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1 This article was downloaded by: [Wuhan University] On: 29 May 2014, At: 21:32 Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: Registered office: Mortimer House, Mortimer Street, London W1T 3JH, UK Marine Geodesy Publication details, including instructions for authors and subscription information: A Simplification of Ria Coastline with Geomorphologic Characteristics Preserved Tinghua Ai a, Qi Zhou a, Xiang Zhang a, Yafeng Huang ab & Mengjie Zhou a a School of Resource and Environment Sciences, Wuhan University, Wuhan, China b Nanjing Research Institute of Electronics Technology, Nanjing, China Published online: 29 May To cite this article: Tinghua Ai, Qi Zhou, Xiang Zhang, Yafeng Huang & Mengjie Zhou (2014) A Simplification of Ria Coastline with Geomorphologic Characteristics Preserved, Marine Geodesy, 37:2, , DOI: / To link to this article: PLEASE SCROLL DOWN FOR ARTICLE Taylor & Francis makes every effort to ensure the accuracy of all the information (the Content ) contained in the publications on our platform. However, Taylor & Francis, our agents, and our licensors make no representations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of the Content. Any opinions and views expressed in this publication are the opinions and views of the authors, and are not the views of or endorsed by Taylor & Francis. The accuracy of the Content should not be relied upon and should be independently verified with primary sources of information. Taylor and Francis shall not be liable for any losses, actions, claims, proceedings, demands, costs, expenses, damages, and other liabilities whatsoever or howsoever caused arising directly or indirectly in connection with, in relation to or arising out of the use of the Content. This article may be used for research, teaching, and private study purposes. Any substantial or systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form to anyone is expressly forbidden. Terms & Conditions of access and use can be found at

2 Marine Geodesy, 37: , 2014 Copyright Taylor & Francis Group, LLC ISSN: print / X online DOI: / A Simplification of Ria Coastline with Geomorphologic Characteristics Preserved TINGHUA AI, 1 QI ZHOU, 1 XIANG ZHANG, 1 YAFENG HUANG, 1,2 AND MENGJIE ZHOU 1 1 School of Resource and Environment Sciences, Wuhan University, Wuhan, China 2 Nanjing Research Institute of Electronics Technology, Nanjing, China 1. Introduction To meet the requirements of multi-scale mapping in maritime applications, marine charts need to be produced at various levels of detail (LOD) using map generalization. As a prominent geographic feature, the coastline has to be generalized considering the geomorphologic characteristics rather than from a pure geometric perspective. Morphologic and domain-specific constraints (e.g., safety) should be incorporated in designing a coastline generalization algorithm. Motivated by the generalization of ria coastlines, this article proposes a simplification algorithm that is specific to coastlines. An analysis of ria coasts results in several morphologic constraints that have to be satisfied in coastline generalization, such as the dendritic pattern of estuaries. To satisfy these constraints, a hierarchical estuary tree model is first established by Delaunay triangulation, which helps to represent the dendritic pattern of ria coastlines. Minor estuaries are then deleted to achieve a reasonable coastline simplification. To imitate manual generalization, an indicator is designed to calculate the importance of estuaries in a context dependent manner. By comparing with a well-known bend simply algorithm, we show that the presented method can maintain dendritic pattern of coastline and is free from self-intersection and also granted for navigation safety. This article also demonstrates that the proposed approach is applicable to coastlines in general. Keywords map generalization, ria coastline, coastline simplification, Delaunay triangulation Marine charts provide a simplified two dimensional representation of coasts and the seabed and are essential for navigation purposes. In GIS or digital chart software, it is possible to view charts with zooming and panning functionalities, but switching between different scales should be supported by map generalization. To clearly present charts at different scales, data have to be simplified, selected, and symbolized. Most of these processes are currently being automated in the field of map generalization. But the generalization of marine charts needs to be further investigated mainly due to a different set of cartographic Received 20 September 2013; accepted 7 March Address correspondence to Xiang Zhang, School of Resource and Environment Sciences, Wuhan University, 129 LuoYu Road, Wuhan , China. xiang.zhang@whu.edu.cn Color versions of one or more of the figures in the article can be found online at 167

3 168 T. Ai et al. Figure 1. Fragment of a chart shown in a ria coast region. requirements. For example, the main structure of a coastline (e.g., a hierarchy of estuaries in Figure 1) and the elongated shape of individual estuaries should be maintained. Then characteristic points (e.g., estuary sources) should also be kept where they may be connected to tributary rivers (Figure 1). These characteristics should be taken into consideration when designing a simplification algorithm for coastlines (this is discussed in detail in the next section). In automated generalization, line generalization has been studied over decades and many algorithms have been proposed or implemented in commercial GIS software (Regnauld and McMaster 2007). For example, Douglas and Peucker (1973) proposed the Douglas-Peucker (DP) algorithm. Although it is regarded as the most commonly used simplification algorithm (Shi and Cheung 2006; Regnauld and McMaster 2007), DP algorithm suffers from self-intersection issues (Saalfeld 1999). Later, DP algorithm was extended for various generalization purposes (Zhang and Tian 1997; Saalfeld 1999; Chen et al. 2010), in which Saafeld (1999) showed how the self-intersection problem could be corrected. However, as a compression algorithm, DP algorithm is not considered appropriate for generalizing cartographic lines such as coastlines (Li 1993; Christensen 1999; Guilbert

