Morphing and averaging

Overview

The spherical morphing of the surfaces accomplished by FreeSurfer can be employed to bring data from different subjects into a common anatomical frame. This chapter describes utilities which make use of the spherical morphing procedure. mne_morph_labels morphs label files between subjects allowing the definition of labels in a one brain and transforming them to anatomically analogous labels in another. mne_average_estimates offers the capability to compute averages of data computed with the MNE software across subjects.

The morphing maps

The MNE software accomplishes morphing with help of morphing maps which can be either computed on demand or precomputed using mne_make_morph_maps , see Precomputing the morphing maps. The morphing is performed with help of the registered spherical surfaces (lh.sphere.reg and rh.sphere.reg ) which must be produced in FreeSurfer . A morphing map is a linear mapping from cortical surface values in subject A (\(x^{(A)}\)) to those in another subject B (\(x^{(B)}\))

\[x^{(B)} = M^{(AB)} x^{(A)}\ ,\]

where \(M^{(AB)}\) is a sparse matrix with at most three nonzero elements on each row. These elements are determined as follows. First, using the aligned spherical surfaces, for each vertex \(x_j^{(B)}\), find the triangle \(T_j^{(A)}\) on the spherical surface of subject A which contains the location \(x_j^{(B)}\). Next, find the numbers of the vertices of this triangle and set the corresponding elements on the j th row of \(M^{(AB)}\) so that \(x_j^{(B)}\) will be a linear interpolation between the triangle vertex values reflecting the location \(x_j^{(B)}\) within the triangle \(T_j^{(A)}\).

It follows from the above definition that in general

\[M^{(AB)} \neq (M^{(BA)})^{-1}\ ,\]

i.e.,

\[x_{(A)} \neq M^{(BA)} M^{(AB)} x^{(A)}\ ,\]

even if

\[x^{(A)} \approx M^{(BA)} M^{(AB)} x^{(A)}\ ,\]

i.e., the mapping is almost a bijection.

About smoothing

The current estimates are normally defined only in a decimated grid which is a sparse subset of the vertices in the triangular tessellation of the cortical surface. Therefore, any sparse set of values is distributed to neighboring vertices to make the visualized results easily understandable. This procedure has been traditionally called smoothing but a more appropriate name might be smudging or blurring in accordance with similar operations in image processing programs.

In MNE software terms, smoothing of the vertex data is an iterative procedure, which produces a blurred image \(x^{(N)}\) from the original sparse image \(x^{(0)}\) by applying in each iteration step a sparse blurring matrix:

\[x^{(p)} = S^{(p)} x^{(p - 1)}\ .\]

On each row \(j\) of the matrix \(S^{(p)}\) there are \(N_j^{(p - 1)}\) nonzero entries whose values equal \(1/N_j^{(p - 1)}\). Here \(N_j^{(p - 1)}\) is the number of immediate neighbors of vertex \(j\) which had non-zero values at iteration step \(p - 1\). Matrix \(S^{(p)}\) thus assigns the average of the non-zero neighbors as the new value for vertex \(j\). One important feature of this procedure is that it tends to preserve the amplitudes while blurring the surface image.

Once the indices non-zero vertices in \(x^{(0)}\) and the topology of the triangulation are fixed the matrices \(S^{(p)}\) are fixed and independent of the data. Therefore, it would be in principle possible to construct a composite blurring matrix

\[S^{(N)} = \prod_{p = 1}^N {S^{(p)}}\ .\]

However, it turns out to be computationally more effective to do blurring with an iteration. The above formula for \(S^{(N)}\) also shows that the smudging (smoothing) operation is linear.

Precomputing the morphing maps

The utility mne_make_morph_maps was created to assist mne_analyze and mne_make_movie in morphing. Since the morphing maps described above take a while to compute, it is beneficial to construct all necessary maps in advance before using mne_make_movie .

The precomputed morphing maps are located in $SUBJECTS_DIR/morph-maps . mne_make_morph_maps creates this directory automatically if it does not exist. If this directory exists when mne_analyze or mne_make_movie is run and morphing is requested, the software first looks for already existing morphing maps there. Also, if mne_analyze or mne_make_movie have to recompute any morphing maps, they will be saved to $SUBJECTS_DIR/morph-maps if this directory exists.

The names of the files in $SUBJECTS_DIR/morph-maps are of the form:

<A> - <B> -morph.fif ,

where <A> and <B> are names of subjects. These files contain the maps for both hemispheres, and in both directions, i.e., both \(M^{(AB)}\) and \(M^{(BA)}\), as defined above. Thus the files <A> - <B> -morph.fif or <B> - <A> -morph.fif are functionally equivalent. The name of the file produced by mne_analyze or mne_make_movie depends on the role of <A> and <B> in the analysis.

If you choose to compute the morphing maps in batch in advance, use mne_make_morph_maps.