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This paper presents a new hybrid split-and-merge image segmentation method based on computational geometry and topology using persistent homology. The algorithm uses edge-directed topology to initially split the image into a set of regions based on the Delaunay triangulations of the points in the edge map. Persistent homology is used to generate three types of regions: p-persistent regions, p-transient regions, and d-triangles. The p-persistent regions correspond to core objects in the image, while p-transient regions and d-triangles are smaller regions that may be combined in the merge phase, either with p-persistent regions to refine the core or with other p-transient and d-triangles regions to potentially form new core objects. Performing image segmentation based on topology and persistent homology guarantees several nice properties, and initial results demonstrate high quality image segmentation.