  We propose a new model-based algorithm solving the inverse rig problem in
facial animation retargeting, exhibiting higher accuracy of the fit and
sparser, more interpretable weight vector compared to SOTA. The proposed method
targets a specific subdomain of human face animation - highly-realistic
blendshape models used in the production of movies and video games. In this
paper, we formulate an optimization problem that takes into account all the
requirements of targeted models. Our objective goes beyond a linear blendshape
model and employs the quadratic corrective terms necessary for correctly
fitting fine details of the mesh. We show that the solution to the proposed
problem yields highly accurate mesh reconstruction even when general-purpose
solvers, like SQP, are used. The results obtained using SQP are highly accurate
in the mesh space but do not exhibit favorable qualities in terms of weight
sparsity and smoothness, and for this reason, we further propose a novel
algorithm relying on a MM technique. The algorithm is specifically suited for
solving the proposed objective, yielding a high-accuracy mesh fit while
respecting the constraints and producing a sparse and smooth set of weights
easy to manipulate and interpret by artists. Our algorithm is benchmarked with
SOTA approaches, and shows an overall superiority of the results, yielding a
smooth animation reconstruction with a relative improvement up to 45 percent in
root mean squared mesh error while keeping the cardinality comparable with
benchmark methods. This paper gives a comprehensive set of evaluation metrics
that cover different aspects of the solution, including mesh accuracy, sparsity
of the weights, and smoothness of the animation curves, as well as the
appearance of the produced animation, which human experts evaluated.
