) K.Wall and P.E.Danielsson,"A fast sequential method for polygonal approximation of digitized Curves," Comput.Vision,Graphics,Image Processing,vol. 28,pp  Home/Search   Document Not in Database   Summary   Related Articles   Check

....from the discrete trace of image contours (i.e. edge pixels) to a continuous or linked set of points through which the curve passes. Essentially, two kinds of encoding and representation schemes of smooth curves have been proposed in the computational vision literature: curve fitting algorithms [33, 32, 38, 8] and orientation chaining algorithms [14, 24, 15, 6, 1, 29, 26] In the first scheme, a set of curves is fit to the contour by minimizing a certain error measure. The points where curves meet are retained as knot points and are essentially arbitrary. Most such techniques employ polygonal ....

K. Wall and P.-E. Danielsson. A fast sequential method for polygonal approximation of digitized curves. CVGIP, 28:220--227, 1984.

....from the discrete trace of image contours (i.e. edge pixels) to a continuous or linked set of points through which the curve passes. Essentially, two kinds of encoding and representation schemes of smooth curves have been proposed in the computational vision literature: curve fitting algorithms [33, 32, 38, 8] and orientation chaining algorithms [14, 24, 15, 6, 1, 29, 26] In the first scheme, a set of curves is fit to the contour by minimizing a certain error measure. The points where curves meet are retained as knot points and are essentially arbitrary. Most such techniques employ polygonal ....

K. Wall and P.-E. Danielsson. A fast sequential method for polygonal approximation of digitized curves. CVGIP, 28:220--227, 1984.

....representing them. Janssen and Vossepoel [13] have studied the different vectorization methods. It is worth noting that all of them, except for the method called separate pixels , require the previous stage for their application. These methods are: The Area method. Used by Wall and Danielson [14]. The criterion used in this method is the maximum area skewness allowed per unit length comprised between vectorization and approximated curve. The last point of the curve not violating this criterion becomes the main node in the vectorization, the process being repeated up to the end of the ....

K. Wall and P.E. Danielsson. "A fast sequential method for polygonal approximation of digitized curves". Compt. Vision Graphics Image Process. Vol. 28. 220-227, 1984.

....analysis is the interpretation and recognition of contours by means of contour tracing procedures. The subsequent analysis of contours can be sought in many different ways, such as contour polygonization in order to obtain a simpler representation of two dimensional discrete contour points [34] [35] and critical point detection in order to extract features of contours [36] 37] 38] 39] 40] As far as the internal characteristics of shape is concerned, the medial axis transformation (MAT) is a typical methodology [41] Thinning technology [42] is also widely applied to reduce the ....

K. Wall and P. Danielsson, "A fast sequential method for polygonal approximation of digitized curves," Computer Vision, Graphics and Image Processing, vol. 28, pp. 220--227, 1984.

.... threshold value from the approximating segments [Ramer 1972] the arc to chord deviation (maximum perpendicular distance of any point on the arc from the chord approximating that arc) Chung et al. 1994] an area deviation threshold, possibly per unit length of an approximating polygon segment [Wall and Danielsson 1984] Each of these criteria can be used in di erent ways too. For instance Ramer [1972] ensures that each point satis es the criteria. Chung et al. 1994] however minimise the total arc to chord deviation between the polygon and the curve. The following criteria are used to detect signi cant ....

K. Wall and P. Danielsson. A fast sequential method for polygonal approximation of digitised curves. Computer Vision, Graphics and Image Processing, 28:220-227, 1984.

....top left extremity of the selected area and walking round the edge in a 4 connective way (i.e. move up, left, down or right) until the initial point is reached again. 3 Fitting Lines to Shape Outlines There has been much written on the approximation of continuous functions by straight lines [5, 13, 17, 18]. The general conclusion is that there is no correct solution to such problems. Methods that have been employed to perform such tasks generally fall into two categories (i) Points of maximum curvature or (ii) Iterative end point fits. In (i) the continuous function is segmented by picking ....

K. Wall and P.E. Danielsson. A fast sequential method for polygonal approximation of digitised curves. Computer Vision, Graphics and Image Processing, 28:220--227, April 1984.