4 Simplification of Ria Coastline 169 and Lin 2007). Other earlier line simplification approaches include Visvalingam-Whyatt algorithm (Visvalingam and Whyatt 1993), Li-Openshaw s natural principle algorithm (Li 1993), and a topology-safe algorithm for simplifying polygons such as land use maps (Berg and van Kreveld 1998). The idea underlying these algorithms is either to select a set of important points from the original curve to depict its main characteristics, or to use a curve with fewer different points to replace the original points (Balboa and López 2000). In our view, their main objective is to simplify lines while keeping positional accuracy and approximating the original shapes as much as possible. In general, these algorithms can be classified as geometry oriented simplification as they are designed from a geometric point of view. A pure geometry based generalization algorithm does not always ensure adequate cartographic quality, and therefore geographical components or higher level semantics should be considered for specific geographical features (Ai 2007; Lüscher, Weibel, and Mackaness 2008; Touya, Duchêne, and Ruas 2010). This also applies to marine charts as geomorphologic properties of coastlines should be kept to obtain satisfactory results. Mainly two types of lines, namely the isobathic lines and coastlines, are involved in chart generalization. To respect complex and domain-specific constraints, optimizationbased techniques such as the snake model were introduced to the generalization of cartographic lines (Burghardt 2005). To generalize isobathic lines on marine charts, Guilbert and his colleagues (Guilbert and Lin 2007; Guilbert and Saux 2008) extended the snake model to further respect navigation safety. The generalization of coastlines, on the other hand, was previously less studied. Wang (1998) made an earlier attempt to coastline generalization based on bend simplification. However, the generalization of coastlines is still challenging because coastlines can take many complex landforms (e.g., estuary systems, rias, fjords) and specific morphologic characteristics (e.g., morphology of estuaries) should be explicitly handled. These characteristics are not adequately considered by the previous approaches. As a result of geomorphologic processes (e.g., wave erosion), bend structures can be observed on coastlines. For example, seaward bends may represent spits, sand bars or peninsulas while landward bends may represent bays, rias, or fjords. This may imply that simplification of bends could be a starting point for coastline generalization. Generalization algorithms that can handle bends were proposed by Li (1993), Wang (1998), Gold and Thibault (2001), and Guilbert and Saux (2008). These algorithms only handle and generalize bends individually and may produce acceptable results where estuaries coincide with individual bends. However, estuaries are usually hidden in deep and hierarchically branched coastlines, and an individual bend does not correspond to an entire estuary (e.g., as in Figure 1). As such, these approaches cannot fully respect the geomorphologic characteristics such as the morphology of estuaries and the dendritic pattern (as demonstrated in our experiments). We take a viewpoint that generalization of geographical features is guided by the characteristics of the specific features (e.g., coastlines in our case). This idea is not new and has become acknowledged in topographic generalization (Ai 2007; Lüscher, Weibel, and Mackaness 2008; Touya, Duchêne, and Ruas 2010). That is, the realization of an algorithm is via the manipulation of low-level geometries (e.g., characteristic points or bends), but the structure recognition and the generalization decision are made on a higher geographic or semantic level (e.g., the geomorphologic properties of coastlines in our case). This article presents an improved coastline simplification algorithm based on the work of Ai, Guo, and Liu (2000) and Ai (2007). Unlike the previous approaches, the proposed algorithm considers bend structures in a hierarchical structure with multi-branches tree

