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55
A BrunnMinkowski inequality for the integer lattice
 TRANS. AMER. MATH. SOC
, 2001
"... A close discrete analog of the classical BrunnMinkowksi inequality that holds for finite subsets of the integer lattice is obtained. This is applied to obtain strong new lower bounds for the cardinality of the sum of two finite sets, one of which has full dimension, and, in fact, a method for com ..."
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Cited by 24 (3 self)
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A close discrete analog of the classical BrunnMinkowksi inequality that holds for finite subsets of the integer lattice is obtained. This is applied to obtain strong new lower bounds for the cardinality of the sum of two finite sets, one of which has full dimension, and, in fact, a method for computing the exact lower bound in this situation, given the dimension of the lattice and the cardinalities of the two sets. These bounds in turn imply corresponding new bounds for the lattice point enumerator of the Minkowski sum of two convex lattice polytopes. A RogersShephard type inequality for the lattice point enumerator in the plane is also proved.
Reconstruction of 4 and 8connected convex discrete sets from row and column projections
, 2001
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Discrete tomography by convex–concave regularization and D.C. programming
, 2005
"... We present a novel approach to the tomographic reconstruction of binary objects from few projection directions within a limited range of angles. A quadratic objective functional over binary variables comprising the squared projection error and a prior penalizing nonhomogeneous regions, is supplemen ..."
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Cited by 17 (3 self)
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We present a novel approach to the tomographic reconstruction of binary objects from few projection directions within a limited range of angles. A quadratic objective functional over binary variables comprising the squared projection error and a prior penalizing nonhomogeneous regions, is supplemented with a concave functional enforcing binary solutions. Application of a primaldual subgradient algorithm to a suitable decomposition of the objective functional into the difference of two convex functions leads to an algorithm which provably converges with parallel updates to binary solutions. Numerical results demonstrate robustness against local minima and excellent reconstruction performance using five projections within a range of 90◦. Our approach is applicable to quite general objective functions over binary variableswith constraints and thus applicable to a wide range of problems within and beyond the field of discrete tomography.
BINARY TOMOGRAPHY BY ITERATING LINEAR PROGRAMS
, 2004
"... A novel approach to the reconstruction problem of binary tomography from a small number of Xray projections is presented. Based on our previous work, we adopt a linear prograrnming relaxation of this combinatorial problem which includes an objective function for the reconstruction, the approximat ..."
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Cited by 13 (0 self)
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A novel approach to the reconstruction problem of binary tomography from a small number of Xray projections is presented. Based on our previous work, we adopt a linear prograrnming relaxation of this combinatorial problem which includes an objective function for the reconstruction, the approximation of a smoothness prior enforcing spatially homogeneous solutions, and the projection constraints. We supplement this problem with an unbiased concave functional in order to gradually enforce binary minimizers. Application of a primaldual subgradient iteration for optimizing this enlarged problem amounts to solve a sequence of linear programs, where the objective function changes in each step, yielding a sequence of solutions which provably converges.
Sums, projections, and sections of lattice sets, and the discrete covariogram
 DISCRETE COMPUT GEOM
"... Basic properties of finite subsets of the integer lattice Z n are investigated from the point of view of geometric tomography. Results obtained concern the Minkowski addition of convex lattice sets and polyominoes, discrete Xrays and the discrete and continuous covariogram, the determination of sym ..."
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Cited by 11 (1 self)
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Basic properties of finite subsets of the integer lattice Z n are investigated from the point of view of geometric tomography. Results obtained concern the Minkowski addition of convex lattice sets and polyominoes, discrete Xrays and the discrete and continuous covariogram, the determination of symmetric convex lattice sets from the cardinality of their projections on hyperplanes, and a discrete version of Meyer’s inequality on sections of convex bodies by coordinate hyperplanes.
A network flow algorithm for reconstructing binary images from discrete Xrays
, 2007
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A linear programming approach to limited angle 3d reconstruction from dsa projections
 Special Issue of Methods of Information in Medicine 4
, 2004
"... Objectives: We investigate the feasibility of binaryvalued 3D tomographic reconstruction using only a small number of projections acquired over a limited range of angles. Methods: Regularization of this strongly illposed problem is achieved by (i) confining the reconstruction to binary vessel / no ..."
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Cited by 8 (5 self)
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Objectives: We investigate the feasibility of binaryvalued 3D tomographic reconstruction using only a small number of projections acquired over a limited range of angles. Methods: Regularization of this strongly illposed problem is achieved by (i) confining the reconstruction to binary vessel / nonvessel decisions, and (ii) by minimizing a global functional involving a smoothness prior. Results: Our approach successfully reconstructs volumetric vessel structures from 3 projections taken within ¢¡¤ £. The percentage of reconstructed voxels differing from ground truth is below ¥§ ¦. Conclusion: We demonstrate that for particular applications – like Digital Substraction Angiography – 3D reconstructions are possible where conventional methods must fail, due to a severly limited imaging geometry. This could play an important role for dose reduction and 3D reconstruction using nonconventional technical setups.
DirectionDependency of a Binary Tomographic Reconstruction Algorithm
"... Abstract. We study how the quality of an image reconstructed by a binary tomographic algorithm depends on the direction of the observed object in the scanner, if only a few projections are available. To do so we conduct experiments on a set of software phantoms by reconstructing them form different ..."
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Cited by 7 (3 self)
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Abstract. We study how the quality of an image reconstructed by a binary tomographic algorithm depends on the direction of the observed object in the scanner, if only a few projections are available. To do so we conduct experiments on a set of software phantoms by reconstructing them form different projection sets using an algorithm based on D.C. programming (a method for minimizing the difference of convex functions), and compare the accuracy of the corresponding reconstructions by two suitable approaches. Based on the experiments, we discuss consequences on applications arising from the field of nondestructive testing, as well.