....1 oe 2 (A0g oe (x Gamma l 0 ) Gamma A1g oe (x Gamma l 1 ) where g oe (x) 1 oe p 2 e Gamma x 2 2oe 2 is the Gaussian filter. This response has two extrema (Fig. 1.b) which generate a double step in 2D. On these contours, an edge following [6] and a polygonal approximation [16, 9] generate two close segments, theoretically parallel, which are the edges of the line contour. However, a double contour can also be generated by a double step, shaped like a staircase (Fig. 1.c) With the same notations, a staircase model can be defined in the following manner [15] E(x) A0H(x ....

K. Wall and P. Danielsson. A Fast Sequential Method for Polygonal Approximation of Digitized Curves. Computer Vision, Graphics and Image Processing, 28:220-- 227, 1984.

....threshold. This latter criteria allows us to discard the chains corresponding to entities which are out of interest, for which the step in the signal is too low. The remaining chains are finally divided into straight line segments using the polygonal approximation algorithm by Wall and Danielsson [14]. These segments represent for the most part the external contours of the blocks of buildings. They now have to be matched with the segments extracted from the cadastral map. 2.2 The Cadastral Map The map must be first transformed into a form suitable to the subsequent processing. The initial ....

K. Wall and P. Danielsson. A Fast Sequential Method for Polygonal Approximation of Digitized Curves. Computer Vision, Graphics and Image Processing, 28:220--227, 1984.

....is defined as a ratio between the maximum deviation and the length of the segment. Recently, Rosin has proposed other possible measures of significance [18] and we plan to explore this further in the coming months. We have also used for many years an iterative method, that of Wall and Danielsson [23], enhanced in our team with a direction change marking procedure to preserve the angular points. The method only needs a single threshold, on the ratio between the algebraic surface and the length of the segments. It is fast and efficient, but not as precise as the former. On the other hand, Rosin ....

....the algebraic surface and the length of the segments. It is fast and efficient, but not as precise as the former. On the other hand, Rosin and West s method tends to split up the lines around a junction into too many small segments (see Fig. 8) This is a direct consequence (a) Wall Danielsson [23] with typical threshold = 20. b) Rosin West [19] Figure 8. Comparison of two polygonal approximation methods applied to Fig. 6. of its preciseness and of the previously mentioned displacement of junction points by skeleton based methods. Based on these experiments, and depending on the ....

K. Wall and P. Danielsson. A Fast Sequential Method for Polygonal Approximation of Digitized Curves. Computer Vision, Graphics and Image Processing, 28:220--227, 1984.

.... r Gamma 1 Gamma cos M Delta 2r sin M = 1 2 tan 2M : Another possibility, used by Rosin and West [27] is to sum the normalised maximum deviations from each line segment, giving E Sigma1=L (M ) 1 2M tan 2M : Rather than normalise the maximum deviation Wall and Danielson [34] normalised the area deviation E 1=L (M ) r Gamma Gamma M 2 sin 2 M Delta 2 sin M : A rather different measure, which was maximised by Sato [30] is the length of the polygon L = 2Mr sin M : However, it can be seen that none of the above criteria provide a FOM that is ....

K. Wall and P.E. Danielsson. A fast sequential method for polygonal approximation of digitized curves. Computer Vision, Graphics and Image Processing, 28:220--227, 1984.

....1D curves which gives good results even for a small number of points. The method is based on Wall and Danielsson s fast, sequential polygonization concept where the break point criterion is the area between the curve and the approximating line segment, divided by the length of the line segment [9]. 2. Dimension by Polygonization We suggest a new method for estimation of fractal dimension from 1D curves obtained by scanning 2D images, based on the polygonization method of Wall and Danielsson [9] Wall Danielsson s method steps from point to point through an ordered sequence of points, ....

....the curve and the approximating line segment, divided by the length of the line segment [9] 2. Dimension by Polygonization We suggest a new method for estimation of fractal dimension from 1D curves obtained by scanning 2D images, based on the polygonization method of Wall and Danielsson [9]. Wall Danielsson s method steps from point to point through an ordered sequence of points, and outputs the previous point as a new break point if the area deviation per unit length of the approximating line segment exceeds a prespeci ed tolerance, TWD . Using the previous break point as the ....

K. Wall and P.E. Danielsson. A Fast Sequential Method for Polygonal Approximation of Digitized Curves. Computer Vision, Graphics, and Image Processing, Vol. 28, 1984, pp. 220-227.

