  In this paper, we present an advanced approach to solving the inverse rig
problem in blendshape animation, using high-quality corrective blendshapes. Our
algorithm introduces novel enhancements in three key areas: ensuring high data
fidelity in reconstructed meshes, achieving greater sparsity in weight
distributions, and facilitating smoother frame-to-frame transitions. While the
incorporation of corrective terms is a known practice, our method
differentiates itself by employing a unique combination of $l_1$ norm
regularization for sparsity and a temporal smoothness constraint through
roughness penalty, focusing on the sum of second differences in consecutive
frame weights. A significant innovation in our approach is the temporal
decoupling of blendshapes, which permits simultaneous optimization across
entire animation sequences. This feature sets our work apart from existing
methods and contributes to a more efficient and effective solution. Our
algorithm exhibits a marked improvement in maintaining data fidelity and
ensuring smooth frame transitions when compared to prior approaches that either
lack smoothness regularization or rely solely on linear blendshape models. In
addition to superior mesh resemblance and smoothness, our method offers
practical benefits, including reduced computational complexity and execution
time, achieved through a novel parallelization strategy using clustering
methods. Our results not only advance the state of the art in terms of
fidelity, sparsity, and smoothness in inverse rigging but also introduce
significant efficiency improvements. The source code will be made available
upon acceptance of the paper.
