Reconstruction of Bicolored Images

In this paper, we present an integer programming approach to estimating a discrete bi-colored image from its two-color horizontal and vertical projections. The two-color projections basically refer to the number of pixels per column having colors $$c_1$$ and $$c_2$$, and likewise for each row as well. The aim of the integer programming approach is to minimize the number of conflict pixels, i.e. the number of pixels that have color $$c_1$$ as well as $$c_2$$. Since the problem is NP-complete, we give a survey of the literature and we propose a new integer programming formulation of this problem.

[1]  G. Herman,et al.  Advances in discrete tomography and its applications , 2007 .

[2]  Fethi Jarray,et al.  An acyclic days-off scheduling problem , 2006, 4OR.

[3]  Dominique de Werra,et al.  A solvable case of image reconstruction in discrete tomography , 2005, Discret. Appl. Math..

[4]  G. Herman,et al.  Discrete tomography : foundations, algorithms, and applications , 1999 .

[5]  Fethi Jarray.  A 4-Day or a 3-Day workweeks Scheduling Problem with a Given workforce Size , 2009, Asia Pac. J. Oper. Res..

[6]  H. Ryser.  Combinatorial Properties of Matrices of Zeros and Ones , 1957, Canadian Journal of Mathematics.

[7]  Attila Kuba,et al.  Advances in Discrete Tomography and Its Applications (Applied and Numerical Harmonic Analysis) , 2007 .

[8]  Maurice Nivat,et al.  Reconstructing Convex Polyominoes from Horizontal and Vertical Projections , 1996, Theor. Comput. Sci..

[9]  Fethi Jarray,et al.  Approximating Bicolored Images from Discrete Projections , 2011, IWCIA.

[10]  Marek Chrobak,et al.  Reconstructing hv-Convex Polyominoes from Orthogonal Projections , 1999, Inf. Process. Lett..

[11]  Andrea Frosini,et al.  A reconstruction algorithm for a subclass of instances of the 2-color problem , 2011, Theor. Comput. Sci..

[12]  Belhassen Zouari,et al.  Local Verification Using a Distributed State Space , 2013, Fundam. Informaticae.

[13]  Gerhard J. Woeginger,et al.  The reconstruction of polyominoes from their orthogonal projections , 2001, Inf. Process. Lett..

[14]  D. Gale.  A theorem on flows in networks , 1957 .

[15]  Geir Dahl,et al.  Optimization and reconstruction of hv-convex (0, 1)-matrices , 2003, Discret. Appl. Math..

[16]  Andrea Frosini,et al.  Solving the Two Color Problem: An Heuristic Algorithm , 2011, IWCIA.

[17]  Andrea Frosini,et al.  Reconstructing a binary matrix under timetabling constraints , 2005, Electron. Notes Discret. Math..

[18]  Elena Barcucci,et al.  Solving Multicolor Discrete Tomography Problems by Using Prior Knowledge , 2013, Fundam. Informaticae.

[19]  Dominique de Werra,et al.  Using graphs for some discrete tomography problems , 2006, Discret. Appl. Math..

[20]  Fethi Jarray,et al.  Reconstruction of binary matrices under adjacency constraints , 2005 .

[21]  Andrea Frosini,et al.  Discrete Tomography: Reconstruction under Periodicity Constraints , 2002, ICALP.

[22]  Martín Matamala,et al.  Reconstructing 3-Colored Grids from Horizontal and Vertical Projections Is NP-hard , 2009, ESA.

[23]  Alain Billionnet,et al.  Reconstructing Convex Matrices by Integer Programming Approaches , 2013, J. Math. Model. Algorithms Oper. Res..