R.M. Haralick, "Digital Step Edges from Zero Crossings of Second Directional Derivatives," IEEE Trans. PAMI 6, pp. 58--68 (1984).  Home/Search   Document Not in Database   Summary   Related Articles   Check

....then evaluates some di#erent corner detectors based on the model parameters and decides that the best one is I xx I yy xy (2.2) where I xx , I yy , and I xy are second order derivatives of the image intensity function. Haralick, Kitchen Rosenfeld, and Nagel ( Haralick and Watson, 1981] [Haralick, 1984], Haralick and Shapiro, 1993] Kitchen and Rosenfeld, 1982] Nagel, 1983] have all developed similar corner detectors based on polynomial expansion models. An incomplete third degree polynomial model is fitted locally to the image: I(x, y) k 1 k 2 x k 3 y k 4 x k 5 xy k 6 ....

.... have been used in a number of image analysis applications including gradient edge detection, zero crossing edge detection, image segmentation, line detection, corner detection, three dimensional shape estimation from shading, and determination of optical flow, see [Haralick and Shapiro, 1993] [Haralick, 1984], Haralick and Watson, 1981] The polynomial model is fitted to a local square shaped neighborhood in the image using non weighted least squares. Recently Farneback has shown that polynomial expansion using weighted least squares with a Gaussian weight function can give much better results on ....

Haralick, R. M. (1984). Digital step edges from zero crossing of second directional derivatives. PAMI-6(1):58--68.

....that fall in between points on the previous surface. As the depth samples become sparser, the interpolation process becomes increasingly ill conditioned. In general, when considering the above elements, two forms of surface reconstruction emerge. The first is feature based surface reconstruction [5, 9, 27, 28, 43, 49, 56]. This approach chooses features in the image that are stable for large motions. Thus, a sparse set of very confident depth estimates is obtained. Difficulties occur when trying to interpolate full surface representation. Often planarity assumptions must be used, or some underlying knowledge of ....

Haralick, R.M., Digital Step Edges from Zero Crossing of Second Directional Derivatives, IEEE Transactions on Pattern Analysis and Machine Intelligence, Vol. 6, No. 1, pp. 58-68, Jan 1984. 84

....strength of normalized convolution is that it combines all of them simultaneously in a well structured and theoretically sound way. The method is a generally useful tool for signal analysis in the spatial domain, which formalizes and generalizes least squares techniques, e.g. the facet model [41, 42, 43, 44], that have been used for a long time. A weakness of the presentation used here is that it is not immediately clear how normalized convolution can be applied to the problem of computing filter responses on uncertain data when the convolution kernels are given. This is discussed in detail in ....

....to have more samples in areas of high contrast. Figure 3.5(a) shows such a test image, only containing 4 of the original pixels. The result of normalized averaging, with applicability given by figure 3.4(b) is shown in figure 3.5(b) 3.9. 2 The Cubic Facet Model In the cubic facet model [42, 43], it is assumed that in each neighborhood of an image, the signal can be described by a cubic polynomial . 3.39) k are determined by an unweighted least squares fit within a square window of some size. A typical application of the cubic facet model is to estimate the image ....

[Article contains additional citation context not shown here]

R. M. Haralick. Digital step edges from zero crossing of second directional derivatives. IEEE Transactions on Pattern Analysis and Machine Intelligence, PAMI-6(1):58--68, January 1984.

....derivatives to locate an edge position. This is a relatively good method, but it has some deficiency in not being based on explicit mathematical models of the edge in real images and hence not giving the values of some parameters and not revealing the ways for improving their quality (see, e.g. [5,6]) Our edge detector is based on direct evaluation of edge probability at given point and it detects an edge only if its probability exceeds a threshold and has also a good localization characteristics. This measure for probability is being used for further evaluation of the probabilities for ....

....a middle point of the slope segment should be chosen for a slope edge model. For some problems it is required to have more precise edge position locations, which can be found by an interpolation method. The first known approach is directional polynomial approximation of the detected edge, [5]. For this purpose it is reasonable to use a third order polynomial and the edge point would be zero crossing of its second order derivative. Another method follows directly from our edge model and consists of a piecewise approx mation of edge segments in the given direction. It implies ....

