Gill Barequet | Technion - Israel Institute of Technology (original) (raw)

Papers by Gill Barequet

The Visual Computer, 2017

Journal of Graph Algorithms and Applications, 2004

We consider the problem of representing size information in the edges and vertices of a planar gr... more We consider the problem of representing size information in the edges and vertices of a planar graph. Such information can be used, for example, to depict a network of computers and information traveling through the network. We present an efficient linear-time algorithm which draws edges and vertices of varying 2-dimensional areas to represent the amount of information flowing through them. The algorithm avoids all occlusions of nodes and edges, while still drawing the graph on a compact integer grid.

Trends in mathematics, 2021

Discrete Mathematics & Theoretical Computer Science, Mar 17, 2016

We study the configuration space of rectangulations and convex subdivisions of n points in the pl... more We study the configuration space of rectangulations and convex subdivisions of n points in the plane. It is shown that a sequence of O(n log n) elementary flip and rotate operations can transform any rectangulation to any other rectangulation on the same set of n points. This bound is the best possible for some point sets, while Θ(n) operations are sufficient and necessary for others. Some of our bounds generalize to convex subdivisions of n points in the plane.

Lecture Notes in Computer Science, 2006

In this paper we show a lower bound for the on-line version of Heilbronn's triangle problem in d ... more In this paper we show a lower bound for the on-line version of Heilbronn's triangle problem in d dimensions. Specifically, we provide an incremental construction for positioning n points in the d-dimensional unit cube, for which every simplex defined by d + 1 of these points has volume Ω(1/n (d+1) ln (d−2)+2).

Computer Vision and Image Understanding, Mar 1, 1996

netic resonance imaging) apparata. These cross sections, In this paper we present a new technique... more netic resonance imaging) apparata. These cross sections, In this paper we present a new technique for piecewise-linear hereafter called slices, are the basis for interpolating the surface reconstruction from a series of parallel polygonal cross boundary surface of the organ. The interpolated object sections. This is an important problem in medical imaging, can then be displayed in graphics applications, or (more surface reconstruction from topographic data, and other applirecently) even manufactured by an NC (numerically concations. We reduce the problem, as in most previous works, to trolled) or an RP (rapid prototyping) machine. Another a series of problems of piecewise-linear interpolation between motivation for this problem is the nondestructive digitizaeach pair of successive slices. Our algorithm uses a partial curve tion of objects: after an object is scanned by an echomatching technique for matching parts of the contours, an graphic or an X-ray apparatus, the obtained slices are used optimal triangulation of 3-D polygons for resolving the unfor the reconstruction of the original object. Yet another matched parts, and a minimum spanning tree heuristic for interpolating between nonsimply connected regions. Unlike motivation is the reconstruction of a three-dimensional previous attempts at solving this problem, our algorithm seems model of a terrain from topographic elevation contours. to handle successfully in practice any kind of data. It allows Many solutions were suggested for the pure raster intermultiple contours in each slice, with any hierarchy of contour polation. These usually handle two raster images, where nesting, and avoids the introduction of counterintuitive bridges each pixel is either white or black, or assigned a gray level between contours, proposed in some earlier papers to handle taken from a fixed range. The interpolation produces one interpolation between multiply connected regions. Experimenor more intermediate raster images, which smoothly and tal results on various complex examples, involving actual medilocally turn the first image into the second one. Then, the cal imaging data, are presented and show the good and robust bounding surface is detected using other methods, such as performance of our algorithm.

Discrete Mathematics & Theoretical Computer Science, 2005

We improve the lower bounds on Klarner's constant, which describes the exponential growth rate of... more We improve the lower bounds on Klarner's constant, which describes the exponential growth rate of the number of polyominoes (connected subsets of grid squares) with a given number of squares. We achieve this by analyzing polyominoes on a different surface, a so-called twisted cylinder by the transfer matrix method. A bijective representation of the "states" of partial solutions is crucial for allowing a compact representation of the successive iteration vectors for the transfer matrix method.

