GIS Project Overview, April July 2009

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1 GIS Project Overview, April July 2009 Marc van Kreveld 1 Introduction This document describes the projects to accompany the GIS course. The idea is to analyze a not so clearly stated problem, derive criteria that influence the quality of a solution, study these criteria, set up an algorithmic problem statement, and solve it. The steps involved in this process are detailed in the next sections. The first section deals completely with the analysis of the problem, and the second deals completely with algorithms design and analysis. The running example of this document will be river name label placement. Assume that the project description asks for a well-shaped river label for a given river, and any other map features like cities and other text can be ignored, see Figure 1. Dordogne Figure 1: A river and its label without any other map features. It is important to study the requirements of a vague problem carefully before working on the algorithms. Otherwise, one runs the risk that a problem is solved in a way that gives useless output, because some criterion that influences the quality was forgotten. It is of course very annoying to have implemented an algorithm, and only then realize that some important criterion wasn t taken into account. Possible criteria are that the label should be close to the river, and that it should not intersect the river. One could easily forget the requirement that the river label should not be too sharply curved (too high curvature). On the other hand, setting up many criteria means that an algorithm won t be able to deal with all of them. So it is important to understand early which criteria are the ones that are most relevant, and which combinations of criteria will give when respected good solutions to the problem. If the curvature of a river label is bounded, then the river label can hardly intersect itself (unless the river name is really long). So generally the no self-intersection requirement can be ignored if the curvature constraint is used. 2 Phase I: Problem analysis 2.1 Literature study & reverse engineering A project starts with learning a lot about the problem to be solved. This is done in three ways. Firstly, related literature should be read, and understood how it applies to the problem to be solved. Secondly, existing examples of solutions (maps) that were hand-generated by an expert (in our case: a cartographer) should be examined carefully. Thirdly, sketches of sufficient complexity and variety should be made that show typical situations. These self-designed examples and possible good solutions must be studied as well to get more intuition. 1

2 When you study river label placement, you should read general texts on label placement. If there are no texts on automated river label placement specifically, you can read point label placement and region label placement texts instead, and see to what extent the criteria used there apply. The study of river labels on maps in atlasses gives rise to questions: Why was this river label placed here and not there? Answers to such questions reveal the criteria used by expert cartographers who made these maps. Also, you may suddenly realize that river label names are sometimes repeated along the river. The sketches should contain typical meandering rivers, rivers as they may occur in mountainous areas, and rivers with tributaries (little side rivers). Sketches should have sufficient size and complexity, but should be realistic as well (see Figure 2). Figure 2: Left, three good examples of typical river shapes. Right, an overly complex, non-realistic river and a too simplistic and small complexity river. 2.2 Statement of the criteria The study of literature and existing maps leads to a list of criteria that are relevant to the problem under consideration. Depending on the problem, four to ten criteria can be stated easily. Most of these should be geometry-related criteria, for the purpose of this project. Again using hand-drawn examples of possible input, the criteria can be examined further. The idea is to draw examples that satisfy the criteria but are still not good as output. This will lead to new criteria that were missed so far. If you cannot find such examples, then your list of criteria is likely to be complete. For river label placement, one can easily come up with criteria like: (i) the label should be close to the river, (ii) the label should not intersect the river or other map features, (iii) the curvature of the label should be bounded, (iv) the label should not self-intersect. Note that the problem statement excluded other map features, so the second part of (ii) can be ignored. When you study the literature on label placement, you will notice that cartographers also do not like lables with many inflection points, which you can verify on maps. Furthermore, you may see that cartographers (v) prefer labels above rivers over labels under rivers. Also, (vi) a more or less straight label is better than a very curvy one, (vii) a horizontal label is better than a vertical one, and (viii) the text should be right-side up. More criteria may relate to tribuaries and repetition of a river label; but we will ignore these in the running example. If other map features would be present, then it is easy to come with many more criteria that deal with the interaction. Luckily, the problem statement says that this aspect can be ignored. 2.3 Dependency of the criteria The criteria listed can be similar or somewhat independent. The next step is to examine whether the criteria enforce each other, or conflict with each other ( enforce should be interpreted as: 2

