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Journal ArticleDOI

A non-parametric sequential method for polygonal approximation of digital curves

01 Feb 1994-Pattern Recognition Letters (Elsevier Science Inc.)-Vol. 15, Iss: 2, pp 161-167
TL;DR: Though the procedure is sequential and one pass, neither does it round off sharp turnings nor does it dislocate the vertices near the other turnings.
About: This article is published in Pattern Recognition Letters.The article was published on 1994-02-01. It has received 42 citations till now. The article focuses on the topics: Polygonal chain & Line segment.
Citations
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Journal ArticleDOI
TL;DR: This work proposes an efficient and deterministic method, TopDom, to identify TDs, along with a set of statistical methods for evaluating their quality, and reveals that the locations of housekeeping genes are closely associated with cross-tissue conserved TDs.
Abstract: Genome-wide proximity ligation assays allow the identification of chromatin contacts at unprecedented resolution. Several studies reveal that mammalian chromosomes are composed of topological domains (TDs) in sub-mega base resolution, which appear to be conserved across cell types and to some extent even between organisms. Identifying topological domains is now an important step toward understanding the structure and functions of spatial genome organization. However, current methods for TD identification demand extensive computational resources, require careful tuning and/or encounter inconsistencies in results. In this work, we propose an efficient and deterministic method, TopDom, to identify TDs, along with a set of statistical methods for evaluating their quality. TopDom is much more efficient than existing methods and depends on just one intuitive parameter, a window size, for which we provide easy-to-implement optimization guidelines. TopDom also identifies more and higher quality TDs than the popular directional index algorithm. The TDs identified by TopDom provide strong support for the cross-tissue TD conservation. Finally, our analysis reveals that the locations of housekeeping genes are closely associated with cross-tissue conserved TDs. The software package and source codes of TopDom are available athttp://zhoulab.usc.edu/TopDom/.

222 citations

Journal ArticleDOI
TL;DR: This work gives a new near-optimal approximation algorithm based on reduced-search dynamic programming that is simple, fast, and it has a low space complexity.

84 citations


Cites methods from "A non-parametric sequential method ..."

  • ...There also exist a number of faster but sub-optimal methods (Douglas and Peucker, 1973; Ray and Ray, 1994; Pikaz and Dinstein, 1995; Rosin, 1997; Yin, 1998; Huang and Sun, 1999; Zhang and Guo, 2001) with time complexities ranging from OðNÞ to OðN 2Þ....

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Journal ArticleDOI
TL;DR: This work presents a fast polygonal approximation algorithm in 2-D space for the GPS trajectory simplification under the so-called integral square synchronous distance error criterion in a linear time complexity and achieves a better approximation result than the existing competitive methods.
Abstract: Recent advances in geopositioning mobile phones have made it possible for users to collect a large number of GPS trajectories by recording their location information. However, these mobile phones with built-in GPS devices usually record far more data than needed, which brings about both heavy data storage and a computationally expensive burden in the rendering process for a Web browser. To address this practical problem, we present a fast polygonal approximation algorithm in 2-D space for the GPS trajectory simplification under the so-called integral square synchronous distance error criterion in a linear time complexity. The underlying algorithm is designed and implemented using a bottom-up multiresolution method, where the input of polygonal approximation in the coarser resolution is the polygonal curve achieved in the finer resolution. For each resolution (map scale), priority-queue structure is exploited in graph construction to construct the initialized approximated curve. Once the polygonal curve is initialized, two fine-tune algorithms are employed in order to achieve the desirable quality level. Experimental results validated that the proposed algorithm is fast and achieves a better approximation result than the existing competitive methods.

76 citations

Journal ArticleDOI
TL;DR: This paper proposes to find the optimal approximation of a cyclically extended closed curve of double size, and to select the best possible starting point by search in the extended search space for the curve.

51 citations

Journal ArticleDOI
TL;DR: An algorithm for joint near-optimal approximation of multiple polygonal curves is proposed, based on iterative reduced search dynamic programming introduced earlier for the min-@eproblem of a singlepolygonal curve.

