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Journal ArticleDOI

Theory of Edge Detection

29 Feb 1980-Proceedings of The Royal Society B: Biological Sciences (Proc R Soc Lond B Biol Sci)-Vol. 207, Iss: 1167, pp 187-217
TL;DR: The theory of edge detection explains several basic psychophysical findings, and the operation of forming oriented zero-crossing segments from the output of centre-surround ∇2G filters acting on the image forms the basis for a physiological model of simple cells.
Abstract: A theory of edge detection is presented. The analysis proceeds in two parts. (1) Intensity changes, which occur in a natural image over a wide range of scales, are detected separately at different scales. An appropriate filter for this purpose at a given scale is found to be the second derivative of a Gaussian, and it is shown that, provided some simple conditions are satisfied, these primary filters need not be orientation-dependent. Thus, intensity changes at a given scale are best detected by finding the zero values of delta 2G(x,y)*I(x,y) for image I, where G(x,y) is a two-dimensional Gaussian distribution and delta 2 is the Laplacian. The intensity changes thus discovered in each of the channels are then represented by oriented primitives called zero-crossing segments, and evidence is given that this representation is complete. (2) Intensity changes in images arise from surface discontinuities or from reflectance or illumination boundaries, and these all have the property that they are spatially. Because of this, the zero-crossing segments from the different channels are not independent, and rules are deduced for combining them into a description of the image. This description is called the raw primal sketch. The theory explains several basic psychophysical findings, and the operation of forming oriented zero-crossing segments from the output of centre-surround delta 2G filters acting on the image forms the basis for a physiological model of simple cells (see Marr & Ullman 1979).
Citations
More filters
Journal ArticleDOI
TL;DR: There is a natural uncertainty principle between detection and localization performance, which are the two main goals, and with this principle a single operator shape is derived which is optimal at any scale.
Abstract: This paper describes a computational approach to edge detection. The success of the approach depends on the definition of a comprehensive set of goals for the computation of edge points. These goals must be precise enough to delimit the desired behavior of the detector while making minimal assumptions about the form of the solution. We define detection and localization criteria for a class of edges, and present mathematical forms for these criteria as functionals on the operator impulse response. A third criterion is then added to ensure that the detector has only one response to a single edge. We use the criteria in numerical optimization to derive detectors for several common image features, including step edges. On specializing the analysis to step edges, we find that there is a natural uncertainty principle between detection and localization performance, which are the two main goals. With this principle we derive a single operator shape which is optimal at any scale. The optimal detector has a simple approximate implementation in which edges are marked at maxima in gradient magnitude of a Gaussian-smoothed image. We extend this simple detector using operators of several widths to cope with different signal-to-noise ratios in the image. We present a general method, called feature synthesis, for the fine-to-coarse integration of information from operators at different scales. Finally we show that step edge detector performance improves considerably as the operator point spread function is extended along the edge.

28,073 citations


Cites background or methods from "Theory of Edge Detection"

  • ...This is similar to the selection criterion proposed by Marr and Hildreth [18] for choosing between different Laplacian of Gaussian channels....

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  • ...In fact, a one-dimensional Marr-Hildreth edge detector is almost identical with the operator we have derived because maxima in the output of a first derivative operator will correspond to zero-crossings in the Laplacian operator as used by Marr and Hildreth....

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  • ...The effect of the window function becomes very marked for large operator sizes and it is probably the biggest single reason why operators with large support were not practical until the work of Marr and Hildreth on the Laplacian of Gaussian....

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  • ...Directional operators very much like the ones we have derived were suggested by Marr [17], but were discarded in favor of the Laplacian of Gaussian [18]....

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  • ...There are also strong links with the Laplacian of Gaussian operator suggested by Marr and Hildreth [18]....

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Journal ArticleDOI
TL;DR: This work uses snakes for interactive interpretation, in which user-imposed constraint forces guide the snake near features of interest, and uses scale-space continuation to enlarge the capture region surrounding a feature.
Abstract: A snake is an energy-minimizing spline guided by external constraint forces and influenced by image forces that pull it toward features such as lines and edges. Snakes are active contour models: they lock onto nearby edges, localizing them accurately. Scale-space continuation can be used to enlarge the capture region surrounding a feature. Snakes provide a unified account of a number of visual problems, including detection of edges, lines, and subjective contours; motion tracking; and stereo matching. We have used snakes successfully for interactive interpretation, in which user-imposed constraint forces guide the snake near features of interest.

18,095 citations

Book ChapterDOI
07 May 2006
TL;DR: A novel scale- and rotation-invariant interest point detector and descriptor, coined SURF (Speeded Up Robust Features), which approximates or even outperforms previously proposed schemes with respect to repeatability, distinctiveness, and robustness, yet can be computed and compared much faster.
Abstract: In this paper, we present a novel scale- and rotation-invariant interest point detector and descriptor, coined SURF (Speeded Up Robust Features). It approximates or even outperforms previously proposed schemes with respect to repeatability, distinctiveness, and robustness, yet can be computed and compared much faster. This is achieved by relying on integral images for image convolutions; by building on the strengths of the leading existing detectors and descriptors (in casu, using a Hessian matrix-based measure for the detector, and a distribution-based descriptor); and by simplifying these methods to the essential. This leads to a combination of novel detection, description, and matching steps. The paper presents experimental results on a standard evaluation set, as well as on imagery obtained in the context of a real-life object recognition application. Both show SURF's strong performance.

