Reconstructing hv-Convex Polyominoes from Orthogonal Projections

  title={Reconstructing hv-Convex Polyominoes from Orthogonal Projections},
  author={M. Chrobak and C. D{\"u}rr},
Abstract We address the problem of reconstructing a discrete 2D object, represented by a set of grid cells, from its orthogonal projections. We focus on objects called hv-convex polyominoes, which are connected objects with the property that the cells in each row and column are consecutive. The main result of this paper is a simple, O(mn min(m2,n2))-time algorithm for reconstructing hv-convex polyominoes. 
Reconstructing hv-convex polyominoes with multiple colours
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This paper focuses on the case where there are multiple disjoint polyominoes that are hv-convex, i.e., any intersection with a horizontal or vertical line is contiguous, and shows that reconstruction of such polyaminoes is polynomial if the number of colours is constant, but NP-hard for an unbounded number of colors. Expand
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An intermediate class is defined --- the class of hv -convex canonical discrete sets --- and a constructive proof is given that the above problem remains computationally tractable for this class, too. Expand
A uniqueness result for reconstructing hv-convex polyominoes from horizontal and vertical projections and morphological skeleton
  • Norbert Hantos, P. Balázs
  • Computer Science, Mathematics
  • 2013 8th International Symposium on Image and Signal Processing and Analysis (ISPA)
  • 2013
This article shows that the uniqueness of the reconstruction in a special class of 4-connected hv-convex images, using two projections and the so-called morphological skeleton depends only on the values of those parameters. Expand
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Since the problem of reconstructing hv-convex binary matrices from few projections is NP-complete, an iterative approximation based on a longest path and a min-cost/max-flow model is provided. Expand
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The well-known problem of reconstructing hv-convex polyominoes is considered from a set of noisy data, and a probabilistic evaluation in the reconstruction algorithm, where different pixels assume different probabilities to be part of the reconstructed image. Expand
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Reconstruction in Different Classes of 2D Discrete Sets (Invited Paper)
It is shown that the reconstruction algorithms used in the class of hv-convex 4-connected sets (polyominoes) can be used, with small modifications, for reconstructing hV- Convex 8- connected sets and the directed h-Convex sets are uniquely reconstructible with respect to the row and column sum vectors. Expand
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A polynomial algorithm for reconstruction of some class of convex three-dimensional polyominoes that has time complexity O(n7 log n) is given. Expand
A Fast Algorithm for Reconstructing hv-Convex Binary Images from Their Horizontal Projection
This paper provides a fast polynomial-time algorithm for reconstructing canonical hv-convex images with given number of 4-connected components and with minimal number of columns satisfying a prescribed horizontal projection and shows that the method gives a solution that is always 8-connected. Expand


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Some operations for recontructing convex polyominoes by means of vectors H's and V's partial sums allows a new algorithm to be defined whose complexity is less than O(n2m2). Expand
The reconstruction of polyominoes from their orthogonal projections
  • G. Woeginger
  • Mathematics, Computer Science
  • Inf. Process. Lett.
  • 2001
It will be proved that it is NP-complete to reconstruct a two-dimensional pattern from its two orthogonal projections H and V, if (1) the pattern has to be connected (and hence forms a so-called polyomino), or if (2) the patterns have to be horizontally and vertically convex. Expand
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In a previous report, we studied the problem of reconstructing a discrete set 𝒮 from its horizontal and vertical projections. We defined an algorithm that decides whether there is a convex polyominoExpand
Medians of polyominoes: A property for reconstruction
A new algorithm whose complexity is less than O (nm ) . Expand
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Matrices of zeros and ones with fixed row and column sum vectors
Abstract Let m and n be positive integers, and let R =( r 1 ,…, r m ) and S =( s 1 ,…, s n ) be nonnegative integral vectors. We survey the combinational properties of the set of all m × n matricesExpand
On the Complexity of Timetable and Multicommodity Flow Problems
A very primitive version of Gotlieb’s timetable problem is shown to be NP-complete, and therefore all the common timetable problems are NP-complete. A polynomial time algorithm, in case all teachersExpand
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A simple constructive algorithm for the evaluation of formulas having two literals per clause, which runs in linear time on a random access machine. Expand
Matrices of zeros and ones with xed row and column sum vectors. Linear Algebra and Applications
  • Matrices of zeros and ones with xed row and column sum vectors. Linear Algebra and Applications
  • 1980