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This paper proposes a new class of fusion operations which are useful for binary image processing. To begin with, an algorithm for the classical fusion operations is presented. They are sequentially composed of two basic operations, that is, expansion and contraction. The strategy of the proposed algorithm is as follows: the first operation is performed with the distance transformation, and the second operation utilizes the distance information obtained in the first operation. Next, a topological contraction and a topological expansion are defined. The topological contraction preserves the Euler number of an input image except that some simply connected components of 1 pixels are erased. The topological expansion preserves the Euler number of an input image except that some simply connected holes are filled. Then, several new fusion operations are obtained by combining the classical contraction and the topological expansion (or, the classical expansion and the topological contraction) sequentially. The algorithms for them use the same strategy as the proposed algorithm for the classical fusion operations. Finally, experimental results show that the new fusion operations and their combinations are effectively used in the binary image processing, such as shape smoothing, merging and separating connected components, and structure analysis.