IEEE Transactions on Pattern Analysis and Machine Intelligence, 1997

In this paper we present a new technique for partial surface and volume matching of images in thr... more In this paper we present a new technique for partial surface and volume matching of images in three dimensions. In this problem we are given two objects in 3-space, each represented as a set of points, and the goal is to find a rigid motion of one object which makes a sufficiently large portion of its boundary lying sufficiently close to a corresponding portion of the boundary of the second object. This is an important problem in pattern recognition and in computer vision, with many industrial, medical, and chemical applications. Our method treats separately the rotation and the translation components of the Euclidean motion that we seek. The algorithm steps through a sequence of rotations, in a steepest-descent style, and uses a novel technique for scoring the match for any fixed rotation. Experimental results on various examples, involving data from industrial applications, medical imaging, and molecular biology, are presented, and show the accurate and robust performance of our algorithm.

Electronic Journal of Combinatorics, Aug 26, 2022

We consider minimum-perimeter lattice animals, providing a set of conditions which are sufficient... more We consider minimum-perimeter lattice animals, providing a set of conditions which are sufficient for a lattice to have the property that inflating all minimumperimeter animals of a certain size yields (without repetitions) all minimum-perimeter animals of a new, larger size. We demonstrate this result on the two-dimensional square and hexagonal lattices. In addition, we characterize the sizes of minimumperimeter animals on these lattices that are not created by inflating members of another set of minimum-perimeter animals.

Lecture Notes in Computer Science, 2019

A polyomino is a set of edge-connected squares on the square lattice. We investigate the combinat... more A polyomino is a set of edge-connected squares on the square lattice. We investigate the combinatorial and geometric properties of minimal-perimeter polyominoes. We explore the behavior of minimal-perimeter polyominoes when they are “inflated,” i.e., expanded by all empty cells neighboring them, and show that inflating all minimal-perimeter polyominoes of a given area create the set of all minimal-perimeter polyominoes of some larger area. We characterize the roots of the infinite chains of sets of minimal-perimeter polyominoes which are created by inflating polyominoes of another set of minimal-perimeter polyominoes, and show that inflating any polyomino for a sufficient amount of times results in a minimal-perimeter polyomino. In addition, we devise two efficient algorithms for counting the number of minimal-perimeter polyominoes of a given area, compare the algorithms and analyze their running times, and provide the counts of polyominoes which they produce.

Lecture Notes in Computer Science, Aug 12, 2008

ABSTRACT A d-D polycube of size n is a connected set of n cells (hypercubes) of an orthogonal d-d... more ABSTRACT A d-D polycube of size n is a connected set of n cells (hypercubes) of an orthogonal d-dimensional lattice, where connectivity is through (d − 1)-dimensional faces of the cells. Computing A d (n), the number of distinct d-dimensional polycubes of size n, is a long-standing elusive problem in discrete geometry. In a previous work we described the generalization from two to higher dimensions of a polyomino-counting algorithm of Redelmeier. The main deficiency of the algorithm is that it keeps the entire set of cells that appear in any possible polycube in memory at all times. Thus, the amount of required memory grows exponentially with the dimension. In this paper we present a method whose order of memory consumption is a (very low) polynomial in both n and d. Furthermore, we parallelized the algorithm and ran it through the Internet on dozens of computers simultaneously. This enables us to find A d (n) for values of d and n far beyond any previous attempt.

Springer eBooks, 2006

A planar polyomino of size n is an edge-connected set of n squares on a rectangular 2-D lattice. ... more A planar polyomino of size n is an edge-connected set of n squares on a rectangular 2-D lattice. Similarly, a d-dimensional polycube (for d ≥ 2) of size n is a connected set of n hypercubes on an orthogonal d-dimensional lattice, where two hypercubes are neighbors if they share a (d − 1)-dimensional face. There are also two-dimensional polyominoes that lie on a triangular or hexagonal lattice. In this paper we describe a generalization of Redelmeier's algorithm for counting twodimensional rectangular polyominoes [Re81], which counts all the above types of polyominoes. For example, our program computed the number of distinct 3-D polycubes of size 18. To the best of our knowledge, this is the first tabulation of this value.