5 170 T. Ai et al. rather than binary tree. This could enable an effective detection of estuaries and their hierarchies in complex coastal landforms such as rias, fjords, or nested bays and peninsulas. Note that the hierarchical structure used in our approach is extended from (Ai, Guo, and Liu 2000; Ai 2007). But the idea of recognizing geographic features from the hierarchy and generalizing the features according to the specific characteristics was not fully explored in their approaches. For example, Ai (2007) used the hierarchical structure mainly for detecting bends and characteristic points for contour generalization. Besides, a binary tree structure was suggested in previous approach, which simplifies bends in isolation. This in our case would prevent estuaries from being kept in their entirety (cf. figure 7 in Ai 2007). This study improves the above two methods in that it proposes an improved hierarchical structure suitable for describing estuary systems as well as an indicator that measures the importance of estuaries in context. The algorithm is designed to integrate the specific characteristics of coastlines, such as the shape and integrity of estuaries, their hierarchical structures and possibly fractal properties. In addition, navigation safety can be guaranteed in our approach (as shown in our experiments). Therefore, it should be better suited for the generalization of irregular and indented coastlines than the reviewed approaches. Note that, although the presented algorithm was motivated by the generalization of ria coastlines, it can also be applied to coastlines in general (as shown in our experiments). This is because in essence our approach deals with bend structures, which is a main concern in coastline generalization (Wang 1998). This article is organized as follows. In the next section, characteristics of ria coastline are analyzed giving insight into the constraints and strategies used in subsequent generalization. In section 3, hierarchical structures of estuaries are established by extracting skeleton network from Delaunay triangulation; a method to detect main (trunk) estuaries from the skeleton network is described; also detected estuary entities at different hierarchy levels are characterized. Section 4 describes a ria coastline simplification process by removing the minor estuaries; the objective of the removal is to keep fractal properties. In the experiments (section 5), our algorithm is compared with Wang and Muller s (WM) algorithm (Wang and Muller 1998) with coastlines of different characteristics to validate our algorithm; properties of both algorithms are also discussed. Finally, conclusions and future work are given. 2. Constraints of Ria Coastline Generalization 2.1 Geomorphologic Characteristics of Ria Coastline A ria is a drowned river valley due to a rise in sea levels, a sink in the local land, or the combination of the two; the valley was formed by river erosion in ancient time and is now partly flooded by the ocean (Bird 2008). An analysis of the main morphological characteristic of ria coastlines shows a significant branching pattern of estuaries. That is, the main estuary splits into the secondary estuaries, which continue to split into the next level estuaries. A typical structure of rias is their hierarchical, treelike, organization of estuaries, which is inherited from the dendritic drainage pattern of the flooded river valleys (Shepard 1937). In addition, some geographical units can be observed in ria coastlines, such as estuary sources, estuary crossings and mouths (Figure 2a). An estuary source is the point of maximum elevation in the whole range of the estuary, with narrow water surface and

6 Simplification of Ria Coastline 171 Figure 2. Related concepts demonstrated by an example estuary system: (a) geographic features in the estuary system; (b)-(d) computational concepts used to derive the estuary hierarchy. shallow body water. A small river or steam may run into an estuary at its source (Figure 1). An estuary crossing is the section where multiple estuaries join (Figure 2a). An estuary mouth is a wide location where water in the dendritic ria meets the ocean. Usually, the water surface of an estuary becomes wider from its source to the crossing or mouth (Figures 1 and 2a).

7 172 T. Ai et al. 2.2 Constraints of Ria Coastline Simplification At least two types of constraints should be satisfied when simplifying ria coastlines. One is the graphic constraint of legibility, which aims to make the marine chart more readable. For example, the bend area and the spacing between adjacent bends should be large enough to avoid visual clutter. The other is the constraint that aims to preserve morphologic properties of the coastline (e.g. the dendritic structure of a ria shoreline and the shapes of the estuaries). We find it useful to divide geographical entities within a ria coastline into three levels. The first level is a dendritic, tree-like, river valley system outlined by the coastline. The second level includes estuaries shaped by bends in the coastline. The third level includes estuary sources or other characteristic points along the coastline. Accordingly, the following constraints should be respected in coastline generalization: Preserving the dendritic pattern of coastlines. This means that important relations (e.g., confluence and hierarchy) between estuaries should be preserved. In addition, fractal properties that are essential in characterizing coastlines should be retained as much as possible. Retaining the shape characteristic that the width of estuaries gradually increases as water runs from the source to the mouth (or the estuary crossing). This ensures that the geomorphologic properties of coastlines are still recognizable after the generalization. Keeping important characteristic points such as estuary sources. Estuary sources are important features of an estuary and are sometimes connected to rivers. Therefore, they should not be eliminated if the estuary is kept. In addition, navigation safety which is perhaps the most important domain-specific constraint for marine charts should always be respected during the generalization process. 2.3 Generalization Strategies In general, the ria coastline generalization aims to remove unnecessary details and to maintain major morphologic characteristics. The generalization result should preserve regional geographical characteristics such as the dendritic structure and the shape of estuaries as discussed previously. To simplify a coastline, big and higher level estuaries should be retained while minor estuaries that are not discernible at the smaller scale can be eliminated. Reasonable removal of small estuaries should be able to preserve or even enhance the dendritic pattern of the coastline. Therefore, minor estuary removal is regarded as a basic operation for coastline generalization. To realize the generalization, two main problems need be addressed. First, how to recognize estuaries from those usually deep and hierarchically branched coastlines, and, second, how to measure and quantify the detected estuaries so that their importance in context can be properly captured. The proposed approach mainly addresses these two problems. In short, estuaries are recognized using principles of visual perception. Estuary selection focuses on maintaining the dendritic pattern (i.e. fractal properties) of coastlines and density contrast of estuary areas. To determine the importance of estuaries, we consider not only the characteristics of individual estuaries (e.g., their length and width) but also the structural and contextual properties among estuaries (e.g., their intervals, densities and hierarchical levels).

