In the case of radiography of a cylindrically symmetric object, the Abel transform is useful for describing the tomographic measurement operator. The inverse of this operator is unbounded, so regularization is required for the computation of satisfactory inversions. We introduce the use of the total variation seminorm for this purpose, and prove existence and uniqueness of solutions of the corresponding variational problem. We illustrate the effectiveness of the total variation regularization with an example and comparison with the unregularized inverse and the H¹ regularized inverse.