Book ChapterDOI

# Efficient Algorithms for Reconstruction of 2D-Arrays from Extended Parikh Images

01 Dec 2008-pp 1137-1146

TL;DR: The concept of a Parikh matrix or an extended Parikh mapping of words introduced by Mateescu et al (2001) is formulated here for two-dimensional (2D) arrays and the problem of reconstructing a 2D-array over {0,1} from its image under the extendedParikh mapping along three or more directions is shown to be NP-hard.

AbstractThe concept of a Parikh matrix or an extended Parikh mapping of words introduced by Mateescu et al (2001) is formulated here for two-dimensional (2D) arrays. A polynomial time algorithm is proposed to reconstruct an unknown 2D-array over { 0,1 } from its image under the extended Parikh mapping along a single direction. On the other hand the problem of reconstructing a 2D-array over { 0,1 } from its image under the extended Parikh mapping along three or more directions is shown to be NP-hard. Also a polynomial time algorithm to reconstruct a 2D-array over {0,1} with a maximum number of ones close to the main diagonal of the array is presented by reducing the problem to Min-cost Max-flow problem.

##### Citations
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Journal ArticleDOI
TL;DR: This work extends the notion of Parikh matrix of a word to a picture array and associates with it two kinds ofParikh matrices, called row ParikhMatrix and column Parikh Matrix, and obtains conditions that ensure M-ambiguity.
Abstract: Parikh matrix mapping or Parikh matrix of a word has been introduced in the literature to count the scattered subwords in the word. Several properties of a Parikh matrix have been extensively investigated. A picture array is a two-dimensional connected digitized rectangular array consisting of a finite number of pixels with each pixel in a cell having a label from a finite alphabet. Here we extend the notion of Parikh matrix of a word to a picture array and associate with it two kinds of Parikh matrices, called row Parikh matrix and column Parikh matrix. Two picture arrays A and B are defined to be M-equivalent if their row Parikh matrices are the same and their column Parikh matrices are the same. This enables to extend the notion of M-ambiguity to a picture array. In the binary and ternary cases, conditions that ensure M-ambiguity are then obtained.

7 citations

### Cites background from "Efficient Algorithms for Reconstruc..."

• ...Among a number of problems of interest related to digitized picture arrays (see for example [4, 9, 11]), investigation of combinatorial properties of arrays has been done (see for example [5, 6, 17])....

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• ...tion of binary images has been extensively studied (See for example [4, 11])....

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Book ChapterDOI
23 May 2011
TL;DR: This work introduces the notion of M-vector of a binary word which allows us to have a linear notation in the form of a unique vector representation of the Parikh matrix of the binary word and extends this notion to a binary image treating it as a binary array over a two-symbol alphabet.
Abstract: Mateescu et al (2001) introduced the notion of Parikh matrix of a word as an extension of the well-known concept of Parikh vector of a word. The Parikh matrix provides more numerical information about a word than given by the Parikh vector. Here we introduce the notion of M-vector of a binary word which allows us to have a linear notation in the form of a unique vector representation of the Parikh matrix of the binary word. We then extend this notion of M-vector to a binary image treating it as a binary array over a two-symbol alphabet. This is done by considering the M-vectors of the words in the rows and columns of the array. Among the properties associated with a Parikh matrix, M-ambiguity or simply ambiguity of a word is one which has been investigated extensively in the literature. Here M-ambiguity of a binary array is defined in terms of its M-vector and we obtain conditions for M-ambiguity of a binary array.

2 citations

Journal ArticleDOI

Abstract: A word is a finite sequence of symbols. Parikh matrix of a word is an upper triangular matrix with ones in the main diagonal and nonnegative integers above the main diagonal which are counts of certain scattered subwords in the word. On the other hand, a picture array, which is a rectangular arrangement of symbols, is an extension of the notion of a word to two dimensions. Parikh matrices associated with a picture array have been introduced, and their properties have been studied. Here, we obtain certain algebraic properties of Parikh matrices of binary picture arrays based on the notions of power, fairness, and a restricted shuffle operator extending the corresponding notions studied in the case of words. We also obtain properties of Parikh matrices of arrays formed by certain geometric operations.

1 citations

### Cites methods from "Efficient Algorithms for Reconstruc..."

• ...,e problem of reconstruction of 2D binary images has been studied [22] based on Parikh matrices....