Haralick R.M.: Digital step edges from zero crossing of second directional derivatives. IEEE qrans. PAMI 6, 1, 58-69. 1986

....faster on large arrays of processors such as the Connection Machine than Warp. This is because no communication is required between distant elements of the array, and the large array of processors can be readily mapped onto the large image array. For example, the computation of an 11xll Laplacian [15] on a 512x512 image, followed by the detection of zero crossings, takes only 3 milliseconds on CM 1, as opposed to 400 milliseconds on Warp. The floating point processors in Warp aid the programmer in ehminating the need for low level algorithmic analysis. For example, the Connection Machine used ....

Haralick, R. M. "Digital Step Edges from Zero Crossings of Second Directional Derivatives". IEEE Transactions on Pattern Analysis and Machine Intelligence 6 (1984), 58-68.

....gradients or Laplacians [5] Early techniques relied on finite differences to estimate these quantities [6] 7] however, these simple operators used on noisy images perform poorly. More recent approaches often depend on the concept of fitting a continuous surface locally to the data [5] 8] [9]. Haralick used local least squares polynomial fits to determine the zero crossing of the directional second derivatives [9] Poggio et al. proposed a smoothing cubic spline technique to improve the estimation of the intensity gradient in the presence of noise [10] 11] These authors showed the ....

....these simple operators used on noisy images perform poorly. More recent approaches often depend on the concept of fitting a continuous surface locally to the data [5] 8] 9] Haralick used local least squares polynomial fits to determine the zero crossing of the directional second derivatives [9]. Poggio et al. proposed a smoothing cubic spline technique to improve the estimation of the intensity gradient in the presence of noise [10] 11] These authors showed the approach to be more or less equivalent to smoothing the image with a Gaussian low pass filter in a preprocessing step. In ....

R. M. Haralick, "Digital step edges from zero crossing of second directional derivatives," IEEE Trans. Part. Anal. Machine lntell., vol. PAMI-6, pp. 58-68, Jan. 1984.

....detection in the retina see [133] Since the Laplacian of Gaussian operator is spatiaUy symmetric, it ignores the asymmetry of edges, which are one dimensional curves with a preferred direction. Some algorithms address this problem by computing second directional derivatives of the input [52]. The computation of the second derivative in the direction of the intensity gradient has been shown optimal for the detection of oriented edges [149] As in stereo, a multi resolution approach (computing the edges at several levels of smoothing) proved useful in edge detection. A widely used ....

R. M. Haralick. Digital step edges from zero crossings of second directional derivatives. IEEE Transactions on Pattern Analysis and Machine Intelligence, 6:58-68, 1984.

....see [Rosenfeld82] Section 11.2.2. The size of this image requires 4 virtual processors per physical processor. Each pixel is mapped into a virtual processor. 2.1.1 Convolution with Laplacian The 11xll sample Laplacian actually corresponds to filtering with a Gaus sian where a is 1. 4, [Haralick84], but see [Grimson85] where it is argued that a much larger mask should be used for reliable results) But, for a mask diameter of 11 pixels, the binomial approximation to the Gaussian, followed by a discrete Laplacian, requires only 3 ms. 2.1.2 Detecting Zero Crossings This takes negligible ....

R.M. Haralick, "Digital step edges from zero crossings of second directional derivatives", IEEE Trans. on Pattern Analtsis and Machine Intelligence 6, 1984, 58-68.

....Matched Percent Correctly Matched Photo of Cat 9109 6527 71.65 5264 57.79 Photo of Cat 1.5 5137 4468 86.98 4318 84.06 indicates that no Gaussian filtering was performed. The argument for the proper choice of the standard deviation (s) for Gaussian filtering remains unsettled [30 34]. The issue here is that an optimal choice of a s for a given scene may not also be optimal for another scene. An optimal choice of s (optimal filtering) in the practical sense, would mean preserving the majority of the feature points while attenuating or eliminating the effect of the majority ....

R. M. Haralick, "Digital Step Edges from Zero Crossing of Second Directional Derivatives," IEEE Transactions on Pattern Analysis and Machine Intelligence, Vol. 6, No. 1, pp. 58-68, January, 1984.

