The Reconstruction of Some 3D Convex Polyominoes from Orthogonal Projections Maciej G¸ ebala Institute of Mathematics, Wroclaw University of Technology Janiszewskiego 14, 50–370 Wroclaw, Poland mgc@im.pwr.wroc.pl Abstract. The reconstruction of discrete two- or three-dimensional sets from their orthogonal projections is one of the central problems in the areas of medical diagnostics, computer-aided tomography, and pattern recognition. In this paper we will give a polynomial algorithm for recon- struction of some class of convex three-dimensional polyominoes that has time complexity . 1 Introduction A unit cube is a cube of volume one, whose centre belongs to and whose vertices belong to the lattice . We do not distinguish between a unit cube and its centre, thus denotes the unit cube of centre .A three-dimensional polyomino is a finite connected union of unit cubes. In this paper we consider polyominoes contained in a finite lattice . The lattice with a polyomino corresponds to the three-dimensional binary matrix , when the 1’s correspond to filling positions in the lattice, and 0’s correspond to empty positions. The slices are two-dimensional sections of the matrix . We define the sli- ces , and by , and , respectively, for . The bars are one-dimensional sections of the matrix . We define the bars , and by , and , respectively, for . For a polyomino in the matrix we define three two-dimensional matrices of orthogonal projections (i.e. the number of 1’s in each bar of matrix ): , and by for . Supported by the KBN grant No. 7 T11C 03220