3 more or less imply ). They could also be independent. Criteria should be examined in pairs. If you have c criteria, then you have to consider ( c 2) pairs of criteria. If c > 5, then it becomes too much work for this project to study all pairs, and the most important 10 pairs of criteria should be discussed. Somestimes, a criterion is enforced by a combination of two other criteria. If you notice such a situation, you should list it too. Many criteria are more of less independent, for example: (v) and (ii) are unrelated. We already noted that criterion (iii) probably helps to satisfy criterion (iv), but vice versa this relation is less clear. Criteria (i) and (vi) are conflicting, because if a river has many curves, it is generally not possible to have a straight label that is close to the river along the whole label. Note that in a sense, pairs of criteria are often conflicting because criteria restrict the output, and satisfying one can make that another cannot be satisfied. However, this type of conflict should not be considered. Only the clearly conflicting pairs should be classified as conflicting. 2.4 Quantification of the criteria The criteria that were found can be made into formal, quantitative requirements in many ways. We will now examine what possibilities there are for each criterion. Usually there is the option of stating a criterion as a binary constraint. This is good if some situation must definitely be avoided, at all cost. It is also possible to define quality in a quantitative manner, and state a quality function that returns a value for a situation, which represents the quality of the situation according to that criterion. A precise formula should be given for such quality functions. The criterion that a river label should not self-intersect is best formalized into a binary constraint. On the other hand, the criterion that the label should not be too curvy can be stated with a quality function: the total angular change along the label (total change in angle of the line tangent to the baseline of the label). It is zero for a straight label, and 2π for a label consisting of a half-circle bending rightward followed by a half-circle bending leftward. One can also state this as a binary criterion: the total angular change may not be more than π. The closeness of the label to the river can be formalized in many different ways: minimum distance from a point on the label to the closest point on the river (Figure 3 shows why this is not the best idea), maximum distance from a point on the label to the closest point on the river, etc. The maximum version makes most sense here; it is also called the directed Hausdorff distance. Ariver Ariver Figure 3: Examples that show why the quantification of criterion (i) by minimum distance between label and river is not the best idea: in both examples, the label has the same closeness quality. Geometric criteria are usually measured by length, area, or angle. Note that the closeness of a label to a river can be measured by an area too, namely the area in between the baseline of the label and a section of the river (that section defined by two points, namely the ones closest to the first and last points of the baseline), see Figure 4. (As an exercise, try to figure out why this measure is also not such a good idea.) 3

4 points on river closest to the start and end point of the baseline of the river label Figure 4: Illustration of the area between a river baseline (dashed fat circular arc) and a river (fat solid polygonal line with vertices shown). 3 Phase II: Algorithm design 3.1 Specification of the input and output Once the criteria have been listed and examined, we should consider and formalize what input is given to the algorithm we will design, and what output types we will generate. The output types are in a sense a restriction of outputs to a certain type. The input will generally consist of a set of geometric objects that can include points, polygonal lines, and polygons, with or without holes, and a meaning of these objects. It could also be that the objects are weighted (points that represent cities could be weighted by their population, for certain applications). Specifying the output type is basically specifying all forms that the output can take, or, the solution space. This involves looking at the possible degrees of freedom that are involved in the problem. For river label placement, the easiest input type is a non-self-intersecting polygonal line P with n vertices, and a text that represents the name of the label. For simplicity we will assume that the text is a single word in a given font, which specifies the height and the width of the label to be placed. For the output, we abstract from the text by seeing the label as a shape (like a curved rectangle). The easiest output type is a horizontal, straight rectangle of the given size. In this case the output is simply an x- and a y-coordinate, for instance of the lower left corner. There are two degrees of freedom. Better is to allow the rectangle to be non-horizontal as well, but still straight. This gives three degrees of freedom: two coordinates and an angle. Even better is to allow the label to be any sector of an annulus (the region between two concentric circles), which is the easiest type of curved label (four degrees of freedom: two coordinates for the center of the circle, one for the radius of the smaller circle, and one for the angle at which the text starts, with respect to the positive x-axis; note that the radius of the larger circle is defined by the radius of the smaller circle and the text height). Notice that this generalizes straight labels, because they are in a sense sectors of an annulus of infinite radius. Even more complex is to allow the label to consist of any sequence of straight rectangles and annular sectors that smoothly continue into each other (no undefined tangents at the label boundary except at the four corners), see Figure 5. This last version is needed for high cartographic-quality labels, but may be ignored in the design of a first algorithm which is what is done in this project. 3.2 Algorithmic problem statement After all this preparation work, it is finally time to define which problem we will actually try to solve algorithmically. We will select the seemingly most important criteria, select the most appropriate quantifications, and state a problem. You must come up with two different problem statements. Choose problem statements that you like and that you can motivate (that the output 4

