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The spherical morphing of BEM surfaces accomplished by FreeSurfer can be employed to bring data from different subjects into a common anatomical frame. This page describes utilities which make use of the spherical :term:`morphing` procedure. :func:`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. :meth:`mne.SourceMorph.apply` offers the capability to transform all subject data to the same space and, e.g., compute averages of data across subjects.
Morphing examples in MNE-Python
Examples of morphing in MNE-Python include :ref:`this tutorial <tut-mne-fixed-free>` on surface source estimation or these examples on :ref:`surface <ex-morph-surface>` and :ref:`volumetric <ex-morph-volume>` source estimation.
Modern neuroimaging techniques, such as source reconstruction or fMRI analyses, make use of advanced mathematical models and hardware to map brain activity patterns into a subject-specific anatomical brain space. This enables the study of spatio-temporal brain activity. The representation of spatio-temporal brain data is often mapped onto the anatomical brain structure to relate functional and anatomical maps. Thereby activity patterns are overlaid with anatomical locations that supposedly produced the activity. Anatomical MR images are often used as such or are transformed into an inflated surface representations to serve as "canvas" for the visualization.
In order to compute group-level statistics, data representations across subjects must be morphed to a common frame, such that anatomically and functional similar structures are represented at the same spatial location for all subjects equally. Since brains vary, :term:`morphing` comes into play to tell us how the data produced by subject A would be represented on the brain of subject B (and vice-versa).
The MNE software accomplishes morphing with help of 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.
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.