## On Edge Detection (1984)

Venue: | IEEE Transactions on Pattern Analysis and Machine Intelligence |

Citations: | 194 - 6 self |

### BibTeX

@ARTICLE{Torre84onedge,

author = {V. Torre and T. Poggio},

title = {On Edge Detection},

journal = {IEEE Transactions on Pattern Analysis and Machine Intelligence},

year = {1984},

volume = {8},

pages = {147--163}

}

### Years of Citing Articles

### OpenURL

### Abstract

Edge detection is the process that attempts to characterize the intensity changes in the image in terms of the physical processes that have originated them. A critical, intermediate goal of edge detection is the detection and characterization of significant intensity changes. This paper discusses this part of the edge d6tection problem. To characterize the types of intensity changes derivatives of different types, and possibly different scales, are needed. Thus, we consider this part of edge detection as a problem in numerical differentiation.

### Citations

956 | Theory of edge detection
- Marr, Hildreth
- 1980
(Show Context)
Citation Context ... they correspond to object boundaries or to changes in surface orientation or material properties (Ballard and Brown, 1982; Binford, 1981, 1982; Brady, 1981; Canny, 1983; Davis, 1975; Hildreth, 1980; =-=Marr and Hildreth, 1980-=-; Pavlidis, 1977; Rosenfeld and Kak, 1976). Three-dimensional edges are often mapped by the imaging process into critical points of the two-dimensional intensity profile formed in the eye or in a came... |

945 |
Solution of Ill-posed Problems
- Tikhonov, Arsenin
- 1977
(Show Context)
Citation Context ... Af(x) is the integral operator f()d, = h(x -- .)f()d: [2.2] where h is the step function. It is well known that inverse linear problems in which g(x) and f(;r) belong to Hilbert space are ill-posed (=-=Tikhonov and Arsenin, 1977-=-; Bertero, 1981). 2.2. Regularization techniques Rigorous methods for transforming ill-posed problems into well-posed problems have been developed over the past years (see especially Tikhonov, 1963; T... |

877 |
Computer Vision
- Ballard, Brown
- 1982
(Show Context)
Citation Context ...surfaces around the viewer). Physical edges are one of the most important properties of objects since they correspond to object boundaries or to changes in surface orientation or material properties (=-=Ballard and Brown, 1982-=-; Binford, 1981, 1982; Brady, 1981; Canny, 1983; Davis, 1975; Hildreth, 1980; Marr and Hildreth, 1980; Pavlidis, 1977; Rosenfeld and Kak, 1976). Three-dimensional edges are often mapped by the imaging... |

598 |
Uncertainty relation for resolution in space, spatial frequency, and orientation optimized by two-dimensional visual cortical filters
- Daugman
- 1985
(Show Context)
Citation Context ...e composed by Hermite functions with low . From the point of view of uncertainty, the optimal even filter is c': 'z-, and the optimal odd filter is ..v:' (the two-dimensional case has been treated by =-=Daugman, 1984-=-a). Anotler class of functions with small oncertainty consists of Gabor functions They are complex functions of a real variable and have uncertainty At; equal to . However, the real and imaginary part... |

567 | Scale-space filtering - Witkin - 1983 |

366 |
A computational theory of human stereo vision
- Marr, Poggio
- 1979
(Show Context)
Citation Context ...everal scales of filtering was realized quite early on, and this was supported by evidence suggesting the presence of filters of several sizes in the human visual system (Rosenreid, 1982; Marr, 1976; =-=Marr and Poggio, 1979-=-; Marr and Hildreth, 1980). More recently, Witkin (1983; see also Stansfield, 1980) introduced a scale-space description of zero-crossings which gives the position of the zero.crossing across a contin... |

328 |
Interpolation and Approximation
- DAVIS
- 1975
(Show Context)
Citation Context ...t properties of objects since they correspond to object boundaries or to changes in surface orientation or material properties (Ballard and Brown, 1982; Binford, 1981, 1982; Brady, 1981; Canny, 1983; =-=Davis, 1975-=-; Hildreth, 1980; Marr and Hildreth, 1980; Pavlidis, 1977; Rosenfeld and Kak, 1976). Three-dimensional edges are often mapped by the imaging process into critical points of the two-dimensional intensi... |