5 Figure 5: Strip consisting of rectangles and annular sectors that smoothly continue into each other and that lies above a river. should be reasonable for realistic inputs). A possible problem statement is the following. Compute the annulus sector where the smaller circle has radius at least r (criterion (iii)), which does not intersect the river (ii), and minimizes the maximum distance of any point on the label to the closest point on the river (i). Note that this problem statement ignores horizontality of the label (vii) and the label being above the river (v), but the correct side up (viii) can still be chosen within the annulus sector. To extend the problem statement so that it includes horizontality somewhat, we could require the annular sector to have at least some part that is horizontal (seeing the annulus sector with the numbers of a clock contained in it, the annulus sector must contain the 6 or the 12 on the clock). There are many other ways of defining horizontality, and also many more ways of defining a reasonable problem. 3.3 Algorithm development Algorithms that attempt to satisfy various constraints can be iterative, like hill-climbing or simulated annealing, or combinatorial, like greedy or genetic approaches. The task is now to develop an algorithm that takes the chosen criteria into account as much as possible, or optimizes the chosen criterion as much as possible while respecting the other chosen criteria. What type of algorithm should be developed depends largely on the problem that was chosen. You may have to sacrifice optimality, because it may not be clear how to achieve it with an efficient algorithm. When developing an algorithm, it is always good to realize what the solution space is, and which parameters need to be computed in a solution. You should specify the parameters of the solution space explicitly. When the solution space is low-dimensional, one can try to compute the true optimum, or use a sampling strategy. Here we try a set of values for every dimension in the solution space and compute whether that solution satisfies the criteria, and what value for any optimization is obtained. For the problem just given, these are the two coordinates of the center of the annulus, the radius of the smaller circle of the annulus, and the angle between 0 and 2π where the label will start (the end angle is specified by the name length in the given font size). So the solution space is 4-dimensional. (One subtlety: we should also specify whether the label goes clockwise or counterclockwise in the annulus. This is not a separate dimension but a binary choice. An algorithmic solution that solves the clockwise version can easily be extended with the same asymptotic efficiency to include the other case as well.) As you can see from the problem statement and solution space, it won t be easy at all to come up with an optimal algorithm for the problem statement, despite the fact that the problem statement seemed restricted at first. To sample the solution space: try every combination of 100 values for each of the four parameters and compute for each choice the distance, while checking the other conditions. This approach will 5

6 be very slow. It is also possible to sample only some of the parameters and optimize of the others. For example, try 36 values for the starting angle and 50 for the radius, and compute the optimal placement for each choice, which is now only specified by an x- and y-coordinate. For every choice of the parameters, we still need an efficient algorithm to determine its quality according to the quantified criteria. Figure 6: A possible river (solid fat polyline), part of the Voronoi diagram of its edges and vertices (solid, thin subdivision), a possible label (gray, in the annulus shown paryially and dashed), the baseline of the label (dashed and fat), and six small arrows indicating which edge or vertex of the river is closest. An idea to achieve some degree of efficiency in the solution for sampling is the following: since we want to know for the whole label what is the closest point on the river many times, we may think of constructing a Voronoi diagram of all line segments of the river (see Figure 6), and preprocessing it for planar point location. Then the distance to the river can be determined in O(log n) time for any point. For a circular arc of the label, we can also discover the point on it that maximizes the shortest distance to the river, but it is beyond the scope of this document to go into details. When the solution space has unbounded dimension, then heuristics, greedy methods and genetic algorithms can be used. These must be examined carefully too. For example, a genetic algorithm needs a population, fitness function, mutation, and cross-over to be defined appropriately. This also asks for algorithms: How do you compute the fitness of a candidate solution efficiently? Discuss which approach you will take and why. When you choose an approach, keep in mind that for these problems, worst-case efficiency and true optimality may be less important than typical running time, and that an approximation of the true optimum may be good enough. 3.4 Efficiency analysis General remarks about the efficiency analysis of the algorithm that you developed cannot be given here. It depends largely on the specific algorithm that was developed. For evolutionary algorithms and simulated annealing, you can analyse how much time one step or one generation of the process takes. The efficiency of any step for which this is possible should be examined. Use different notation for different aspects of the input. For example, if the problem has an administrative subdivision as the input, you typically define it to have n vertices in total, and m faces. You can then analyze the running time and expresss it in n and m. A running time of O(mn log n) is better in practice than a running time of O(n 2 ), because for administrative subdivisions we have m n (typically, m is much smaller than n). 6

