## Edge Detection with Embedded Confidence (2001)

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Venue: | IEEE Trans. Pattern Anal. Machine Intell |

Citations: | 70 - 1 self |

### BibTeX

@ARTICLE{Meer01edgedetection,

author = {Peter Meer and Bogdan Georgescu},

title = {Edge Detection with Embedded Confidence},

journal = {IEEE Trans. Pattern Anal. Machine Intell},

year = {2001},

volume = {23},

pages = {1351--1365}

}

### Years of Citing Articles

### OpenURL

### Abstract

Computing the weighted average of the pixel values in a window is a basic module in many computer vision operators. The process is reformulated in a linear vector space and the role of the different subspaces is emphasized. Within this framework well known artifacts of the gradient based edge detectors, such as large spurious responses can be explained quantitatively. It is also shown that template matching with a template derived from the input data is meaningful since it provides an independent measure of confidence in the presence of the employed edge model. The widely used three-step edge detection procedure: gradient estimation, nonmaxima suppression, hysteresis thresholding; is generalized to include the information provided by the confidence measure. The additional amount of computation is minimal and experiments with several standard test images show the ability of the new procedure to detect weak edges. Keywords: edge detection, performance assessment, gradient estimation, window operators 1

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Citation Context ...te manifold, however, the two-dimensional subspace of the gradient operator will be used. Most papers in the vision literature treat the optimality of image differentiation (and edge detection) e.g., =-=[1, 17, 31]-=-, as well as the arising artifacts e.g., [6], in the continuous domain. The discrete nature of the input was also not taken into account when linear differentiation operators were combined with Boolea... |

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Citation Context ...mated discontinuity. Hummel [15] extended the approach by deriving the basis functions from the Karhunen-Lo eve expansion of the local image structure. The edge detector proposed by Nalwa and Binford =-=[24]-=- also made extensive use of model fitting, after an initial edge hypothesis was obtained from the gradient. The 1D profile of the edge was refined by a sequence of linear (cubic and quadratic polynomi... |

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Citation Context ...rs extended the method beyond the original 3 \Theta 3 window or significantly modified the original idea. The recently proposed parametric eigenspace based feature detection technique of Baker et al. =-=[3]-=- belongs to both the model fitting and the vector space paradigms. A set of templates is represented in the subspace of their most significant eigenvectors as a manifold parametrized by the variables ... |

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Citation Context ...ine such a confidence measure and integrate it into all three steps of the edge detection procedure. The confidence measure is based on two ideas popular for edge detection in the late 1970s. Hueckel =-=[14]-=- was probably the first in the vision literature to use least squares fitting of an ideal 2D step-edge model to the data. The estimation process was implemented with orthogonal basis functions and the... |

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Citation Context ...edge model to the data. The estimation process was implemented with orthogonal basis functions and the presence of an edge was determined based on the step-size of the estimated discontinuity. Hummel =-=[15]-=- extended the approach by deriving the basis functions from the Karhunen-Lo eve expansion of the local image structure. The edge detector proposed by Nalwa and Binford [24] also made extensive use of ... |

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Citation Context ...tained only if they already have at least one neighbor in the edge map. This step is repeated till convergence. The last two steps (postprocessing) are critical for the quality of the edge map, e.g., =-=[8]-=-, and the gradient based edge detectors in the literature differ mostly through the details of the postprocessing [13]. The above described edge detection procedure uses the magnitude of the gradient ... |

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Citation Context ...te manifold, however, the two-dimensional subspace of the gradient operator will be used. Most papers in the vision literature treat the optimality of image differentiation (and edge detection) e.g., =-=[1, 17, 31]-=-, as well as the arising artifacts e.g., [6], in the continuous domain. The discrete nature of the input was also not taken into account when linear differentiation operators were combined with Boolea... |

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Smoothed differentiation filters for images
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Citation Context ...pressions in these coefficients. If orthogonal polynomials defined over a discrete interval are employed, all the computational steps can be replaced by an a priori computed differentiation mask. See =-=[23]-=- for a detailed technical presentation, and Appendix B for a short summary. A large family of differentiation masks are separable, the weights being obtained from the outer product of two one-dimensio... |

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Citation Context ...eration in low level computer vision with a plethora of techniques, belonging to several distinct paradigms, having been published. See, for example, [4] for an extensive review of older methods, and =-=[2, 13]-=- for the current state-of-the-art. The optimality of an edge detector, however, can only be assessed in the context of a well defined task [29]. That is, the quality of the edge map is directly relate... |

16 |
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Citation Context ...f the gradient operator will be used. Most papers in the vision literature treat the optimality of image differentiation (and edge detection) e.g., [1, 17, 31], as well as the arising artifacts e.g., =-=[6]-=-, in the continuous domain. The discrete nature of the input was also not taken into account when linear differentiation operators were combined with Boolean logic to validate the extracted local stru... |

