) +ΔxΔyI M (x +1, y +1)

37 The optimizer helps us to find the optimal set of transformation parameters Different optimization strategies, e.g. Gradient Descent Optimization is usually an iterative process The optimizer needs a metric to tell it how well the s are aligned

38 Optimization metrics Intra-model intensity metrics Mean-squared differences (MSD; cost function) Absolute differences (MAD; cost function) Normalized correlation (NCC; similarity function) Inter-modal intensity metrics Mutual information (MI; similarity function) Normalized mutual information (NMI; cost function) Correlation ratio (CR; similarity function)

39 Optimization metrics MSD MSD = 1 N j ( I Fj I ) 2 Tj F = T = ed moving N = Number of voxels in the overlapping region H(.) = Marginal entropy H(.,.) = Joint entropy

40 Optimization metric MI MI = H(I F )+ H(I T ) H(.I F, I T ) H(I F ) = p F ( f )log p F ( f ) [ ] H(I T ) = p T ( f )log p T ( f ) H(I F, I T ) = F = T = ed moving [ ] p FT ( f,t)log p FT ( f,t) [ ] N = Number of voxels in the overlapping region H(.) = Marginal entropy H(.,.) = Joint entropy

41 Multi-resolution Improves speed Improves robustness Improves accuracy Registration Registration Registration

42 Nonlinear registration Basis functions Splines Elastic-solid Viscous fluid Smoothed displacement fields

43 Nonlinear registration - FFD

44 Nonlinear registration - FFD Active region