47 citations

References
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Book
01 Jan 1973
TL;DR: In this article, a unified, comprehensive and up-to-date treatment of both statistical and descriptive methods for pattern recognition is provided, including Bayesian decision theory, supervised and unsupervised learning, nonparametric techniques, discriminant analysis, clustering, preprosessing of pictorial data, spatial filtering, shape description techniques, perspective transformations, projective invariants, linguistic procedures, and artificial intelligence techniques for scene analysis.
Abstract: Provides a unified, comprehensive and up-to-date treatment of both statistical and descriptive methods for pattern recognition. The topics treated include Bayesian decision theory, supervised and unsupervised learning, nonparametric techniques, discriminant analysis, clustering, preprosessing of pictorial data, spatial filtering, shape description techniques, perspective transformations, projective invariants, linguistic procedures, and artificial intelligence techniques for scene analysis.

13,647 citations

Book
01 Aug 1981
TL;DR: This chapter discusses Graphics, Image Processing, and Pattern Recognition, and the Reconstruction techniques used in this program, as well as some of the problems faced in implementing this program.
Abstract: 1: Introduction.- 1.1 Graphics, Image Processing, and Pattern Recognition.- 1.2 Forms of Pictorial Data.- 1.2.1 Class 1: Full Gray Scale and Color Pictures.- 1.2.2 Class 2: Bilevel or "Few Color" pictures.- 1.2.3 Class 3: Continuous Curves and Lines.- 1.2.4 Class 4: Points or Polygons.- 1.3 Pictorial Input.- 1.4 Display Devices.- 1.5 Vector Graphics.- 1.6 Raster Graphics.- 1.7 Common Primitive Graphic Instructions.- 1.8 Comparison of Vector and Raster Graphics.- 1.9 Pictorial Editor.- 1.10 Pictorial Transformations.- 1.11 Algorithm Notation.- 1.12 A Few Words on Complexity.- 1.13 Bibliographical Notes.- 1.14 Relevant Literature.- 1.15 Problems.- 2: Digitization of Gray Scale Images.- 2.1 Introduction.- 2.2 A Review of Fourier and other Transforms.- 2.3 Sampling.- 2.3.1 One-dimensional Sampling.- 2.3.2 Two-dimensional Sampling.- 2.4 Aliasing.- 2.5 Quantization.- 2.6 Bibliographical Notes.- 2.7 Relevant Literature.- 2.8 Problems.- Appendix 2.A: Fast Fourier Transform.- 3: Processing of Gray Scale Images.- 3.1 Introduction.- 3.2 Histogram and Histogram Equalization.- 3.3 Co-occurrence Matrices.- 3.4 Linear Image Filtering.- 3.5 Nonlinear Image Filtering.- 3.5.1 Directional Filters.- 3.5.2 Two-part Filters.- 3.5.3 Functional Approximation Filters.- 3.6 Bibliographical Notes.- 3.7 Relevant Literature.- 3.8 Problems.- 4: Segmentation.- 4.1 Introduction.- 4.2 Thresholding.- 4.3 Edge Detection.- 4.4 Segmentation by Region Growing.- 4.4.1 Segmentation by Average Brightness Level.- 4.4.2 Other Uniformity Criteria.- 4.5 Bibliographical Notes.- 4.6 Relevant Literature.- 4.7 Problems.- 5: Projections.- 5.1 Introduction.- 5.2 Introduction to Reconstruction Techniques.- 5.3 A Class of Reconstruction Algorithms.- 5.4 Projections for Shape Analysis.- 5.5 Bibliographical Notes.- 5.6 Relevant Literature.- 5.7 Problems.- Appendix 5.A: An Elementary Reconstruction Program.- 6: Data Structures.- 6.1 Introduction.- 6.2 Graph Traversal Algorithms.- 6.3 Paging.