203 | The Curvature Primal Sketch
- Asada, Brady
- 1983
(Show Context)
Citation Context ...mitives, that capture the whole information in the signal and characterize it uniquely. Subsequent processes can therefore work on this more compact representation instead of the original signal (see =-=Asada and Brady, 1984-=-). The second theorem has theoretical interest in that it answers the question of what information is conveyed by the edges identified with zero- and level-crossings of multiscale Gaussian fiitered si... |

195 | Entire Functions - Boas - 1954 |

180 | Contributions to the problem of approximation of equidistant data by analytic functions—part B: on the problem of osculatory interpolation. A second class of analytic approximation formulae - Schoenberg - 1946 |

153 | T.A.: Scaling theorems for zero crossings - Yuille, Poggio - 1986 |

146 | Uniqueness of the gaussian kernel for scale-space filtering - Babaud, Witkin, et al. - 1986 |

140 |
Structural Pattern Recognition
- Pavlidis
- 1977
(Show Context)
Citation Context ...t boundaries or to changes in surface orientation or material properties (Ballard and Brown, 1982; Binford, 1981, 1982; Brady, 1981; Canny, 1983; Davis, 1975; Hildreth, 1980; Marr and Hildreth, 1980; =-=Pavlidis, 1977-=-; Rosenfeld and Kak, 1976). Three-dimensional edges are often mapped by the imaging process into critical points of the two-dimensional intensity profile formed in the eye or in a camera. The ultimate... |

130 |
The Fourier Integral and its Applications
- Papoulis, Athanasios
- 1962
(Show Context)
Citation Context ...here +oo --2 2 X -- f_ (x-- x) J (x)dx [3.10] n ' = ("' - [3.12] 1,'() is the Fourier transform of /() and Notice that )" is proportional to the density of of zero-crossings for Gaussian whi=-=te noise (Papoulis, 1962-=-; Papoulis, 1965, p. 487). It is well.known that the Gaussian function r'; - is the real function / L /?(!)') that minimizes the uncertainty All. On these grounds it has been proposed by Marr & Hildre... |

115 |
Quelques proprietes globales des varietes dierentiables
- Thom
- 1954
(Show Context)
Citation Context ...pace whose dimension is as small as possible. From this definition, it follows that the surface ,5'j. - (:c, .v, .f(x, !/)) meets the surface So -- (x., in (, .) Igrad/ls0. [5.2] The isotopy theorem (=-=Thom, 1954-=-) shows that transversal intersections are structurally stable. The converse is also true in that non-transversal intersections are structurally unstable. Transversality (and the implicit function the... |

97 |
Transversal Mappings and Flows
- Abraham, Robbin
- 1967
(Show Context)
Citation Context ...oints defined by 2O Torre, Poggio On Edge Detection where )(;,.v) is the filtered image and /)'* can be a RID or a DD operator. We first recall in the next two sections the notions of transversaltry (=-=Abraham and Robbin, 1967-=-, Poston and Stewart, 1976) and of Morse functions (Poston and Stewart, 1976). In section 5.4 we will classify the types of zero-crossings contours that can appear in images. 5.1. Transversality and z... |

87 |
Inferring Surfaces from Images
- Binford
- 1981
(Show Context)
Citation Context ...r). Physical edges are one of the most important properties of objects since they correspond to object boundaries or to changes in surface orientation or material properties (Ballard and Brown, 1982; =-=Binford, 1981-=-, 1982; Brady, 1981; Canny, 1983; Davis, 1975; Hildreth, 1980; Marr and Hildreth, 1980; Pavlidis, 1977; Rosenfeld and Kak, 1976). Three-dimensional edges are often mapped by the imaging process into c... |

69 | Spline Functions and the Problem of Graduation - Schoenberg - 1964 |

49 |
Survey of Model-Based Image Analysis Systems
- Binford
- 1982
(Show Context)
Citation Context ...o the use of a directional operator but to the distortion introduced by a too large width of the operator. The concomitant use of several directional derivatives has been proposed by several authors (=-=Binford 1982-=-; Canny 1983). Since in . the directional derivative in any arbitrary direction can be expressed in terms of 7 and ,,, it is evident that in a noise-free image the use of more than two-directional der... |