7 4 Assignment Projects are done in groups of two students. Topics for the projects are: 1. Flow maps, with arrows between the regions of an administrative subdivision. 2. Label placement for a group of islands. 3. Time-space maps. 4. Non-contiguous area cartograms. 5. Selection of roads in road networks. 6. City block generalization. 7. Zoning, or political redistricting. 8. Relative position of regions. 9. Designing touristic routes through national parks. 10. Determining ridges and valleys and their prominence. For some projects, the various steps in the study outline described before cannot be followed precisely. In this case, a step can be skipped or interpreted differently. This is true for the projects that are less cartographic in nature. Date Event Tasks prepared by group To bring April 28 (Thu) Distribution of projects May 6 (Wed) First meeting sections 2.1, 2.2 All notes and sketches May 20 (Wed) Second meeting sections 2.1, 2.2, 2.3, 2.4 All notes and sketches May 28 (Thu) Hand In & Presentations sections 2.1, 2.2, 2.3, 2.4 All notes and sketches June 4/5 (Thu/Fri) Third meeting section 3.1, 3.2 All notes and sketches June 17 (Wed) Optional fourth meeting section 3.3 All notes and sketches June 25 (Thu) Project presentations July 5 (Fri) Hand In sections 3.1, 3.2, 3.3, 3.4 All notes and sketches Both hand-ins should be words in length, and should be carefully prepared. Figures with good examples are very important. They should illustrate the point that is being made in a sufficiently complex but not overly complex situation. All figures must be self-made; no copies of images found on the web or in books may be used. Figures should have good, descriptive captions. The quality of the figures (choice of example and attention to detail) will influence the score for your project. An introduction to the problem should be kept very short, and it doesn t count in the words. 5 Project descriptions 5.1 Flow maps Flow maps show arrows between regions of an administrative subdivision to represent flow or movement, for example, migration or trade. Chapter 12 of the book by Dent: Cartography: Thematic map design is all about flow. Read it carefully, and study flow maps that you find in atlasses or on the Internet. Versions of the problem include straight arrows only or curved arrows too, and different thicknesses of the arrows or unit thickness (in which case only the presence of flow is shown, not the quantity). Sometimes arrows approximately follow the actual path of flow (e.g., transportation over the sea), and sometimes they just represent a relationship between two regions (e.g., immigration). 7

8 5.2 Label placement for groups of islands There are many texts about label placement, but hardly any about label placement for islands. Check atlasses to find out how cartographers place labels for islands. Also study the paper by Imhof: Positioning names on maps to get a general idea of good placements. When you have a group of islands, each one should get a label in such a way that it is always clear which label belongs to which island. Furthermore, the group may have a name too, which should also receive a label in a way that makes it clear that the label belongs to the whole group. There are other criteria that play a role and deal more with aesthetics. You may assume that no other map features are present. For example, there is no mainland to consider, and the islands do not have features like cities, etc. 5.3 Time-space maps Time-space maps show the map space distorted to show how well the reachability of some location is from all other locations. The central location, a transportation network, and a set of cities form the input. After computing the travel time from all cities to the central location, the deformed map with cities and boundaries can be defined and computed. Two examples of time-space maps are given in the thesis of Jan Robert Ritsema van Eck (page 39). One of the most challenging aspects is defining the deformation for the outer boundary (the Netherlands itself), because there is no actual travel time data from the coast itself to the specified center location. The deformation should somehow follow the displacement of nearly locations. 5.4 Non-contiguous area cartograms In a cartogram, a geographic variable like population per province is represented as the size of the province. For example, a province A with twice as many inhabitants as province B is shown twice as large in the cartogram. This means that either the shape of the provinces must be changed, or the connectivity of the provinces must be sacrificed. In non-contiguous area cartograms, the latter option is chosen. All provinces are scaled in such a way that they do not overlap each other, and have the correct ratio of areas (as specified by the geographic variable). Read Chapter 11 of Dent s book on cartograms. Also look at the cartogram animation of Keith Clarke accessible from Besides showing the scaled regions, you must also show at least the outer boundary of the original region in its normal shape. 5.5 Selection of roads in road networks In a road data set of a country, for example Sweden, there are regions with lots of roads and regions with hardly any road. In a region with many roads, small local roads are not important, but in the far North a small local road may be the only road for kilometers around. We see that the importance of a road is determined by the number of roads (or: by the number of larger or equally large roads) in the neighborhood. For point selection (also called settlement selection), road selection, and area selection, several possibilities have been suggested. For the road selection problem, geometric criteria have not been studied much. The objective in this project is to assign to roads an importance, and to perform a selection that preserves the character of the original road network. 8

