A Functional Regression Approach to Facial Landmark Tracking.

Linear regression is a fundamental building block in many face detection and tracking algorithms, typically used to predict shape displacements from image features through a linear mapping. This paper presents a Functional Regression solution to the least squares problem, which we coin Continuous Regression, resulting in the first real-time incremental face tracker. Contrary to prior work in Functional Regression, in which B-splines or Fourier series were used, we propose to approximate the input space by its first-order Taylor expansion, yielding a closed-form solution for the continuous domain of displacements. We then extend the continuous least squares problem to correlated variables, and demonstrate the generalisation of our approach. We incorporate Continuous Regression into the cascaded regression framework, and show its computational benefits for both training and testing. We then present a fast approach for incremental learning within Cascaded Continuous Regression, coined iCCR, and show that its complexity allows real-time face tracking, being 20 times faster than the state of the art. To the best of our knowledge, this is the first incremental face tracker that is shown to operate in real-time. We show that iCCR achieves state-of-the-art performance on the 300-VW dataset, the most recent, large-scale benchmark for face tracking.