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Citation Context ...uous domain. The discrete nature of the input was also not taken into account when linear differentiation operators were combined with Boolean logic to validate the extracted local structure [16]. In =-=[7]-=- Canny's continuous optimization criteria were translated into the discrete domain to introduce an optimal discrete filter. Such an approach, however, is not equivalent with analyzing the behavior of ... |

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Citation Context ...ocal feature extraction processes. APPENDIX A COMPENDIUM ON MATRICES In this appendix, the matrix properties employed throughout the paper are reviewed. For more background on introductory topics see =-=[30]-=-, on advanced topics [10], and on matrix calculus [11]. Let A be an n p matrix having rank r. Without loss of generality will assume r p n. The trace of the matrix is trace‰AŠ ˆ Xp iˆ1 aii ˆ Xr iˆ1 i;... |

7 |
A Three-Module Strategy for Edge Detection
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Citation Context ...carried by the local pattern is employed, the influence of the amplitude of the discontinuity is eliminated in the angle computation. The vector space approach of Frei and Chen was also considered in =-=[18, 22, 26]-=- but none of these papers extended the method beyond the original 3 \Theta 3 window or significantly modified the original idea. The recently proposed parametric eigenspace based feature detection tec... |

6 |
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Citation Context ...h an approach, however, is not equivalent with analyzing the behavior of the operator in 3 the discrete domain. Only rarely are low-level vision operators defined directly on the sampling grid, e.g., =-=[20]-=-. In this paper edge detection is approached exclusively in the discrete domain as an operation over data defined on the regular sampling lattice. The paper is organized as follows. In Section 2 the c... |

5 |
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Citation Context ...dge map output is then defined on the same sampling lattice as the input. Subpixel accuracy (if desired) can be achieved by analyzing the gray level values in the neighborhood of an edge pixel, e.g., =-=[19]-=-. However, to assure that the edge pixels are correctly located, it is of interest to investigate the influence of an offset on the estimated gradient vector and on the confidence measuresj, (6). To g... |

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Citation Context ...n Edge detection is arguably the most important operation in low level computer vision with a plethora of techniques, belonging to several distinct paradigms, having been published. See, for example, =-=[4]-=- for an extensive review of older methods, and [2, 13] for the current state-of-the-art. The optimality of an edge detector, however, can only be assessed in the context of a well defined task [29]. T... |

4 |
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Citation Context ...te manifold, however, the two-dimensional subspace of the gradient operator will be used. Most papers in the vision literature treat the optimality of image differentiation (and edge detection) e.g., =-=[1, 17, 31]-=-, as well as the arising artifacts e.g., [6], in the continuous domain. The discrete nature of the input was also not taken into account when linear differentiation operators were combined with Boolea... |

4 |
An operator which locates edges in digital pictures,"JACM
- Hueckel
- 1971
(Show Context)
Citation Context ...ine such a confidence measure and integrate it into all three steps of the edge detection procedure. The confidence measure is based on two ideas popular for edge detection in the late 1970s. Hueckel =-=[14]-=- was probably the first in the vision literature to use least-squares fitting of an ideal 2D step-edge model to the data. The estimation process was implemented with orthogonal basis functions and the... |

2 |
Edge detection by associative mapping
- Meer, Wang, et al.
- 1989
(Show Context)
Citation Context ...carried by the local pattern is employed, the influence of the amplitude of the discontinuity is eliminated in the angle computation. The vector space approach of Frei and Chen was also considered in =-=[18, 22, 26]-=- but none of these papers extended the method beyond the original 3 \Theta 3 window or significantly modified the original idea. The recently proposed parametric eigenspace based feature detection tec... |

2 |
Comments on ‘A three-module strategy for edge detection
- Park, Choi
- 1990
(Show Context)
Citation Context ...carried by the local pattern is employed, the influence of the amplitude of the discontinuity is eliminated in the angle computation. The vector space approach of Frei and Chen was also considered in =-=[18, 22, 26]-=- but none of these papers extended the method beyond the original 3 \Theta 3 window or significantly modified the original idea. The recently proposed parametric eigenspace based feature detection tec... |

2 |
ªPerformance Assessment through Bootstrap,º
- Cho, Meer, et al.
- 1997
(Show Context)
Citation Context ...se (and many other) images obtained with all the state-of-the-art techniques. The cameraman is probably one of the oldest test images in the field, being often used in edge detection papers including =-=[5]-=- in a context similar to the present work. The 512 512 basket image (Fig. 8) was processed with a 7 7 gradient operator. (In all the experiments the data is weighted with binomial weights.) The image ... |

2 |
ªOn Detecting Edges,º
- Nalwa, Binford
- 1986
(Show Context)
Citation Context ...mated discontinuity. Hummel [15] extended the approach by deriving the basis functions from the Karhunen-LoeÁve expansion of the local image structure. The edge detector proposed by Nalwa and Binford =-=[24]-=- also made extensive use of model fitting, after an initial edge hypothesis was obtained from the gradient. The 1D profile of the edge was refined by a sequence of linear (cubic and quadratic polynomi... |

