LINE SIMPLIFICATION ALGORITHMS Dr. George Taylor Work on the generalisation of cartographic line data, so that small scale map data can be derived from larger scales, has led to the development of algorithms for a variety of generalisation operations, including line simplification. A major objective when capturing linear digital map data is the creation of accurate representation of naturally occurring lines, using a minimum of processing time and storage. When paper maps are converted into a digital representation using manual digitising methods, one time saving method used is that of stream mode digitising. In stream mode the digitiser samples continuously as the cursor is moved along the manuscript line. Co-ordinate density in stream mode is usually determined by time interval, which can lead to the generation of surplus co-ordinate data. For a number of practical considerations this superfluous data should be eliminated. These considerations include: 1. Reduced storage space: this may reduce a data set by up to 75% (Jenks 1989), which will result in faster data retrieval and management. 2. Faster vector processing: for example, a simplified polygon boundary will reduce the number of boundary segments to be checked for shading or point in polygon algorithms. 3. Reduced plotting time. For these and other reasons, a multitude of line simplification (conditioning) algorithms have been developed. There are many studies that have discussed the merits of different algorithms, these include McMaster (1983), McMaster (1987) and Zhilin (1988). These studies contain detailed descriptions of many of these algorithms. A brief discussion is given here of the major types of simplification algorithm. Two algorithms, described in detail below, were implemented in the AutoLISP programming language and tested in AutoCAD, a computer aided design package (CAD). AutoCAD, like many CAD packages, is widely used for surveying and mapping applications. It is also an integrated component in a number of commercial Geographical Information Systems (GIS), for example; ArcCAD by AutoDesk and Environmental Systems Research Institute (ESRI); Geo/SQL by Generation 5 Technology. SLIMPAC, a mapping information system, is the particular AutoCAD application in which the described Douglas Peuker algorithm is used. SLIMPAC merges AutoCAD and one of the market's leading database packages dbaseiv. No functionality is removed from either package but many map handling operations have been added, of which line simplification is one. Simplification algorithms may be divided into a number of classes: 1. Independent point routines. 2. Localised processing routines. 3. Extended local processing routines (with constraints). 4. Extended local processing routines (without constraints). 5. Global routines.
(McMaster 1987) Independent point routines Algorithms in the first category are very simple in nature and do not take into account the mathematical relationship of neighbouring co-ordinate points. An example of this is the nth point routine where every nth co-ordinate pair (e.g. 2nd, 4th, 6th etc.) is retained. These routines are very computationally efficient, but they are unacceptable for accurate mapping applications. Localised processing routines The second category utilises the characteristics of the immediate neighbouring co-ordinate points in deciding whether to retain co-ordinate pairs. An example of this category uses the minimum Euclidean distance between two co-ordinate pairs. If the distance between two coordinate pairs is less than a predefined minimum, one pair is rejected. Extended local processing routines The third and fourth categories search beyond the immediate neighbouring co-ordinate pairs and evaluate sections of the line. The algorithms are often constrained by the number of coordinates and a distance parameter (tolerance). An example algorithm of this type was first described by Lang (1969) and was reported by Douglas and Peuker (1973) "as producing an acceptable result". Douglas and Peuker extended the idea of distance tolerance to produce a modified algorithm, described later. Figure 1 Perpendicular distance method Another algorithm, reported by Jenks (1989), calculates the perpendicular distance from a line connecting two co-ordinate pairs to an intermediate co-ordinate pair. The method is illustrated in Figure 1.