In this video, we survey some results concerning polyominoes, which are sets of connected cells o... more In this video, we survey some results concerning polyominoes, which are sets of connected cells on the square lattice, and specifically, minimal-perimeter polyominoes, that are polyominoes with the minimal-perimeter from all polyominoes of the same size

arXiv (Cornell University), Feb 9, 2013

We prove a conjecture of Aanjaneya, Bishnu, and Pal that the minimum number of diffuse reflection... more We prove a conjecture of Aanjaneya, Bishnu, and Pal that the minimum number of diffuse reflections sufficient to illuminate the interior of any simple polygon with n walls from any interior point light source is n/2 − 1. Light reflecting diffusely leaves a surface in all directions, rather than at an identical angle as with specular reflections.

International Journal of Computational Geometry and Applications, Aug 1, 2013

Computational Geometry: Theory and Applications, Oct 1, 2021

Springer eBooks, 2010

The triangle-perimeter 2-site distance function defines the “distance” from a point x to two othe... more The triangle-perimeter 2-site distance function defines the “distance” from a point x to two other points p,q as the perimeter of the triangle whose vertices are x,p,q. Accordingly, given a set S of n points in the plane, the Voronoi diagram of S with respect to the triangle-perimeter distance, is the subdivision of the plane into regions, where the region of the pair p,q ∈ S is the locus of all points closer to p,q (according to the triangle-perimeter distance) than to any other pair of sites in S. In this paper we prove a theorem about the perimeters of triangles, two of whose vertices are on a given circle. We use this theorem to show that the combinatorial complexity of the triangle-perimeter 2-site Voronoi diagram is O(n 2 + ε ) (for any ε> 0). Consequently, we show that one can compute the diagram in O(n 2 + ε ) time and space.

Computational Geometry: Theory and Applications, Feb 1, 2013

In 1994 Grünbaum [2] showed, given a point set S in R 3 , that it is always possible to construct... more In 1994 Grünbaum [2] showed, given a point set S in R 3 , that it is always possible to construct a polyhedron whose vertices are exactly S. Such a polyhedron is called a polyhedronization of S. Agarwal et al. [1] extended this work in 2008 by showing that a polyhedronization always exists that is decomposable into a union of tetrahedra (tetrahedralizable). In the same work they introduced the notion of a serpentine polyhedronization for which the dual of its tetrahedralization is a chain. In this work we present an algorithm for constructing a serpentine polyhedronization that has vertices with bounded degree of 7, answering an open question by Agarwal et al. [1].

Computational Geometry: Theory and Applications, Feb 1, 1999

In this paper we present a new technique for partial surface and volume matching of images in thr... more In this paper we present a new technique for partial surface and volume matching of images in three dimensions. In this problem, we are given two objects in 3-space, each represented as a set of points, scattered uniformly along its boundary or inside its volume. The goal is to find a rigid motion of one object which makes a sufficiently large portion of its boundary lying sufficiently close to a corresponding portion of the boundary of the second object. This is an important problem in pattern recognition and in computer vision, with many industrial, medical, and chemical applications. Our algorithm is based on assigning a directed footprint to every point of the two sets, and locating all the pairs of points (one of each set) whose undirected components of the footprints are sufficiently similar. The algorithm then computes for each such pair of points all the rigid transformations that map the first point to the second, while making the respective direction components of their footprints coincide. A voting scheme is employed for computing transformations which map significantly large number of points of the first set to points of the second set. Experimental results on various examples are presented and show the accurate and robust performance of our algorithm.

Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms

A polycube is a face-connected set of cubical cells on Z 3. To-date, no formulae enumerating poly... more A polycube is a face-connected set of cubical cells on Z 3. To-date, no formulae enumerating polycubes by volume (number of cubes) or perimeter (number of empty cubes neighboring the polycube) are known. We present a few formulae enumerating polycubes with a fixed deviation from the maximum possible perimeter.

