`Reconstructing binary matrices from their row, column, diagonal, and antidiagonal sums (also called projections) plays a central role in discrete tomography. One of the main difficulties in this task is that in certain cases the projections do not uniquely determine the binary matrix. This can yield an extremely large number of (sometimes very different) solutions. This ambiguity can be reduced by having some prior knowledge about the matrix to be reconstructed. The main challenge here is to find classes of binary matrices where ambiguity is drastically reduced or even completely eliminated. The goal of this paper is to study the class of $hv$-convex matrices which have decomposable configurations from the viewpoint of ambiguity. First, we give a negative result in the case of three projections. Then, we present a heuristic for the reconstruction using four projections and analyze its performance in quality and running time.`