....some false edge extensions. 7 work well on edges of high curvature. In [14] and [56] similar analytical approaches to that of Canny are taken, resulting in efficient algorithms which have exact recursive implementations. These algorithms give results which are very similar to Canny s. In [22], Haralick proposes the use of zero crossings of the second directional derivative of the image brightness function. This is theoretically the same as using maxima in the first directional derivatives, and in one dimension is the same as the LoG filter. The zero crossings are found by fitting two ....

R.M. Haralick. Digital step edges from zero crossing of second directional derivatives. IEEE Trans. on Pattern Analysis and Machine Intelligence, 6(1):58--68, January 1984.

....and white blocks are set at 10 and 240 (Top) EV RIST segmentation (see text) at 90 (middle) EV RIST segmentation at 91 (bottom) The first image we deal with is the chessboard shown in Fig. 3 (Top) This case is very interesting, since it allows us, by means of the Haralick s metrics [Haral84], to get objective measures under noise on a relatively complicate image containing Brodatz textures [Broda66] along with black and white blocks. EVRIST achieves a correct segmentation both on the original chessboard (w 1 = 2; w 2 = 25; j 1 = 40; j 2 = 22) and on the same image with additive ....

R. M. Haralick, Digital Step Edges from Zero Crossing of Second Directional Derivatives, IEEE Trans. on PAMI, 6, pp.58-68, 1984.

....most algorithms for computational reasons do work on this kind of data. It should also be noticed that even if we should have signals of unlimited size, we are often interested in analyzing only a limited neighborhood of the signal at a time. 6. 2 The Cubic Facet Model In the cubic facet model [5], it is assumed that in each neighborhood of an image, the signal can be described by a cubic polynomial f(x; y) k 1 k 2 x k 3 y k 4 x 2 k 5 xy k 6 y 2 k 7 x 3 k 8 x 2 y k 9 xy 2 k 10 y 3 : 65) The coecients fk i g are determined by a least squares t within a ....

....of an image, the signal can be described by a cubic polynomial f(x; y) k 1 k 2 x k 3 y k 4 x 2 k 5 xy k 6 y 2 k 7 x 3 k 8 x 2 y k 9 xy 2 k 10 y 3 : 65) The coecients fk i g are determined by a least squares t within a square window of some size 8 . In [5] coe cients are rst computed with respect to an orthogonal subspace basis and then transformed to the desired subspace basis. The orthogonal subspace basis is built by a Gram Schmidt process in one dimension and a tensor product construction to get to two dimensions. Incidentally this ....

R. M. Haralick. Digital step edges from zero crossing of second directional derivatives. IEEE Transactions on Pattern Analysis and Machine Intelligence, PAMI-6(1):58-68, January 1984.

....y : 2) Zuniga and Haralick [75] proposed a corner detector that is very similar to Kitchen and Rosenfeld s, with a corner being defined as a significant change in curvature along an edge curve. Rather than the quadratic polynomial used by Kitchen and Rosenfeld, they used Haralick s facet model [18, 19] to fit a least squares bi cubic curvature are only equivalent at critical points. CHAPTER 2. REVIEW OF 2D FEATURE DETECTION 14 polynomial surface to each pixel. In fact, Shah and Jain [65] showed that if Zuniga and Haralick s bi cubic polynomial is used to approximate the surface, then the only ....

R. M. Haralick. Digital step edges from zero-crossings of second directional derivatives. IEEE Transactions on Pattern Analysis and Machine Intelligence, 6(1):58--68, 1984. BIBLIOGRAPHY 112

....one of the fundamental operations in computer vision with numerous approaches to it. In an historical paper, Marr and Hildreth [11] introduced the theory of edge detection and described a method for determining the edges using the zero crossings of the Laplacian of Gaussian of an image. Haralick [6] determined edges by fitting polynomial functions to local image intensities and finding the zero crossings of the second directional derivative of the functions. Canny [2] determined edges by an optimization process and proposed an approximation to the optimal detector as the maxima of gradient ....

R. Haralick, "Digital Step Edges from Zero Crossing of Second Directional Derivatives," IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 6, 1984, pp. 58--68.

