Supervised Descent Method and Its Applications to Face Alignment

TLDR

Nonlinear optimization underpins many computer vision tasks, yet conventional second‑order descent methods falter when functions are non‑differentiable or Hessians are large and indefinite. The paper introduces the Supervised Descent Method to overcome these limitations in minimizing nonlinear least‑squares functions. SDM is trained on sampled NLS functions to learn a sequence of descent directions, which are then applied at test time to minimize the objective without computing Jacobian or Hessian. Experiments on synthetic and real data show SDM achieves state‑of‑the‑art performance for facial feature detection. Code is available at www.humansensing.cs and cmu.edu/intraface.

Abstract

Many computer vision problems (e.g., camera calibration, image alignment, structure from motion) are solved through a nonlinear optimization method. It is generally accepted that 2nd order descent methods are the most robust, fast and reliable approaches for nonlinear optimization of a general smooth function. However, in the context of computer vision, 2nd order descent methods have two main drawbacks: (1) The function might not be analytically differentiable and numerical approximations are impractical. (2) The Hessian might be large and not positive definite. To address these issues, this paper proposes a Supervised Descent Method (SDM) for minimizing a Non-linear Least Squares (NLS) function. During training, the SDM learns a sequence of descent directions that minimizes the mean of NLS functions sampled at different points. In testing, SDM minimizes the NLS objective using the learned descent directions without computing the Jacobian nor the Hessian. We illustrate the benefits of our approach in synthetic and real examples, and show how SDM achieves state-of-the-art performance in the problem of facial feature detection. The code is available at www.humansensing.cs. cmu.edu/intraface.