  We propose a method to non-rigidly align a three-dimensional (3D) volumetric
image with a two-dimensional (2D) planar image representing a projection of the
deformed volume. The application in mind comes from biological studies in which
2D intravital microscopy videos of living tissue are recorded, after which the
tissue is excised and a more detailed 3D volume microscopy is performed.
Coregistration of both data sets allows to combine the temporal (but 2D)
information with more detailed spatial 3D information.
  Our approach is variational and uses a hyperelastic deformation
regularization, as is appropriate for biological material. As a particular
feature, the out of plane deformation is estimated based on the out of focus
blur inside the 2D microscopy image. The approach becomes computationally
feasible through the use of a coarse-to-fine optimization strategy and higher
order optimization methods.
