Statistics of displacement magnitude for each data set.

<p>Statistics of displacement magnitude for each data set.</p

A Combined Approach to Cartographic Displacement for Buildings Based on Skeleton and Improved Elastic Beam Algorithm

<div><p>Scale reduction from source to target maps inevitably leads to conflicts of map symbols in cartography and geographic information systems (GIS). Displacement is one of the most important map generalization operators and it can be used to resolve the problems that arise from conflict among two or more map objects. In this paper, we propose a combined approach based on constraint Delaunay triangulation (CDT) skeleton and improved elastic beam algorithm for automated building displacement. In this approach, map data sets are first partitioned. Then the displacement operation is conducted in each partition as a cyclic and iterative process of conflict detection and resolution. In the iteration, the skeleton of the gap spaces is extracted using CDT. It then serves as an enhanced data model to detect conflicts and construct the proximity graph. Then, the proximity graph is adjusted using local grouping information. Under the action of forces derived from the detected conflicts, the proximity graph is deformed using the improved elastic beam algorithm. In this way, buildings are displaced to find an optimal compromise between related cartographic constraints. To validate this approach, two topographic map data sets (i.e., urban and suburban areas) were tested. The results were reasonable with respect to each constraint when the density of the map was not extremely high. In summary, the improvements include (1) an automated parameter-setting method for elastic beams, (2) explicit enforcement regarding the positional accuracy constraint, added by introducing drag forces, (3) preservation of local building groups through displacement over an adjusted proximity graph, and (4) an iterative strategy that is more likely to resolve the proximity conflicts than the one used in the existing elastic beam algorithm.</p></div

Adjusting proximity graph with building grouping information.

<p>Building groups are represented by group nodes in the adjusted proximity graph, and related proximity edges are deleted or collapsed.</p

Data set partitioning.

<p>The data set partitioning method has four major steps: (1) buffering of buildings, (2) merging of buffering polygons, (3) conducting an overlap with street network, and (4) processing a point-in-polygon operation to assign each building to the corresponding partition.</p

Comparative test results of existing elastic beam algorithm (partition A₃).

<p>(A) <i>E</i> = 50 000, <i>A</i> = 1 and <i>I</i> = 1; (B) <i>E</i> = 200 000, <i>A</i> = 1 and <i>I</i> = 1; and (C) <i>E</i> = 1 000 000, <i>A</i> = 1 and <i>I</i> = 1.</p

Extraction of the CDT skeleton.

<p>Extracted skeleton graph (left) in a map partition and detailed part (right) of the extracted skeleton graph to illustrate the concepts of skeleton arc, super-arc, and sub-arc.</p

Testing data sets and partitions.

<p>(A) Partitions of data set A, and (B) partitions of data set B.</p

Visual result of data set B.

<p>(A) Initial situation (1∶25 000), (B) resultant situation (1∶25 000), and (C) comparing the initial and final states (1∶25 000 enlarged to 1∶12 500).</p

Positional accuracy constraint and drag force.

<p><i>O</i> is the original position of the building’s centroid, <i>O′</i> is the new position of the building’s centroid after displacement, and <i>r</i>_max is the threshold of positional accuracy. If , a drag force, , will be introduced to drag the building back to the acceptable area again.</p

Construction of initial proximity graph from the skeleton graph.

<p>Each skeleton arc is corresponding to a proximity edge in the constructed proximity graph. There was a one-to-one correspondence between building and building node and a one-to-<i>n</i> (<i>n</i> ≧1) correspondence between boundary line segment and boundary node in the graph. The number of the boundary nodes on each boundary segment depends on the number of involved proximity relationships.</p