Euler Principal Component Analysis

Abstract

Principal Component Analysis (PCA) is perhaps the most prominent learning tool for dimensionality reduction in pattern recognition and computer vision. However, the 2 -norm employed by standard PCA is not robust to outliers. In this paper, we propose a kernel PCA method for fast and robust PCA, which we call Euler-PCA (e-PCA). In particular, our algorithm utilizes a robust dissimilarity measure based on the Euler representation of complex numbers. We show that Euler-PCA retains PCA's desirable properties while suppressing outliers. Moreover, we formulate Euler-PCA in an incremental learning framework which allows for efficient computation. In our experiments we apply Euler-PCA to three different computer vision applications for which our method performs comparably with other stateof-the-art approaches. Electronic supplementary material The online version of this article

A Comprehensive Performance Evaluation of Deformable Face Tracking “In-the-Wild”

“…Rigid and deformable tracking of faces and facial features have been a very popular topic of research over the past twenty years (Black and Yacoob 1995;Lanitis et al 1995;Sobottka and Pitas 1996;Essa et al 1996Essa et al , 1997Oliver et al 1997;Decarlo and Metaxas 2000;Jepson et al 2003;Xiao et al 2004;Patras and Pantic 2004;Kim et al 2008;Ross et al 2008;Papandreou and Maragos 2008;Amberg et al 2009;Kalal et al 2010a;Koelstra et al 2010;Tresadern et al 2012;Tzimiropoulos and Pantic 2013;Xiong and De la Torre 2013;Liwicki et al 2013;Smeulders et al 2014;Asthana et al 2014;Li et al 2016a;Xiong and De la Torre 2015;Snape et al 2015;Tzimiropoulos 2015). In this section we provide an overview of face tracking spanning over the past twenty years up to the present day.…”

“…For example, La Cascia et al (2000) formulate the tracking task as an image registration problem in the cylindrically unwrapped texture space and Sung et al (2008) combine active appearance models (AAMs) with a cylindrical head model for robust recovery of the global rigid motion. Currently, rigid face tracking is generally treated along the same lines as general model free object tracking (Jepson et al 2003;Smeulders et al 2014;Liwicki et al 2013Liwicki et al , 2012bRoss et al 2008;Li et al 2016a). An overview of model free object tracking is given in Sect.…”

A New Approach of Matrix Factorization on Complex Domain for Data Representation

“…Fitch et al [15] introduced the fast robust correlation and expressed a robust matching surface as a series of correlations to improve the performance of matching. Liwicki et al [9] adopted this cosine-based dissimilarity measure as follows:…”

“…The robustness of cosine dissimilarity in the real domain has been found. It is able to suppress outliers [9].…”

“…Motivated by the interesting works of Liwicki et al [9] and Duong et al [10], [11], which showed the effectiveness of projecting pixel intensity values to the complex plane, this work develops a new approach for data representation, named simplicial complex matrix factorization (siCMF). Herein, the cosine dissimilarity, used on the reconstruction error function, is exploited judiciously by transferring data from the real-valued space to the complex field.…”