Gaussian-Based Edge-Detection Methods — A Survey
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paper_title: Theory of edge detection

paper_content:

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 ∇ 2 G(x, y) * I(x, y) for image I, where G(x, y) is a two-dimen­sional Gaussian distribution and ∇ 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 localized. 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 ∇ 2 G filters acting on the image forms the basis for a physiological model of simple cells (see Marr & Ullman 1979).
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paper_title: Digital Picture Processing

paper_content:

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.
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paper_title: Towards the unification of band-limited derivative operators for edge detection

paper_content:

Abstract Edge detection has been an important subject in image processing. Many linear operators for edge detection in noisy images, including some ‘optimal’ ones, have been proposed. Note that most of them were deduced from the mono-edge model, but in real images there exist always many edge points. So it will be interesting to analyse the problem of multi-edge detection in images. In the present paper, we present at first the optimal filter for mono- and multi-edge detection and show that an optimal filter for noise removing in edge detection is the symmetric exponential filter of an infinite size (ISEF). In more general multi-edge cases, the optimal filter is a linear combination of the output from different stages of a cascade of symmetric exponential filters. Based on the ISEF cascade for multi-edge detection, we analyse then some band-limited linear edge detectors, such as Gaussian filter, Gaussian cascade technique, box technique, fast filter transform, Canny's filter, DRF method, box difference and Gabor's filter used for edge detection. We show that these filters can be explained and unified by the cascade of exponential filters presented for multi-edge detection.
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paper_title: Uniqueness of the Gaussian Kernel for Scale-Space Filtering

paper_content:

Scale-space filtering constructs hierarchic symbolic signal descriptions by transforming the signal into a continuum of versions of the original signal convolved with a kernal containing a scale or bandwidth parameter. It is shown that the Gaussian probability density function is the only kernel in a broad class for which first-order maxima and minima, respectively, increase and decrease when the bandwidth of the filter is increased. The consequences of this result are explored when the signal?or its image by a linear differential operator?is analyzed in terms of zero-crossing contours of the transform in scale-space.
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paper_title: Scaling Theorems for Zero Crossings

paper_content:

We characterize some properties of the zero crossings of the Laplacian of signals¿in particular images¿filtered with linear filters, as a function of the scale of the filter (extending recent work by Witkin [16]). We prove that in any dimension the only filter that does not create generic zero crossings as the scale increases is the Gaussian. This result can be generalized to apply to level crossings of any linear differential operator: it applies in particular to ridges and ravines in the image intensity. In the case of the second derivative along the gradient, there is no filter that avoids creation of zero crossings, unless the filtering is performed after the derivative is applied.
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paper_title: Theory of edge detection

paper_content:

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 ∇ 2 G(x, y) * I(x, y) for image I, where G(x, y) is a two-dimen­sional Gaussian distribution and ∇ 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 localized. 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 ∇ 2 G filters acting on the image forms the basis for a physiological model of simple cells (see Marr & Ullman 1979).
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paper_title: A Computational Approach to Edge Detection

paper_content:

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.
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paper_title: Edge Focusing

paper_content:

Edge detection in a gray-scale image at a fine resolution typically yields noise and unnecessary detail, whereas edge detection at a coarse resolution distorts edge contours. We show that ``edge focusing'', i.e., a coarse-to-fine tracking in a continuous manner, combines high positional accuracy with good noise-reduction. This is of vital interest in several applications. Junctions of different kinds are in this way restored with high precision, which is a basic requirement when performing (projective) geometric analysis of an image for the purpose of restoring the three-dimensional scene. Segmentation of a scene using geometric clues like parallelism, etc., is also facilitated by the algorithm, since unnecessary detail has been filtered away. There are indications that an extension of the focusing algorithm can classify edges, to some extent, into the categories diffuse and nondiffuse (for example diffuse illumination edges). The edge focusing algorithm contains two parameters, namely the coarseness of the resolution in the blurred image from where we start the focusing procedure, and a threshold on the gradient magnitude at this coarse level. The latter parameter seems less critical for the behavior of the algorithm and is not present in the focusing part, i.e., at finer resolutions. The step length of the scale parameter in the focusing scheme has been chosen so that edge elements do not move more than one pixel per focusing step.
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paper_title: Authenticating Edges Produced by Zero-Crossing Algorithms

paper_content:

Abslruct-It is shown that zero-crossing edge detection algorithms can produce edges which do not correspond to significant image intensity changes. Such edges are called phantom or spurious. A method for classifying zero crossings as corresponding to “authentic” or “phantom” edges is presented. The contrast of an authentic edge is shown to increase with decreasing filter scale (and hence higher resolution), and the contrast of phantom edges is shown to decrease with a decrease in the filter scale. Thus, a phantom edge is truly a phantom edge in that the closer one examines it the weaker it becomes. The results of applying the classification schemes described in this paper to synthetic and authentic signals in one and two dimensions are given. The significance of the phantom edges is examined with respect to their frequency and strength relative to the authentic edges, and it is seen that authentic edges are denser and stronger, on the average, than phantom edges. Index Terms-Edge authentication, edge detection, phantom edges, scale space.
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paper_title: Edge Detection and Ridge Detection with Automatic Scale Selection

paper_content:

When computing descriptors of image data, the type of information that can be extracted may be strongly dependent on the scales at which the image operators are applied. This article presents a systematic methodology for addressing this problem. A mechanism is presented for automatic selection of scale levels when detecting one-dimensional image features, such as edges and ridges. ::: ::: A novel concept of a scale-space edge is introduced, defined as a connected set of points in scale-space at which: (i) the gradient magnitude assumes a local maximum in the gradient direction, and (ii) a normalized measure of the strength of the edge response is locally maximal over scales. An important consequence of this definition is that it allows the scale levels to vary along the edge. Two specific measures of edge strength are analyzed in detail, the gradient magnitude and a differential expression derived from the third-order derivative in the gradient direction. For a certain way of normalizing these differential descriptors, by expressing them in terms of so-called γ-normalized derivatives, an immediate consequence of this definition is that the edge detector will adapt its scale levels to the local image structure. Specifically, sharp edges will be detected at fine scales so as to reduce the shape distortions due to scale-space smoothing, whereas sufficiently coarse scales will be selected at diffuse edges, such that an edge model is a valid abstraction of the intensity profile across the edge. ::: ::: Since the scale-space edge is defined from the intersection of two zero-crossing surfaces in scale-space, the edges will by definition form closed curves. This simplifies selection of salient edges, and a novel significance measure is proposed, by integrating the edge strength along the edge. Moreover, the scale information associated with each edge provides useful clues to the physical nature of the edge. ::: ::: With just slight modifications, similar ideas can be used for formulating ridge detectors with automatic selection, having the characteristic property that the selected scales on a scale-space ridge instead reflect the width of the ridge. ::: ::: It is shown how the methodology can be implemented in terms of straightforward visual front-end operations, and the validity of the approach is supported by theoretical analysis as well as experiments on real-world and synthetic data.
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paper_title: A semidiscrete nonlinear scale-space theory and its relation to the Perona-Malik paradox

paper_content:

