Specifying the transformations within and between geographic data models

Transactions in GIS, 1996, vo/. 7 , no. 2, p. 137

Specifying the transformations within and between geographic data models

MARK GAHEGAN

Department of G I s , Curtin University, PO BOX U 1987, Perth 6001, Western Australia. email: [email protected]

Geographic information is acquired according to several different underlying models of geographic space. Any meaning inherent wirhin a dataset is intrinsically connected to the model by which it was captured. A design is presented for an inregrated geographic information system offering a number of distinct views onto geographic space, of varying degrees of abstraction, to suppom the needs of an expanded user base. An architecture for such a system is developed by an extension of the traditional three layer architecture used in database design. T h e movement of data from one level of abstraction to another is formalized by a series of dataset and model transformations operating between four different geographic data models. From this formalism, a functional taxonomy of GIS operations is developed.

Introduction

I t is possible to regard the geographic world from a number of differenr philosophical perspectives, for example as a grid of quanrized and multi-valued cells or a collection of identified and discrete objects. When formalized, these become different geographic dnra models. Data is routinely gathered using a variecy of these different views of realiry, wirh different packages (and in some cases different branches of science) having developed to specifically address rhe handling and modelling requirements that these views impose. However, many problems wirhin the geosciences require chat data from these various views be integrated. For example, in land use analysis or geo-exploration there is often a need to combine data gathered as point samples, images, thematic maps and discrete polygons.

The firsr step towards full integration of geographic data, regardless of its underlying view, is to provide an architecture whereby the different types of data and different data structures used in each nor only co-exist but can also interact freely (e.g. Ehlers er al 1991, Bruegger 1995). The Open GIS Foundarion (OGF) is currently leading a combined research iniriative, which includes many of rhe major GIS vendors, ro address this problem (OGF 1336). Indeed, one of the main aims of the

OGF is to achieve an interoperability environment where data and functionality may be openly shared between the products offered by different GIS vendors. Accordingly, they have developed a Virtual Geodata Model (Buehler 1994) to divorce the data from the underlying models and data structures imposed by these producrs. T h e proposals of the SQW multi-media group S Q L / M M 3 (ANSI 1994) also address similar issues, bur from the lower-level perspective of database management. These provide a much needed integrated database environment for many of the different spatial data strucrures that are used in GIS (Sulima 1995).

T h e building of all-embracing spatial data structures has obvious advantages in terms of data exchange and interoperability. However, it requires that particular care be given to the process of data abstracrion, since within a single application, data may differ not only by data structure but also by implied meaning (semanric content). A geographic data set is captured according to some underlying model of space. For example, data may originate from point measurements in the field (field view), as a remotely-sensed scene (image view), or as digitized polygons (object or fearure view). When combining data gathered according to different underlying philosophies we must ensure that we do not exceed the meaning assorltrted rcvitll the data within the model

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u n h which it was captured. Simple integration that is primarily concerned with data structure compatibility will nor offer this protection. If data from different

by the featurization process can be given.

development of a Fully inregrated GIS is desirable: Below are listed several reasons why the

views is combined then there is considerable risk that the results will be misleading. For example, it would be naive to assume that a pixel taken in isolation from a satellite image indicates the presence of a forest object simply because its spectral response is closest to that of the forest training set. Consequently, there is a need formally to characterize the transformation of data sets between models as well as between data structures. The integration of data structures is in itself a good thing, but if unchecked it offers yet another means to misuse the data. Where misuse may reasonably be foreseen, then there is a duty to impose measures to guard against or warn of potential problems.

This paper proposes an architecture and a theoretical basis by which various geographical data models may co-exist and freely exchange data. As an example, transformations between field, image, thematic, and feature data are described. Integration is built upon on a single generic descripcion of a geographic data set that spans all of the defined views. T h e user can then be presented with logical (abstracted) concepts that are based around the operations to be performed, rather than the data types or data models by which the data is represented.

The geo-objects (here termed fearurer) that are used as the model basis in many commercial GIS are often derived from lower forms of data, such as remotely-sensed imagery, by applying a process of extraction which may be manual or automated (e. g. van Cleynenbreugel et al 1990). Most current systems do not have an integrated model to describe the transformations used in extraction; so the outcome of each step is treated as an entirely new and separate entity. Consequently, there are no logical dependencies or relationships captured. Two notable exceptions are the work of Lanter (1991), whose GEOLINEUS tool keeps track of the lineage of a dacaset within ARC/INFO, and Kuhn (1994), whose high level semantics are defined in terms of source data. Of particular importance is the ability to maintain details of the process by which any features are formed for the following reasons (Gahegan 1996): (i) their derivation can be subject to analysis, and possibly refinement; (ii) justification can be given as to the suitability of a feature for a given task and (iii) a comprehensive statement of uncertainty introduced

