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Of the many proposed image segmentation methods, region growing has been one of the most popular. Research on region growing, however, has focused primarily on the design of feature measures and on growing and merging criteria. Most of these methods have an inherent dependence on the order in which the points and regions are examined. This weakness implies that a desired segmented result is sensitive to the selection of the initial growing points. We define a set of theoretical criteria for a subclass of region-growing algorithms that are insensitive to the selection of the initial growing points. This class of algorithms, referred to as symmetric region growing algorithms, leads to a single-pass region-growing algorithm applicable to any dimensionality of images. Furthermore, they lead to region-growing algorithms that are both memory- and computation-efficient. Results illustrate the method's efficiency and its application to 3D medical image segmentation.