2DLDA as matrix-variate formulation of a separable 1DLDA

Necessary and sufficient conditions for separability of 1DLDA are derived.The commonly used separable scatter model is proved as a special case.Separability of the MVLDA operator is proved.1DLDA and 2DLDA solutions are theoretically related and compared.Zigzag sorting procedure is proposed for row & column-sorted 2DLDA features. Two-directional (2D) variants of the linear discriminant analysis (LDA) algorithm have been widely used to extract features of matrix-variate signals. This paper derives the theoretical relationship between 2DLDA and one-directional (1D) LDA based on the separable transformation framework. Separable transforms such as separable 2DDCT are widely used for image compression in the JPEG standard; therefore, a similar framework for 2DLDA provides the corresponding parallel foundation for separable image feature extraction. There are existing 2DLDA methods providing a separable transformation, however they are not directly related to the 1DLDA solution. We will derive a 2DLDA method as a matrix-variate representation of a separable 1DLDA operator. Furthermore, we derive the necessary and sufficient conditions for separability of 1DLDA. These conditions will be helpful to clarify both limitations and advantages of 2DLDA. Also, a 2DLDA framework in parallel to 2DDCT allows us to exploit related techniques developed for 2DDCT, such as the feature selection procedure.

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