This paper proposes a new row-action type iterative algorithm which is appropriate to reconstruct sparse objects from a limited number of projections. The main idea is to use the L1 norm to pick up a sparse solution from a set of feasible solutions to the measurement equation. By perturbing the linear program to a quadratic program, we use the duality of nonlinear programming to construct a row-action type iterative algorithm to find the solution. We also prove that the algorithm converges to the solution under mild assumptions. We show that this method works well in the 3-D blood-vessel reconstruction and its computation time is shorter than those of our previous method and MART method. Furthermore, we apply the method to real data measured with the Micro-CT device developed at Marquette University.