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 2
nd
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, 2
nd
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-ofthe-art performance in the problem of facial feature detection. The code is available at www.humansensing.cs.
cmu.edu/intraface.
1. Introduction
Mathematical optimization has a fundamental impact in
solving many problems in computer vision. This fact is
apparent by having a quick look into any major conference in computer vision, where a significant number of papers use optimization techniques. Many important problems in computer vision such as structure from motion, image alignment, optical flow, or camera calibration can be
posed as solving a nonlinear optimization problem. There
are a large number of different approaches to solve these
continuous nonlinear optimization problems based on first
and second order methods, such as gradient descent [1] for
dimensionality reduction, Gauss-Newton for image alignment [22, 5, 14] or Levenberg-Marquardt for structure from
motion [8].
“I am hungry. Where is the
apple? Gotta do Gradient
descent”
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