SEGMENTATION OF LINES BY MEANS OF DOUGLAS PEUCKER APPLIED TO EFFECTIVE AREAS OF THE VISVALINGAM AND WHYATT ALGORITHM

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1 SEGMENTATION OF LINES BY MEANS OF DOUGLAS PEUCKER APPLIED TO EFFECTIVE AREAS OF THE VISVALINGAM AND WHYATT ALGORITHM Francisco J. Ariza López (*); José L. García Balboa (*), Juan F. Reinoso Gordo (*) (*) Grupo de Investigación en Ingeniería Cartográfica. Dpto. de Ingeniería Cartográfica, Geodésica y Fotogrametría. Universidad de Jaén. Campus Las Lagunillas s/n Jaén (Spain). fjariza@ujaen.es. Tel: (+34) jlbalboa@ujaen.es. Tel.: (+34) jreinoso@ugr.es. ABSTRACT: A line segmentation methodology is developed, based on a sinuosity criteria measured by means of the effective area as derived from the Visvalingam-Whyatt algorithm. Sections are determined by applying the Douglas-Peucker algorithm to a length/effective-area space representation. An experiment is carried out with four real line examples and results seems to be adequate. 1. INTRODUCTION Lines are the most abundant elements in cartography (Ariza, 2002). Lines are present as linear elements and also as perimeters of areal elements. Usually cartographic lines are not homogeneous, mainly being a succession of sections with different geometric and visual properties. Cartographic generalization is one of the main difficulties when dealing with digital cartography, but nowadays it is also one of the most important cartographic processes. The first studies of automatic generalization of linear elements were mainly devoted to simplification algorithms, their main target being to reduce the number of coordinates to store. The performance of a generalization algorithm varies with its control parameter and also depends very much on the line (homogeneity, sinuosity, and so on) to which it is applied. For this reason it would be useful to obtain an automatic segmentation and enrichment of lines in order to apply to each section the best algorithm and the appropriate parameter. Because some processes and algorithms are more adequate to some typologies of lines, the search of optimal automatic classification and segmentation methodologies is a first goal in line generalization. Some authors note the importance of the segmentation process; Jasinski (1990) indicates as an essential goal the identification of the graphical structure of the line; Buttenfield (1991) and Dutton (1999) propose splitting the line in homogeneous sections for a better algorithm and parameters selection; for Visvalingam and Williamsom (1995) automatic segmentation is a pending topic for research; Richardson and Mackaness (1999) highlight the importance of segmentation techniques because of the sensibility of generalization methods to different line types; and for Plazanet (1997) segmentation is a process that must take place before generalization in order to take into account local characteristics of the feature. For Müller (1986) cartographers have always tended to preserve the degree of complexity of each perceived section of a line. Skopelity and Tsoulos (1999) consider that segmentation can be achieved by using the same measures as those being used to characterize a line. The segmentation of a line is related to: Homogeneity: The conservation of certain characteristics along a section (Skopelity and Tsoulos, 1999). Sinuosity: The quality of a line having many changes of direction or specific curvature behaviour such as spirals, hairpins, and so on (Plazanet et al, 1995). Complexity: The quality of a line being more sinuous, less homogenous or more dense (Plazanet et al, 1995). Critical points: Those that mark changes in a line and can be used to limit sections. Line segmentation methodologies developed in previous work can be classified as:

2 Measure based methods: The idea is to use sound measures for detecting homogeneous sections. A example of this is the work of Skopeliti and Tsoulos (1999), which applies the fractal dimension for natural linear elements segmentation. Point removal methods: Simplification algorithms are used so that, after the removal, the remaining points indicate section limits. Buttenfield (1991), searching for the subdivision of a line in portions by using the Douglas-Peucker algorithm, does not speak explicitly about segmentation but her portions can be considered as sections. Visvalingam and Williamson (1995) point out the possibility of using the Visvalingam and Whyatt (1993) algorithm in a segmentation process. Inflection points methods: There is a search for inflection points within the line because a non homogeneous spatial distribution implies different sections. Plazanet (1995, 1996, 1997) and Plazanet et al. (1995, 1998) propose a hierarchical segmentation method based on the human perception studies of Mokhtarian and Mackworth (1986, 1992). Intersection with tendency methods: The distribution of intersections between the original line and its tendency (obtained by smoothing) is analyzed. It is understood that a non homogeneous spatial distribution intersection implies different sections. Burghardt (2002) develops such a kind method as a previous step for smoothing. Wavelets-based methods: Line filtering in a length-curvature space by wavelets coefficients allows us to detect changes in the curvature behaviour of the line. When using this method for line simplification, García and Ariza (2000) point out the possible use of wavelets for line segmention. In this study we develop and apply a segmentation methodology based on a sinuosity criteria measured by means of the effective area as derived from the Visvalingam-Whyatt algorithm. The paper contents are divided into three sections: i) methodology, including an introduction, and a method overview and tuning sections, ii) results obtained for four cases being analyzed and, iii) a general conclusions section. 2. METHODOLOGY This section is divided into three subsections. The first one is an introduction to the basis and hypothesis of the method and also presents data which will be used in the experiment; the second subsection presents a brief overview of the steps of the method; and the third subsection explains the tuning needed for the parameters that intervene in the developed process. 2.1 Introduction Taking into account previous studies of line segmentation, our proposal uses well known algorithms like Visvalingam and Whyatt (1990, 1993) and Douglas-Peucker (Douglas and Peucker, 1973) not for line simplification (García, 1998; García et al, 1999; Reinoso et al, 2001) but for line segmentation. The process can be summarized briefly: a Douglas- Peucker point selection algorithm is applied to an effective area diagram calculated by the Visvalingam and Whyatt algorithm. The base concept of the process is thus the effective area A of the Visvalingam and Whyatt algorithm. This A can be defined for each point P i of a curve C such as the area of the triangle {P i-1, P i, P i+1 }. The meaning of A i is the shift of the line when removing the P i point from the line curve C. The Visvalingam and Whyatt is applied in successive steps eliminating only one point at each time. Out hypotheses are: More sinuous line sections will have A values greater than less sinuous line sections. The Douglas-Peucker algorithm is able to detect a change in sinuosity when applied to an effective area diagram. It is important to note here that the proposed sections are preliminary sections and perhaps a merging process of similar and contiguous sections would be necessary. Here this process is not addressed; our work is devoted only to obtaining an adequate segmentation of the line. The described method is applied to a sample of lines of the Mapa Topográfico Nacional 1: (MTN25) from the Instituto Geográfico Nacional de España (IGN) (1996a, 1996b, 1996c, 1997). The sample is formed by four curves, belonging to the communication layer, and these are of different categories and characteristics: N-323 (motorway), A- 316 (A-road), Almadén (mountain track) and Jabalcuz (mountain track). Figure 1 shows a general view of the selected elements.

3 Figure 1 - General view of the four line sample used in the experiment 2.2. Method overview Such as it will be presented in the following sections the process is: 1. The Visvalingam and Whyatt algorithm is applied to derive the effective area A i of each point of a line curve C. 2. The original line C, which extends in an X,Y space, is substituted by a function F(D i, A i ) which represents the A i (OY vertical axe) versus line length (OX horizontal axe) accumulated up to the point P i of the line C. 3. For each point P i effective areas are integrated for a fixed neighborhood of distance L and the vertical axe OY is changed to a logarithmic scale: F(D i, Log(A i )). 4. A gaussian filter G with parameter is applied to the previous function: G(F(D i, Log(A i )); ). 5. The Douglas-Peucker algorithm with a DP parameter is applied to the function G(F(D i, Log(A i )); ) in order to determine the points which bound homogeneous sections. The first step of the method consists of applying the Visvalingam and Whyatt algorithm to each curve. As an example, Figure 2 shows the behaviour of the effective area versus the length along curve for the case of the A-316 curve. Here it can be appreciated that the greater portion of the total effective area is concentrated at a reduced set of points (peaks in the figure), those that will be the only ones remaining after a strong filtering with this algorithm. This graph is not enough for a segmentation taking into account sinuosity, but is the basis for further operations. Effective area [Ha] Length (m) Figure 2 - Effective area diagram for the A-316 curve Because the idea in mind is to associate sinuous sections with greater values of effective area, for each point P i we consider a neighborhood of L meters and accumulate the A j(l) values of this set of points: SLog ( Ai ) Log ( A j ) 10 j / j i ; dis( i, j) L j Values are added in a logarithmic scale because otherwise peak points, such as those of Figure 3, will dominate the representation. Figure 3 shows the new representation of the A-316 curve. Here we can interpret the sinuosity behaviour of the curve:

4 A horizontal behaviour can be associated with a section where sinuosity is stable because the sum of effective areas is nearly constant. An incremental tendency represents a progressive increase of sinuosity because each following point accumulates more effective area than the previous. A decremental tendency represents a progressive decrease of sinuosity because each following point accumulates less effective area than the previous Slog(Ai) Length (m) Figure 3 - Accumulated effective area diagram (D i, Log(A i )) for A-316 curve, with a vertical log-scale and L = 1500 m Because of the interpretations given above, which are common to the four curves been analyzed and also corroborated by a detailed inspection of original curves, we believe that the representation of a curve in a (D i, Log(A i )) space is appropriate for detecting sinuosity sections. Therefore we propose the application of the Douglas-Peucker algorithm (Douglas and Peucker, 1973) in order to determine such sections. We look for the detection of critical points which determine a change in the tendency in a space representation like those of Figure 4. Here the process depends on the Douglas-Peucker tolerance or parameter DP which allows detection of more or less critical points. Figure 4 shows the application of the proposed method to the A-316 curve. Previous to the Douglas-Peucker point selection algorithm we can also apply a gaussian filter (of parameter ) in order to smooth the (D i, Log(A i )) representation Slog(Ai) Longitud (m) Figure 4 - Example of critical point detection by means of Douglas-Peucker algorithm (above) and corresponding segmentation of the A-316 curve (below)

5 2.3. Method tuning The method described in the previous section depends on three parameters: L = distance [meter] previous to a point, taken into account for accumulating effective areas. L = neighborhood of distance L /2 [meter] of a point that is taken into account for gaussian smoothing of the accumulated effective area representation. DP = Douglas-Peucker parameter [meter, log(meter 2 )] which is used to detect critical points on the accumulated effective area representation. All three parameters must be tuned in order to achieve an adequate performance, but in the interests of simplification, in a first step we have fixed L and L, so DP is the final control parameter of the process. A complete discussion of the tuning method can be found in Ariza and García (2004). Here we present only the results and some explanatory remarks about them. L parameter can not be too small because peaks points (remember Figure 2) would thus dominate the accumulation of effective areas, but neither can L be too large because the possibility of detecting small sinuosity sections would be lost. The L parameter also has a positive relation to the standard deviation of the effective area (Ariza and García, 2004), which means that a greater deviation corresponds to a greater L value. By means of a heuristic and iterative process with L values in the range [1000 m, 5000 m], and using analytical measures and a perceptive evaluation of the results, the optimized values are: L A-316 = 3000m, L Almadén = 2000m, L Jabalcuz = 1500m, L N-323 = 5000m. Gaussian filtering has no great importance in the process. It can be applied or not, depending on the appearance of the (D i, Log(A i )) representation, but final results are very similar for commonly used control values. This filter is controlled by a parameter which corresponds to the size of the neighborhood to be considered, measured in point units. Here we do not consider the size of the neighborhood in point units but in distance along the curve, from its mean point density, and because of this we call the parameter L instead of. But the controlling behaviour is the same for both, the greater the value the greater the smoothing. Here we have considered L = 1000m for all the curves, which indeed implies a larger neighborhood, measured in points, for curves with more points density. Once L and L are fixed it is the DP parameter which allows the detection of more or less detail for the homogeneous sections. It is important to remember that the Douglas-Peucker algorithm performs a recursive approach when working (Ballard, 1981; Buttenfield, 1991), which means that each detected section is included within other sections of a higher level, all of these being part of a binary tree of sections. In other words, we can fix a resolution level for the segmentation tree. Extreme cases occur when the whole curve is a unique section and when all segments of consecutive points of the original curve are considered as sections. This process is controlled by the DP parameter, and should be oriented to obtaining an adequate simplification/segmentation of the curve, in order to achieve the desired purpose Slog(Ai) Length A-316 Almadén Jabalcuz N-323 Figure 5 - Smoothed accumulated effective area diagram for each curve of the sample The geometric interpretation of the DP parameter is a little more difficult than L and L because it is applied to a distance/log(effective area) space, which is not a common space. Let us consider two extreme cases to further understand its meaning. When the representation shows a predominantly vertical behaviour, for instance a great step in the function, the base line of the Douglas-Peucker also shows a predominantly vertical behaviour, so the DP parameter has to be analyzed taking into account the horizontal axis which corresponds to a distance axis. When the representation shows a predominantly horizontal behaviour the base line of the Douglas-Peucker also shows a predominantly horizontal behaviour, so the DP parameter has to be analyzed taking into account the vertical axis which corresponds to the effective area accumulation. This is expressed in the form of the exponents of the base of the logarithm. Intermediate behaviours of the function result in a mixed distance-logarithmic exponent interpretation of the DP parameters values being applied. Nevertheless, the application of the Douglas-Peucker algorithm is only one way of

