Topic
Line fitting
About: Line fitting is a research topic. Over the lifetime, 501 publications have been published within this topic receiving 9662 citations.
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Book•
1 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.
1,407 citations
4 Jun 2008
TL;DR: In this paper, a robust and real-time approach to lane marker detection in urban streets is presented, which is based on generating a top view of the road, filtering using selective oriented Gaussian filters, using RANSAC line fitting to give initial guesses to a new and fast RANAC algorithm for fitting Bezier Splines, which was then followed by a post-processing step.
Abstract: We present a robust and real time approach to lane marker detection in urban streets. It is based on generating a top view of the road, filtering using selective oriented Gaussian filters, using RANSAC line fitting to give initial guesses to a new and fast RANSAC algorithm for fitting Bezier Splines, which is then followed by a post-processing step. Our algorithm can detect all lanes in still images of the street in various conditions, while operating at a rate of 50 Hz and achieving comparable results to previous techniques.
814 citations
TL;DR: A robust and real time approach to lane marker detection in urban streets based on generating a top view of the road, filtering using selective oriented Gaussian filters, using RANSAC line fitting to give initial guesses to a new and fast RansAC algorithm for fitting Bezier Splines, which is then followed by a post-processing step.
Abstract: We present a robust and real time approach to lane marker detection in urban streets. It is based on generating a top view of the road, filtering using selective oriented Gaussian filters, using RANSAC line fitting to give initial guesses to a new and fast RANSAC algorithm for fitting Bezier Splines, which is then followed by a post-processing step. Our algorithm can detect all lanes in still images of the street in various conditions, while operating at a rate of 50 Hz and achieving comparable results to previous techniques.
782 citations
TL;DR: This procedure is an extension and improvement of the circle-finding concept sketched by Duda and Hart as an extension of the Hough straight-line finder.
Abstract: We describe an efficient procedure for detecting approximate circles and approximately circular arcs of varying gray levels in an edge-enhanced digitized picture. This procedure is an extension and improvement of the circle-finding concept sketched by Duda and Hart [2] as an extension of the Hough straight-line finder [6].
667 citations
TL;DR: The proposed PEaRL combines model sampling from data points as in RANSAC with iterative re-estimation of inliers and models’ parameters based on a global regularization functional and converges to a good quality local minimum of the energy automatically selecting a small number of models that best explain the whole data set.
Abstract: Geometric model fitting is a typical chicken-&-egg problem: data points should be clustered based on geometric proximity to models whose unknown parameters must be estimated at the same time. Most existing methods, including generalizations of RANSAC, greedily search for models with most inliers (within a threshold) ignoring overall classification of points. We formulate geometric multi-model fitting as an optimal labeling problem with a global energy function balancing geometric errors and regularity of inlier clusters. Regularization based on spatial coherence (on some near-neighbor graph) and/or label costs is NP hard. Standard combinatorial algorithms with guaranteed approximation bounds (e.g. ?-expansion) can minimize such regularization energies over a finite set of labels, but they are not directly applicable to a continuum of labels, e.g. ${\mathcal{R}}^{2}$ in line fitting. Our proposed approach (PEaRL) combines model sampling from data points as in RANSAC with iterative re-estimation of inliers and models' parameters based on a global regularization functional. This technique efficiently explores the continuum of labels in the context of energy minimization. In practice, PEaRL converges to a good quality local minimum of the energy automatically selecting a small number of models that best explain the whole data set. Our tests demonstrate that our energy-based approach significantly improves the current state of the art in geometric model fitting currently dominated by various greedy generalizations of RANSAC.
355 citations
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| Year | Papers |
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| 2022 | 1 |
| 2021 | 19 |
| 2020 | 23 |
| 2019 | 38 |
| 2018 | 27 |
| 2017 | 27 |