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Fossil single-celled marine organisms known as foraminifera are widely used in oceanographic research. The identification of species is one of the most common tasks when analyzing ocean samples. One of the primary criteria for species identification is their morphology. Automatic segmentation of images of foraminifera would aid on the identification task as well as on other morphological studies. We pose this problem as an edge detection task for which capturing the correct topological structure is essential. Due to the presence of soft edges and even unclosed segments, state-of-the-art techniques have problems capturing the correct edge structure. Standard pixel-based loss functions are also sensitive to small deformations and shifts of the edges penalizing location more heavily than actual structure. Hence, we propose a homology-based detector of local structural difference between two edge maps with a tolerable deformation. This detector is employed as a new criterion for the training and design of data-driven approaches that focus on enhancing these structural differences. Our approaches demonstrate significant improvement on morphological segmentation of foraminifera when considering region-based and topology-based metrics. Human ranking of the quality of the results by marine researchers also supports these findings.