This paper describes general purpose algorithms for segmenting boundary images. Intersecting and incomplete boundaries commonly occur in line-art images, and also in natural images depicting translucent objects. Humans have little difficulty in segmenting such boundaries into sets corresponding to the perceptually significant regions in the image. Many existing machine vision algorithms, however, have difficulty in processing images which contain intersecting or incomplete boundaries. However, Walters' segmentation algorithm based on the p-space representation of oriented edges will correctly segment images with non-acutely intersecting lines and boundaries. This paper suggests a non-iterative, parallel algorithm which will fill large gaps and segment acutely intersecting boundaries. These algorithms are useful in a variety of applications including fake color separation, character recognition, and also in segmenting images depicting translucent objects.