8 Simplification of Ria Coastline Pattern Analysis and Feature Extraction by Delaunay Triangulation Recognizing the dendritic pattern of coastlines is crucial for ria coastline simplification. In doing so, we transform the two-dimensional representation of dendritic branches in an estuary system into the one-dimensional representation, which we term skeletons of the estuary system. The hierarchical organization of estuaries in the system is established based on these skeletons. Delaunay triangulation is used in our approach to extract skeletons as it proves to be a powerful tool for skeletonization and for contextual analysis in cartographic applications (Jones, Bundy, and Ware 1995; Poorten and Jones 1999; Ai, Guo, and Liu 2000; Gold and Thibault 2001). We describe the process in detail in the following sections. These include the construction of constraint Delaunay triangulation, the exaction of skeleton network, the recognition of estuary hierarchy and the characterization of detected estuaries. 3.1 Constructing Constrained Delaunay Triangulation The points and segments that represent a ria coastline are used to calculate the Delaunay triangulation. To avoid the appearance of narrow triangles, we apply a linear interpolation to insert more points to long line segments (Ai, Guo, and Liu 2000). This, in effect, results in a constrained Delaunay triangulation (CDT) with the points as input and the segments as constrained lines. But our computation is more efficient than the original CDT. Note that we only consider the triangles on the sea side of the coastline for the purpose of estuary recognition and structuring. Skeletons are extracted on top of a classification of triangles in the triangulation. Ai, Guo, and Liu (2000) proposed a way to classify triangles into types I, II, and III according to the number of neighboring (adjacent) triangles. This approach only considers the relationship of spatial adjacency from a pure geometric point of view. In our approach, triangles are classified into four types according to their geographical meaning implied in a ria coastline: Mouth triangles are those which are connected to the non-enclosed external area that is outside the extent of the triangulation (e.g., the green triangle in Figure 2b). Estuary crossing triangles are those which have three adjacent valid triangles (e.g., the blue triangles in Figure 2b). Estuary source triangles are those which are adjacent to only one valid triangle (e.g., the red triangles in Figure 2b). Connecting triangles are the remaining valid triangles which are adjacent to two other triangles (e.g., the white triangles in Figure 2b). Figure 2 demonstrates a strong correlation between the triangles in our classification and the geographical entities in a ria coastline. 3.2 Extracting Skeleton Network This section extracts a skeleton network consisting of network nodes and skeleton segments from a coastline. This is done by a tracing procedure that starts from a network node and ends with another node. Note that mouth, crossing and source triangles are treated as nodes in this procedure. Three types of paths can be traced by walking in the triangulation network. The first type of path is traced from a mouth triangle to a crossing triangle. The second type of path is traced between two immediate crossing triangles. The third type of

9 174 T. Ai et al. Figure 3. Two graphic representations of the estuary hierarchy in Figure 2(d): (a) the tree of estuary skeletons (numbers are triangle indices in the triangulation network; colors are estuary levels), and (b) the tree of estuary bends (numbers are vertex indices in the coastline). path is delineated between a crossing and a source triangle. Midpoints along the triangle boundaries are used to depict the skeleton segments, except that for source triangles a triangle vertex is used to end a skeleton, and that for crossing triangles a centroid is added to connect the three outgoing skeletons. The tracing procedure traverses all triangles and produces a network of skeletons (Figure 2c). A detailed account of the tracing procedure is described in Ai, Guo, and Liu (2000). 3.3 Establishing Estuary Hierarchies The skeleton network constructed above consists of skeleton segments, showing a dendritic structure. Not every skeleton segment, however, corresponds to a complete estuary feature. Some estuaries may be structured by several connected skeletons. For example, the biggest estuary in Figure 2d stretches from the mouth on the bottom to the bend on the top (see also Figure 4a). Therefore, it is necessary to group related skeleton segments such that meaningful estuaries at different hierarchy levels can be formed. In (Ai 2007) a similar hierarchy is represented by binary tree for contours. Our approach did not use binary tree (Figure 3) because it does not allow for the recognition of trunk and tributary estuaries. Either the longest skeleton connection or the maximum connecting angle could be used as a principle to group skeletons that correspond to meaningful estuaries. We used the longest skeleton connection rule because the maximum connecting angle rule is highly sensitive to local data noise in our case. The process that identifies meaningful sets of skeletons and organizes them in a hierarchy is described as follows. 1. The process searches all paths in the skeleton network starting from the root node (i.e., the mouth triangle) to all leaf nodes (i.e., source triangles). The length of each path is recorded during the search. The longest path (the red path in Figure 2d) is selected as the first level estuary skeleton. 2. For each skeleton segment connected to the first level estuary skeleton, it searches all paths starting from the crossing node (i.e., the estuary crossing triangles) to all the leaf nodes in this branch. Again, the longest path in this branch is selected as the second level estuary skeletons (the blue paths in Figure 2d). Repeating this process from a parent level estuary results in estuary skeletons of the next level, which in the end forms a hierarchy of estuary skeletons.

