Inequality-Constrained and Robust 3D Face Model Fitting
Fitting 3D morphable models (3DMMs) on faces is a well-studied problem, motivated by various industrial and research applications. 3DMMs express a 3D facial shape as a linear sum of basis functions. The resulting shape, however, is a plausible face only when the basis coefficients take values within limited intervals. Methods based on unconstrained optimization address this issue with a weighted $ll_2$ penalty on coefficients; however, determining the weight of this penalty is difficult, and the existence of a unique weight that works universally is questionable. We propose a new formulation that does not require the tuning of any weight parameter. Specifically, we formulate 3DMM fitting as an inequality-constrained optimization problem, where the primary constraint is that basis coefficients should not exceed the interval that is learned when the 3DMM is constructed. We employ additional constraints to exploit sparse landmark detectors by enforcing the facial shape to be within the error bounds of a reliable detector. To enable operation ``in-the-wild'', we use a robust objective function, namely gradient correlation. Our approach performs comparably with deep learning (DL) methods on ``in-the-wild'' data, which have inexact ground truth, and better than DL methods on more controlled data with exact ground truth. Since our formulation does not require any learning, it enjoys a versatility that allows it to operate with multiple frames of arbitrary sizes. This study's results encourage further research on 3DMM fitting with inequality-constrained optimization methods, which remained unexplored compared to unconstrained methods."