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Ability to divorce derails of implementation and storage from any logical relationships in the data, implying the hiding of these details from the user (logical abstraction). Thus the GIS may be constructed around the tasks that the user might wish to carry out, and not around a particular data structure that happens to be used to represent the data (Albrecht 1994). Although it takes a lot of effort to solve the deep semantic issues that this approach raises, it also presents the opportunity to design a geographic data model that is at last logically independent of the underlying data structures. Data Exchange. Following on from the above, data exchange can focus on the data sets themselves, rather than the storage structures used to represent them. Inreroperability. With mechanisms in place for data exchange, it then becomes possible to pass requests and instructions between GIS in addition to data sets. This type of embedded agent technology has been available within desktop applications for some time. Ability to take in and operate o n data from a number of views, including in primary (raw) form and in feature (processed) form. There may be sound reasons why it is not desirable to change all the data used in a project into a single geographical model. Consequently, simultaneous support must be provided for several models, along with mechanisms for their interaction. Ability to &scribe the formation of all derived objects, such as features. All derivations should be repeatable and communicable. Many of the feature descriptions used within GIS (and also in computer vision) can unfortunately be described as unrepeatable, unrefutable and subjective science, in that little is known concerning the reliability or applicability of the features produced. There is movement in some circles (Haralick 1994, Prechelr 1994) to make such experiments repeatable, possibly by others, so offering a much broader basis for evaluation and comparison. All too often, however the meta-ahfa that is required for the correct and Jppropriate application of data is lacking (Davis and Simoncrt 139 1 ) .

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Some alternative views of geographical space

To begin, four alrernarive views of geographical space are briefly described and contrasted. These descriprions are nor meant to be definitive and more derailed accounts have been given by many aurhors (e.g. Goodchild 1992, Jeon and Landgrebe 1992, Cressie 1993). The purpose of introducing them here is so that the operations within and between each view may be defined. Other views are also possible, such as a triangulated surface view. Specifically, the retrieval operation is considered, and the various qualifiers chat may be applied to data within each view are given. For simplicity, the temporal and error characteristics of the dara are for the present ignored.

The original morivarion for the development of rhese parricular views was to support the integration of remote sensing and GIS (Wilkinson 1996, Gahegan and Flack 1996), specifically to govern the transformarions from image dara to fearure form as shown in Figure 1. Indeed, :i

common roure for rhe rxtracrion of features from remotely-sensed imagery, via a specrral classification followed by objecr formation, reflects these views closely. Orher geographical data models that may be relevmr to differenr applications could be added. T h e pre-requisites are of course that their relationships ro other views musr be explicitly defined. otherwise i t is nor possible to manage the process of view transformation explicirly. For example, a qualitative view onto the fearure model might be included by building a qualitative spatial reasoner as an inrerface ro the feature view (i.e. at the exrernal layer) (Shartna er al 1994).

Mosr: commercially available GIS offer only a subser of these views to the user, wirh poorly defined relationships between rhe views; if indeed they are defined ar all. However, whilst these views are

image zzta I)

certainly different, [hey are not necessarily conflicting and rherefore nor mutually exclusive: the selection of one view over the ocher is nor then the issue. Each one may be useful in irs turn for parricular types of analysis. In the next major section an architecture is described which supports these views as points along a cont inuum of abstracrion, moving from unprocessed source data ar one end to recognized geographical features at the ocher.

Field model of space I n the field model i t is assumed rhar a region of inrerest is composed of an infinire number of points, each wich a vector of real variables describing the values of some real-world phenomenon at rhar point. T h e model assumes that a variable changes continuously, forming a surface. In practice, it is not possible to measure values at all points, so a finite (and hopefiilly represencarive) set is chosen. There are often restrictions on rhe measurements taken, in terms of the phenomena chat may be measured and the dynamic range of measurement. So, whilst a poinr in the field model may rheorerically possess values for all phenomena of interest, a measured point will usually have only a subser of rhese. Thus rhe view provides measurements of data across a finite number of concurrent surfaces in the form of a poinr with a vector of atrribures drawn from the measured domains. For example, a wearher starion may gather rainfall, barometric pressure, and humidity values. To form a continuous rainfall surface, data from samples must be interpolated across the region, using one of many specifically designed techniques (e.g. Webster and Oliver 1990, Forheringham and Rogerson 1994).

T h e field view is restricted to data defined by a specific geometry (the point) and rhe property values occurring ar a point. With the geometry of data imposed, the only free variables are location

/-\ Figure 1. Continuum I Qualitative I

interface

m Geographic I features

- of abstraction from image data ro feature dara.

interiacz J

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and value. The data has no higher level identity imposed upon it, operations being limited by the lack of feature identifiers or class labels on any of the geometric primitives. Operations that act on the field view may only address the spatial and attribute properties of a point (set of poinrs). For example, the function window() addresses the spatial properties, whereas the function histogram- equalise() addresses the actual values themselves. Since individual features or classes are not defined within this view, the higher spatial operators such as perimeter() or area() have n o meaning.

Image model of space T h e image model may be regarded as a specialization of the more general field model. An image model is given by restricting the number of sample points and the number of domains measured. A grid pattern is imposed to define the (square) region described by each data vector (e.g. Sonka ec al 1993). T h e values captured at the sensor are usually an average reflectance over an approximately circular area, the diameter of which is defined by the point spread function of the sensing device. The approach taken by image processing is to project the sample values onto a grid. In practice the sensed area is either only a part of a pixel, or alternatively, sensed areas overlap into adjacent pixels. It is recognized that the physics of the sensors that provide image data is more complex than is modelled by simply imposing the data on to a grid (e.g. Barrett and Curtis 1992: chapters 2 , 5, 9).