10 Simplification of Ria Coastline 175 To be more specific, the red, blue, green, and yellow paths in Figure 2d represent estuary skeletons at the first, second, third and fourth levels, respectively. Basically, the hierarchical structure can be formalized using the graph representation with two numbering systems (the italic numbers with gray background and the bold numbers in Figure 2d). In the first representation, a set of skeletons representing a trunk estuary is modeled as a node, and the confluence relationship between estuaries is modeled as an edge. Each node is labeled by a sequence of numbers (these numbers are indices of mouth, crossing and source triangles from which a set of skeletons that defines a trunk estuary is traced). For example, the trunk estuary defined by triangles is confluent into the estuary by This establishes a tree of estuary skeletons which describes the hierarchy of the estuary features (see Figure 3a). In the second representation, portions of the coastline that define estuary bends are modeled as nodes. The relationships that a lower level bend is included into a higher level bend are modeled as edges in the tree (Figure 3b). The numbers in the tree indicate the indices of the start and end vertices on the coastline that define an estuary bend. The result is a tree of coastline segments describing the boundaries of estuary bends. The tree is established firstly by resolving the indices of the vertices at which the mouth or crossing triangles split the coastline into bends of different sizes, and then by substituting the numbers in the estuary tree (Figure 3a) with their corresponding split vertex indices. For instance, the estuary bend corresponding to the skeletons is enclosed by vertices 2 7 which subsume two additional estuary bends (Figures 3a and 3b). These hierarchical representations allows for easy access to estuaries, their skeletons, and bend coordinates by computers, which is useful in the subsequent calculation and manipulation of estuary entities for coastline generalization. 3.4 Characterizing Estuary Entities This section calculates several parameters of previously detected estuary entities based on the above-mentioned tree representations. These parameters include estuary length and width, which are required for the proposed coastline simplification. Estuary length Let E be an estuary, Skeleton(E) is its corresponding skeletons. Let E T be the trunk of the estuary E (e.g. the areas in green in Figure 4), Skeleton(E T ) is the set of skeletons corresponding to E T (e.g. the skeletons in blue in Figure 4). These relations can be derived from the tree in Figure 3a. The length of an estuary can be given by the length of skeletons that constitute the trunk estuary in Eq. (1): Len(E) = Len(E T ) = Len(Skeleton(E T )) (1) Estuary width Let Bend(E) be the bend of estuary E. For example, the estuary delineated by vertices 2 7 in Figure 2d is the union of bends A, B, and C in Figure 4b. Let child or lower level estuaries of an estuary E be denoted as E c, the bends of all lower level estuaries can be denoted by the set: {Bend(E c ) 1 c n}. An Estuary bend, Bend(E), always consists of one trunk estuary E T (e.g., A) and sometimes of several lower level estuaries E c (e.g., B

11 176 T. Ai et al. Figure 4. Example of estuary selection (colors indicate trunk estuaries at different hierarchy levels): (a) trunk estuaries in the original coastline, and (b) simplification by estuary removal. and C), so the area of the trunk estuary Area(E T ) is given by Eq. (2): Area(E T ) = Area(Bend(E)) n Area(Bend(E c )) (2) As estuaries are commonly elongated shapes, the width of an estuary E can be approximately calculated by the average width of its trunk estuary in Eq. (3): Width(E) = Area(E Area(Bend(E)) n Area(Bend(E c )) T ) Len(E T ) = c=1 Len(Skeletion(E T )) c=1 (3) 4. Ria Coastline Simplification by Estuary Selection After the estuary recognition, this section presents an estuary selection/removal process based on the removal of minor estuaries. This is important for the geo-oriented ria coastline generalization. The objective is to maintain major estuaries and the dendritic fractal characteristic of the ria coastline. In the following, we discuss two related problems: (1) how to measure the importance of estuaries and (2) how to determine the number of selected estuaries for a target scale. When measuring the importance of estuaries, one should consider not only the characteristics of estuaries (e.g., estuary level, length and width) but also the characteristic of their geographical context, such as the spacing between adjacent estuaries. We thereby define a composite indicator of an estuary E by the sum of the area of the trunk estuary E T and the area of its child estuaries Child i (E) at lower levels. This indicator (Complex Area)isgiven