48 | Theory of Approximation - Achieser - 1956 |

45 | Discrete representation of signals - Oppenheim, Johnson - 1972 |

29 |
Differential Geometry
- Lipschutz
- 1969
(Show Context)
Citation Context ... X = (% y, z), where z ----- S(x, y) with f E U'(D), Ds' and r > 2. The mean curvature of the surface X is where t=l+f F=ff, G=l+f, [4.7] are the coefficients of the first fundamental form l(dx, dy) (=-=Lipschutz, 1969, Pog-=-orelov, 1965), and L-- f M- f N--- f--Y-, [4.7] g g g with " = I 4-/ +/, are the coefficients of the second fundamental form l/(dx, d.v). We use equations [4.2], [4.3] and [4.8] and the property ... |

22 |
l~inding Edges and Lines
- Canny
- 1983
(Show Context)
Citation Context ...most important properties of objects since they correspond to object boundaries or to changes in surface orientation or material properties (Ballard and Brown, 1982; Binford, 1981, 1982; Brady, 1981; =-=Canny, 1983-=-; Davis, 1975; Hildreth, 1980; Marr and Hildreth, 1980; Pavlidis, 1977; Rosenfeld and Kak, 1976). Three-dimensional edges are often mapped by the imaging process into critical points of the two-dimens... |

22 | ªThe Structure of TwoDimensional Scalar Fields with Applications to Vision,º - KoeÈnderink, Doorn - 1979 |

21 | Computational approaches to image understanding - Brady - 1982 |

20 | Edge and region analysis for digital image data - HARALICK - 1980 |

20 |
Some asymptotic expansions for prolate spheroidal wave functions
- Slepian
- 1965
(Show Context)
Citation Context ... the former are band-limited and fall as :, while the latter fall off faster--somewhat too fast to be band-lirnited It has been shown, however, that a crude approximation of O,,l), whensis large (see =-=Slepian 1965-=-) is where D,, is a Weber parabolic cylinder function. Now [3.21 [3.22] where II,,(:r) are the usual Hermite polynomials. This approximation fails for large ao, where /,.,,(:r) falls off assand I),,(:... |

11 |
Six formal properties of two-dimensional anisotropic visual filters: Structural principles and frequency/orientation selectivity
- Daugman
- 1983
(Show Context)
Citation Context ...e composed by Hermite functions with low . From the point of view of uncertainty, the optimal even filter is c': 'z-, and the optimal odd filter is ..v:' (the two-dimensional case has been treated by =-=Daugman, 1984-=-a). Anotler class of functions with small oncertainty consists of Gabor functions They are complex functions of a real variable and have uncertainty At; equal to . However, the real and imaginary part... |

7 | An optimal frequency domain filter for edge detection in digital pictures - Shanmugam, Dickey, et al. - 1979 |

6 | On Boundary Detection - Herskovitz, Binford - 1970 |

5 |
Expansion and Catastrophes
- Poston, Stewart, et al.
- 1976
(Show Context)
Citation Context ... Poggio On Edge Detection where )(;,.v) is the filtered image and /)'* can be a RID or a DD operator. We first recall in the next two sections the notions of transversaltry (Abraham and Robbin, 1967, =-=Poston and Stewart, 1976-=-) and of Morse functions (Poston and Stewart, 1976). In section 5.4 we will classify the types of zero-crossings contours that can appear in images. 5.1. Transversality and zero-crossing (z.c.) A curv... |

4 | The Theory of Splines and their - Ahlberg, Nilson, et al. - 1967 |

4 | The interpretation of geometric structure from image boundaries - Lowe, Binford - 1981 |

4 | The asymptotic Optimal Frequency Domain Filter for Edge Detection - Lunscher - 1983 |

4 |
Conclusions from the commodity expert project
- Stansfield
- 1980
(Show Context)
Citation Context ...nce suggesting the presence of filters of several sizes in the human visual system (Rosenreid, 1982; Marr, 1976; Marr and Poggio, 1979; Marr and Hildreth, 1980). More recently, Witkin (1983; see also =-=Stansfield, 1980-=-) introduced a scale-space description of zero-crossings which gives the position of the zero.crossing across a continuum of scales, i.e., sizes of the Gaussian filter (parametrized by thesof the Gaus... |