6 detecting critical points on the function's representation space, so no physical sense is considered in this case. Taking into account the considerations of the DP parameter we decided to apply the Douglas-Peucker algorithm to the representation of each curve function, as shown in Figure 5. After giving the DP parameter values in the interval [50, 450] some selected results are shown in Figure 6. DP Line A-316 Line Almadén Line Jabalcuz Line N sections 11 sections 14 sections 21 sections sections 6 sections 7 sections 6 sections sections 4 sections 5 sections 4 sections sections 3 sections 1 sección 3 sections sections 1 sección 1 sección 3 sections Figure 6 - Segmentation results of the sample for different values of the DP parameter in the range [50, 450]

7 3. RESULTS The results of the application of the proposed methodology are shown in Figure 6. It can be observed that the variation of DP allows us to obtain a greater or lesser number of sections. Taking into account only an ad oculi criteria some of the results can also be considered as adequate solutions, for instance the A-316 (DP = 250), Almadén and Jabalcuz (DP = 50) and N-323 (DP = 350 or DP = 450). Analyzing the original curves and the place where the critical points are detected, we can considered that the method is sensible to sinuosity changes. Here the hypothesis is that given a curve which has a particular geometry, this curve can be segmented in n sections of uniform sinuosity depending on the degree of sinuosity being considered. Thus, the evolution profiles of Figure 7 represent the sinuosity sections detected when applying different sinuosity criteria by means of the DP parameter values. In order to determine how many sections make up the best segmentation of a curve we propose to analyze the representation shown in Figure 7 where we can observe the evolution of the number of detected sections (vertical axis) versus the DP [50, 450] values (horizontal axis). Here two evolution patterns can be observed, those of curves A-316 and N-323 and those of curves Almadén and Jabalcuz. The first evolution pattern can be associated with less sinuous curves and the presence of a stability zone for the A-316 (n = 5, 4, 3) and the N-323 (n = 6, 4, 3) curves is very clear. The second pattern can be associated with more sinuous cases like the Almadén and Jabalcuz curves. Here the profiles have greater drops and the stability zone is not so clear (but it can be observed that Jabalcuz seems to stabilize at n = 5 and Almadén at n = 4, 3. Line A-316 Line N-323 Line Jabalcuz Line Almadén Figure 7 - Detected number of sections in each curve of the example for different values of the DP parameter in the range [50, 450]

8 In order to evaluate results they can be compared with those derived from the method suggested by Plazanet (Plazanet, 1995, 1996, 1997; Plazanet et al, 1995, 1998) when searching for a similar number of sections. The Plazanet method is very different to ours because it searches for sections in the set of inflection points. If we consider those solutions indicated at the beginning of this section, Figure 8 allows us a visual analysis of proposed sections. Both results are quite different. It is difficult to say which is the best for curve A-316 but Plazanet's result seems to be more logical. For the Almadén curve our proposal is capable of detecting sections of different sinuosity, but the application of Plazanet's method proposes some sections which seem to be capricious. The results of the application of the two methods to Jabalcuz and N-323 are different, but it is difficult to choose between them. Method Line A-316 Line Almadén Line Jabalcuz Line N-323 Ours 5 sections 11 sections 14 sections 3 sections Plazanet s 5 sections 11 sections 14 sections 3 sections Figure 8 - Segmentation results with our method and Plazanet s. 4. CONCLUSIONS The segmentation of curves is one of the more interesting aspects of automatic generalization because it is supposed that a good segmentation will facilitate the adaptation of generalization algorithms (an algorithm or its controlling parameter) to each section of a curve. In this study we have developed and tested a segmentation methodology based on sinuosity by applying the bases of the well known Visvalingam-Whyatt and Douglas-Peucker algorithms. The basic hypotheses are: The more sinuous a section is, the greater the effective area. The Douglas-Peucker algorithm is able to detect sinuosity changes in an effective area space. Our methodology has two main steps: i) calculation, by means of the Visvalingam-Whyatt algorithm, of the effective area of each point in a curve; and ii) the selection of points by means of the Douglas-Peucker algorithm applied to a space of effective areas. An analysis has been carried out of four curves, and the visual analysis of the results seems quite encouraging for the continuation of this research. While our results are similar in quality to those obtained by applying the segmentation method proposed by Plazanet, we have only studied four cases, so the sample size must be increased in order to obtain a deeper insight. For this reason these results can only be considered as a first approximation to this problem when using the effective area approach.

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