Minimum $L_i$ norm estimation is a robust procedure ins the sense that it leads to an estimator which has greater statistical eficiency than the least squares estimator in the presence of outliers. And the $L_1$ norm estimator has some desirable statistical properties. In this paper a new computational procedure for $L_1$ norm estimation is proposed which combines the idea of reweighted least squares method and the linear programming approach. A modification of the projective transformation method is employed to solve the linear programming problem instead of the simplex method. It is proved that the proposed algorithm terminates in a finite number of iterations.