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In this paper we propose a novel approach for representing facial expressions based on a quadratic deformation model applied to muscle regions. The non-linear nature of muscle deformations can be captured for each expression, by subdividing the face into 16 facial regions and using the most general rubber-sheet transformation of second degree. The deformation parameters are derived using a least-square minimization technique, and used to construct a facial deformation table (FDT) to mathematically represent each expression. The generalized nature of the transformations allows us to easily map expressions from one model to another, and employ the FDTs in facial expression applications such as facial recognition and animations. The paper presents experimental results using the smile expression.