Endomicroscopy techniques such as confocal, multi-photon, and wide-field imaging have all been demonstrated using coherent fiber-optic imaging bundles. While the narrow diameter and flexibility of fiber bundles is clinically advantageous, the number of resolvable points in an image is conventionally limited to the number of individual fibers within the bundle. We are introducing concepts from the compressed sensing (CS) field to fiber bundle based endomicroscopy, to allow images to be recovered with more resolvable points than fibers in the bundle. The distal face of the fiber bundle is treated as a low-resolution sensor with circular pixels (fibers) arranged in a hexagonal lattice. A spatial light modulator is located conjugate to the object and distal face, applying multiple high resolution masks to the intermediate image prior to propagation through the bundle. We acquire images of the proximal end of the bundle for each (known) mask pattern and then apply CS inversion algorithms to recover a single high-resolution image. We first developed a theoretical forward model describing image formation through the mask and fiber bundle. We then imaged objects through a rigid fiber bundle and demonstrate that our CS endomicroscopy architecture can recover intra-fiber details while filling inter-fiber regions with interpolation. Finally, we examine the relationship between reconstruction quality and the ratio of the number of mask elements to the number of fiber cores, finding that images could be generated with approximately 28,900 resolvable points for a 1,000 fiber region in our platform.