We study X-ray tomograqphic reconstruction using statistical methods. The problem is expressed in cylindrical coordinates, which yield significant computational and memory savings, with nonnegativity bounds. A change of variables involving a Fourier matrix attempts to improve the conditioning of the Hessian but introduces linear inequality constraints. The scale and density of the problem call for factorization-free methods. We argue that projections into the feasible set can be computed efficiently by solving a bound-constrained linear least-squares problem with a fast operator. This motivates our interest towards projection-based active-set methods for the reconstruction problem, namely a spectral projected gradient method and a trust-region projected Newton method that we generalize to our specific scenario. For the projection subproblem, we consider several projection-based methods for bound-constrained problems. We assess the performance of several algorithm combinations on the reconstruction problem using synthetic data. Our results show that the projected Newton method combined with efficient projection strategies applied to the problem in cylindrical coordinates with linear inequality constraints is competitive in terms of run time with a limited-memory BFGS applied to the problem in cartesian coordinates with simple bounds, but reduces the memory footprint by a factor of about 233 on a 2D problem with 512x512 pixels.