The number of convex polyominoes reconstructible from their orthogonal projections

AbstractMany problems of computer-aided tomography, pattern recognition, image processing and data compression involve a reconstruction of bidimensional discrete sets from their projections. [3–5,10,12,16,17]. The main difficulty involved in reconstructing a set Λ starting out from its orthogonal projections (V,H) is the ‘ambiguity’ arising from the fact that, in some cases, many different sets have the same projections (V,H). In this paper, we study this problem of ambiguity with respect to convex polyominoes, a class of bidimensional discrete sets that satisfy some connection properties similar to those used by some reconstruction algorithms. We determine an upper and lower bound to the maximum number of convex polyominoes having the same orthogonal projections (V,H), with V ∈ Nn and H ∈ Nm. We prove that under these connection conditions, the ambiguity is sometimes exponential. We also define a construction in order to obtain some convex polyominoes having the same orthogonal projections

Generation and reconstruction of hv-convex 8-connected discrete sets

An algorithm is given to generate 2-dimensional hv-convex 8-connected discrete sets uniformly. This algorithm is based on an extension of a theory previously used for a more special class of hv-convex discrete sets. The second part of the paper deals with the reconstruction of hv-convex 8-connected discrete sets. The main idea of this algorithm is to rewrite the whole reconstruction problem as a 2SAT problem. Using some a priori knowledge we reduced the number of iterations and the number of clauses in the 2SAT expression which results in reduction of execution time

Reconstructing hv-convex polyominoes with multiple colours

This thesis examines the problem of reconstructing multiple discrete 2D objects, represented by a set of cells arranged in an m × n grid, from their projections. The objects being constructed are disjoint, hv-convex polyominoes, each of which has a separate colour. The main results presented here are two algorithms for unordered C-colour reconstruction that have time complexities of O(C^2n^{2C +1}m^{2C +1}) and O(C^2 min(n^{2C}, m^{2C})nm), an ordered C-colour reconstruction algorithm that is O(Cmin(n^{2C}, m^{2C})nm), and an NP-completeness proof when the number of colours is unbounded