13,011 citations

Journal ArticleDOI
TL;DR: A review of recent as well as classic image registration methods to provide a comprehensive reference source for the researchers involved in image registration, regardless of particular application areas.

6,842 citations


Cites methods from "Theory of Edge Detection"

  • ...Dreschler and Nagel [43] searched for the local extrema of the Gaussian curvature....

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  • ...Kumar et al. [102] combined different types of pyramids (Laplacian, Gaussian) with different similarity measures (CC, sum of squared differences) to register aerial video sequences....

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  • ...Multiquadrics, reciprocal multiquadrics, Gaussians, Wendland’s functions, and thin-plate splines are several examples of the radial basis functions used in image registration....

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  • ...In general, this coarse-to-fine hierarchical strategy applies the usual registration methods, but it starts with the reference and sensed images on a coarse resolution (generated using Gaussian pyramids, simple averaging or wavelet transform coefficients, among others)....

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  • ...6), quadratic splines [42,191], cubic Bsplines [89], higher-order B-splines [108], Catmull–Rom cardinal splines [100,184], Gaussians [7], and truncated sinc functions [182] belong to the most commonly used interpolants....

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Journal ArticleDOI
TL;DR: This paper investigates two fundamental problems in computer vision: contour detection and image segmentation and presents state-of-the-art algorithms for both of these tasks.
Abstract: This paper investigates two fundamental problems in computer vision: contour detection and image segmentation. We present state-of-the-art algorithms for both of these tasks. Our contour detector combines multiple local cues into a globalization framework based on spectral clustering. Our segmentation algorithm consists of generic machinery for transforming the output of any contour detector into a hierarchical region tree. In this manner, we reduce the problem of image segmentation to that of contour detection. Extensive experimental evaluation demonstrates that both our contour detection and segmentation methods significantly outperform competing algorithms. The automatically generated hierarchical segmentations can be interactively refined by user-specified annotations. Computation at multiple image resolutions provides a means of coupling our system to recognition applications.

5,068 citations


Cites background or methods from "Theory of Edge Detection"

  • ...Marr and Hildreth [20] use zero crossings of the Laplacian of Gaussian operator....

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  • ...Our gPb detector [3] performs significantly better than other algorithms [2], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28] across almost the entire operating regime....

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References
More filters
Journal ArticleDOI
TL;DR: This method is used to examine receptive fields of a more complex type and to make additional observations on binocular interaction and this approach is necessary in order to understand the behaviour of individual cells, but it fails to deal with the problem of the relationship of one cell to its neighbours.
Abstract: What chiefly distinguishes cerebral cortex from other parts of the central nervous system is the great diversity of its cell types and interconnexions. It would be astonishing if such a structure did not profoundly modify the response patterns of fibres coming into it. In the cat's visual cortex, the receptive field arrangements of single cells suggest that there is indeed a degree of complexity far exceeding anything yet seen at lower levels in the visual system. In a previous paper we described receptive fields of single cortical cells, observing responses to spots of light shone on one or both retinas (Hubel & Wiesel, 1959). In the present work this method is used to examine receptive fields of a more complex type (Part I) and to make additional observations on binocular interaction (Part II). This approach is necessary in order to understand the behaviour of individual cells, but it fails to deal with the problem of the relationship of one cell to its neighbours. In the past, the technique of recording evoked slow waves has been used with great success in studies of functional anatomy. It was employed by Talbot & Marshall (1941) and by Thompson, Woolsey & Talbot (1950) for mapping out the visual cortex in the rabbit, cat, and monkey. Daniel & Whitteiidge (1959) have recently extended this work in the primate. Most of our present knowledge of retinotopic projections, binocular overlap, and the second visual area is based on these investigations. Yet the method of evoked potentials is valuable mainly for detecting behaviour common to large populations of neighbouring cells; it cannot differentiate functionally between areas of cortex smaller than about 1 mm2. To overcome this difficulty a method has in recent years been developed for studying cells separately or in small groups during long micro-electrode penetrations through nervous tissue. Responses are correlated with cell location by reconstructing the electrode tracks from histological material. These techniques have been applied to

12,923 citations


"Theory of Edge Detection" refers result in this paper

  • ...This state of affairs is consistent with the neurophysiology since Hubel & Wiesel (1962) originally defined simple cells by the linearity of their response, and they reported many bar-shaped receptive fields....