2 | Linear and circular prolate functions - Frieden - 1971 |

2 | The digital edge - Haralick - 1981 |

2 |
Prolate spherical wave functions, Fourier analysis and uncertainty-It
- Landau, Pollack
- 1961
(Show Context)
Citation Context ...s where D,,, is the operator defined as These results show clearly the difference between ff, o() nd sinc(:). They are both band-limited, but ,,(:c) f11s off more rapidly than sinc(::) (see Fig. 2 of =-=Landau and Pollack 1961-=-). On the other hand, the strictly support-limited function, which has the minimal spread of frequencies, is not a Haar function or a Difference Of Boxes filter (see later) but is Oscillations in the ... |

2 |
Regularization of incorrectly-posed problems
- Tikhonov
- 1963
(Show Context)
Citation Context ...and Arsenin, 1977; Bertero, 1981). 2.2. Regularization techniques Rigorous methods for transforming ill-posed problems into well-posed problems have been developed over the past years (see especially =-=Tikhonov, 1963-=-; Tikhonov and Arsenin, 1977; and Nashed, t974, 1976). Regularization of the ill-posed problem of finding z from the data y, such that A: := ,; requires the choice of suitable norms il'II, (usually qu... |

2 |
The heat equation, Acad
- Widder, V
- 1975
(Show Context)
Citation Context ...a solution at an appropriate time t == - of the two dimensional heat equation Ox -- + iysOr' [3.24] with the initial condition: y, o) = y). This is because the "source solution" of the heat =-=equation (Widder, 1975-=-) is -(2 y,t) - c ,, [3.26] with time playing the role of the variance, that is, a 2 = 2t. [3.27] From Theorem 4.1 of Widder (1975) the solutions of the heat equation are entire functions ofsand #. In... |

1 | Accuracy of Laplacian edge detectors - Bet, V - 1984 |

1 | Correlation and Convolution algebras - Borsellino, Poggio - 1973 |

1 | On Edge Detection Davis, L. "A survey of edge detection techniques - Torre - 1975 |

1 | Introduction to spline functions", in Theory and Applications of Spline Functions - Greville - 1969 |

1 | Transmission of information by orthogonal functions - Hermuth - 1972 |

1 | Implementation of a theory of edge detection - Flildreth - 1980 |

1 |
The Binford-Horn edge finder
- Horn
- 1972
(Show Context)
Citation Context ...terature. Our review is neither intended to be exhaustive nor does it aim to present edge detectors in full detail. 7.1. Difference of boxes (DOB) Binford and coworkers (Herskovitz and Binford, 1970; =-=Horn, 1972-=-; Binford, 1981) have suggested the use of simpport-limited filters in the filtering step of edge detection. They have used the Haar function [3.8D'] in directional filtering or a difference of functi... |

1 |
Elements de la theode des functions et de I'analse fonctionelle
- Kolmogorov, Fomine
- 1974
(Show Context)
Citation Context ...tion This filter has already been proposed by Herskovitz and Binford (1970) and is commonly called DOE] (Difference of E]oxes). It is also a Haar function (see Fig.19 of Flarmuth 1972 and page 399 of =-=Kolmogorov and Fomine, 1974-=-). The system of Haar functions is complete and constitutes a basis for all square integrable functions on a bounded interval. This property may have some relevance in the context of image processing ... |

1 | A regularized solution to edge detection", in preparation - Poggio, Voorhees, et al. - 1984 |

1 |
Quadtrees and Pyramids: hierarchical representation of images
- Rosenreid
- 1982
(Show Context)
Citation Context ...on, the advantages of using several scales of filtering was realized quite early on, and this was supported by evidence suggesting the presence of filters of several sizes in the human visual system (=-=Rosenreid, 1982-=-; Marr, 1976; Marr and Poggio, 1979; Marr and Hildreth, 1980). More recently, Witkin (1983; see also Stansfield, 1980) introduced a scale-space description of zero-crossings which gives the position o... |