The Visual Computer, 2017

Journal of Graph Algorithms and Applications, 2004

We consider the problem of representing size information in the edges and vertices of a planar gr... more We consider the problem of representing size information in the edges and vertices of a planar graph. Such information can be used, for example, to depict a network of computers and information traveling through the network. We present an efficient linear-time algorithm which draws edges and vertices of varying 2-dimensional areas to represent the amount of information flowing through them. The algorithm avoids all occlusions of nodes and edges, while still drawing the graph on a compact integer grid.

Trends in mathematics, 2021

Discrete Mathematics & Theoretical Computer Science, Mar 17, 2016

We study the configuration space of rectangulations and convex subdivisions of n points in the pl... more We study the configuration space of rectangulations and convex subdivisions of n points in the plane. It is shown that a sequence of O(n log n) elementary flip and rotate operations can transform any rectangulation to any other rectangulation on the same set of n points. This bound is the best possible for some point sets, while Θ(n) operations are sufficient and necessary for others. Some of our bounds generalize to convex subdivisions of n points in the plane.

Lecture Notes in Computer Science, 2006

In this paper we show a lower bound for the on-line version of Heilbronn's triangle problem in d ... more In this paper we show a lower bound for the on-line version of Heilbronn's triangle problem in d dimensions. Specifically, we provide an incremental construction for positioning n points in the d-dimensional unit cube, for which every simplex defined by d + 1 of these points has volume Ω(1/n (d+1) ln (d−2)+2).

Computer Vision and Image Understanding, Mar 1, 1996

netic resonance imaging) apparata. These cross sections, In this paper we present a new technique... more netic resonance imaging) apparata. These cross sections, In this paper we present a new technique for piecewise-linear hereafter called slices, are the basis for interpolating the surface reconstruction from a series of parallel polygonal cross boundary surface of the organ. The interpolated object sections. This is an important problem in medical imaging, can then be displayed in graphics applications, or (more surface reconstruction from topographic data, and other applirecently) even manufactured by an NC (numerically concations. We reduce the problem, as in most previous works, to trolled) or an RP (rapid prototyping) machine. Another a series of problems of piecewise-linear interpolation between motivation for this problem is the nondestructive digitizaeach pair of successive slices. Our algorithm uses a partial curve tion of objects: after an object is scanned by an echomatching technique for matching parts of the contours, an graphic or an X-ray apparatus, the obtained slices are used optimal triangulation of 3-D polygons for resolving the unfor the reconstruction of the original object. Yet another matched parts, and a minimum spanning tree heuristic for interpolating between nonsimply connected regions. Unlike motivation is the reconstruction of a three-dimensional previous attempts at solving this problem, our algorithm seems model of a terrain from topographic elevation contours. to handle successfully in practice any kind of data. It allows Many solutions were suggested for the pure raster intermultiple contours in each slice, with any hierarchy of contour polation. These usually handle two raster images, where nesting, and avoids the introduction of counterintuitive bridges each pixel is either white or black, or assigned a gray level between contours, proposed in some earlier papers to handle taken from a fixed range. The interpolation produces one interpolation between multiply connected regions. Experimenor more intermediate raster images, which smoothly and tal results on various complex examples, involving actual medilocally turn the first image into the second one. Then, the cal imaging data, are presented and show the good and robust bounding surface is detected using other methods, such as performance of our algorithm.

Discrete Mathematics & Theoretical Computer Science, 2005

We improve the lower bounds on Klarner's constant, which describes the exponential growth rate of... more We improve the lower bounds on Klarner's constant, which describes the exponential growth rate of the number of polyominoes (connected subsets of grid squares) with a given number of squares. We achieve this by analyzing polyominoes on a different surface, a so-called twisted cylinder by the transfer matrix method. A bijective representation of the "states" of partial solutions is crucial for allowing a compact representation of the successive iteration vectors for the transfer matrix method.