....directions [12] There are also a lot of polygonal approximation algorithms that do not try to minimize the vertex count. Ramer s recursive subdivision algorithm is a classic example [20] Wall and Danielsson s method is fast and popular in application domains such as optical character recognition [29]. It chooses the next output vertex by keeping track of the area between the segment and corresponding part of the input polygon. None of the algorithms cited above impose grid constraints except in a very theoretical context or by restricting the output vertices to be a subset of the input ....

Karin Wall and Per-Erik Danielsson. A fast sequential method for polygonal approximation of digitized curves. Computer Vision Graphics and Image Processing, 28(2):220--227, November 1984.

....1D curves which gives good results even for a small number of points. The method is based on Wall and Danielsson s fast, sequential polygonization concept where the break point criterion is the area between the curve and the approximating line segment, divided by the length of the line segment [9]. 2. Fractal Dimension by Polygonization We suggest a new method for estimation of fractal dimension from 1D curves obtained by scanning 2D images, based on the polygonization method of Wall and Danielsson [9] Wall Danielsson s method steps from point to point through an ordered sequence of ....

....curve and the approximating line segment, divided by the length of the line segment [9] 2. Fractal Dimension by Polygonization We suggest a new method for estimation of fractal dimension from 1D curves obtained by scanning 2D images, based on the polygonization method of Wall and Danielsson [9]. Wall Danielsson s method steps from point to point through an ordered sequence of points, and outputs the previous point as a new break point if the area deviation per unit length of the approximating line segment exceeds a prespeci ed tolerance, T WD . Using the previous break point as the ....

K. Wall and P.E. Danielsson. A Fast Sequential Method for Polygonal Approximation of Digitized Curves. Computer Vision, Graphics, and Image Processing, Vol. 28, 1984, pp. 220-227.

....intersection method of Williams [35, 36] Sklansky and Gonzalez [30] Leung and Yang [17] and Badi i and Peikari [4] All but [4] run in linear time; 36] is unique in that the output vertices are not restricted to be a subset of the input. Other linear time algorithms include Wall and Danielsson [34] which measures error by keeping track of the area between the input polygon and the approximation. Algorithms by Davis [6] Hemminger and PomalazaR aez [12] and Ansari and Delp [2] provide little control over the approximation error. None of the linear time algorithms achieves an optimal ....

....approximated by simpler, less noisy polygons that can be processed more efficiently. The algorithm presented here is an ideal way to do this. Henry Baird of AT T Bell Laboratories plans to incorporate our implementation into his OCR system. He is currently using the Wall and Danielsson algorithm [34]; he plans to continue to use that algorithm for easy cases, and to switch to our more elaborate algorithm when the system requires more accuracy. Figures 10 and 11 in Section 6 give examples of how the polygons produced by our algorithm compare to those from faster algorithms: they appear much ....

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K. Wall and P.-E. Danielsson. A fast sequential method for polygonal approximation of digitized curves. Comput. Vision Gr. and Image Proc., 28(2):220--227, Nov. 1984.

....a labelled skeleton and we can later retrieve the thickness of each line. ffl The pixels of the skeleton image are chained using the Freeman chain code [4] ffl The chains yielded by the previous step are converted to a list of vectors with Wall and Daniellson s polygonal approximation algorithm [19], which performs in linear time on the chain codes. At this stage, the thickness criterion which was computed during thinning is taken into account as an additional stop factor (we start a new segment if the thickness deviates too much from the current one, and an average thickness is associated ....

K. Wall and P. Danielsson. A Fast Sequential Method for Polygonal Approximation of Digitized Curves. Computer Vision, Graphics and Image Processing, 28:220--227, 1984.

....Thus the input to the segment intersection algorithm is a set of polygonal approximations to map features as shown in Figure 7b. For testing purposes, the polygonal approximations were done with the Wall and Danielsson algorithm with the maximum average error set at 10 Gamma4 degrees latitude [26]. a) b) Figure 7: a) Roads from the map database with data points marked by dots; b) the same roads with polygonal approximations that bypass explicit intersections. The region shown is about 330 meters by 220 meters. This application is interesting because the map data contains degeneracies, ....