- 6.4 Pyramids or Quad Trees.- 6.4.1 Creating a Quad Tree.- 6.4.2 Reconstructing an Image from a Quad Tree.- 6.4.3 Image Compaction with a Quad Tree.- 6.5 Binary Image Trees.- 6.6 Split-and-Merge Algorithms.- 6.7 Line Encodings and the Line Adjacency Graph.- 6.8 Region Encodings and the Region Adjacency Graph.- 6.9 Iconic Representations.- 6.10 Data Structures for Displays.- 6.11 Bibliographical Notes.- 6.12 Relevant Literature.- 6.13 Problems.- Appendix 6.A: Introduction to Graphs.- 7: Bilevel Pictures.- 7.1 Introduction.- 7.2 Sampling and Topology.- 7.3 Elements of Discrete Geometry.- 7.4 A Sampling Theorem for Class 2 Pictures.- 7.5 Contour Tracing.- 7.5.1 Tracing of a Single Contour.- 7.5.2 Traversal of All the Contours of a Region.- 7.6 Curves and Lines on a Discrete Grid.- 7.6.1 When a Set of Pixels is not a Curve.- 7.6.2 When a Set of Pixels is a Curve.- 7.7 Multiple Pixels.- 7.8 An Introduction to Shape Analysis.- 7.9 Bibliographical Notes.- 7.10 Relevant Literature.- 7.11 Problems.- 8: Contour Filling.- 8.1 Introduction.- 8.2 Edge Filling.- 8.3 Contour Filling by Parity Check.- 8.3.1 Proof of Correctness of Algorithm 8.3.- 8.3.2 Implementation of a Parity Check Algorithm.- 8.4 Contour Filling by Connectivity.- 8.4.1 Recursive Connectivity Filling.- 8.4.2 Nonrecursive Connectivity Filling.- 8.4.3 Procedures used for Connectivity Filling.- 8.4.4 Description of the Main Algorithm.- 8.5 Comparisons and Combinations.- 8.6 Bibliographical Notes.- 8.7 Relevant Literature.- 8.8 Problems.- 9: Thinning Algorithms.- 9.1 Introduction.- 9.2 Classical Thinning Algorithms.- 9.3 Asynchronous Thinning Algorithms.- 9.4 Implementation of an Asynchronous Thinning Algorithm.- 9.5 A Quick Thinning Algorithm.- 9.6 Structural Shape Analysis.- 9.7 Transformation of Bilevel Images into Line Drawings.- 9.8 Bibliographical Notes.- 9.9 Relevant Literature.- 9.10 Problems.- 10: Curve Fitting and Curve Displaying.- 10.1 Introduction.- 10.2 Polynomial Interpolation.- 10.3 Bezier Polynomials.- 10.4 Computation of Bezier Polynomials.- 10.5 Some Properties of Bezier Polynomials.- 10.6 Circular Arcs.- 10.7 Display of Lines and Curves.- 10.7.1 Display of Curves through Differential Equations.- 10.7.2 Effect of Round-off Errors in Displays.- 10.8 A Point Editor.- 10.8.1 A Data Structure for a Point Editor.- 10.8.2 Input and Output for a Point Editor.- 10.9 Bibliographical Notes.- 10.10 Relevant Literature.- 10.11 Problems.- 11: Curve Fitting with Splines.- 11.1 Introduction.- 11.2 Fundamental Definitions.- 11.3 B-Splines.- 11.4 Computation with B-Splines.- 11.5 Interpolating B-Splines.- 11.6 B-Splines in Graphics.- 11.7 Shape Description and B-splines.- 11.8 Bibliographical Notes.- 11.9 Relevant Literature.- 11.10 Problems.- 12: Approximation of Curves.- 12.1 Introduction.- 12.2 Integral Square Error Approximation.- 12.3 Approximation Using B-Splines.- 12.4 Approximation by Splines with Variable Breakpoints.- 12.5 Polygonal Approximations.- 12.5.1 A Suboptimal Line Fitting Algorithm.- 12.5.2 A Simple Polygon Fitting Algorithm.- 12.5.3 Properties of Algorithm 12.2.- 12.6 Applications of Curve Approximation in Graphics.- 12.6.1 Handling of Groups of Points by a Point Editor.- 12.6.2 Finding Some Simple Approximating Curves.- 12.7 Bibliographical Notes.- 12.8 Relevant Literature.- 12.9 Problems.- 13: Surface Fitting and Surface Displaying.