We introduce in this work a novel algorithm for volume reconstruction from data acquired on an irregular grid, e.g., from multiple co-registered images. The algorithm, which is based on an inverse interpolation formalism, is superior to other methods in particular when the input images have lower spatial resolution than the reconstructed image. Local intensity bounds are enforced by an L-BFGSB optimizer, regularize the reconstruction problem, and preserve the intensity distribution of the input images. We demonstrate the usefulness of our method by applying it to retrospective motion correction in interleaved MR images.