This Letter presents a framework for computational imaging (CI) in fiber-bundle-based endoscopy systems. Multiple observations are acquired of objects spatially modulated with different random binary masks. Sparse-recovery algorithms then reconstruct images with more resolved pixels than individual fibers in the bundle. Object details lying within the diameter of single fibers are resolved, allowing images with 41,663 resolvable points to be generated through a bundle with 2,420 fibers. Computational fiber bundle imaging of micro- and macro-scale objects is demonstrated using fluorescent standards and biological tissues, including in vivo imaging of a human fingertip. In each case, CI recovers details that conventional endoscopy does not provide.