Although much effort has been spent in the recent decade to establish a theoretical foundation of certain partial differential equations (PDEs) as scale-spaces, it is almost never taken into account that, in practice, images are sampled on a fixed pixel grid1. For nonlinear PDE-based filters, usually straightforward finite difference discretizations are applied in the hope that they reflect the nice properties of the continuous equations. Since scale-spaces cannot perform better than their numerical discretizations, however, it would be desirable to have a genuinely discrete nonlinear framework which reflects the discrete nature of digital images. In this paper we discuss a semidiscrete scale-space framework for nonlinear diffusion filtering. It keeps the scale-space idea of having a continuous time parameter, while taking into account the spatial discretization on a fixed pixel grid. It leads to nonlinear systems of coupled ordinary differential equations. Conditions are established under which one can prove existence of a stable unique solution which preserves the average grey level. An interpretation as a smoothing scale-space transformation is introduced which is based on an extremum principle and the existence of a large class of Lyapunov functionals comprising for instance p-norms, even central moments and the entropy. They guarantee that the process is not only simplifying and information-reducing, but also converges to a constant image as the scale parameter t tends to infinity.
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paper_title: Uniqueness of the Gaussian Kernel for Scale-Space Filtering

paper_content:

Scale-space filtering constructs hierarchic symbolic signal descriptions by transforming the signal into a continuum of versions of the original signal convolved with a kernal containing a scale or bandwidth parameter. It is shown that the Gaussian probability density function is the only kernel in a broad class for which first-order maxima and minima, respectively, increase and decrease when the bandwidth of the filter is increased. The consequences of this result are explored when the signal?or its image by a linear differential operator?is analyzed in terms of zero-crossing contours of the transform in scale-space.
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paper_title: Image enhancement using a human visual system model

paper_content:

Abstract In this paper we report the result of a set of computer experiments carried out to enhance digital images. We use a special line weight function (LWF) which is a combination of zero- and second-order Hermite functions. We are motivated by the physiological evidence reported in R. A. Young, Spatial Vision 2 (4), 273–293 (1987), that visual receptive fields are shaped like the sum of a Gaussian function and its Laplacian. This function can also be derived mathematically when the contrast sensitivity experiments in psychophysics are posed as an eigenvalue problem (A. L. Stewart and R. Pinkham, Biol. Cybernetics 64 , 373–379 (1991). Analyses of the edge location error show that the proposed function has extremely good localization capability (i.e. the points marked by the operator is as close as possible to the center of the true edge). We also show that the LWF does not detect phantom edges which do not correspond to significant image intensity changes.
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paper_title: Scale space approach to directional analysis of images

paper_content:

In this paper a new technique for directional analysis of linear patterns in images is proposed based on the notion of scale space. A given image is preprocessed by a sequence of filters which are second derivatives of 2-D Gaussian functions with different scales. This gives a set of zero crossing maps (the scale space) from which a stability map is generated. Significant linear patterns are detected from measurements on the stability map. Information regarding orientation of the linear patterns in the image and the area covered by the patterns in specific directions is then computed. The performance of the method is illustrated through applications to synthetic patters and to scanning electron microscope images of collagen fibrils in rabbit ligaments.
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paper_title: Nonlinear image filtering with edge and corner enhancement

paper_content:

The authors describe a model of nonlinear image filtering for noise reduction and edge enhancement using anisotropic diffusion. The method is designed to enhance not only edges, but corners and T junctions as well. The method roughly corresponds to a nonlinear diffusion process with backward heat flow across the strong edges. Such a process is ill posed, making the results depend strongly on how the algorithm differs from the diffusion process. Two ways of modifying the equations using simulations on a variable grid are studied. >
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paper_title: Edge contours using multiple scales

paper_content:

Abstract An algorithm for finding a single good path through the set of edge points detected by gradient of Gaussian operator is discussed. First, an algorithm for finding contours at one scale is presented, then an extension of that algorithm which uses multiple scales to produce improved detection of weak edges is presented. The set of possible edge points is placed on a priority queue with the edge point having largest magnitude at the top. The strongest edge point that is not already on a contour is retrieved from the queue. The point in the computed direction is examined first, then in those in the adjacent directions on either side of it. Each branch is followed to the end and a weight assigned at each point based on four factors: a measure of noisiness, a measure of curvature, contour length, and the gradient magnitude. The point with the largest average weight is chosen. After searching from the initial point in one direction, a similar search is conducted in the oppositedirection unless a closed contour has been formed. In the algorithm for multiple scales the search for a contour proceeds as for the single scale, using the largest scale, until a best partial contour at that scale has been found. Then the next finer scale is chosen and the neighborhood around the end points of the contour are examined to determine possible edge points in a direction similar to the end point of the contour. The original algorithm is then followed for each of the points satisfying the above condition, and the best is chosen as an extension to the original edge. Further, in order to determine the size neighborhood that should be searched when attempting to pick up an edge at a smaller scale, a theoretical analysis of the movement of idealized edges is performed. This analysis examines two adjacent step edges having the same parity (a staircase) and opposite parity (a pulse). It is determined that maximum movement for both cases is σ, where σ is the standard deviation of the Gaussian used. This maximum movement occurs for the staircase when the two nearby edges have the same step size and are at a distance of 2σ apart. However, for edges closer or farther away, maximum movement decreases rapidly. For a pulse, maximum movement occurs when the two edges have the same step size and are very close together. Again the movement decreases rapidly as the edges become farther apart. Movement also decreases in both cases when the relative strengths of the two edges are not equal.
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paper_title: Accuracy of laplacian edge detectors

paper_content:

The sources of error for the edge finding technique proposed by Marr and Hildreth (D. Marr and T. Poggio, Proc. R. Soc. London Ser. B204, 1979, 301–328; D. Marr and E. Hildreth, Proc. R. Soc. London Ser. B.207, 1980, 187–217) are identified, and the magnitudes of the errors are estimated, based on idealized models of the most common error producing situations. Errors are shown to be small for linear illuminations, as well as for nonlinear illuminations with a second derivative less than a critical value. Nonlinear illuminations are shown to lead to spurious contours under some conditions, and some fast techniques for discarding such contours are suggested.
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paper_title: Local Scale Control for Edge Detection and Blur Estimation

paper_content:

Selecting the appropriate spatial scale for local edge analysis is a challenge for natural images, where blur scale and contrast may vary over a broad range. While previous methods for scale adaptation have required the global solution of a non-convex optimization problem [8], it is shown that knowledge of sensor properties and operator norms can be exploited to define a unique, locally-computable minimum reliable scale for local estimation. The resulting method for local scale control allows edges spanning a broad range of blur scales and contrasts to be reliably localized by a single system with no input parameters other than the second moment of the sensor noise. Local scale control further permits the reliable estimation of local blur scale in complex images where the conditions demanded by Fourier methods for blur estimation break down.
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paper_title: Theory of edge detection

paper_content:

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 ∇ 2 G(x, y) * I(x, y) for image I, where G(x, y) is a two-dimen­sional Gaussian distribution and ∇ 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 localized. 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 ∇ 2 G filters acting on the image forms the basis for a physiological model of simple cells (see Marr & Ullman 1979).
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paper_title: Scale-space and edge detection using anisotropic diffusion

paper_content:

A new definition of scale-space is suggested, and a class of algorithms used to realize a diffusion process is introduced. The diffusion coefficient is chosen to vary spatially in such a way as to encourage intraregion smoothing rather than interregion smoothing. It is shown that the 'no new maxima should be generated at coarse scales' property of conventional scale space is preserved. As the region boundaries in the approach remain sharp, a high-quality edge detector which successfully exploits global information is obtained. Experimental results are shown on a number of images. Parallel hardware implementations are made feasible because the algorithm involves elementary, local operations replicated over the image. >
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paper_title: The structure of images

paper_content:

In practice the relevant details of images exist only over a restricted range of scale. Hence it is important to study the dependence of image structure on the level of resolution. It seems clear enough that visual perception treats images on several levels of resolution simultaneously and that this fact must be important for the study of perception. However, no applicable mathematically formulated theory to deal with such problems appears to exist. In this paper it is shown that any image can be embedded in a one-parameter family of derived images (with resolution as the parameter) in essentially only one unique way if the constraint that no spurious detail should be generated when the resolution is diminished, is applied. The structure of this family is governed by the well known diffusion equation (a parabolic, linear, partial differential equation of the second order). As such the structure fits into existing theories that treat the front end of the visual system as a continuous stack of homogeneous layers, characterized by iterated local processing schemes. When resolution is decreased the images becomes less articulated because the extrem ("light and dark blobs") disappear one after the other. This erosion of structure is a simple process that is similar in every case. As a result any image can be described as a juxtaposed and nested set of light and dark blobs, wherein each blob has a limited range of resolution in which it manifests itself. The structure of the family of derived images permits a derivation of the sampling density required to sample the image at multiple scales of resolution.(ABSTRACT TRUNCATED AT 250 WORDS)
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paper_title: An adaptive Gaussian filter for noise reduction and edge detection

paper_content:

Gaussian filtering has been intensively studied in image processing and computer vision. Using a Gaussian filter for noise suppression, the noise is smoothed out, at the same time the signal is also distorted. The use of a Gaussian filter as pre-processing for edge detection will also give rise to edge position displacement, edges vanishing, and phantom edges. Here, the authors first review various techniques for these problems. They then propose an adaptive Gaussian filtering algorithm in which the filter variance is adapted to both the noise characteristics and the local variance of the signal. >
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paper_title: The primary raster: a multiresolution image description

paper_content:

The primary raster is a multiresolution image description. This description has the form of a raster where each pixel (edgel) has four characteristics: the detection scale, the blurring scale, the local contrast, and an edgel type. The detection scale is the finest resolution where the edgel appears, while the blurring scale is the coarsest resolution where the edgel is still present. The local contrast is the difference of the mean intensities taken from each side of the edgel. The edgel type depends on the evolution of the gradient from the finest to the coarsest resolution. Experimental results on the primary raster are presented. It is concluded that the primary raster provides a basis for a higher-level image description. >
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paper_title: A Computational Approach to Edge Detection

paper_content:

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.
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paper_title: Digital Step Edges from Zero Crossing of Second Directional Derivatives

paper_content:

We use the facet model to accomplish step edge detection. The essence of the facet model is that any analysis made on the basis of the pixel values in some neighborhood has its final authoritative interpretation relative to the underlying gray tone intensity surface of which the neighborhood pixel values are observed noisy samples. With regard to edge detection, we define an edge to occur in a pixel if and only if there is some point in the pixel's area having a negatively sloped zero crossing of the second directional derivative taken in the direction of a nonzero gradient at the pixel's center. Thus, to determine whether or not a pixel should be marked as a step edge pixel, its underlying gray tone intensity surface must be estimated on the basis of the pixels in its neighborhood. For this, we use a functional form consisting of a linear combination of the tensor products of discrete orthogonal polynomials of up to degree three. The appropriate directional derivatives are easily computed from this kind of a function. Upon comparing the performance of this zero crossing of second directional derivative operator with the Prewitt gradient operator and the Marr-Hildreth zero crossing of the Laplacian operator, we find that it is the best performer; next is the Prewitt gradient operator. The Marr-Hildreth zero crossing of the Laplacian operator performs the worst.
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paper_title: Adaptive determination of filter scales for edge detection

paper_content:

The authors suggest a regularization method for determining scales for edge detection adaptively for each site in the image plane. Specifically, they extend the optimal filter concept of T. Poggio et al. (1984) and the scale-space concept of A. Witkin (1983) to an adaptive scale parameter. To avoid an ill-posed feature synthesis problem, the scheme automatically finds optimal scales adaptively for each pixel before detecting final edge maps. The authors introduce an energy function defined as a functional over continuous scale space. Natural constraints for edge detection are incorporated into the energy function. To obtain a set of optimal scales that can minimize the energy function, a parallel relaxation algorithm is introduced. Experiments for synthetic and natural scenes show the advantages of the algorithm. In particular, it is shown that this system can detect both step and diffuse edges while drastically filtering out the random noise. >
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paper_title: The Perona-Malik paradox

paper_content:

The Perona--Malik equation is a formally ill-posed parabolic equation for which simple discretizations are nevertheless numerically found to be stable. After discussing the background of this paradox in computer vision, this paper shows the nonexistence of weak solutions even in those cases where computations are successful, and introduces a notion of generalized solutions for this equation, which do evolve smoothly and possess many of the features of numerical calculations.
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paper_title: On edge focusing

paper_content:

Abstract An algorithm is described that can track edges determined by the Laplacian of Gaussian operator from low to high resolution. The algorithm does not require a precomputed scale step size, but rather determines the step size adaptively by using the image content. A binary search is carried out to find the topology of the scale-space image, which is then used to track the edges. The proposed algorithm guarantees correct edge tracking with step sizes considerably larger than has been previously used.
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paper_title: Optimal parameters for edge detection

paper_content:

Bennamoum, and Masoud, Bennamoum and Bayoumi (1991), suggested a robust edge detector which relaxes the trade-off between robustness against noise and accurate localization of the edges. This hybrid detector separates the tasks of localization and noise suppression between two sub-detectors. In this paper, we present an extension to this hybrid detector to determine its optimal parameters, independently of the scene. This extension defines a probabilistic cost function using for criteria the probability of missing an edge buried in noise and the probability of detecting false edges. The optimization of this cost function allows the automatic selection of the parameters of the hybrid edge detector given the height of the minimum edge to be detected and the variance of the noise, /spl sigma//sub n//sup 2/. The results were applied to the 2D case and the performance of the adaptive hybrid detector was compared to other detectors.
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paper_title: Optimal edge detection in two-dimensional images

paper_content:

This paper presents a new edge detection scheme that detects two-dimensional (2-D) edges by a curve-segment-based detection functional guided by the zero-crossing contours of the Laplacian-of-Gaussian (LOG) to approach the true edge locations. The detection functional is shown to be optimal in terms of signal-to-noise ratio (SNR) and edge localization accuracy; it also preserves the nice scaling property held uniquely by the LOG in scale space.
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paper_title: Representations Based on Zero-Crossings in Scale-Space

paper_content:

Abstract Using the Heat Equation to formulate the notion of scale-space filtering, we show that the evolution property of level-crossings in scale-space is equivalent to the maximum principle. We briefly discuss filtering over bounded domains. We then consider the completeness of the representation of data by zero-crossings, and observe that for polynomial data, the issue is solved by standard results in algebraic geometry. For more general data, we argue that gradient information along the zero-crossings is needed, and that although such information more than suffices, the representation is still not stable. We give a simple linear procedure for reconstruction of data from zero-crossings and gradient data along zero-crossings in both continuous and discrete scale-space domains.
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Title: Gaussian-Based Edge-Detection Methods — A Survey

Section 1: INTRODUCTION
Description 1: Describe the purpose and the scope of the survey, mentioning the importance of edge detection in image processing and the prevalence of Gaussian-based methods.

Section 2: PROBLEMS OF EDGE DETECTION
Description 2: Discuss the various challenges faced during edge detection, including noise, edge localization, and the impact of smoothing operations on edge precision.

Section 3: SIGNIFICANCE OF THE GAUSSIAN FILTER
Description 3: Explain why the Gaussian filter is widely used in edge detection, highlighting its properties and contributions to the advancement of edge-detection techniques.

Section 4: GAUSSIAN-BASED EDGE DETECTION TECHNIQUES
Description 4: Present and analyze various techniques based on the Gaussian filter for edge detection, including Marr-Hildreth and Canny detectors, among others, and discuss their advantages and drawbacks.

Section 5: SUMMARY
Description 5: Summarize the survey, emphasizing the Gaussian filter's features and the main findings related to edge-detection techniques, their challenges, and their computational aspects.