Remote Sensing Systems (RSS) usually operate on an image view of space, where fearures are not explicitly defined. T h e user operates on source data values (for example pixel reflectance values) and typically may proceed as far as classifying these data. With respect to GIS tasks, data in image form is much harder to work with since its meaning is unspecified. From a semantic perspective the image is ‘content free’; it must be processed before higher level objects can be recognized or imposed. Its use explicitly demands an in-depth understanding of the sensor and the physical properties of any ‘targets’, and hence is not a suitable task for the non expert.

Like the field view, the image view is restricted to spatial information defined by a specific geometry, in this c . w the gridcell (or pixel), and the property values occurring therein. Geometry is

further constrained by the gridding process. operations that act on the image view may address the spatial and attribute properties of a pixel (set of pixels).

Thematic model of space T h e thematic model is based on classes. It is more abstract in nature than the image model, since each constituent class has some meaning or interpretation imposed upon it. An overlay represents a view of a region according to some predefined classification scheme. Such an overlay might be created by performing a spectral classification on each pixel in a satellite image. T h e overlay has a theme or purpose, hence the term thematic mapping. T h e individual classes each have their own spatial representation (which may be geometrically disjoint) and these define the only spatial properties available within the view.

Operations are restricted to those acting on the class label and those acting on the spatial description of a class (set of classes). In contrast to the previous views, the geometry of a class is unrestricted bur its data value is fixed to that of a single label, replacing the original data values. Note that there is now no flexibility as to how the graphical primitives can be selected by value, since an interpretation scheme has already been imposed. There is no restriction on the origins of data to be classified, so this view can accept data from any of the other views, changing their data for a class label, and merging their spatial properties into class sets. New operations become possible by way of the thematic nature of the data; for example, selecting a class, merging two classes, and calculating the area occupied by a class.

Many types of Operations, common to computer vision and image processing, produce data at the thematic level of abstraction. For example, an edge detector labels the components of an image with values related to the likelihood of a component being part of an edge (e.g. Canny 1986). This results in a thematic layer representing a classification of edge strength. Again, individual objects are not explicitly recognized, and further processing is required to form a meaningful segmentation of the scene.

Feature model of space In the feature model. the user inreracrs wirh

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Specifying the tronsformctions within and between geographic data mcdels

delineated and labelled spatial objects. These objects are, to all intents and purposes, uniform in nature. In fact, all of the spatial properties of the feature have been reduced to just two components: a geometric description (shape) and a class or feature label. A weakness inherent with this treatment of data (if conducted in isolation from other views) is that all the variance and subtlety in the original data from which the features were formed is disregarded, or at best may be statistically summarized in a coarse fashion. This approach is still very useful, where features are well-defined by precise boundaries, but problems arise when feature descriptions are derived from lower data, or are defined more loosely (e.g. Raper 1992). Feature descriptions may be useful only at one spatial scale, and perhaps only for a specific number of casks (Burrough 1987: 137). Worse still, feature descriptions may not convey all of these attributes, so the user may not be aware of what they are!

Whilst there is a loss of precision in the formation of features, chis is often regarded as being offset by the increased utility that feature descriptions can provide. Features are conceptually simple entities, which aids the user since they appear to be both unambiguous and homogeneous. Their homogeneity also ensures that manipulation and combination is straightforward in an algorithmic sense, and that results can be readily interpreted. GIS operations take as their operands geographic features that could not easily be described in terms of image data. I t is not surprising then that most GIS operations require that data be in feature form before they may be carried out.

The feature view manipulates explicitly defined geographical features. It provides a view where the data values or class labels have been replaced with feature identifiers. Operations can address the underlying geometry of a feature (set of features) or any other recorded property of the feature, via the feature identifier. The feature identifier provides access to any additional

descriptive data which may then be used to impose retrieval qualification. Note that there is no flexibility as to the identity (value) or the shape (spatial description) of the feature, since both are now defined explicitly.

Correspondence between views To summarize the above descriptions, the field model represents what is measurably there without the imposition of any form of classification or identification scheme. The image model provides a quantized, restricted view onto the field model. The thematic model offers a view where data is classified according to (artificial) criteria whereas the fpature model views the world as sets of individually delineated objects. Table 1 provides a summary of the attributes available for operations within the four models described above.

In practice, the distinctions berween these four views can become blurred. For example, point samples gathered for a soil or geology survey are likely to be recorded according to a pre-defined classification scheme, so by the definitions above belong in the thematic view. Likewise, there are strong similarities between the thematic and feature view in cases where each class can be described by a single polygon.

