Reconstructing (h,v)-convex 2-dimensional patterns of objects from approximate horizontal and vertical projections

The problem of reconstructing a pattern of an object from its approximate discrete orthogonal projections in a 2-dimensional grid, may have no solution because the inaccuracy in the measurements of the projections may generate an inconsistent problem. To attempt to overcome this difficulty, one seeks to reconstruct a pattern with projection values having possibly some bounded differences with the given projection values and minimizing the sum of the absolute differences.

This paper addresses the problem of reconstructing a pattern with a difference at most equal to +1 or −1 between each of its projection values and the corresponding given projection value. We deal with the case of patterns which have to be horizontally and vertically convex and the case of patterns which have to be moreover connected, the so-called convex polyominoes. We show that in both cases, the problem of reconstructing a pattern can be transformed into a Satisfiability (SAT) Problem. This is done in order to take advantage of the recent advances in the design of solvers for the SAT Problem. We show, experimentally, that by adding two important features to CSAT (an efficient SAT solver), optimal patterns can be found if there exist feasible ones. These two features are: first, a method that extracts in linear time an optimal pattern from a set of feasible patterns grouped in a generic pattern (obtaining a generic pattern may be exponential in the worst case) and second, a method that computes actively a lower bound of the sum of absolute differences that can be obtained from a partially defined pattern. This allows to prune the search tree if this lower bound exceeds the best sum of absolute differences found so far.