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Journal ArticleDOI
TL;DR: Algorithms are developed for solving binary image reconstruction problems given some numerical information on the image and correctness of the algorithms are discussed.
Abstract: The problem of reconstruction of a binary image in the field of discrete tomography is a classic instance of seeking solution applying mathematical techniques. Here two such binary image reconstruction problems are considered given some numerical information on the image. Algorithms are developed for solving these problems and correctness of the algorithms are discussed.
Dissertation
01 Jan 2012
TL;DR: This work considers reconstruction of a binary image with some prescribed numerical information on the rows of the binary image treated as a binary matrix of 0's and 1's.
Abstract: Discrete tomography deals with the reconstruction of images, in particular binary images, from their projections. A number of binary image reconstruction methods have been considered in the literature, using different projection models or additional constraints. Here, we will consider reconstruction of a binary image with some prescribed numerical information on the rows of the binary image treated as a binary matrix of 0's and 1's. The problem involves information, referred to as row projection, on the number of 1's and the number of subword 01's in the rows of the binary image to be constructed. The algorithm proposed constructs one among the many binary images having the same numerical information on the number of 1's and the number of subword 01. This proposed algorithm will also construct the image uniquely for a special kind of a binary image with its rows in some specific form.

##### References
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Book
01 Jan 1999
Abstract: Preface Contributors Part I. Foundations Discrete Tomography: A Historical Overview \ Attila Kuba, Gabor T. Herman Sets of Uniqueness and Additivity in Integer Lattices \ Peter C. Fishburn, Lawrence A. Shepp Tomopgraphic Equivalence and Switching Operations \ T. Yung Kong, Gabor T. Herman Uniqueness and Complexity in Discrete Tomography \ Richard J. Gardner, Peter Gritzmann Reconstruction of Plane Figures from Two Projections \ Akira Kaneko, Lei Huang Reconstruction of Two-Valued Functions and Matrices \ Attila Kuba Reconstruction of Connected Sets from Two Projections \ Alberto Del Lungo, Maurice Nivat Part II. Algorithms Binary Tomography Using Gibbs Priors \ Samuel Matej, Avi Vardi, Gabor T. Herman, Eilat Vardi Probabilistic Modeling of Discrete Images \ Michael T. Chan, Gabor T. Herman, Emanuel Levitan Multiscale Bayesian Methods for Discrete Tomography \ Thomas Frese, Charles A. Bouman, Ken Sauer An Algebraic Solution for Discrete Tomography \ Andrew E. Yagle Binary Steering of Nonbinary Iterative Algorithms \ Yair Censor, Samuel Matej Reconstruction of Binary Images via the EM Algorithm \ Yehuda Vardi, Cun-Hui Zhang Part III. Applications CT-Assisted Engineering and Manufacturing \ Jolyon A. Browne, Mathew Koshy 3D Reconstruction from Sparse Radiographic Data \ James Sachs, Jr., Ken Sauer Heart Chamber Reconstruction from Biplane Angiography \ Dietrich G.W. Onnasch, Guido P.M. Prause Discrete Tomography in Electron Microscopy \ J.M. Carazo, C.O. Sorzano, E. Rietzel, R. Schroeder, R. Marabini Tomopgraphy on the 3D-Torus and Crystals \ Pablo M. Salzberg, Raul Figueroa A Recursive Algorithm for Diffuse Planar Tomography \ Sarah K. Patch From Orthogonal Projections to Symbolic Projections \ Shi-KuoChang Index