The views as described above are all intimately connected, not surprisingly since they each offer a different means of regarding a single region of interest. To illustrate their relationship, consider the example of a forest. In the field view a forest is not explicitly defined, bur a surveyed point might indicate the presence of trees. In the image view a subset of pixels might show a spectral response typical of a forested region. Within the thematic view, the forest is recognized as such by the labelling of graphical primitives, such labels may be directly supplied or derived from the image view by pixel classification. The same forest in the feature view is represented by an object that has the defined type ‘forest’ which does not rely on any set of graphical

Table 1. Summary of valid retrieval qualifiers

Value qualifiers

Field view a vector of real values individual point geometry within views. Image view a vector of quantized values individual pixel geometry Thematic view a single class label geometry of a class Feature view a feature identifier geometry of a feature

(+ other properties linked to this)

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primitives or overlay for its existence. I t is recognized as a forest by virtue of its gpe, as opposed to any raw data value or arbitrary classification label. Smith and Smith (1977) recognize this kind of object relativity as being necessary to support mulciple abstractions of the same data. As the process of feature formation is carried out, the level of meaning char is encapsulated by a particular region increases. For example, a region extracted from an image to represent a forest receives a more precise (specialized) definition and associated behaviour than does che unlabelled region in the original image. This is generally known as semantic abstraction or extraction (Smith er a1 1992).

Class and feature formarion are also closely coupled to structural abstraction; where objects are described in simple terms within the data model, wirh internal complexity hidden from the user (e.g. Worboys 1994). Intimately connected to the notions of extraction and abstraction is that of introduced uncertainty. As the definition of a region in space is made more precise (certain), the confidence wirh which the definition applies often decreases. This is due to the generalization that has been applied to the data in order to perform

exrraccion; heterogeneous data is replaced by class labels or object identifiers. It is interesting to note that the opposite is often implied; that is, the definition is precise rherpfore the accuracy must be high. The amount of uncertainty that is introduced is determined by the degree to which generalization is required. For example when extracting features, highly homogeneous regions and regions wirh ‘discrete’ boundaries (such as paddocks) will be more accurately described than poorly delineated heterogeneous regions (such as remnant vegetation).

As an example of chree different views of a region, Figure 2 shows a fragment of an agricultural scene as it appears in the image, thematic and feature view. T h e lowest layer is a fragment of an unprocessed Landsar TM scene. T h e middle layer is the result of applying a classifier to the image data, different shades represent distinct land use classes. T h e top layer represencs features that have been extracted from the classified image. (Shading has not been used on che feature view so chat the underlying layers can be seen more clearly.)

A class represents a partitioning of data objects, often according to some predefined scatistical model, such as the Maximum Likelihood

Figure 2 . Three different views of an agriculcural scene.

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Classifier. As can be seen from Figure 2, rhe presence of geographic features is nor logically implied by rhe class labels and cannot be assumed from any groups of connecred pixels chat share the same label. Image noise, lack of resolurion and edge effects all conspire to ensure rhat rhe spectral response of a pixel does nor always equate co class membership and char class membership does nor always imply rhar a darum is part of a specific feature, whose type is rhe class in question. SO, the conversion from classes to fearures is complex since ir must rake accounr of the natural variance within features (Ton ec a1 1331). For certain applicarions, rhe final srep of fearure exrracrion from a thematic view is simply too resrricrive and artificial. This commonly occurs where the resulrs of a classificacion are not broadly homogeneous or sparially clustered, so rhar the formarion of fearures is likely to introduce an unacceptable lowering of precision. In other words, rhe producrion of significantly sized fearures \\,odd involve roo great a generalizarion of rhe data.

Nowhere in the above descriprions is any mention made ofdar:i srruccures. All views mJy use as their basis nny of rhe supporred spatial data strucrures. In practice, the preferred types might be pointsets for rhe field model, rmter for the image model, q r d r r c r for the cheniaric model, and topologird L'ei'tor for rile fe:iturc model, bur the user need not be concerned \virh this as i t does not impinge on their inreraccion wirh the system.

Transformations between views

One of the difficulties in formulating a model describing rhe transformarion of geographic dara is the complexity of rhe tasks involved. Figure 3 shows a single example of each type of transformation rhat mighr be supported for the inregrarion of rhe four views defined above.

As can be seen, some of rhese transformations not only change rhe data set, bur also involve rhe migrarion of the data set from one geographic model to another. For example, the classificarion operation labels the primary (image) data 1-0 give rhemaric data in the form of overlays. From rhese overlays, geographic features may be formed by some kind of labelling procedure. Alrernarively, primary data may be rransformed direcrly ro features via a process of feature exrracrion. In the reverse direction, geographic features can be transformed inro classified dara by aggregarion (by class) to produce a rhernaric overlay. Classes and fearures may be inrerpolared or exrrapolaced to produce an estimated primary data source; rhar is, projecred back to rhe image or field view. Where the classes and fearures are derived from primary data, then rheir reducrion back ro primary data shorild be error free, since ir represenrs the inverse of their formarion (provided lineage derails and rhe original dnra are available). Where rhe classes and fearures have been supplied ready made, [hen rheir projection to field or image views may involve a great deal of uncertainty.

Figure 3. Some example transformarions benveen geographic data models.

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Architectural support for many simultaneous views of space

T h e first issue to be addressed is that of a suitable architecture to support a number of simultaneous views onto the data. Then the effects of the various transformations must be modelled within this architecture, in terms of both the schema dependent and the abstract (schema independent) properties of a geographic dataset.