Perpendicular distance algorithm First, the three points being considered are selected (a, b, c) and the tolerance, t, is set for simplification. Second, a line is defined between points a and c, the first and third coordinate pair. A perpendicular is calculated from this line to the point b, the intermediate coordinate pair. If the length of this perpendicular is greater than the tolerance t, the point is retained and becomes the first of the next three points selected (b, c, d). If the length is less than t, the point b is rejected and the next three points considered are a, c and d. This algorithm is efficient in computer processing time and generally produces good results in application, but not in every case. Unsatisfactory line simplification occurs with this algorithm when it is applied to curving lines consisting of many co-ordinate pairs. Parallelism of lines may be lost, an extreme case is where a circular line feature is reduced to a triangle, an example is given in Figure 3. A change of tolerance for certain line features can overcome this fault, at the cost of retaining more co-ordinate pairs than necessary. This algorithm was tested for use in a line simplification procedure written for AutoCAD. A simple modification was made to the algorithm to overcome its aforementioned fault and allow fully automated line smoothing. This modification counted the number of co-ordinate pairs in a line before and after simplification, if this number was reduced by more than 75%, the tolerance was temporarily reduced by half and the line processed again. Global routines The first four categories describe algorithms which process a line piece by piece, from beginning to end. The final category describes a different, holistic approach to simplification. A global routine considers the line in its entirety while processing. The only global simplification algorithm commonly used in cartography was developed by Douglas and Peuker, now Poiker, in 1973, although there are others (Zhilin 1988). This algorithm is both mathematically and perceptually superior, it produces the best results in terms of both vector displacement and area displacement (McMaster 1983). "Many cartographers consider it to be the most accurate simplification algorithm available, while others think that it is too slow and costly in terms of computer processing time" (Jenks 1989). This is the algorithm now used for the line simplification procedures used by SLIMPAC commands and translation programs, it is described below. Douglas Peuker algorithm The Douglas Peuker Algorithm tries to preserve directional trends in a line using a tolerance factor which may be varied according to the amount of simplification required. The method is illustrated in Figure 2. The first point a is selected as the anchor point and the last point b is selected as the float point. The maximum perpendicular distance, say fc, from the digitised points to the line ab is compared with the tolerance distance (td). If fc>td, then c becomes the new float point. Then, the maximum perpendicular distance from points to ac, say ed, is compared with td. If ed>td, then d becomes the new floating point, c is saved in the first position on a stack, or else d is selected as the new anchor point and all points between a and d are discarded. In this fashion the program processes the entire line, backing up onto the stack as necessary, until all intermediary points are within the corridor, or there are no intermediary points.
Figure 2 Douglas Peuker method The last float point now becomes the anchor and the first co-ordinate pair is taken from the stack to become the new float point. The process described in the previous paragraph is now repeated. This recursive method continues until the last co-ordinate pair is taken from the stack. A stack may be considered as a list that is accessed from only one end. Items are taken from a stack in reverse to the order in which they are placed on it, i.e. last in first out (LIFO). An example of line simplification, in AutoCAD, using this algorithm is given in Figure 3. The choice of a correct tolerance value for a specific data set may be quite difficult, with little or no guide-lines in the literature on the work. One value may not work correctly for a whole data set, particularly with the perpendicular distance algorithm described earlier. Experience gained during testing indicates that when simplifying an OS 1:1250 scale digital map sheet a tolerance value of 0.25m will give acceptable results, retaining detail accurate to scale. Line simplification on the OS 1:1250 scale data using this algorithm reduced the number of co-ordinate points by up to 40%. An example of line simplification An example of the effect of line simplification in AutoCAD, using the two routines discussed, is given in Figure 3. This figure displays the result of simplification on two closed polylines (a), of the perpendicular distance algorithm (b) and the Douglas Peuker algorithm (c). The inner line has a diameter of 6 units, the outer line 10 units and the tolerance is 0.25 units. Each line contains 400 co-ordinate pairs before simplification. The perpendicular distance algorithm reduces both lines to 3 points each, while the Douglas Peuker algorithm reduces the outer line to 16 points and the inner line to 8 points. If (a) was a line feature taken from a 1:1250 scale map and the units were metres, (c) would be an acceptable result. The above example, although extreme, indicates the superiority of the Douglas Peuker algorithm in obtaining consistently acceptable results. The perpendicular distance algorithm will yield an acceptable result in this example if the tolerance is reduced to less than 0.01 units. However, a tolerance of 0.01 with this latter algorithm, would produce insignificant simplification on a whole map sheet at 1:1250 scale.