....8614 2740 3833 77 3.42 4.87 47.6 47.8 122.0 122.2 10 Gamma4 10829 2912 4560 77 4.25 5.90 47.6 47.8 122.2 122.4 10 Gamma4 12752 12046 15171 115 5.88 8. 55 Table 1: Results of test runs on the indicated portions of the map database as preprocessed by the Wall and Danielsson algorithm [26]. The grid spacing is ffl degrees latitude, and the n, k, and k 0 columns give the number of segments, the number of intersection points, and the number of tolerance square intersections. The sweepline column gives the average number of segments on the sweepline, and the last two columns give ....

Karin Wall and Per-Erik Danielsson. A fast sequential method for polygonal approximation of digitized curves. Computer Vision Graphics and Image Processing, 28(2):220--227, November 1984.

....especially those based on the distance transform. We have therefore implemented a skeletonization method, first proposed by Sanniti di Baja [1] and based on the 3 4 distance transform. This is followed by polygonal approximation; after having used Wall and Danielsson s iterative algorithm [10] for many years, we have now switched to an algorithm which essentially works without any threshold, that of Rosin and West [6] based on a previous algorithm proposed by Lowe, and which also allows us to recognize arcs subsequently. We are currently developping a post processing method to better ....

K. Wall and P. Danielsson. A Fast Sequential Method for Polygonal Approximation of Digitized Curves. Computer Vision, Graphics and Image Processing, 28:220--227, 1984.

....same way, the configuration Arc Arc generates a hypothesis of a circle or ellipse ( in the case of a convex curve ) or a general curve. Figure 5 shows an example of segmentation of a grouping into segments and an arc. The algorithm of segmentation is inspired by a method of polygonal approximation [18]. Each grouping is evaluated sequentially and divided according to the most important points of interest ( that is, the points corresponding to important changes of curvature and orientation ) Other cues can be used in this step to improve the segmentation ; for example, we also used corner ....

K. Wall and P.-E. Danielson. A fast sequential method for polygonal approximation of digitized curves. In CVGIP - Image Understanding, volume 28, pages 220--227, Feb 1984.

....of the approximation of a curve by a line segment. Four types of error measures have been investigated. P Q l(x) x h A B Figure 4.1: The approximation of a curve by a line segment. See the text for an explanation. 64 Model Based Bottom Up Grouping ffl The measure proposed by Wall and Danielsson [108]. The curve and the line segment cut out regions from the plane. The difference between the area of the regions to the right of the line segment and those to its left is called the signed area between the line and the curve. The error measure proposed by Wall and Danielsson is the absolute value ....

....an advantage, because of the special role of spatial structure in such applications. In the application of our method to polygonal approximation of curves, various measures of fit were tested, some of them known from the literature. It was shown that the measure proposed by Wall and Danielsson [108] and the 159 maximal distance measure produce the best results. In the application of our method to image segmentation, we used the constant grey level model, which is the simplest possible choice. We also considered the linear grey level model for regions. It was explained why the segmentation ....

K. Wall and P.-E. Danielsson. A fast sequential method for polygonal approximation of digitized curves. Computer Vision, Graphicsand Image Processing, 28:220--227, 1984.

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) K.Wall and P.E.Danielsson,"A fast sequential method for polygonal approximation of digitized Curves," Comput.Vision,Graphics,Image Processing,vol. 28,pp

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) K.Wall and P.E.Danielsson,"A fast sequential method for polygonal approximation of digitized Curves," Comput.Vision,Graphics,Image Processing,vol. 28,pp

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K. Wall and P.-E. Danielsson. A fast sequential method for polygonal approximation of digitized curves. Computer Vision, Graphicsand Image Processing, 28:220--227, 1984.

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K. Wall and P. Danielsson. A Fast Sequential Method for Polygonal Approximation of Digitized Curves. Computer Vision, Graphics and Image Processing, 28:220--227, 1984.

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K. Wall, and P. Danielsson, "A fast sequential method for polygonal approximation of digitized curves", Comput. Vision Gr.Image Process. 28 pp 220-227 (1984).

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K. Wall and P. Danielsson. A Fast Sequential Method for Polygonal Approximation of Digitized Curves. Computer Vision, Graphics and Image Processing, 28:220227, 1984.

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