The Reconstruction of Convex Polyominoes from Horizontal and Vertical Projections ⋆ Maciej G¸ ebala Institute of Computer Science, University of Wroc law Przesmyckiego 20, 51–151 Wroc law, Poland mgc@ii.uni.wroc.pl Abstract. The problem of reconstructing a discrete set from its hori- zontal and vertical projections (RSP) is of primary importance in many different problems for example pattern recognition, image processing and data compression. We give a new algorithm which provides a reconstruction of convex polyominoes from horizontal and vertical projections. It costs atmost O(min(m, n) 2 · mn log mn) for a matrix that has m × n cells. In this paper we provide just a sketch of the algorithm. 1 Introduction 1.1 Definition of the problem Let R be a matrix which has m × n cells containing “0”s and “1”s. Let S be a set of cells containing “1”s. Given S we put h i (S) which is the number of cells containing “1” in the ith row of S and we put v j (S) which is the number of cells containing “1” in the j th column of S. We call h i (S) the ith row projection of S and v j (S) the j th column projection of S. We consider the different properties of a set S. We say that a set S of cells satisfies the properties p, v and h if p: S is a polyomino i.e. S is a connected finite set. v: every column of S is a connected set i.e. a column in R containing “0” between two different “1”s does not exist. h: every row of S is a connected set i.e. a row in R containing “0” between two different “1”s does not exist. The set S belongs to class (x)(S ∈ (x)) iff it satisfies the properties x. We can now define the problem of reconstructing a set S from its pro- jections: Given two assigned vectors H =(h 1 ,h 2 ,...,h m ) ∈{1,...,n} m and V =(v 1 ,v 2 ,...,v n ) ∈{1,...,m} n we examine whether the pair (H, V ) is satis- fiable in class (x). It is satisfiable if there is at least one set S ∈ (x) such that h i (S)= h i , for i =1,...,m, and v j (S)= v j , for j =1,...,n. We also say that S satisfies (H, V ) in (x). We define a set S as a convex polyomino if S ∈ (p, v, h). ⋆ Supported by KBN grant No 8 T11C 029 13