Figure 3 Line simplification Line simplification problems There are a number of undesirable effects, Figure 3, dependent on the structure of the data set, which may occur when line simplification algorithms are applied. 1. Lines that originally join may not do so after simplification. 2. Lines may cross after simplification, which before did not. (Webb 1989) If spaghetti data is simplified 1 and 2 may occur, if link and node data is simplified, using the Douglas Peuker algorithm, 1 will not occur as line end points (nodes) are always preserved. A consequence of 2 may be that extra sliver polygons and extra network spurs are created in the simplified data set. The importance of such additions is dependent on the use of the simplified data set. In a topological system, using spatial analysis operations on the data, additional polygons and spurs will distort the true topology of the data. All such errors must, therefore, be corrected. A system which does not maintain a topological data structure will not itself be affected, user mis-interpretation being the only possible error. This article has described some of the various methods and algorithms used for the spatial data process of line simplification. Manually digitised map data should be line conditioned so that it may be processed more efficiently. This may be achieved without loss of accuracy or
fidelity with the Douglas Peuker method and careful choice of tolerance. Line simplification may also be used to drastically reduce data quantity where loss of accuracy is of no consequence. References Jenks, G.F., 1989, Geographic logic in line generalisation, Cartographica, Vol. 26, No. 1, pp. 27-42. McMaster, R.B., 1983, A mathematical evaluation of simplification algorithms, Proceedings, Sixth International Conference on Automated Cartography AutoCarto 6, pp. 267-276. McMaster, R.B., 1987, Automated line generalization, Cartographica, Vol. 24, No. 2, pp. 74-111. Zhilin, L., 1988, An algorithm for compressing digital contour data, The Cartographic Journal, Vol 25, pp. 143-146. Lang, T., 1969, Rules for robot draughtsmen, Geographical Magazine, Vol 42, pp. 50-51. Douglas, D. and T. Peuker, 1973, Algorithms for the reduction of the number of points required to represent a digitised line or its caricature, The Canadian Cartographer, Vol 10, pp. 112-122. Webb, G., 1989, Change only updates for land information systems: a trial implementation and feasibility study, M.Phil Thesis (unpublished), Department of Surveying, University of Newcastle upon Tyne.
George Taylor is a member of the Mapping Information Science (MIS) research group in The Department of Surveying, University of Newcastle upon Tyne. This group's interests include the collection, categorisation, structuring, transfer and integration of map based data from the wide range of data collectors and users. Consideration is given to the storage of this data in computer database structures independent of application, and the development of toolboxes of analysis algorithms applicable to the complete range of potential uses. Much of the work is focused on very large scale map data which are appropriate to local authority, public utility and engineering applications. However, current projects also include investigations into the use of Geographical Information Systems (GIS) for archaeology, pipeline route planning, coastal zone management and geodemographics. Work is also carried out in collaboration with a number of commercial and government organisations and a variety of products and services are marketed. The Mapping Information Technology Unit (MITU) has recently been formed to manage much of the group's commercial activity. MITU is both an Ordnance Survey licensed developer and an AutoCAD registered applications developer. The groups most recent software products include: AutoNTF; a reader for NTF format map data directly into AutoCAD. This includes all Ordnance Survey basic scale map data and other OS digital products. The program can handle data at levels 1,2 and 3 and maintains topology at level 3. SLIMPAC for Windows: SLIMPAC, a Land Information System (LIS), integrates the two standard microcomputer software applications AutoCAD and dbaseiv to create a procedural and recording mapping information system. SLIMPAC is designed to manage large scale mapping applications, such as the recording of land ownership and leasing for property terriers, and the management of facilities for industrial estates and other building complexes. Recent work, has investigated the development of a spatial indexing system for buildings. SLIMPAC is now available for AutoCAD for Windows combined with dbase IV Ver. 2.0.