1 |
ªConsistent Gradient Operators,º
- Ando
- 2000
(Show Context)
Citation Context ...te manifold, however, the two-dimensional subspace of the gradient operator will be used. Most papers in the vision literature treat the optimality of image differentiation (and edge detection) e.g., =-=[1]-=-, [17], [31], as well as the arising artifacts e.g., [6], in the continuous domain. The discrete nature of the input was also not taken into account when linear differentiation operators were combined... |

1 |
ªParametric Feature Detection,º Int'l
- Baker, Nayar, et al.
- 1998
(Show Context)
Citation Context ...se papers extended the method beyond the original 3 3 window or significantly modified the original idea. The recently proposed parametric eigenspace-based feature detection technique of Baker et al. =-=[3]-=- belongs to both the model fitting and the vector space paradigms. A set of templates is represented in the subspace of their most significant eigenvectors as a manifold parametrized by the variables ... |

1 |
ªLocalization and Noise in Edge Detection,º
- Micheli, Caprile, et al.
- 1989
(Show Context)
Citation Context ...e gradient operator will be used. Most papers in the vision literature treat the optimality of image differentiation (and edge detection) e.g., [1], [17], [31], as well as the arising artifacts e.g., =-=[6]-=-, in the continuous domain. The discrete nature of the input was also not taken into account when linear differentiation operators were combined with Boolean logic to validate the extracted local stru... |

1 |
ªSome Defects in Finite-Difference Edge Finders,º
- Fleck
- 1992
(Show Context)
Citation Context ...tained only if they already have at least one neighbor in the edge map. This step is repeated till convergence. The last two steps (postprocessing) are critical for the quality of the edge map, e.g., =-=[8]-=- and the gradient-based edge detectors in the literature differ mostly through the details of the postprocessing [13]. The above described edge detection procedure uses the magnitude of the gradient v... |

1 |
ªFeature Detection Using Basis Functions,º Computer
- Hummel
- 1979
(Show Context)
Citation Context ...edge model to the data. The estimation process was implemented with orthogonal basis functions and the presence of an edge was determined based on the step-size of the estimated discontinuity. Hummel =-=[15]-=- extended the approach by deriving the basis functions from the Karhunen-LoeÁve expansion of the local image structure. The edge detector proposed by Nalwa and Binford [24] also made extensive use of ... |

1 |
ªLogical/Linear Operators for Image Curves,º
- Iverson, Zucker
- 1995
(Show Context)
Citation Context ...he continuous domain. The discrete nature of the input was also not taken into account when linear differentiation operators were combined with Boolean logic to validate the extracted local structure =-=[16]-=-. In [7], Canny's continuous optimization criteria were translated into the discrete domain to introduce an optimal discrete filter. Such an approach, however, is not equivalent with analyzing the beh... |

1 |
ªA Three-Module Strategy for Edge Detection,º
- Lacroix
- 1988
(Show Context)
Citation Context ...carried by the local pattern is employed, the influence of the amplitude of the discontinuity is eliminated in the angle computation. The vector space approach of Frei and Chen was also considered in =-=[18]-=-, [22], [26] but none of these papers extended the method beyond the original 3 3 window or significantly modified the original idea. The recently proposed parametric eigenspace-based feature detectio... |

1 |
ªEdge Detection by Associative Mapping,º
- Meer, Wang, et al.
- 1989
(Show Context)
Citation Context ...d by the local pattern is employed, the influence of the amplitude of the discontinuity is eliminated in the angle computation. The vector space approach of Frei and Chen was also considered in [18], =-=[22]-=-, [26] but none of these papers extended the method beyond the original 3 3 window or significantly modified the original idea. The recently proposed parametric eigenspace-based feature detection tech... |

1 |
ªSmoothed Differentiation Filters for Images,º
- Meer, Weiss
- 1992
(Show Context)
Citation Context ...pressions in these coefficients. If orthogonal polynomials defined over a discrete interval are employed, all the computational steps can be replaced by an a priori computed differentiation mask. See =-=[23]-=- for a detailed technical presentation and Appendix B for a short summary. A large family of differentiation masks are separable, the weights being obtained from the outer product of two onedimensiona... |

1 |
ªComments on `A Three-Module Strategy for Edge Detection',º
- Park, Choi
- 1990
(Show Context)
Citation Context ...he local pattern is employed, the influence of the amplitude of the discontinuity is eliminated in the angle computation. The vector space approach of Frei and Chen was also considered in [18], [22], =-=[26]-=- but none of these papers extended the method beyond the original 3 3 window or significantly modified the original idea. The recently proposed parametric eigenspace-based feature detection technique ... |

1 |
ªOn Edge Detection,º
- Torre, Poggio
- 1986
(Show Context)
Citation Context ..., however, the two-dimensional subspace of the gradient operator will be used. Most papers in the vision literature treat the optimality of image differentiation (and edge detection) e.g., [1], [17], =-=[31]-=-, as well as the arising artifacts e.g., [6], in the continuous domain. The discrete nature of the input was also not taken into account when linear differentiation operators were combined with Boolea... |