12 Simplification of Ria Coastline 177 by Eq. (4): n Complex Area(E) = Area(E T ) + Area(Child i (E)) (4) i=1 In general, a longer estuary involving more child estuaries will yield a larger value for this index and is of greater significance. The use of Complex Area in estuary selection should improve the coastline generalization. There are several reasons. First, as the composite area of an estuary is necessarily larger than the total area of its child estuaries, no dangling estuaries would be introduced. Second, the use of this indicator may reflect the fact that estuaries with more child estuaries are more important. For example, the estuaries A and B (marked in green in Figure 4a) are of similar width and length. With the composite indicator, A is considered more important than B as areas of C and D are also involved in A. This imitates the result of manual generalization. The two would otherwise be regarded as being equally important if estuary width and length were used for the selection. Ideally, the dendritic pattern of a coastline can be characterized by its fractal dimension. As a result, if the dendritic pattern of a coastline is retained after generalization, its fractal dimension should be similar to the one before generalization. By fixing the fractal dimension, we are able to calculate the length of the coastline at a target scale using the equation proposed by Wang and Wu (1998): L MA = L MB ( MA M B ) 1 D (5) where M B and M A are the scale denominators before and after generalization, respectively; L MB and L MA are the length of curves before and after generalization; and D is the fractal dimension of the original curve. Note that we used the method given by Richardson and later by Mandelbrot (Richardson 1961; Mandelbrot 1967; Buttenfield 1989) to calculate fractal dimensions for our coastline data. The equation suggests that if a curve is generalized to length L MA with appropriate generalization, its fractal dimension after the generalization is similar to the original curve. Note that the empirically formulated Radical Law (Töpfer and Pillewizer 1966) bears a similar form to Eq. (5) which is derived from the fractal theory. Such a similarity may not be a coincidence as was implied by Jiang, Liu, and Jia (2013) in their research toward a universal rule for map generalization. They suggested that the Radical Law may resemble the underlying idea of the fractal property of geographic space (Jiang, Liu, and Jia 2013). Eq. (5) can indicate which objects to select based on a ranking of the objects. In our case, we generalize a coastline to the target length L MA by iteratively removing minor estuaries in an ascending order of their composite areas calculated by Eq. (4). This process proceeds until the length of the resulting curve approaches to L MF. More specifically, we denote f(t) as the coastline length after removing the t th (1 t n) estuary. Clearly, f(t) is a monotonic decreasing function (Figure 5). The larger the t, the faster f(t) decreases. Before the iteration, f(t) L MF ; when f(t i ) L MA the iteration stops. Then we compare the value of f(t-1)-l MA and f(t)- L MA. If the former is smaller the number of deleted estuaries equals to t-1 (i.e. we backtrack to the previous iteration); otherwise t estuaries should be removed. Figure 4a illustrates a source coastline with a number of 14 estuaries. The simplified coastline after removing seven minor estuaries using our approach is depicted in Figure 4b.

13 178 T. Ai et al. Figure 5. The relationship between the length of generalized curve and the number of removed estuaries. 5. Experiments and Discussion 5.1 Experimental Setting The proposed algorithm was implemented with C ++ using Esri s ArcObjects, which provide robust implementation of many geometric computations including Delaunay triangulation. Our algorithm is validated mainly by applying the algorithm to the generalization of a portion of Chesapeake Bay (Figure 6a), which is at a scale of approximately 1:100K. Chesapeake Bay is a typical ria coastline that lies off the Atlantic Ocean, surrounded by Maryland and Virginia in the United States. As a comparison, we evaluated our algorithm against WM Bend Simplify algorithm (available in the toolbox of ArcGIS, 9.3). This is because WM algorithm (Wang and Muller 1998) is also based on a bend detection technique and can generate comparable results. Both qualitative and numerical evaluations are analyzed to obtain more insights into the algorithms. Additionally, our algorithm was tested against other datasets such as Kodiak Islands, Alaska (scale 1:1M) to show its potential in generalizing other coastlines in general. The data sets used are downloaded from NOAA. 1 A comparison between the two algorithms generally requires that both results are at the same (or similar) map scale. This, however, cannot be strictly met because the parameters of both algorithms differ greatly. In our algorithm, the generalized curve length (and hence the number of estuaries removed) is controlled explicitly by the target scale using Eq. (5). This makes it easy to relate our results to map scales. WM algorithm, on the other hand, uses the length of a reference bend baseline (or the diameter of a half circle) to reason whether a bend is removed or kept. But this does not necessarily mean that a bend with a baseline shorter than the reference baseline will be removed (Wang and Muller 1998). This makes it more difficult to be related to a specific scale. Therefore, the results of WM algorithm were tuned to achieve a target curve length that is similar to the results of our approach at different scales. 1 NOAA National Geophysical Data Center, Coastline extracted (e.g. WVS, GSHHG), Date Retrieved,

14 Simplification of Ria Coastline 179 Figure 6. The process of ria coastline generalization illustrated with part of Chesapeake Bay, United States (source and generalized lines in (e) and (f) are in red and black, respectively).