Traditional architectures for information systems From a computing perspective, a GIS may be regarded as a highly specialized database management system. Accordingly, GIS may be designed around the proven ANWSPARC three layer database architecture (e.g. Korrh and Silberschan 1991). An overview of this architecture is given in Figure 4. A single conceptual data model is mapped onto several external views which each present a subset of the conceptual schema to a user or application program. T h e internal layer is responsible for the physical organization and retrieval of data from the storage media. T h e binding between the various layers occurs at runtime, thus providing a large degree of logical and physical data independence.

An architecture for full integration Based on the above, an integrated architecture can now be developed. I t must support interaction with data within each of the four views described above, so all of these views must be represented by their own conceptual schema. Users or applications a t the

Figure 4. T h e ANSI/SPARC three layer database

l - 7 Conceptual Layer L1"

internal

loyer

external layer may choose to work only with features (or any other view of the data) but are not forced to do so. The required extensions to the architecture of Figure 4 are shown in Figure 5. Note that the conceptual layer is composed of four distinct schernas (representing the four views), and must also contain functionality to provide conversion between these views, as shown in Figure 3. In practice, the conceptual layer may be broken down Further to separate out high level semantics from lower level concerns such as geometry, resulting in a semantic and a logical layer (Hadzilacos and Tryfona 1996, Roberts and Gahegan 1993). T h e internal layer is shown here as consisting of three different components, each specializing in a particular spatial data structure. Other types of storage structures will be required in addition, for storing non-spatial attribute data, and possibly for storing spatio- temporal data (e.g. Worboys 1992).

The different concepiual views need not be physically par t of a single piece of software. Provided that integration berween views can be achieved at the conceptual layer as opposed to the external layer, then there is no reason why the views may not be separately implemented or be part of distinct (even remote) systems (although in practice there may be some nerwork bandwidth limitations).

A formal basis for transformations within and between views Pascoe and Penny (1995) introduce a useful formalism for describing the movement of data sets between GIs , with the aim of exchanging data sets across a communications network. Here, their original ideas are adapted to describe the movement of data between the different geographic models and the different physical data structures defined above. T h e resulting formalism may then be used to describe the various transformations that occur within and berween views, as shown in Figure 3.

Within the architecture shown in Figure 5, a data set may exist at one of four levels of abstraction: field, image, thematic, and feature. These views are described by database schemas at the external and conceptual layer, the internal layer describes only the properties of the data set that relate to storage and retrieval policy. T h e conceptual layer (C) supports all the views 11s {Co,Cl,Cl,C~t where C,, =

feature, Cl = thematic, C, = image, C, = field. The

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External Layer

Conceptual Layer

Internal Layer

......

Oblect Thematic Image Field

Raster Quadtree Vector Struclures Structures Slruclures

R Q V

Figure 5. An archirecrure for full data incegrarion.

individual views defined ar rhe external level ( E ) may supporr only some of these bur may inrroduce additional, more specialized interfaces as menrioned above. Ar the inrernal (physical) layer ( P ) , several lower level views onto rhe dara are provided in rerms of rhe physical dam organization (dara strucrures) and access merhods. I n rhe examples shown here, rhree types of sparial data srrucrure are used: {PI,, P p P v ] where PI, = rnsrer data, PQ = quadrree data and Pv = vector dara. So, within the darabase, a data ser may be described by the following expression:

A(c,, C,,, fz), w h e r e x E { O , T , I , F ,... I , y E 10. T. I . Fl: z E {R, Q, Vt.

Transformarions on A may now be described by a befire and a$er stare:

Thar is, a rransformarion rakes a data ser A and produces a (modified) dara set A', possibly represented in a different manner ar the exrernal, conceprual and internal layers.

For example, the transformation from raster image dara to vecror fearure dara as shown in Figure 1 , via a classified overlay, represenred by a quadrree, is described by rhe following expression:

A(.. , c,, PR) -) A'( ..., c , , PQ) + A"( ..., c,,, f"). In rhe case of a system based 3olely on features ar rhe exrernal level, all rransformarions must ulrimarely map rhe data ro the fearure view:

A(€,.C,.. f,, + ... -+ AYE(,, c,,, f,)

Characterizing the effects of transformations on a data set When considered independently from any sysrem implernentarion, a geographic data ser may be described according ro a number of absrracr properties (e.g. Flowerdew 199 1). For simplicity, we shall restricr rhe discussion ro rhe following properties: the data values, their spacial exrenrs (Roberrs er al 1931), their temporal exrenrs (Langran 1992), associared uncerrainty (Veregin 1995) and lineage informarion (Lanrer, 1391). So, independent from any database schema, a dara ser may be described as:

A(D. S, 7; U, W, in which D represenrs rhe data values, S rhe sparial extents and T the temporal exrenrs. U represenrs some measure of uncerrainty associated wirh D, S and T, and V represenrs the lineage or derivation hisrory of A in terms of the operations, and rransforrnarions by which A was formed. Transformarions on A can now be expressed by schema dependenr and independenr terms, where the firsr rerm represenrs the absrracr properties of the data set, and rhe second rerm represents rhe implementation schema for a given archirecture as:

Generally, rhe uncerrainry associared wirh rhe resulr may be defined by the uncerrainty already in A plus the uncerrainty associared wirh rhe rransformarion (Y) :

U '= U 8 unce rmin ry (Y

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I r is recognized char the adding together of uncertainty values is a far from trivial exercise (e.g. Veregin 1995). The aim here is ro show how uncerrainry may be supported and propagated through rhe rransformarions described and nor to consider bow it might be calculated. T h e plus operaror should be regarded as a complex (and polymorphic) method to be supplied by a domain expert. If Y' does nor change any of D, S or T, then U, = U, that is, rhe uncerrainry remains fixed. In a similar manner, rhe lineage of rhe resulr is given by rhe lineage of A plus a parametric description of 'f':

v'= V O Y .

