Approximating hv-Convex Binary Matrices and Images from Discrete Projections Fethi Jarray 1,2,⋆ , Marie-Christine Costa 2 , and Christophe Picouleau 2 1 Gabes University of Sciences 6072 Gabes, Tunisia 2 Laboratoire CEDRIC, 292 rue Saint-Martin, 75003 Paris, France fethi jarray@yahoo.fr, {costa,chp}@cnam.fr Abstract. We study the problem of reconstructing hv-convex binary matrices from few projections. We solve a polynomial time case and we determine some properties of the hv-convex matrices. Since the problem is NP-complete, we provide an iterative approximation based on a longest path and a min-cost/max-flow model. The experimental results show that the reconstruction algorithm performs quite well. Keywords: Discrete Tomography; hv-convex; Image Reconstruction. 1 Introduction Discrete tomography deals with the reconstruction of discrete homogeneous ob- jects regarded as binary matrices from their projections. The problem of recon- structing a m × n binary matrix from its orthogonal projections H and V is the following [12]: given H =(h 1 ,...,h m ) and V =(v 1 ,...,v n ) two nonnegative inte- ger vectors find a binary matrix such that the number of ones in every row i (resp. column j ) equals h i (resp. v j ). Ryser [12] gives necessary and sufficient conditions for the existence of a solution. However, the problem is usually highly underde- termined and a large number of solutions may exist [13]. The reader is referred to the book of Herman and Kuba [9] for an overview on discrete tomography. In many applications such as image processing and electron microscopy, the orthogonal projections alone are not sufficient to uniquely determine matrices or objects. Fortunately, objects that occur in practical applications usually ex- hibit certain properties. Hence we seek to reconstruct binary matrices under additional constraints like connectivity or convexity for instance. Woeginger [14] prove that the consistency problem for polyominoes (connected sets) is NP- complete. The consistency problem for h-convex objects (polyominoes or not) is also NP-complete [1]. The above result extends to the v-convex objects. The consistency problem for hv-convex objects is NP-complete [1,14]. Therefore, the reconstruction can be solved in polynomial time only for the hv-convex polyomi- noes objects [1,5]. In the present paper, we will deal with the problem of reconstructing hv- convex matrices. Since the problem is NP-complete, one way to solve it is to ⋆ The corresponding author. D. Coeurjolly et al. (Eds.): DGCI 2008, LNCS 4992, pp. 413–422, 2008. c  Springer-Verlag Berlin Heidelberg 2008