Ancient Herculaneum papyrus scrolls, hopelessly charred in the 79 A.D. Vesuvius eruption, contain valuable writings of the Greek philosophers of the day, including works of the Epicurean Philodemus. X-ray phase contrast tomography has recently begun unlocking their secrets. However, only small portions of the text hidden inside the scroll have been recover. One of the challenging tasks in Herculaneum papyri investigation is their virtual unfolding because of their highly complicated structure and three-dimensional arrangement. Although this procedure is feasible, problems in segmentation and flattening hinder the unrolling of a large portion of papyrus. We propose a computational platform for the virtual unfolding procedure, and we show the results of its application on two Herculaneum papyrus fragments. This work paves the way to a comprehensive survey and to further interpretation of larger portions of text hidden inside the carbonized Herculaneum papyri.