John D. Hobby. Polygonal approximations that minimize the number of inflections. In Proceedings of the Fourth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 93--102, January 1993.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....maximum distance between the points and l is minimized. The method is generalized for polygonal curves, not only closed polygons, and di#erent constraints are imposed according to the definition and specifications of the problem. Another method that minimizes the inflections is proposed by Hobby [7]. In this method a special data structure is designed for the set of points so that the direction and length of the approximating line between points can be determined. The main use of such a method may be approximating the scanned bitmap images to smooth polygons. Polygon simplification ....

J.D. Hobby. Polygonal approximations that minimize the number of inflections. In Proceedings of Fourth ACM-SIAM Symposium on Discrete Algorithms, pages 93--102, 1993.

.... 1 Introduction The line simplification problem is a well studied problem in various disciplines including geographic information systems [Buttenfield 85, Cromley 88, Douglas Peucker 73, Hershberger Snoeyink 92, Li Openshaw 92, McMaster 87] digital image analysis [Asano Katoh 93, Hobby 93, Kurozumi Davis 82] and computational geometry [Chan Chin 92, Eu Toussaint 94, Guibas et al. 93, Imai Iri 88, Melkman O Rourke 88] Often the input is a polygonal chain and a maximum allowed error ffl, and methods are described to obtain another polygonal chain with fewer ....

J.D. Hobby, Polygonal approximations that minimize the number of inflections. Proc. 4th ACM-SIAM Symp. on Discrete Algorithms (1993), pp. 93--102.

....increases its complexity by a logarithmic factor, which is tight. We also look briefly at the algorithmic issues involved. These are closely related to rounding techniques for robust handling of arrangements of line segments, although we ask slightly different questions. See papers by Hobby [11, 12, 13], Milenkovic [17, 18, 19, 20] and Guibas et al. 7, 8] for more on arrangements of segments. 2 Exact minimum link paths Under the real RAM model of computation, a minimum link path from s to t in P can be computed by a simple greedy algorithm. We let (P; s; t) denote the number of links in ....

J. D. Hobby. Polygonal approximations that minimize the number of inflections. In Proceedings of the Fourth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 93--102, 1993.

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John D. Hobby. Polygonal approximations that minimize the number of inflections. In Proceedings of the Fourth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 93--102, January 1993.

....Minimization and Grid Constraints 2 polygonal approximation algorithm can actually improve the outlines by eliminating noise induced jaggies. The algorithm that generated Figure 1b tries to do this by emphasizing smoothness and quality of fit with little regard to reducing the vertex count [7]. The goal of the present work is to retain the advantages of smooth, well fitting outlines, but simplify them enough so that they can be stored compactly. What are these advantages Consider an electronic library project involving scanned document images as described by O Gorman [14] When a ....

....aligned with the scanner s pixel grid. It is important to correct the skew because small angle rotations lead to annoying image defects, especially when viewed on a computer screen. Agazzi, Church, and Gale [1] have found that a good way to do the required rotations is to use the algorithm from [7] to generate outlines, then rotate the outlines and scan convert them to get another binary image. It would be much better to retain the outlines, since the page images need to be rescaled to cope with a wide range of output devices as explained in [1] In other words, we need simpler outlines to ....

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John D. Hobby. Polygonal approximations that minimize the number of inflections. In Proceedings of the Fourth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 93--102, January 1993.

....input page as a binary image, we first perform skew detection and page layout analysis up to character segmentation [1] 9] The bitmaps of the segmented character images are then sent to a clustering procedure, which produces an initial set of clusters. Bitmap averaging, by smooth shading as in [5, 7], is then applied to each of these clusters to obtain an outline character that represents the cluster. Individual character bitmaps are then compared to the cluster average, and mismatches are rejected. The rejected images are compared to other cluster averages to find possible matches. After ....

J. D. Hobby. Polygonal approximations that minimize the number of inflections. In Proceedings of the Fourth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 93--102, Jan. 1993.

....of as a collection of black blobs on a white background, a more effective way is to find an outline description and apply a smoothing algorithm to the outlines. The goal of the smoothing algorithm should be to determine the most likely underlying shape that could lead to the observed input. See [6,7] for details. Once the image is in outline form, it can easily be rescaled or rotated. A scan conversion algorithm can convert the outlines to pixels at the desired resolution. In this correspondence, we describe a method that combines the outline method with another technique, adaptive ....

....to obtain better outlines as in Figure 1c, and then use a polygon scan conversion algorithm to convert the outlines to pixels at the desired resolution. The precise definition of better is beyond the scope of this correspondence but it should involve a reduction in high frequency noise. See [7] for a discussion of the goals and [6,7] for appropriate polygon smoothing algorithms. The outlines of the black regions can be found via a well known sweep line algorithm that updates lists of outline vertices each time it considers a new row of the input image. See [8] for details. The final ....

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J.D. Hobby", "Polygonal Approximations that Minimize the Number of Inflections", Proc. of the Fourth Annual ACM-SIAM Symp. on Discrete Algorithms, Jan. 1993, pp. 93-102. - 16 -

....goal of smoothing out the jaggies caused by noise from the scanning process. The algorithm actually used to generate Figure 1b achieves such smoothness by minimizing the number of inflections Space Efficient Outlines from Image Data via Vertex Minimization and Grid Constraints 2 in the output [6]; i.e. vertices can be classified as left turns or right turns and the number of alternations between the two is minimized subject to a bound on the approximation error. See [6] for a complete description of this algorithm and discussions of competing polygonal approximation algorithms. The ....

....of inflections Space Efficient Outlines from Image Data via Vertex Minimization and Grid Constraints 2 in the output [6] i.e. vertices can be classified as left turns or right turns and the number of alternations between the two is minimized subject to a bound on the approximation error. See [6] for a complete description of this algorithm and discussions of competing polygonal approximation algorithms. The only other approach that minimizes inflections is Montanari s idea of minimizing the perimeter subject to the error bounds [7] See also Sklansky [12, 13, 14] Unfortunately, this ....

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John D. Hobby. Polygonal approximations that minimize the number of inflections. In Proceedings of the Fourth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 93--102, January 1993.

....outlines and produce output that fits the input as closely as possible while smoothing out wobbles up to some specified magnitude. There should be no attempt to minimize the number of vertices in the output at the expense of quality of fit. A good choice of smoothing algorithm is Hobby s algorithm [8]. This algorithm minimizes the number of inflections in the resulting polygonal outlines subject to a bound on the deviation. It produces a description of a class of outlines that obey these criteria so that the output can be chosen to fit the input as well as possible. The post processing step ....

....the most successful preprocessing strategies. 2.1 Preprocessing by Smoothing the Outlines One approach to preprocessing is to take each rasterized input image and try to guess the underlying shape that generated it. This can be thought of as inverse rasterization, and the algorithm of [8] can do it as shown in Figure 2. a) b) Figure 2: a) A comparison of the original outlines (thin lines) with the smoothed versions (thick lines) for one of input images from Figure 1; b) the smoothed outlines filled in. The smoothing algorithm has an error tolerance parameter ffl that must be ....

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John D. Hobby. Polygonal approximations that minimize the number of inflections. In Proceedings of the Fourth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 93--102, January 1993.

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J.D. Hobby. Polygonal approximations that minimize the number of inflections. Proc. 4th Annual ACM-SIAM Symp. on Disc. Algorithms, 1993, 93--102. 10

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