.... of the two methods commonly used for edge detection, the suppression of the local non maxima of the magnitude of the gradient of image intensity in the direction of this gradient [8] also called NMS) the other one being to consider edges as the zero crossings of the Laplacian of image intensity [10, 9]. NMS consists of: 1. Let a point #x; y#,wherex and y are integers and I#x; y# the intensity of pixel #x; y#. 2. Calculate the gradient of image intensity and its magnitude in #x; y#. 3. Estimate the magnitude of the gradient along the direction of the gradient in some neighborhood around #x; ....

Robert Haralick. Digital step edges from zero crossing of second directional derivatives. IEEE Transactions on Pattern Analysis and Machine Intelligence, 6(1):58--68, January 1984.

....systems for image processing and analysis. 11 Chapter 2 Tools for Image Analysis Steerable Filters 2. 1 Introduction Oriented filters are used in many vision and image processing tasks, such as texture analysis, edge detection, image data compression, motion analysis, and image enhancement [70, 27, 20, 43, 89, 33, 45, 4, 38, 57, 60]. In many of these tasks, it is necessary to apply filters of arbitrary orientation under adaptivecontrol, and to examine the filter output as a function of both orientation and phase. We will discuss techniques that allow the synthesis of a filter at arbitrary orientation and phase, and develop ....

....consider the 2 dimensional, circularly symmetric Gaussian function, G, written in Cartesian coordinates, x and y: G(x# y) e ; x 2 y 2 ) # (2:1) where scaling and normalization constants have been set to 1 for convenience. The directional derivative operator is steerable as is well known [25, 31, 43, 54, 60, 61, 62, 63, 73, 85]. Let us write the nth derivative of a Gaussian in the x direction as G n . Let ( represent the rotation operator, such that, for any function f(x# y) f (x# y)isf(x# y) rotated through an angle about the origin. The first x derivative of a Gaussian, G 0 ffi 1 ,is G 0 ffi 1 = ....

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R. M. Haralick. The digital step edge from zero crossings of second directional derivatives. IEEE Pat. Anal. Mach. Intell., 6(1):58--68, 1984.

....than in the Plane Fit Algorithm. In our approach, the lines are computed from only one set of support regions, in manner similar to the Principal Axis Algorithm. We evaluate the output quality of these algorithms with the help of a performance evaluation method from the University of Washington [7, 12, 13]. Experiments were run on three diverse image sets; the groundtruth having been received from the University of Washington for one of them. For the sake of classifying the algorithm with respect to quality and timing, the same experiments were also run for the Plane Fit and the Principal Axis ....

....Approaches We review some of the more widely referenced line algorithms in this section and explain the choice of the three that are used in this study. Most other line extraction algorithms base their approaches on the linking of edge pixels from the edge extraction algorithms of either [16] [5, 7] or [10] We briefly review these edge extractors here. The Marr Hildreth operator of [16] for example, convolves the image with the Laplacian of a Gaussian, and then declares step edges to occur at the zero crossings of the convolved image. The Canny operator [5] first smoothes the image by ....

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R.M. Haralick. Digital step edges from zero crossings of second directional derivatives. IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. PAMI6, pp. 58-68, Jan. 1984.

....edges represent important information about the objects in an image. The accuracy in the locations of detected edges will influence the performance of higher level processes such as object recognition. Because of the importance of edge detection, much work has been done in the last 30 years [1, 2, 5, 6, 7, 12, 16, 21, 23, 24, 28, 29, 30]. Intuitively, edges correspond to abrupt intensity changes. Formally, a function f(t) is said to have a discontinuity of order k (k 0) at t 0 , if the left and right kth derivatives at t 0 are different, i.e. f (k) t 0 ) 6= f (k) t 0 Gamma) where f (k) t 0 ) denotes limit ffl 0 f ....

....that the surface fitting approach is an effective way to detect edges, since it can detect local features as well as suppress the noise. The following sections discuss in detail three surface fitting schemes that form the basis of this proposed research. 19 2.4. 1 Polynomial Fitting Haralick [16] proposes that the underlying gray tone intensity function f takes the parametric form of a polynomial in the row and column coordinates. The polynomials used are the discrete Chebyshev polynomials, denoted in the one dimensional case by fP i (r) i = 0; 1; N Gamma 1g, where r 2 R, and R ....

