  Advances in the development of largely automated microscopy methods such as
MERFISH for imaging cellular structures in mouse brains are providing spatial
detection of micron resolution gene expression. While there has been tremendous
progress made in the field Computational Anatomy (CA) to perform diffeomorphic
mapping technologies at the tissue scales for advanced neuroinformatic studies
in common coordinates, integration of molecular- and cellular-scale populations
through statistical averaging via common coordinates remains yet unattained.
This paper describes the first set of algorithms for calculating geodesics in
the space of diffeomorphisms, what we term Image-Varifold LDDMM,extending the
family of large deformation diffeomorphic metric mapping (LDDMM) algorithms to
accommodate the "copy and paste" varifold action of particles which extends
consistently to the tissue scales. We represent the brain data as geometric
measures, termed as {\em image varifolds} supported by a large number of
unstructured points, % (i.e., not aligned on a 2D or 3D grid), each point
representing a small volume in space % (which may be incompletely described)
and carrying a list of densities of {\em features} elements of a
high-dimensional feature space. The shape of image varifold brain spaces is
measured by transforming them by diffeomorphisms. The metric between image
varifolds is obtained after embedding these objects in a linear space equipped
with the norm, yielding a so-called "chordal metric."