We now have a formalism powerful enough ro characrerize the behaviour of the rransformations inrroduced above, Some examples are given below. The main emphasis of currenr CIS functionality is rowards the value and spatial domains, and rhe examples reflecr chis bias. Also, we are nor concerned here wirh the interface by which the darn is presenred ro the user, so rhe exrernal layer is lefr unspecified.

A conversion from quadrree to raster (or vice versa) changes the physical format of the dam, but none of rhe absrracr properries of the data set (D,S, or T ) , or the conceprual data model, which remain fixed. T h e rransformarion is given by:

A(D, 5, 7; u. V ) : (.. .,cj,, f Q ) -) A'(D. s, 7; u, V): ( . . . ,CY, PI{).

By conrrasr, raster or quadrree ro vector conversion may also change the spacial properries of the data in a subtle manner due ro quantization and smoorhing, rhus adding in further uncerrainry. However, the data values themselves (and rhe data model) remain unchanged:

A(D, 5, 7; u, V ) : (.. .,CY, P,) -) A'(D, s: 7; u: v'): (..., cu, f " ) . ( 2 )

Poinr to grid inrerpolarion changes both rhe data ser and the conceptual model:

A ( D . 5. 7; u, V ) : ( . . . , c,, P,) -)

A Y D ; 5 : 7; u: L"): f . __ , c,. f L ) . ( 3 )

Image quanrizarion is an example of a rransfor- mation wirhin the image view. Data values associated with individual pixels are changed:

A@, s, 7; u, V): (..., c,, f,, -9

A'(D: s, 7: u: V ) : (.. .,c,. P,). ( 4 )

Cartographic (spatial) generalization (Muller 199 1) is an example of a transformation wirhin the feature view where the spatial description of individual features is simplified:

A(D, s, 7; u, V ) : (..., co, f , ) -+ A'@, s: 7; u; V'): (..., c,,, f,). ( 5 )

Temporal interpolation of data. Although not widely available in CIS, there is a need to transform data sets in order that [hey share some common rernporal basis. Only one conceptual model for rime is used, so unlike irs sparial counrcrparr, remporal inrerpolarion does nor move the data from one geographic model ro another:

A(D, s, 7; u, V ) : (..., C", f,, -9

A'(D. s, T: u: V ' ) : i ..., C),, P,). (6)

T h e path from image dara (raster) via a classified overlay (quadrree) ro distinct features (vector) as shown in Figure 1 involves transformation via rwo srrs of value and sparial properries and is described by:

A(D, s, 7; u, V ) : ( . . . I c,, f i t ) -)

A'(D: 5 : 7; u: V ' ) : (...,C,., Po) -+ A"(D': s': 7; u'; V"): (...,C(), f " ) . (7)

In some circumsrances data musr be integrated, with several inpur data sers being processed rogerher ro produce a resulring data set A,. Examples are mulri-band image classificarion and multiple map overlay operations. Inregrarion may be described by Equarion 8 (see below).

Some properry of rhe data ser musr be used as the basis of inregracion, usually space, ro which rhe remaining properries are mapped. For example, if space is rhe inregrator, as in 3 multi-b.ind image

146

SpcciFying rhc tronsfomctions within and betwcen gcogrophic &to iiiodels

classificarion cask, rhen A , , A,, ..., A,, represent the n inpur layers which musr be spatially compatible (aligned grids). In rhis case rhe following consrraincs also apply:

D, = 'I' (Dl, D2 ,... D,), s, = s2 ... = s,,, c, = c2 . . . = c,,.

I f che example is carried through, [hen C,, C,, ..., C,, represenr rhe image view (CJ; P I , Pl, ..., P,, represenr rasrer dara srrucrures (PR); and che resulring view (C,.) is chemaric. Also, S,, Sz, ..., S,, represenr pixel geometry 5, and mighr be a maximum-likelihood classifier. Substituting these consrrainrs inro Equacion 8 gives Equation 3 (see below).

T h e remporal properries of A, musr represenr some form of union of TI, T,, ..., T, from rhe lefr- hiind side of Equarion 3. Additionally, [he uncerrainry of rhe resulr U,. now becomes the zrnion of all conrriburing uncerrainry plus the uncerrainry introduced by rhe fiiiiccion 'i' used ro derermine D,. Finall>: [he lineage of rhe resulc V, is a node rhat links rhr sources of A , , A?, ..., A,, via a description of 'i'. Note &.o rhat rhe physical darn scrucrure used ro represrnr rhr resulring darn ser (P,) is nor consrrainrd ro be raster, bur is free ro rake on any form.

