Transform Facial Expression Space to Euclidean Space Using Riemann Normal Coordinates and Its Applications

Abstract

It is reported recently in [1] on construction of the psychophysical space of facial expressions through measurement of JND thresholds in the space. It is shown that the facial expression space is, in fact, not a Euclidean space but a Riemann space of which the Riemann metric is defined by the JND thresholds. In this paper, we shown how to transform the facial expression space to Euclidean space in a way to preserve geometry such as distances and angles in the Riemannian space. We build the Riemann normal coordinate system in the facial expression space, which can be regarded as a generalized polar coordinate system consisted of geodesics emanating from the origin and concentric circles with the radius measured by geodesic distances. Results by applying the above method to the JND data in [1] are shown together with application to expression recognition.