9 5.6 City block generalization When you study maps at scales 1:25,000 and 1:50,000, you will notice that some blocks inside urban areas are completely filled, and some show the separate houses. The latter is especially done for blocks that have free houses on large properties. Of course, it also depends on the scale of the map. Assume that data on individual houses and flats (buildings in general) in a city are given. Give a method to decide for each block whether it should be shown as a completely built block, or with separate houses, or a bit of both (one part aggregated to a built-up block, one part with individual houses). Study topographic maps of the scales mentioned carefully, and try to find properties of such maps. Use geometric characteristics, but also keep the global picture in mind. 5.7 Zoning, or political redistricting In the zoning problem, a subdivision into many regions must be redefined into a subdivision into fewer ones by grouping them. For example, assume that the Netherlands is changing to a system where regional political representatives are also chosen into the Tweede Kamer, say, 30 out of the 150 seats are taken by regional representatives. Then the 129 municipalities should be grouped so that 30 groups remain. To represent local matters, the grouping should take geometry into account too; an elongated group is not so good, for example, because people living at the one end may have different (local!) interests than people living at the other end of the group. Zoning is also called redistricting. Search for these terms on the Internet to get a feel for the types of requirements that apply to zoning/redistricting. Concentrate on the geometric ones, and think of various possible ways to formalize listed or not-listed geometric requirements. See e.g Relative position of regions Researchers in GIS have studied modeling the relative position of regions with respect to each other. For example, can one say that Luxembourg is to the South of Belgium, or to the East? Or both, or neither? For two points in the plane it is easy to define their relative position by taking the direction of the vector between those points. But for regions the situation is less clear. The same is true for close and far. You could claim that Norway is equally far from Sweden as from Finland, since Norway is adjacent to both. But intuitively, everyone will say that Sweden is closer. In this project the objective is to give and compare possible formalizations of relative directions and distances between two regions. To this end, start by reading the papers by Goyal and Egenhofer: Similarity of cardinal directions (Advances in Spatial and Temporal Databases, 7th International Symposium, SSTD 2001, pages in LNCS 2121) and by Hong, Egenhofer and Frank: On the robustness of qualitative distance- and direction-reasoning (Proceedings Twelfth Int. Symposium on Computer-Assisted Cartography pages , 1995). You need not read them carefully; they just give an idea of the issues. You may consider other papers too that discuss cardinal directions, relative positions, etc. 5.9 Designing touristy routes through national parks When a nature region becomes a National Park, good roads through the park are needed. These roads should be such that good views of scenic natural wonders are incorporated. At the same time, the road should still provide relatively fast transportation from one point to another. Many National Parks are in mountainous areas, which makes road planning more difficult. In this project we assume that a region is given with its elevation model. The objective is to design the best road, taking into account the terrain (steepness, for example), the scenic value 9

10 (passing by interesting points, good views of other interesting points), and the transportation value. To get an idea of roads through national parks, consult for instance although you cannot see the good views or reasons for roads on the maps Determining ridges and valleys and their prominence Ridges and valleys are linear features of mountainous regions that are highest, respectively lowest, in a certain sense. They have definitions that apply to grid models and TIN models for terrains, but the common definitions are very local. The prominence of a ridge or valley is determined by its elevation, by the elevation of the region around it, by its length, and by (the prominence of) other ridges or valleys nearby. A country like Switzerland has many ridges and valleys. Some are more prominent than others. The most prominent ridges and valleys are the ones that will be shown even when considerable map generalization is performed. Sometimes it is not clear whether something is one single ridge, or two separate ridges that basically lie in each other s extension. Valleys seem to have a slightly different structure. In this project the objective is to determine a set of ridges and valleys from a TIN elevation model, to define their prominence suitably, and to design efficient algorithms for the computation of the prominence of ridges and valleys. This project can start by examining various definitions of ridges and valleys that exist in dictionaries of physical geography, geomorphology, etc. These definitions are descriptive, rather than precise. Part of the quantification within this project is about making such definitions precise. 10