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R. M. Haralick. Digital step edges from zero crossing of second directional derivatives. IEEE Trans. on Pattern Analysis and Machine Intelligence, 6(1):58--68, 1984.

....to feature detection. One way to approximate the partial derivatives needed to compute the differential invariants is to first fit a surface, and to then use the partial derivatives of the surface as estimates of the partial derivatives in the underlying image. One example of such an approach is [Haralick 84] Another example is [Meer and Weiss, 92] where both unweighted and Gaussian weighted L 2 norms are used to define and then find the closestfitting low order polynomial surface. 1.2 Summary The remainder of this paper is organized as follows. In Section 2 we introduce the modelfitting ....

R.M. Haralick, "Digital Step Edges from Zero Crossing of Second Directional Derivatives," IEEE Transactions on Pattern Analysis and Machine Intelligence, 6:58-- 68, 1984.

....of coherent edges suitable for use as a main structural framework for feature recognition and extraction has proven to be a goal directed task. Several approaches have been proposed in the literature. Some of the most frequently cited examples include the works of Marr and Hildreth [9] Haralick [10], and Canny [11] Since different edge extractors produce different results, there has been a great deal of argument in computer vision circles as to which approach is the optimal [12] but, it appears that performance is application dependent. In other words, edges cannot be considered in ....

R. Haralick, "Digital step edges from zero--crossings of second directional derivatives," IEEE Trans. on Patt. Anal. Mach. Intell., vol. PAMI--6, pp. 58--68, 1984.

....quantitative evaluation of di erent algorithms. The conditional probability of an assigned edge pixel given an ideal (true) edge pixel P( AE IE ) the conditional probability of a true edge pixel given an assigned edge pixel P( IE AE ) and the error distance were calculated for each case [11]. The binarization threshold was chosen so that P( AE IE ) P( IE AE ) in each case. The probabilistic measures P( AE IE ) and P( IE AE ) are the measures of false rejection and false detection, respectively, whereas, error distance is a measure of deviation from true edges. Thus edge ....

R. M. Haralick,\Digital step edges from zero crossing of second directional derivatives," IEEE Trans., Pattern Anal. Machine Intell., vol. PAMI-6, pp. 58-68, Jan. 1984.

....of the Roberts [33] Hueckel [19, 20] and Beaudet [3] operators. The progress has occurred in the reliability of detection and in the accuracy of edgel localization. Although the list is far from exhaustive, we feel that significant advances in edgel detection have been made through the works of [7, 9, 16, 24, 26, 29, 41], and have brought us to the current standard of edgel extraction. In our view, this line of development is now mature. The central issue for future development in intensity event detection is completeness of description rather than noise or localization performance. Illustrative of this new ....

Haralick, R,M. "Digital Step Edges from Zero Crossing of Second Directional Derivatives," PAMI, Vol. 6, p.58-68, 1984.

....proposed in order to extract the edges from an image and most of them consider estimates of the first or second derivative over some support as the appropriate quantity to characterize step edges. Respectively, peak and zero crossings detections are then performed for the extraction step [10] 2] [7]. This section deals with the application of the approximating operators proposed in the previous sections to the problem of edge detection. 7 EXPERIMENTAL RESULTS 15 6.1 Edges from the zero crossings By the derivative rule of convolution, smoothing the input image with the Gaussian filter ....

R.M.Haralick. Digital step edge from zero-crossing of second directional derivatives. IEEE Transactions on Pattern Analysis and Machine Intelligence, 6(1):58--68, January 1984.

....of all false jump points detected by the jump detection procedure and jAnBj jAj is the percentage of all true jump points missed by the procedure. The measurement dQ (A; B) is their average. Apparently, we could also use the weighted . x . 0 0.2 0. 6 1 0 0.1 0.5 1 (a) x y 0 0.2 0.6 1 0 0.2 0.5 1 (b) Figure 4: a) The detected jumps are the design points on line y = 5 and a point ( 8; 1) b) the detected jumps include the design points on line y = 2 only. The true JLC is the line y = ....