15 180 T. Ai et al. In addition to the horizontal comparison, it is also useful to inspect the vertical behavior of both algorithms, i.e., how the generalized results change as scale decreases. 5.1 Intermediate Processes on Real Data Figure 6 demonstrates the intermediate steps of the Chesapeake coastline generalization using our algorithm. The main steps are visualized, including the construction of Delaunay triangulation and classification of triangles (Figure 6b), extraction of estuary skeletons (Figure 6c), organization of hierarchy of estuary skeletons (Figure 6d), ria coastline simplification by estuary selection/removal (the solution is approximately at 1:150K, Figure 6e). Figure 6f shows a solution obtained by WM algorithm with a small parameter value (i.e., baseline length) applied. Note that the triangulation is always constructed on the sea side of the coastline to ensure the detection and removal of estuary bends. If there are several separated rias or estuary systems in the coastline, our algorithm constructs several trees (or a forest). This is demonstrated in Figure 6d where two red trunk estuaries were detected. For closed coastlines such as Kodiak Islands, a forest of trees is constructed. 5.2 Qualitative Evaluation Figure 7 shows the generalization results obtained by our algorithm and by WM algorithm side by side. Our algorithm generalized the source data at 1:200K, 1:250K, and 1:500K, respectively (Figures 7a c). Wang and Muller s algorithm simplified lines with similar curve lengths to our algorithm (Figures 7d f). Although the results in the same row of Figure 7 are not necessarily generalized at exactly the same levels of detail, this comparison still gives a good reference for identifying some of the important properties of the algorithms. In general, both algorithms can explicitly handle bend structures. Figures 7a c demonstrate clearly how the main (or trunk) estuaries and the structure of the ria coastline become discernable as scale decreases. Although WM algorithm resembles our results at smaller baseline thresholds (see, e.g., Figures 6e and 6f), WM algorithm does not well preserve the geomorphologic characteristics of the coastline as scale becomes smaller (Figures 7d f). Several observations can be made from a more detailed comparison. First, WM algorithm does not ensure estuary bends being kept in their entirety (highlighted by the squares in Figures 7d f, where portions correspond to estuary sources were cut out). On the contrary, our algorithm recognized geographic features (i.e., estuaries) embedded in geometries and hence provided a better solution. That is, when an estuary is kept, the area from the estuary source to its crossing or the mouth is also maintained (Figures 7a c). Second, the shape of estuaries was better preserved in our approach than in WM algorithm. That is, our algorithm retains the trend that an estuary becomes wider as water runs from its source to its crossing or the mouth. This characteristic was not respected by WM algorithm in several places as highlighted by circles in Figures 7d f. In the highlighted areas, the original sharp estuary bends were generalized as rounded, bay-like shapes, largely distorting the morphologic characteristic of the coastline. The third observation is that our algorithm simplifies coastlines by eliminating detected estuary bends in the hierarchy, and as a result, lines were generalized toward water areas. In addition, no smoothing or sampling process was used before or after the simplification. Both settings in our algorithm can ensure navigation safety a constraint that cannot be guaranteed by WM algorithm (see also circles in Figure 7). These reasons also reduce the loss of positional accuracy in our results (i.e., the positional accuracy is the same for the remained vertices).

16 Simplification of Ria Coastline 181 Figure 7. Chesapeake coastline, United States (1:100K) generalized at different levels of detail by the two algorithms: (a)-(c) results by our algorithm; (d)-(f) results by Wang and Muller s algorithm. Squares highlight areas where estuaries are not kept in their entirety estuary sources are cut. Circles indicate where the shapes of estuaries and safety constraint are not preserved. Triangles indicate where self-intersections may occur.

17 182 T. Ai et al. Figure 8. Kodiak Islands, United States (1:1M) generalized at different levels of detail by the two algorithms: (a)-(c) results by our algorithm; (d)-(f) results by Wang and Muller s algorithm. Source and generalized lines are in red and black, respectively. Insets show how self-intersections were created by Wang and Muller s algorithm but avoided by our algorithm. A closer look at Figure 8 confirms the above observations. In this test the algorithms were applied to Kodiak Islands (Alaska, USA), a coastline that does not have a significant dendritic pattern or hierarchical estuary structure. This test exemplifies that our algorithm is also applicable to the generalization of general coastlines. This is because essentially

