General knowledge on geoinformatics, spatial relations and spatially referenced data

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Geoinformatics is a new branch of information technology, which uses both graphical and database knowledge. In the 1600's, mathematics has become the basis for almost every natural science just as information technology in our days. Based on high standards, professional software, database technologies and graphical representations are widely used and applied by almost every discipline. By keeping the primary goals of geoinformatics, information technologies have been applying their newest achievements in geoinformation systems. New methodologies have been introduced for representing and analyzing spatial-based data.

Concepts and theoretical ideas used by geoinformatics will be introduced in the following chapters. Mathematical methods, principles and algorithms as well as software are being discussed in detail. We think it is crucial for students attending practice-oriented classes that they should become familiar with methods and principles in addition to geoinformation systems along with their routinish user level and problem solving practices.

Further goals of this material are to introduce a rapidly expanding world of open source technologies. Although only theoretical concepts will be discussed, further materials will rely on this knowledge when introducing problems solved by open source tools and programming libraries. The world of open source is relying on modern theoretical and computer science background, which we will discuss in the following chapters. Interestingly, the world of open source is progressing in a much faster pace than that of commercial software, which is essentially aiming for higher profit; thus, their software concept only changes when it is inevitable due to losing profit. That is not how the open source world operates: they would rather focus on technical challenges.

While editing this material on open source technologies, the authors were bearing in mind that not only theoretical knowledge, but also fully functional GIS applications and programming libraries concepts, which do not require existing spatial software (but rather functionalities), should come from the users’ experience.

By first glance, the earth is spherical, which leads to the use of a polar-coordinate system, having lambda and theta angles with r radius giving the coordinates for any P point. (Let us accept this abstract approach, although we know that the earth is not spherical for projections and the reference surface is not spherical either).

You can open the (larger) image in new window.1_foldgomb_full.jpgFig.1. Description of the globe surface with polar coordinate system (Klinghammer)

One of the most important tasks of a map is to express spatiality by representing the earth on a spherical surface. Whether you are using a traditional, spatially flat paper map or a digital map on a monitor, both products needed some projection from a spherical body to a plane, which resulted the distortion of the original image. A spherical surface cannot be projected to a planar surface without being distorted. Projection theory is dealing with similar problems.

Let us assume that projected planar maps are available before building a spatial information system. The only exception is data measured by GPS, which are fundamentally geographical coordinates (although transformation is applicable into planar surface).

Vectorized digital maps have been projected onto planar surface by analogue source maps. Plane to plane transformations are needed when changing projection systems, calibration and georeferencing operations are done. As an example of importing a digitized map and re-project it into a new projection system, a plane to plane transformation is needed.

In the case of remote sensed images, ground control coordinates define the reference points between the relative coordinates of source images and the real coordinates of transformed images. In the following sections, we will discuss some of the mostly used planar transformations in geoinformation systems.

Consider the case when we have a paper map in a given projection system. Our task is to vectorize it. During vectorization, we may keep the source projection system, although we could transform it into another projection system if required.

Generally, raster-based digitized maps are scanned in a relative coordinate system (upper-left coordinate is the origin, sizes are measured in pixels), and transformation (georeferencing) is needed into a real coordinate system.

Let us look at Figure 2, which describes the possible plane transformations like, congruent, isomorphic, affine, projective and topologic. Invariants, which remain constant during transformation, are related for every single transformation.

You can open the (larger) image in new window.2_topo_full.jpgFig.2. The possible transformations: 1: congruent, 2: isomorf, 3: affin, 4: projectív, 5: topological transformation (Mortenson)

Possible plane transformations are as follows: 1 Congruent, 2 Isomorphic, 3 Affine, 4 Projective, 5 Topological

During transformation, neither its size nor its shape will change.

The feature size does change during isomorphic transformation, but its shape does not. This is usually named Conform transformation, because the angles are not modified during transformation (Figure 3.).

You can open the (larger) image in new window.3_conf1_full.jpgFig.3 Examples for conform transformations -- orientation preserving conform transformation, 2 -- orientation reversing conform transformation, (a): original shape, (b): transformed shape

This figure shows examples of conform transformation: 1 Orientation preserved conform transformation, 2 Conform transformation with inverted orientation a) source feature b) transformed feature.

During affine transformation, both its size and shape are changed, but parallelism is preserved. Note that included angles are not invariants.

You can open the (larger) image in new window.4_affin_trans1_full.jpgFig.4 Affin transformation: 1: original shape, 2: transformed shape. A'B' <> AB, and A' angle <> A angle, but A'D' || B'C', and every point of AD fit to A'D'

Affine transformation 1 Original shape, 2 Transformed shape. As shown in Figure 4, A'B' <> AB and A' angle <> A angle, although A'D' || B'C'; in addition, every each point on AD lies on A'D'.

Equation for affine transformation:

*x' = ax + by*

*y' = cx + dy*

The following table summarizes the properties of the image of affine transformations. Affine transformation is one of the mostly used transformation methods in Geoinformatics. Georeferencing is applied when the source map is a scanned raster image; namely, the relative coordinates are correlated to referenced coordinates: this process is an affine transformation. Most of the coordinate system transformations are based on an affine transformation method.

You can open the (larger) image in new window.tabla_1_full.jpgTable 1. The properties of the image of affine transformations

During projective transformation, all of the following properties are changed: size, shape and parallelism, but straight lines remain straight. This type of transformation is scarcely used on vector-based data types, although projective transformation may come handy for solving the representation of 3-dimensional data, where perspective representation is needed.

During topological transformation angles, distances and parallelism are changed, however, continuity, ordinality and vicinity properties are intact. This type of transformation becomes relevant when affine transformation did not give correct result due to the deformation of raster-based data during georeferencing.

Deformation may occur in the course of scanning a raster map, because the paper transfer rollers of the scanner may slid the paper. In these cases, even correct georeferencing does not give correct transformation. In spite of correct ground control points and available known points with correct coordinates, which are detectable on the scanned map, they are not in their correct position. In cases like this, affine transformation is performed location by location and adding as many ground control points as possible.

A Társadalominformatika: moduláris tananyagok, interdiszciplináris tartalom- és tudásmenedzsment rendszerek fejlesztése az Európai Unió támogatásával, az Európai Szociális Alap társfinanszírozásával, az ELTE TÁMOP 4.1.2.A/1-11/1-2011-0056 projekt keretében valósult meg.

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