....jAj is the percentage of all true jump points missed by the procedure. The measurement dQ (A; B) is their average. Apparently, we could also use the weighted . x . 0 0.2 0.6 1 0 0.1 0.5 1 (a) x y 0 0.2 0. 6 1 0 0.2 0.5 1 (b) Figure 4: a) The detected jumps are the design points on line y = 5 and a point ( 8; 1) b) the detected jumps include the design points on line y = 2 only. The true JLC is the line y = 5. average: w jBnAj j Aj (1 w) jAnBj jAj ; where the weight 0 w 1 ....

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Haralick, R.M. (1984), \Digital step edges from zero crossing of second directional derivatives," IEEE Transactions on Pattern Analysis and Machine Intelligence 6, 58-68.

....work for this particular approach, our purpose is then not to develop it, but to study an alternative approach that is to consider surfaces as a set of voxels rather than a set of faces. This second definition comes from the classical edge detection widely used in the computer vision field (see [15, 16, 17]) edges are either maxima of the gradient norm in the direction of the gradient or zero crossings of the laplacian. In that case, surfaces are constituted by voxels. Morgenthaler and Rosenfeld proposed such a definition of a 3 D surface in 1981 [18] Their article defined the notion of simple ....

R.M. Haralick. Digital step edges from zero crossing of second directional derivatives. IEEE Transactions on Pattern Analysis and Machine Intelligence, 6:58--68, January 1984.

.... of the two methods commonly used for edge detection, the suppression of the local non maxima of the magnitude of the gradient of image intensity in the direction of this gradient [8] also called NMS) the other one being to consider edges as the zero crossings of the Laplacian of image intensity [10, 9]. NMS consists of: 1. Let a point (x; y) where x and y are integers and I(x; y) the intensity of pixel (x; y) 2. Calculate the gradient of image intensity and its magnitude in (x; y) 3. Estimate the magnitude of the gradient along the direction of the gradient in some neighborhood around (x; ....

Robert Haralick. Digital step edges from zero crossing of second directional derivatives. IEEE Transactions on Pattern Analysis and Machine Intelligence, 6(1):58--68, January 1984.

....The fitness of the genetically derived edge detector must be evaluated with respect to the Sobel operator previously described. Since the output image will be one bit (either a pixel is an edge or it is not) there are a number of methods of comparing binary edge outputs from edge detectors[3, 2], although in general, such methods are mathematically complex and too computationally intensiveto preform practically in siumulation. An alternative method, described here, is a simplified minimisation of under and over detection of edge pixels. In essence, the results of the application of a ....

R.M. Haralick. Digital step edges from zero crossing of second directional derivatives. IEEE Trans., Pattern Anal. Machine Intell.,PAMI-6:58--68, Jan. 1984.

....Figure 4: a) Original Image (b) Sobel Output 5.2 Fitness Evaluation The fitness of the genetically derived edge detector needs to be evaluated with respect to the Sobel operator described above. There are a number of methods commonly used to compare the performance of edge detectors [Heath97, Haralick84] although in general such methods are mathematically complex and too computationally intensive to perform practically in simulation. An alternative method, described here, is a simplified minimisation of under and over detection of edge pixels. In essence, the results of the application of a ....

Haralick, R.M., (1984) `Digital Step Edges from Zero Crossing of Second Directional Derivatives' IEEE Trans. on Pattern Analysis and Machine Intelligence, Vol PAMI6, pp 58-68

....a dimension does indeed reduce the performance of a detector. 2.2 Di#erential Invariant Feature Detectors The second major class of feature detectors consists of those based upon di#erential invariants. Well known examples include the Deriche corner detector [30] the Haralick step edge detector [44], and the Marr Hildreth step edge detector [70] As indicated by the name, these detectors base their detection decisions upon di#erential invariants estimated from the image data. For example, the Deriche detector is based upon the Hessian determinant, the Haralick detector upon the second ....

....data, almost any differential invariant can be computed, be it the gradient, Laplacian, or higher order 12 invariant. For example, Haralick proposed an edge detector that detects edges at negatively slopped zero crossings of the second directional derivative, taken in the direction of the gradient [44]. These zero crossings correspond to local maxima of the first order directional derivative taken in the direction of the gradient and are computed from the best fitting cubic surface. In a related paper [43] Haralick used a surface fitting approach to estimate the zero crossings of the first ....

