In this paper, we introduce a new image segmentation scheme that is based on bidirectional labeling and registration and prove that its segmentation performance is equivalent to that of the conventional watershed segmentation algorithm. The proposed bidirectional labeling and registration scheme, which we refer to as bidirectional labeling and registration scheme (BIDS), involves only linear scans of image pixels. It uses one-dimensional operations rather than the queues that are used in traditional segmentation algorithms, which are two-dimensional problems. BIDS also provides unique labels for individual homogeneous regions. In addition to achieving the same segmentation results, BIDS is four times less computationally complex than the conventional watershed by immersion technique.