- 13.1 Introduction.- 13.2 Some Simple Properties of Surfaces.- 13.3 Singular Points of a Surface.- 13.4 Linear and Bilinear Interpolating Surface Patches.- 13.5 Lofted Surfaces.- 13.6 Coons Surfaces.- 13.7 Guided Surfaces.- 13.7.1 Bezier Surfaces.- 13.7.2 B-Spline Surfaces.- 13.8 The Choice of a Surface Partition.- 13.9 Display of Surfaces and Shading.- 13.10 Bibliographical Notes.- 13.11 Relevant Literature.- 13.12 Problems.- 14: The Mathematics of Two-Dimensional Graphics.- 14.1 Introduction.- 14.2 Two-Dimensional Transformations.- 14.3 Homogeneous Coordinates.- 14.3.1 Equation of a Line Defined by Two Points.- 14.3.2 Coordinates of a Point Defined as the Intersection of Two Lines.- 14.3.3 Duality.- 14.4 Line Segment Problems.- 14.4.1 Position of a Point with respect to a Line.- 14.4.2 Intersection of Line Segments.- 14.4.3 Position of a Point with respect to a Polygon.- 14.4.4 Segment Shadow.- 14.5 Bibliographical Notes.- 14.6 Relevant Literature.- 14.7 Problems.- 15: Polygon Clipping.- 15.1 Introduction.- 15.2 Clipping a Line Segment by a Convex Polygon.- 15.3 Clipping a Line Segment by a Regular Rectangle.- 15.4 Clipping an Arbitrary Polygon by a Line.- 15.5 Intersection of Two Polygons.- 15.6 Efficient Polygon Intersection.- 15.7 Bibliographical Notes.- 15.8 Relevant Literature.- 15.9 Problems.- 16: The Mathematics of Three-Dimensional Graphics.- 16.1 Introduction.- 16.2 Homogeneous Coordinates.- 16.2.1 Position of a Point with respect to a Plane.- 16.2.2 Intersection of Triangles.- 16.3 Three-Dimensional Transformations.- 16.3.1 Mathematical Preliminaries.- 16.3.2 Rotation around an Axis through the Origin.- 16.4 Orthogonal Projections.- 16.5 Perspective Projections.- 16.6 Bibliographical Notes.- 16.7 Relevant Literature.- 16.8 Problems.- 17: Creating Three-Dimensional Graphic Displays.- 17.1 Introduction.- 17.2 The Hidden Line and Hidden Surface Problems.- 17.2.1 Surface Shadow.- 17.2.2 Approaches to the Visibility Problem.- 17.2.3 Single Convex Object Visibility.- 17.3 A Quad Tree Visibility Algorithm.- 17.4 A Raster Line Scan Visibility Algorithm.- 17.5 Coherence.- 17.6 Nonlinear Object Descriptions.- 17.7 Making a Natural Looking Display.- 17.8 Bibliographical Notes.- 17.9 Relevant Literature.- 17.10 Problems.- Author Index.- Algorithm Index.

1,395 citations

Journal ArticleDOI
Urs Ramer1
TL;DR: An approximation algorithm is presented which uses an iterative method to produce polygons with a small—but not minimum—number of vertices that lie on the given curve that justifies the abandonment of the minimum-vertices criterion.

1,323 citations

Journal ArticleDOI
TL;DR: A new fast algorithm is proposed which allows for a variable number of segments iniecewise approximation as a way of feature extraction, data compaction, and noise filtering of boundaries of regions of pictures and waveforms.
Abstract: Piecewise approximation is described as a way of feature extraction, data compaction, and noise filtering of boundaries of regions of pictures and waveforms. A new fast algorithm is proposed which allows for a variable number of segments. After an arbitrary initial choice, segments are split or merged in order to drive the error norm under a prespecified bound. Results of computer experiments with cell outlines and electrocardiograms are reported.

589 citations