IEEE Transactions on Pattern Analysis and Machine Intelligence, 1997

In this paper we present a new technique for partial surface and volume matching of images in thr... more In this paper we present a new technique for partial surface and volume matching of images in three dimensions. In this problem we are given two objects in 3-space, each represented as a set of points, and the goal is to find a rigid motion of one object which makes a sufficiently large portion of its boundary lying sufficiently close to a corresponding portion of the boundary of the second object. This is an important problem in pattern recognition and in computer vision, with many industrial, medical, and chemical applications. Our method treats separately the rotation and the translation components of the Euclidean motion that we seek. The algorithm steps through a sequence of rotations, in a steepest-descent style, and uses a novel technique for scoring the match for any fixed rotation. Experimental results on various examples, involving data from industrial applications, medical imaging, and molecular biology, are presented, and show the accurate and robust performance of our algorithm.

Electronic Journal of Combinatorics, Aug 26, 2022

We consider minimum-perimeter lattice animals, providing a set of conditions which are sufficient... more We consider minimum-perimeter lattice animals, providing a set of conditions which are sufficient for a lattice to have the property that inflating all minimumperimeter animals of a certain size yields (without repetitions) all minimum-perimeter animals of a new, larger size. We demonstrate this result on the two-dimensional square and hexagonal lattices. In addition, we characterize the sizes of minimumperimeter animals on these lattices that are not created by inflating members of another set of minimum-perimeter animals.

Lecture Notes in Computer Science, 2019

A polyomino is a set of edge-connected squares on the square lattice. We investigate the combinat... more A polyomino is a set of edge-connected squares on the square lattice. We investigate the combinatorial and geometric properties of minimal-perimeter polyominoes. We explore the behavior of minimal-perimeter polyominoes when they are “inflated,” i.e., expanded by all empty cells neighboring them, and show that inflating all minimal-perimeter polyominoes of a given area create the set of all minimal-perimeter polyominoes of some larger area. We characterize the roots of the infinite chains of sets of minimal-perimeter polyominoes which are created by inflating polyominoes of another set of minimal-perimeter polyominoes, and show that inflating any polyomino for a sufficient amount of times results in a minimal-perimeter polyomino. In addition, we devise two efficient algorithms for counting the number of minimal-perimeter polyominoes of a given area, compare the algorithms and analyze their running times, and provide the counts of polyominoes which they produce.

Lecture Notes in Computer Science, Aug 12, 2008

ABSTRACT A d-D polycube of size n is a connected set of n cells (hypercubes) of an orthogonal d-d... more ABSTRACT A d-D polycube of size n is a connected set of n cells (hypercubes) of an orthogonal d-dimensional lattice, where connectivity is through (d − 1)-dimensional faces of the cells. Computing A d (n), the number of distinct d-dimensional polycubes of size n, is a long-standing elusive problem in discrete geometry. In a previous work we described the generalization from two to higher dimensions of a polyomino-counting algorithm of Redelmeier. The main deficiency of the algorithm is that it keeps the entire set of cells that appear in any possible polycube in memory at all times. Thus, the amount of required memory grows exponentially with the dimension. In this paper we present a method whose order of memory consumption is a (very low) polynomial in both n and d. Furthermore, we parallelized the algorithm and ran it through the Internet on dozens of computers simultaneously. This enables us to find A d (n) for values of d and n far beyond any previous attempt.

Springer eBooks, 2006

A planar polyomino of size n is an edge-connected set of n squares on a rectangular 2-D lattice. ... more A planar polyomino of size n is an edge-connected set of n squares on a rectangular 2-D lattice. Similarly, a d-dimensional polycube (for d ≥ 2) of size n is a connected set of n hypercubes on an orthogonal d-dimensional lattice, where two hypercubes are neighbors if they share a (d − 1)-dimensional face. There are also two-dimensional polyominoes that lie on a triangular or hexagonal lattice. In this paper we describe a generalization of Redelmeier's algorithm for counting twodimensional rectangular polyominoes [Re81], which counts all the above types of polyominoes. For example, our program computed the number of distinct 3-D polycubes of size 18. To the best of our knowledge, this is the first tabulation of this value.

