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This paper presents a mathematical formulation of shape-adaptive (SA) two-dimensional (2-D) transforms. A generic SA transform is represented by a concatenation of three linear operators that characterize the one-dimensional (1-D) vertical transforms, the alignment of vertical coefficients, and the 1-D horizontal transforms. We state and prove several powerful properties of these operators. The mathematical formulation described may serve as an important theoretical basis for future developments in the area of object-based video coding.