In this paper we examine the problem of reconstructing a discrete two-dimensional set from its two orthogonal projection (H,V) when the set satisfies some convexity conditions. We show that the algorithm of the paper [Int. J. Imaging Systems and Technol. 9 (1998) 69] is a good heuristic algorithm but it does not solve the problem for all (H,V) instances. We propose a modification of this algorithm solving the problem for all (H,V) instances, by starting to build the {\textquoteleft}{\textquoteleft}spine{\textquoteright}{\textquoteright}. The complexity of our reconstruction algorithm is O(mnˇlog(mn)ˇmin{m2,n2}) in the worst case. However, according to our experimental results, in 99\% of the studied cases the algorithm is able to reconstruct a solution without using the newly introduced operation. In such cases the upper bound of the complexity of the algorithm is O(mnˇlog(mn)). A systematic comparison of this algorithm was done and the results show that this algorithm has the better average complexity than other published algorithms. The way of comparison and the results are given in a separate paper [Linear Algebra Appl. (submitted)]. Finally we prove that the problem can be solved in polynomial time also in a class of discrete sets which is larger than the class of convex polyominoes, namely, in the class of 8-connected convex sets. ` `