In this video, we survey some results concerning polyominoes, which are sets of connected cells o... more In this video, we survey some results concerning polyominoes, which are sets of connected cells on the square lattice, and specifically, minimal-perimeter polyominoes, that are polyominoes with the minimal-perimeter from all polyominoes of the same size

arXiv (Cornell University), Feb 9, 2013

We prove a conjecture of Aanjaneya, Bishnu, and Pal that the minimum number of diffuse reflection... more We prove a conjecture of Aanjaneya, Bishnu, and Pal that the minimum number of diffuse reflections sufficient to illuminate the interior of any simple polygon with n walls from any interior point light source is n/2 − 1. Light reflecting diffusely leaves a surface in all directions, rather than at an identical angle as with specular reflections.

International Journal of Computational Geometry and Applications, Aug 1, 2013

Computational Geometry: Theory and Applications, Oct 1, 2021

Springer eBooks, 2010

The triangle-perimeter 2-site distance function defines the “distance” from a point x to two othe... more The triangle-perimeter 2-site distance function defines the “distance” from a point x to two other points p,q as the perimeter of the triangle whose vertices are x,p,q. Accordingly, given a set S of n points in the plane, the Voronoi diagram of S with respect to the triangle-perimeter distance, is the subdivision of the plane into regions, where the region of the pair p,q ∈ S is the locus of all points closer to p,q (according to the triangle-perimeter distance) than to any other pair of sites in S. In this paper we prove a theorem about the perimeters of triangles, two of whose vertices are on a given circle. We use this theorem to show that the combinatorial complexity of the triangle-perimeter 2-site Voronoi diagram is O(n 2 + ε ) (for any ε> 0). Consequently, we show that one can compute the diagram in O(n 2 + ε ) time and space.

Computational Geometry: Theory and Applications, Feb 1, 2013

In 1994 Grünbaum [2] showed, given a point set S in R 3 , that it is always possible to construct... more In 1994 Grünbaum [2] showed, given a point set S in R 3 , that it is always possible to construct a polyhedron whose vertices are exactly S. Such a polyhedron is called a polyhedronization of S. Agarwal et al. [1] extended this work in 2008 by showing that a polyhedronization always exists that is decomposable into a union of tetrahedra (tetrahedralizable). In the same work they introduced the notion of a serpentine polyhedronization for which the dual of its tetrahedralization is a chain. In this work we present an algorithm for constructing a serpentine polyhedronization that has vertices with bounded degree of 7, answering an open question by Agarwal et al. [1].

Computational Geometry: Theory and Applications, Feb 1, 1999

In this paper we present a new technique for partial surface and volume matching of images in thr... more In this paper we present a new technique for partial surface and volume matching of images in three dimensions. In this problem, we are given two objects in 3-space, each represented as a set of points, scattered uniformly along its boundary or inside its volume. The goal is to find a rigid motion of one object which makes a sufficiently large portion of its boundary lying sufficiently close to a corresponding portion of the boundary of the second object. This is an important problem in pattern recognition and in computer vision, with many industrial, medical, and chemical applications. Our algorithm is based on assigning a directed footprint to every point of the two sets, and locating all the pairs of points (one of each set) whose undirected components of the footprints are sufficiently similar. The algorithm then computes for each such pair of points all the rigid transformations that map the first point to the second, while making the respective direction components of their footprints coincide. A voting scheme is employed for computing transformations which map significantly large number of points of the first set to points of the second set. Experimental results on various examples are presented and show the accurate and robust performance of our algorithm.

Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms

A polycube is a face-connected set of cubical cells on Z 3. To-date, no formulae enumerating poly... more A polycube is a face-connected set of cubical cells on Z 3. To-date, no formulae enumerating polycubes by volume (number of cubes) or perimeter (number of empty cubes neighboring the polycube) are known. We present a few formulae enumerating polycubes with a fixed deviation from the maximum possible perimeter.