If rhe input darn sen do noc share ;I common sparial basis and physical view rhrn [hey musr be made common by some sorr of projection before incegrarion can be occur. A transformarion to achieve this must have rhe form:

A( ..., 5 ,.... ) : ( ..., CY' f,) -3

A'( .... 5,l ,... ):( ..., c,, fR). Alrernatively, rime may be used as rhe inregracor, with an expression similar ro Equation 8 or 3, excepr no\v T musr be held consranr. In rhis case the resulting shape is defined by the union of S,, S,, ..., S,, and all inpur dara must share a common remporal basis. Finally, space-rime may also be used as the inregraror, requiring all inpur dara ro be projecred ro some fixed values for both S a n d T.

Suppose now thar rhe user requires to apply a certain operarion (Y), the behaviour of which is characrerized by an expression as described above. The left side of rhis expression (given hy Z,) describes the srrucrure that all input data sets ro '4' must have. Each input dara ser may be characterized by che expression A,. If, for any inpur dara ser, A, # Z,, then one or more projections must be applied ro rhat dara ser before the operation can begin.

If the rarger schemas are given as C and P rhen rransformarions for dara projection are described in general by the following three expressions:

Integraring over space. All inpur data sets musr share a common (sparial) basis denored by s, c, P :

A( ..., s, ,... ):( ..., cy. r,, --f

A ' ( . . .s,. . .):(. . . .c, P). (10)

Inregracing over rime. All inpur data sets must firsr share a common (remporal) basis, denored by T, C, P:

A( __., 7; ,._. ):( ..., c),, f,) -3

A'( ..., T,. . .):(...'*C, P). (1 1)

Inregracing over space and rime. Both the sparial and remporal characrerisrics of inpric dara sers musr be common (S, T, C, P):

A( _... s;. 7; .... ):( ..., c-, fz) -3

A' ( ..., S , T , ___ ):( ..., C, P). (12)

Transformations may be described easily using a declarative language such as PROLOG or in an experr sysrem shell. I r rhen becomes possible ro provide a guided (or even auromared) parh rhrough rhe various cransformarions rhar may be necessary before dara can be combined, by using expressions similar ro Equations 10, 11 and 12. This is achieved by simple marching on rhe lefr and righc side of expressions ro form a chain of rransformarions rhac rakes the source dara ro a form suitable for the currenr operation. I t is also a simple matrer ro inform the user of rhe likely effects chat various cransformarions have on rhe dara by examining

147

M Gahegon

which properties of the data set are affected, i.e. those properties that differ between the lefi and right side of expressions.

A transformation taxonomy Figure 6 represents a more general view of Figure 3, showing the transformations that are required within and between the four defined views. It is a fully connected directional graph of in-degree four and out-degree four where each arc represents a family of transformations. Note that all rypes of transformations except 1, 6, 11 and 16 cause the data set to be moved to a different view.

Table 2 summarizes the transformarion types shown in Figure 6 as a series of expressions, along with some further examples of operations. The expressions offer a means of classifying the various transformations, and form a useful taxonomic basis by which to characterize GIS functionality, according to its effect on a data set. The syntax [D, 5, Tldenotes at least one of D, S and T must change as a result of applying a transformation of this type. The expressions show the minimum effect of each transformation; some transformations may change additional properties. For example, the quantization of image values and simple selection of image data by value are both characterized by arc 6. The quantization introduces additional uncertainty, whereas the selection does not.

In a similar manner, Table 3 shows the transformations that apply only at the physical layer benveen the supported data structures. The starred transformations (to and from vector format) may

Figure 6 . State transition diagram for geographic data.

subtly alter the geometry and so (unfortunately) have an effect on the schema independent properties of the data set (e.g. Laurini and Thompson 1992).

By studying the expressions resulting from the transformations in Figure 6, Tables 1 to 3, and the examples given above, it is evident that behaviour may be characterized according to the following:

1 Data Preserving. Some transformations (e.g. Equation 1, raster to quadtree conversion) do not change the abstract data set at all; the value, spatial and temporal properties, and by implication the uncertainty remain the same after transformation, changes are restricted to the physical schema alone (D' = D, S' = S, T = 7; U' = U, Cy, = Cy, P,, # P,). So, a change in data structure does not necessarily imply a change in the data. More formally this is stated as:

2

PI# P=b D'# D, S'#S, T # 7; U'# U

Schema Preserving. Some transformations change the abstract data set properties, whilst retaining the same data base schema (i.e. any of D, S, and T may change withour affecting C or P). Examples are overlay and buffering operations and Equation 6. These types of operations form the majority of the functionality in existing systems (U'# U, Cy. =

Cy, P' = P). So a change in the data need nor imply a change in the schema:

D'# D, S '# S , T # T a C'# C, P'# P.

In this group there are a subset of transformations which alter the data in a subtle way, by changing the spatial or temporal properties. Examples are conversion to and from vector formats, quantization and cartographic generalization Equations 2, 4 and 5 . These operations aim at preserving the nature of the data set intact, but are sometimes forced to compromise for practical reasons, such as resolution limits.

