Leadfield normalization for LCMV beamformer

The LCMV beamformer is still lacking the option of leadfield normalization, see #5365 and #4659. The DICS beamformer has two options of leadfield normalization implemented, column wise or Frobenius norm. Leadfield normalization is controlled by the parameter normalized_fwd, while the type of leadfield normalization is controlled by inversion, which also controls the inversion during the computation of the spatial filter itself. This issue is to serve as a discussion point for how we want to implement the leadfield normalization in LCMV and whether those options should remain like that in DICS.

ping @agramfort @wmvanvliet @sarangnemo

[britta-wstnr]

@agramfort is right, leadfield normalization is crucial with unit-gain LCMVs (if no condition contrast is done) since otherwise there is a strong bias to the head center in the output (the output is basically meaningless). So yes, we do need this option in case someone wants to run the unit-gain and not the unit-noise-gain variant of the LCMV.

I am personally not aware of publications on those inversion options for the LCMV, so I would be hesitant to implement them there without exhaustive testing.

[larsoner]

I’ve started looking into using the new unified _prepare_forward for leadfield normalization (#5947) in LCMV and DICS. In theory it should reduce code duplication and put all our leadfield normalization code in one place.

One other thing I noticed, though, is that proj is applied with potentially more channels in the forward operator than it is to Cm:

proj, ncomp, _ = make_projector(info['projs'], fwd_ch_names)
if info['projs']:
     G = np.dot(proj, G)
G = G[picks_forward]
proj = proj[np.ix_(picks_forward, picks_forward)]

This reduced proj is returned by _prepare_beamformer_input, and then applied with Cm = np.dot(proj, np.dot(Cm, proj.T)) (at least for LCMV).

It seems a bit weird to apply one operator to G, then a subselected one to Cm. Simplifying it to the following, which is more consistent with what (implicitly) happens in the minimum norm and mixed norm code:

proj, ncomp, _ = make_projector(info['projs'], fwd_ch_names)
proj = proj[np.ix_(picks_forward, picks_forward)]
G = np.dot(proj, G[picks_forward])

At least does not break any tests. @britta-wstnr is this change okay for you?