R.M. Haralick. Digital step edges from zero crossing of second directional derivatives. IEEE Transactions on Pattern Analysis and Machine Intelligence, 6(1):58--68, January 1984.

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R.M. Haralick, "Digital Step Edges from Zero Crossings of Second Directional Derivatives," IEEE Trans. PAMI 6, pp. 58--68 (1984).

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R.M. Haralick, "Digital step edges from zero crossing of second directional derivatives," IP, vol. 11, no. 1, pp. 58--68, Jan. 1984.

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R. Haralick. Digital step edges from zero crossing of second directional derivatives. IEEE Transactions on Pattern Analysis and Machine Intelligence, 6:58--68, 1984.

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R.M. Haralick, Digital steps edges from zero crossing of second directional derivatives, IEEE Trans. Pattern Anal. Machine Intell. 6 (1) (1984) 58--68.

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R. M. Haralick, Digital step edges from zero crossings of second directional derivatives, IEEE rans. Pattern Anal. Mach. Intelligence 6, 58 --- 68 (1984).

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Haralick, R.M. (1984) "Digital Step Edges from Zero Crossings of Second Directional Derivatives," IEEETransacTJS--- on Pattern Analysis andMac#L# Intelligenc , Vol. 6, No. 1, pp. 113--129, February.

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R.M. Haralick. The digital step edge from zero crossings of second directional derivatives. IEEE PAMI, 6(1):58--68, 1984.

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R.M. Haralick, "Digital step edges from zero crossing of second directional derivatives," IEEE Transactions on PAMI, Vol. 11, No. 1, pp. 58-68, Jan. 1984.

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R. Haralick. Digital step edges from zero crossing of second directional derivatives. IEEE Transactions on Pattern Analysis and Machine Intelligence, 6(1):58-68, January 1984.

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R. Haralick. Digital step edges from zero crossing of second directional derivatives. IEEE Transactions on Pattern Analysis and Machine Intelligence, 6:58--68, 1984.

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R.M. Haralick, "Digital Step Edges from Zero Crossings of Second Directional Derivatives," IEEE Trans. PAMI 6, pp. 58--68 (1984).

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R.M. Haralick. Digital step edges from zero-crossings of second directional derivatives. IEEE Transactions on Pattern Analysis and Machine Intelligence, 6(1):58-68, 1984.

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R. M. Haralick, "Digital step edges from zero-crossings of second directional derivatives", IEEE Trans. PAMI, 6, No. 1, pp. 58-68, 1984.

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R. M. Haralick, "Digital step edges from zero crossing of second directional derivatives," IEEE Transactions Pattern Analysis and Machine Intelligence, vol. 6, no. 1, pp. 58--68, January 1984.

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R. Haralick. Digital step edges from zero crossing of second directional derivatives. IEEE Transactions on Pattern Analysis and Machine Intelligence, 6:58--68, 1984.

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R.M.Haralick, "Digital Step Edges from Zero Crossing of Second Directional Derivatives",IEEE Trans. on Pattern Analysis and Machine Intelligence, vol.PAMI-6, no.1, pp.5868, 1984.

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R. M. Haralick, "Digital step edges from zero crossing of second directional derivatives," IEEE Trans. Pattern Anal. Machine Intell, vol. PAMI-6, pp. 58--68, Jan. 1984.

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R. Haralick, "Digital step edges from zero crossing of second directional derivatives," IEEE Trans. Pattern Anal. Machine Intell., vol. PAMI-6, pp. 58--68, 1984.

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R. M. Haralick, "Digital step edges from zero crossing of second directional derivatives," vol. PAMI-6, no. 1, pp. 58--68,

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R. M. Haralick. Digital step edges from zero crossing of second directional derivatives. IEEE Transactions on Pattern Analysis and Machine Intelligence, PAMI-6(1):58--68, January 1984.

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R. M. Haralick, "Digital step edges from zero crossing of second directional derivatives," IEEE Trans. Pattern Anal. Machine Intell., vol. PAMI--6, no. 1, pp. 58--68, Jan. 1984.

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