Annals of Mathematical Sciences and Applications

Volume 7 (2022)

Number 1

An optimal transportation-based recognition algorithm for 3D facial expressions

Pages: 49 – 96



Tiexiang Li (School of Mathematics, Southeast University, Nanjing, China; and Nanjing Center for Applied Mathematics, Nanjing, China)

Pei-Sheng Chuang (Department of Applied Mathematics, National Yang Ming Chiao Tung University, Hsinchu, Taiwan)

Mei-Heng Yueh (Department of Mathematics, National Taiwan Normal University, Taipei, Taiwan)


Facial expression recognition (FER) is an active topic that has many applications. The development of effective algorithms for FER has been a competitive research field in the last two decades. In this paper, we propose a fully automatic 3D FER method based on the sparse approximation of 2D feature images. For a prescribed feature defined on the 3D facial surface, we apply a parameterization that not only maps the facial surface onto the unit disk but also locally preserves the feature. To ensure the uniqueness of the solution, some aligning constraints are further taken into account while computing the desired parameterization. The facial surface associated with the feature is then converted into the 2D image of the parameter domain. To recognize the expression of a test facial image, we apply an existing 2D expression recognition model, which is built upon sparse representation. Numerical experiments indicate that the accuracy of the proposed FER algorithm reaches 71.42% on a benchmark facial expression database, which is promising for practical applications.


facial expression recognition, optimal mass transportation, projected gradient descent method

2010 Mathematics Subject Classification

Primary 52C26, 68U05. Secondary 65D18.

This work was partially supported by the National Centre of Theoretical Sciences (NCTS), by S.-T. Yau Center in Taiwan, and by the Shing-Tung Yau Center and Big Data Computing Center in Southeast University.

T. Li was supported in part by the National Natural Science Foundation of China (NSFC) 11971105.

M.-H. Yueh was partially supported by the Ministry of Science and Technology (MoST) grants 109-2115-M-003-010-MY2 and 110-2115-M-003-014, and by the Ministry of Education (MoE).

Received 17 January 2022

Accepted 5 February 2022

Published 7 April 2022