Co-registration and Spatial Normalisation

1. Co-registration and Spatial Normalisation Nazanin Derakshan Eddy Davelaar School of Psychology, Birkbeck University of London

2. What is Spatial Normalisation? • It is a registration method that allows us to warp images from a number of individuals into roughly the same standard space: this allows signal averaging across individuals. • It is useful for determining what happens generically over individuals. • The method results in ‘spatially normalised images’.

3. Why are spatially normalised images useful? • They are useful because activation sites can be reported according to their Euclidian coordinates within a standard space (Fox, 1995). • The most commonly adopted coordinate system within the brain imaging community is that described by Talairach & Toumoux (1988).

4. How does the method work? • Normalisation usually begins by matching the brains to a template image using transformations. This is then followed by introducing nonlinear deformations described by a number of smooth basis functions (Friston et al.1995a).

5. So the method… • Warps the images such that functionally homologous regions from different subjects are as close together as possible • Problems: • No exact match between structure and function • Different brains are organised differently • Computational problems (local minima, not enough information in the images, computationally expensive) • Compromise by correcting gross differences followed by smoothing of normalised images

6. Affine and Non-linear Registration

7. Affine registration • The objective is to fit the source image f to a template image g, using a twelve parameter affine transformation. The images may be scaled quite differently, so an additional intensity scaling parameter is included in the model. • Make sure you have plenty of slices

8. When the error for a particular fitted parameter is known to be large, then that parameter will be based more upon the prior information. • In order to adopt this approach, the prior distribution of the parameters should be known. • This can be derived from the zooms and shears determined by registering a large number of brain images to the template.

9. Affine Registration • The first part is a 12 parameter affine transform • 3 translations • 3 rotations • 3 zooms • 3 shears • Fits overall shape and size • Algorithm simultaneously minimises • Mean-squared difference between template and source image • Squared distance between parameters and their expected values (regularisation)

10. Affine Registration • Minimise mean squared difference from template image(s) Affine registration matches positions and sizes of images. Affine registration

11. Non-Linear Spatial Normalisation • Assumes that the image has already been approximately registered with the template according to a twelve-parameter affine registration. • It is used when the parameters describing global shape differences are not accounted for by affine registration.

12. The model for defining nonlinear warps uses deformations consisting of a linear combination of low-frequency periodic basis functions.

13. Affine + Non-linear Size and global shape of the brain is normalised. Non-linear registration

14. Regularisation Without regularisation, the non-linear spatial normalisation can introduce unnecessary warps.

15. Regularization is achieved by minimizing the sum of squared difference between the template and the warped image, while simultaneously minimizing some function of the deformation field. The principles are Bayesian and make use of the MAP (Maximum A Posteriori) scheme

16. Over-fitting Affine registration (2 = 472.1) Template image Non-linear registration using regularisation. (2 = 302.7) Non-linear registration without regularisation. (2 = 287.3)

17. Deformations consist of a linear combination of smooth basis functions These are the lowest frequencies of a 3D discrete cosine transform (DCT) Algorithm simultaneously minimises • Mean squared difference between template and source image • Squared distance between parameters and their known expectation

18. Co-registration • Matching of two images of different modalities (e.g., T1 with T2) by finding the transformation parameters • Why co-registration? • Realigning functional images can be greatly facilitated by having high-res structural images • Allows a more precise spatial normalization as the warps computed from structural images can be applied to the functional images.

19. co-registration maximizes the mutual information between two images • MI is a measure of the dependence between images

20. Now a tiny bit more technical

22. Spatial normalization: procedure that warps images from a number of individuals into roughly the same standard space to allow signal averaging across subjects.

23. Purpose of spatial normalization is to maximize the sensitivity to neuro-physiological change elicited by experimental manipulation of sensorimotor or cognitive state •  may imply that condition-dependent effects should be incorporated in the optimization procedure

24. Techniques I • Label-based techniques: identify homologous features in the source and reference images and find the transformations that best superpose them. • Labels: (discrete) points, lines, surfaces • Identified manually, time-consuming, subjective

25. Techniques II • Intensity-based techniques identify a spatial transformation that optimizes some voxel-similarity measure between a source and reference image, where both are treated as unlabelled continuous processes. • Hybrid approaches: combine intensity based methods with matching user-defined features (typically sulci)

26. Warping: high-dimensional problem, but much of the spatial variability can be captured using just a few parameters. • Warping transformations are arbitrary and regularization schemes are necessary to ensure that voxels remain close to their neighbors.

27. Regularization is often incorporated in a Bayesian scheme, using maximum a posteriori (MAP) or minimum variance estimate (MVE). (elastic: convolving a deformation field is a form of linear regularization) • An alternative to Bayesian methods is using a viscous fluid model to estimate the warps. (plastic: not the deformation field is regularized, but the increments to the deformations at each iteration)

28. Bayesian registration scheme to obtain MAP estimate of registration parameters •  uses prior knowledge of variability in brain size/shapes

29. Bayes here as well? - Yes.

30. Assumptions • qMAP=mode[p(q|b)], p(q|b) ~ p(b|q)p(q) • All D(p) approx. multi-normal distributions • equal variance for each observation: estimated from SSE from current iteration • Exact form of p(q) is known •  not strictly correct, but close enough

31. Affine registration • Determine 9 or 12 parameter affine transformation that registers images together by minimizing some mutual function • Aim is to fit source image f to template g • … with using our prior knowledge about those q-parameters (courtesy of nice people giving their knowledge)

32. Nonlinear registration/spatial normalization (i.e., doing the curvy stuff) • Assumes image already approx registered with template • Model for defining nonlinear warps uses deformations consisting of linear combinations of low-frequency periodic basis functions (because HF is lost during smoothing) • Discrete (co)sine transform • Optimize q-parameters that weight the various basis functions

33. Lowest basis functions of a 2D DCT Different boundary conditions (DST, DCT/DST, DCT) Getting the curvy stuff

34. Linear regularization • Regularization achieved by minimizing SSE between template and warped image, while minimizing some function of the deformation field. • MAP approach assumes prior estimate with mean zero. Choice of prior affects energy • Membrane, bending, linear-elastic energy (read more in Chapter 2 by Ashburner & Friston)

35. Can you read this now??? Chapter 1, Figure 2 by Friston

36. Cool, however… • Fitting method not optimal when there is no linear relationship between images, e.g., intensities vary (across modalities) • By taking intensity into account, many reference images can be used for registration • Co-registration: matching different modalities on the corresponding templates

37. Caveats of MAP • No guarantee to get the global optimum • No one-to-one match for small structures • MVE may be more appropriate: is the average of all possible solutions, weighted by their individual posterior probabilities • If errors are Normal, MAP=MVE. This is partially satisfied by smoothing before registering

39. Further references • Friston, K. J. Introduction: Experimental design and statistical parametric mapping • Ashburner, J., & Friston, K. J. Chapter 2 Rigid body registration. • Ashburner, J., & Friston, K. J. Chapter 3 Spatial normalization using basis functions.