Sorry to bug you yet another time. I had a look at the deformation package and it seems that that focuses on the part where one applies the transformation and not the registration part of it. Or is that the part where BlockRegistration plugs in and solves the inverse problem?
Sorry, just tried it our and I seem to got it. The ϕs in the cookbook is actually the deformed grid and your link just was the documentation on the GridDeformation type.