18 Simplification of Ria Coastline 183 Table 1 Length (L) and fractal dimension (D) of the Chesapeake coastline generalized at different scales by our algorithm vs. Wang and Muller s (WM) Measure 1:100K (Original) 1:150K 1:200K 1:250K 1:500K Method L (km) Our algorithm D L (km) WM D our algorithm deals with bend structures. Our algorithm organizes bends in hierarchies and recognizes geographic features beyond individual bends, so it can deals with curves with hierarches of bends (e.g., ria or fjord coastlines) better than WM algorithm. Furthermore, Figure 8 shows that topological consistency (e.g., self-intersection) was maintained in our algorithm (see insets in Figures 8a c). Wang and Muller (1998) showed that self-intersections may occur in WM algorithm when replacing a bend by its baseline, and it is possible to correct it. However, such a correction in WM algorithm is only limited to individual bends. It fails to correct intersections between different bends. This is clearly demonstrated in Figures 8d f, where bends on both sides of the narrow channel intersect. Our algorithm, however, can avoid such self-intersections. The reason is that any removed bend is replaced by a crossing triangle edge, which by definition does not intersect the coastline boundary (see Figure 4b). Therefore, in our algorithm no intersection was created at the narrow channel in Figure 8, no matter at which scale the data was generalized. 5.3 Numerical Analysis From the above qualitative evaluation, our approach maintained the dendritic pattern of ria (or fjord) coastlines as well as the shape of geographic features such as estuaries at different scales. The main (or trunk) estuaries (geographic features) rather than individual bends (geometric structures) are maintained, covering the areas from estuary sources to estuary crossings (or the mouth). These help to recognize morphologic characteristics of the coastlines at smaller scales. Numerical analysis was carried out that further shows that in our algorithm the geographical characteristics (e.g. the dendritic pattern) of the coastlines were better preserved. Tables 1 and 2 show the change of fractal dimensions of both Chesapeake and Kodiak Table 2 Length (L) and fractal dimension (D) of Kodiak Islands generalized at different scales by our algorithm vs. Wang and Muller s (WM) Measure 1:1M(Original) 1:2M 1:4M 1:5M Method L (km) Our algorithm D L (km) WM D

19 184 T. Ai et al. coastlines as they were generalized in Figures 6 8. We claim that with appropriate generalization the fractal dimension (or complexity) of a coastline could remain the same or decrease slowly. Table 1 indicates that in our algorithm the fractal dimension of the Chesapeake coastline stays more or less the same as the coastline becomes shorter. This is probably because Chesapeake Bay is highly dendritic (initial dimension 1.414). For the Kodiak coastline, a slight drop in its fractal dimension when generalized by our algorithm can be observed (Table 2). For both coastlines generalized by WM algorithm, their fractal dimensions drop much faster than our algorithm as scale decreases, although the difference in progression of fractal values in the Kodiak case is not as significant as in the Chesapeake case. This may be explained by the fact that the Kodiak coastline (initial dimension 1.297) is not as complex as the Chesapeake coastline, and that the generalization degree for the Kodiak coastline is larger. To summarize, the numerical analysis suggests that our algorithm can better preserve the morphologic characteristics of coastlines in terms of fractal dimensions (this can also be confirmed by looking at the generalization results in Figures 6 8). 6. Conclusion and Outlook This article presents an algorithm for ria coastline simplification. The algorithm employs a hierarchical structure that effectively detects estuaries involved in the irregular and indented coastline. A composite indicator that captures the importance of an estuary in context is proposed. Based on this indicator, minor estuaries can be identified in the hierarchy of estuaries. This eventually realizes the coastline generalization through minor estuary removal. This algorithm was evaluated against a well-known bend simplify algorithm (Wang and Muller 1998). The results show that our algorithm is able to preserve the geomorphologic characteristics of coastlines (e.g., the dendritic pattern of the coastline and the shape of estuaries). Also, it can respect the constraint for navigation safety and is free from self-intersection, two major concerns in generalizing marine charts. In addition to the generalization of rias, our experiments show that the presented algorithm is also applicable to other coastlines in general. Currently, only selection or removal is used in the generalization mainly because navigation safety has to be satisfied. It is possible to further process our generalized data with the smoothing algorithm by (Guilbert and Lin 2007) as their algorithm also respects navigation safety. In applications where the safety constraint can be relaxed, it is possible to integrate other operators such as displacement (e.g. to enlarge local areas that are too narrow), typification and merge. This deserves to be further investigated for applications other than marine charts. Acknowledgements The authors thank the reviewers for their valuable comments, which gave us the possibility to improve the paper significantly. Funding This research was supported by the National High-Tech Research and Development Plan of China under Grant No. 2012AA12A404 and 2012BAJ22B The corresponding author was partly supported by National Natural Science Foundation of China (Grant No ) and partly by China Postdoctoral Science Foundation (Grant No. 2013M531742).

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