474 citations

BookDOI
01 Jan 2007
Abstract: ANHA Series Preface Preface List of Contributors Introduction / A. Kuba and G.T. Herman Part I. Foundations of Discrete Tomography An Introduction to Discrete Point X-Rays / P. Dulio, R.J. Gardner, and C. Peri Reconstruction of Q-Convex Lattice Sets / S. Brunetti and A. Daurat Algebraic Discrete Tomography / L. Hajdu and R. Tijdeman Uniqueness and Additivity for n-Dimensional Binary Matrices with Respect to Their 1-Marginals / E. Vallejo Constructing (0, 1)-Matrices with Given Line Sums and Certain Fixed Zeros / R.A. Brualdi and G. Dahl Reconstruction of Binary Matrices under Adjacency Constraints / S. Brunetti, M.C. Costa, A. Frosini, F. Jarray, and C. Picouleau Part II. Discrete Tomography Reconstruction Algorithms Decomposition Algorithms for Reconstructing Discrete Sets with Disjoint Components / P. Balazs Network Flow Algorithms for Discrete Tomography / K.J. Batenburg A Convex Programming Algorithm for Noisy Discrete Tomography / T.D. Capricelli and P.L. Combettes Variational Reconstruction with DC-Programming / C. Schnoerr, T. Schule, and S. Weber Part III. Applications of Discrete Tomography Direct Image Reconstruction-Segmentation, as Motivated by Electron Microscopy / Hstau Y. Liao and Gabor T. Herman Discrete Tomography for Generating Grain Maps of Polycrystals / A. Alpers, L. Rodek, H.F. Poulsen, E. Knudsen, G.T. Herman Discrete Tomography Methods for Nondestructive Testing / J. Baumann, Z. Kiss, S. Krimmel, A. Kuba, A. Nagy, L. Rodek, B. Schillinger, and J. Stephan Emission Discrete Tomography / E. Barcucci, A. Frosini, A. Kuba, A. Nagy, S. Rinaldi, M. Samal, and S. Zopf Application of a Discrete Tomography Approach to Computerized Tomography / Y. Gerard and F. Feschet Index

248 citations

Journal ArticleDOI
TL;DR: It turns out that for all d ⩾ 2 and for a prescribed but arbitrary set of m ⩽ 2 pairwise nonparallel lattice directions, the problems are solvable in polynomial time if m = 2 and are NP-complete (or NP-equivalent) otherwise.
Abstract: We study the computational complexity of various inverse problems in discrete tomography. These questions are motivated by demands from the material sciences for the reconstruction of crystalline structures from images produced by quantitative high resolution transmission electron microscopy. We completely settle the complexity status of the basic problems of existence (data consistency), uniqueness (determination), and reconstruction of finite subsets of the d-dimensional integer lattice γd that are only accessible via their line sums (discrete X-rays) in some prescribed finite set of lattice directions. Roughly speaking, it turns out that for all d ⩾ 2 and for a prescribed but arbitrary set of m ⩾ 2 pairwise nonparallel lattice directions, the problems are solvable in polynomial time if m = 2 and are NP-complete (or NP-equivalent) otherwise.

162 citations

Journal ArticleDOI
TL;DR: A sharpening of the Parikh map- ping is introduced and an interesting in- terconnection between mirror images of words and inverses of matrices is established.
Abstract: In this paper we introduce a sharpening of the Parikh map- ping and investigate its basic properties. The new mapping is based on square matrices of a certain form. The classical Parikh vector appears in such a matrix as the second diagonal. However, the matrix prod- uct gives more information about a word than the Parikh vector. We characterize the matrix products and establish also an interesting in- terconnection between mirror images of words and inverses of matrices. Mathematics Subject Classification. 68Q45, 68Q70.

122 citations

### "Efficient Algorithms for Reconstruc..." refers background in this paper

• ...The extended Parikh mapping of words (also called Parikh matrix mapping) is defined in [ 1 ], wherein a particular triangular matrix is associated with each symbol in the alphabet....

[...]

• ...Mateescu et al. [ 1 ] introduced an extension of the Parikh mapping of words, called extended Parikh mapping or Parikh matrix mapping of words which gives more information than the Parikh mapping....

[...]

Journal ArticleDOI
01 Oct 2005
TL;DR: This paper presents an evolutionary algorithm for finding the reconstruction which maximises an evaluation function, representing the ''quality'' of the reconstruction, and shows that the algorithm can be successfully applied to a wide range of evaluation functions.
Abstract: One of the main problems in discrete tomography is the reconstruction of binary matrices from their projections in a small number of directions In this paper we consider a new algorithmic approach for reconstructing binary matrices from only two projections This problem is usually underdetermined and the number of solutions can be very large We present an evolutionary algorithm for finding the reconstruction which maximises an evaluation function, representing the ''quality'' of the reconstruction, and show that the algorithm can be successfully applied to a wide range of evaluation functions We discuss the necessity of a problem-specific representation and tailored search-operators for obtaining satisfactory results Our new search-operators can also be used in other discrete tomography algorithms

72 citations