We present a novel method for model-based segmentation of extended, blob-like objects on curved surfaces. Our method addresses several challenges arising when imaging curved biological membrane, such as out-of-membrane signal and geometry-induced background variations. We use a particle-based reconstruction of the membrane geometry, moment-conserving intensity interpolation from pixels to surface particles, and model-based in-surface segmentation. Our method denoises and deconvolves images, corrects for background variations, and quantifies the number, size, and intensity of segmented objects. We benchmark the accuracy of the method and present two applications to (1) neuroepithelial focal adhesion sites during optic cup morphogenesis in zebrafish and (2) reconstituted membrane domains bearing the small GTPase Rab5 on spherical beads.