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Journal ArticleDOI
TL;DR: The striate cortex was studied in lightly anaesthetized macaque and spider monkeys by recording extracellularly from single units and stimulating the retinas with spots or patterns of light, with response properties very similar to those previously described in the cat.
Abstract: 1. The striate cortex was studied in lightly anaesthetized macaque and spider monkeys by recording extracellularly from single units and stimulating the retinas with spots or patterns of light. Most cells can be categorized as simple, complex, or hypercomplex, with response properties very similar to those previously described in the cat. On the average, however, receptive fields are smaller, and there is a greater sensitivity to changes in stimulus orientation. A small proportion of the cells are colour coded. 2. Evidence is presented for at least two independent systems of columns extending vertically from surface to white matter. Columns of the first type contain cells with common receptive-field orientations. They are similar to the orientation columns described in the cat, but are probably smaller in cross-sectional area. In the second system cells are aggregated into columns according to eye preference. The ocular dominance columns are larger than the orientation columns, and the two sets of boundaries seem to be independent. 3. There is a tendency for cells to be grouped according to symmetry of responses to movement; in some regions the cells respond equally well to the two opposite directions of movement of a line, but other regions contain a mixture of cells favouring one direction and cells favouring the other. 4. A horizontal organization corresponding to the cortical layering can also be discerned. The upper layers (II and the upper two-thirds of III) contain complex and hypercomplex cells, but simple cells are virtually absent. The cells are mostly binocularly driven. Simple cells are found deep in layer III, and in IV A and IV B. In layer IV B they form a large proportion of the population, whereas complex cells are rare. In layers IV A and IV B one finds units lacking orientation specificity; it is not clear whether these are cell bodies or axons of geniculate cells. In layer IV most cells are driven by one eye only; this layer consists of a mosaic with cells of some regions responding to one eye only, those of other regions responding to the other eye. Layers V and VI contain mostly complex and hypercomplex cells, binocularly driven. 5. The cortex is seen as a system organized vertically and horizontally in entirely different ways. In the vertical system (in which cells lying along a vertical line in the cortex have common features) stimulus dimensions such as retinal position, line orientation, ocular dominance, and perhaps directionality of movement, are mapped in sets of superimposed but independent mosaics. The horizontal system segregates cells in layers by hierarchical orders, the lowest orders (simple cells monocularly driven) located in and near layer IV, the higher orders in the upper and lower layers.

6,388 citations

Book
01 Jan 1965
TL;DR: In this paper, the authors provide a broad overview of Fourier Transform and its relation with the FFT and the Hartley Transform, as well as the Laplace Transform and the Laplacian Transform.
Abstract: 1 Introduction 2 Groundwork 3 Convolution 4 Notation for Some Useful Functions 5 The Impulse Symbol 6 The Basic Theorems 7 Obtaining Transforms 8 The Two Domains 9 Waveforms, Spectra, Filters and Linearity 10 Sampling and Series 11 The Discrete Fourier Transform and the FFT 12 The Discrete Hartley Transform 13 Relatives of the Fourier Transform 14 The Laplace Transform 15 Antennas and Optics 16 Applications in Statistics 17 Random Waveforms and Noise 18 Heat Conduction and Diffusion 19 Dynamic Power Spectra 20 Tables of sinc x, sinc2x, and exp(-71x2) 21 Solutions to Selected Problems 22 Pictorial Dictionary of Fourier Transforms 23 The Life of Joseph Fourier

5,714 citations


"Theory of Edge Detection" refers background in this paper

  • ...They are, in fact, related by the uncertainty principle, which states that Ax Aw > an (see, for example, Bracewell 1965, pp. 160-163)....

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Book
01 Jan 1976
TL;DR: The rapid rate at which the field of digital picture processing has grown in the past five years had necessitated extensive revisions and the introduction of topics not found in the original edition.
Abstract: The rapid rate at which the field of digital picture processing has grown in the past five years had necessitated extensive revisions and the introduction of topics not found in the original edition.

4,231 citations

Journal ArticleDOI
TL;DR: The contrast thresholds of a variety of grating patterns have been measured over a wide range of spatial frequencies and the results show clear patterns of uniformity in the response to grating noise.
Abstract: 1. The contrast thresholds of a variety of grating patterns have been measured over a wide range of spatial frequencies.2. Contrast thresholds for the detection of gratings whose luminance profiles are sine, square, rectangular or saw-tooth waves can be simply related using Fourier theory.3. Over a wide range of spatial frequencies the contrast threshold of a grating is determined only by the amplitude of the fundamental Fourier component of its wave form.4. Gratings of complex wave form cannot be distinguished from sine-wave gratings until their contrast has been raised to a level at which the higher harmonic components reach their independent threshold.5. These findings can be explained by the existence within the nervous system of linearly operating independent mechanisms selectively sensitive to limited ranges of spatial frequencies.

3,073 citations


"Theory of Edge Detection" refers background in this paper

  • ...The experiments of Hubel & TNiesel (1962) and of Campbell & Robson (1968) introduced two rather distinct notions of the function of early information processing in higher visual systems....

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  • ...Recent psychophysical work by Wilson & Giese (1977), FVilson & Bergen (1979) (see also Macleod & Rosenfeld 1974), has led to a precise quantitative model of the orientation-dependent spatial-frequency-tuned channels discovered by Campbell & Robson (1968)....

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  • ...This requirement, together with the findings of Campbell & Robson (1968), leads to the basic idea, illustrated in figure 1, in which one first takes local averages of the image a t various resolutions and then detects the changes in intensity that occur at each one....

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