We propose a unified variational approach for registration of gene expression data to neuroanatomical mouse
atlas in two dimensions. The proposed energy (minimized in the unknown displacement u) is composed of three
terms: a standard data fidelity term based on L2 similarity measure, a regularizing term based on nonlinear
elasticity (allowing larger smooth deformations), and a geometric penalty constraint for landmark matching. We
overcome the difficulty of minimizing the nonlinear elasticity functional by introducing an auxiliary variable v
that approximates ∇u, the Jacobian of the unknown displacement u. We therefore minimize now the functional
with respect to the unknowns u (a vector-valued function of two dimensions) and v (a two-by-two matrix-valued
function). An additional quadratic term is added, to insure good agreement between v and ∇u. In this way,
the nonlinearity in the derivatives of the unknown u no longer exists in the obtained Euler-Lagrange equations,
producing simpler implementations. Several satisfactory experimental results show that gene expression data are
mapped to a mouse atlas with good landmark matching and smooth deformation. We also present comparisons
with the biharmonic regularization. An advantage of the proposed nonlinear elasticity model is that usually no
numerical correction such as regridding is necessary to keep the deformation smooth, while unifying the data
fidelity term, regularization term, and landmark constraints in a single minimization approach.
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