3 Data set Restricting. Some transformations change a data set by reducing the number of elements within it. without changing these elements or the darabase schema (LI' = U, Cv. =

Cr P' = P). Examples are retrieval operations applied to the different views such as windowing or selection, as described in Table 1.

148


#

1

2

3

4

5

6

7

8

9

10

1 1

12

13

14

15

16

Description

Field to Field

Field to Image

Field to Thematic

Field to Feature

Image to Field

Image to Image

Image to Thematic

Image to Feature

Thematic to Field

Themotic to Image

Example

windowing

quantization onto grid

point classification

pattern recognition

interpolation

image windowing

pixel classification

oblect recognition

interpolation to points

interpolation to grid

Thematic to Thematic classification [super classes]

Themotic to Feature feature formation, connected

component labelling

interpolation to paints Feature to Field

Feature to Image interpolation to grid

Feature to Thematic feature aggregation by type

Feature to Feature feature selection by ospatial property

Table 2. Transformation expressions within and between geographic data models.

4 Schema Changing. Some transformations change the schema from one level of abstraction to another Equations 3, 7 and 9. In the views defined, this must inevitably change the value and spatial properties of the data set, and therefore also the uncertainty. As previously argued, each view has a different spatial basis, so

Schema transformation

A([D, S, r ] ,..., V ] : [ ..., CF ,... 1 --f A' [ [D' , s', T' l , . . . , V ) : [ ..., CF,.. ]

A(. . .,S, . . ., U, V ] : [ . . . ,CF,. . . ] + A ' [ ..., S', ..., U',V]:( ._. , C, ,... ]

A(D,S ,..., U,V) : ( ..., CF ,.._ ] + A ' [D',S', ..., U' ,V ) : [ ._. , Cr ,.., ]

A[D,S,.. . ,U,V] : ( . .,CF,. ..)

-+A' [D' ,S' ,..., U' ,V ) : ( ..., Co , . _ . ]

A[. . .,S,. . ., U, V ] : ( . . . ,Ci,. . .] + A ' [ ..., s' ,..., u' ,V] : ( ..., CF, . . ]

A([D, S, T ] ,..., Vl:[ ..., CI ,. . . 1 + A' [ [D' , S', J'I,..., V ] : ( ..., Ci ,... 1 A[D,S ,..., U , V ] : [ ..., C,,.. 1 + A'[D' ,S' , ..., U' ,V ) : [ . . . , Cr , _ . . ]

A(D,S,. . ., U , V ) : [ . . ., Ci, . . .) + A'(D',S', ..., U',\'']:( _ . . , Co ,. . . ]

A(D,S ,..., U,Vl:l ..., CT , . . . I + A'(D' ,S ' , ..., u',V]:( . . . , CF, ..]

A[D,S,. . , u, v ] : ( . . . , CT,. . . ] + A'[D',S' ,..., U' ,V ) : ( ..., Cl , . . . ]

A([D, S, r ] , . . , V ] : [ ..., CT ,.. . 1 -+ A' ([D', s', , . . . , v w ., cr ,... 1 A[ D, S, . . , U, V ] . ( . . . , Cr, . . ] + A'[D' ,S' , . . ,U ' ,V ] : [ . . , Co ,... 1 A[D,S ,..., U,V) : [ . . . , Co,.. ] -+ A' [D ' ,S ' , , . . , U ' , V ] : ( ..., CF , . . . ]

A(D,S,. . . ,U,V]. [ . ..,Co,. . . ] + A' [D',S',.. ,U ' ,V] : ( . . . , Ci ,... ]

A[D,S,.. . ,U ,V] : [ . . .,Co , ._ . 1 + A'[D' ,S' , ..., U ' , V ] : [ . ., C, ,... 1 A[[D, S, r ] ,..., V l : [ . . . , C, ,... 1 + A' [[D', S', TI , . ._, V ) : [ ..., Co, ]

moving between views must change at least this spatial basis, and will therefore change the data as well, due to interpolation, labelling, and so forth.

C'# C+ D ' # D, S '# S, U p # U, Cy. # Cy

5 Accumulation of Uncertainty. All transformations that in any way estimate (interpolate) or summarize the properties D, S and T, or move from one schema to another

149

M Gahegan

4 5 6

7

a

9

Name

Raster to Raster

Raster to Quadtree

Raster to Vector'

Quadtree to Raster

Quadtree to Quadtree

Quadtree to Vector'

Vector to Raster'

Vector to Quadtree'

Vector to Vectoi

Data structure transform

A(. . ., . . . ,PJ + A[ . . ., . . . ,4] A( ...,. . . ,PI) + A( . .., . . .,Po) A(. . .S,. . ., U, V ] : ( . . .,... Pi) + A(. . . ,S,. . ., U', V ] : ( . . ., . . . ,Pv)

A(. . . , . . , PQ) + A(. . ., . . . ,PI ) A( ..., . . .,Po) + A( ..., . . .,Pa) A(. . .S,. . .,U, V ) : ( . . . , . . . PQ) + A [ . . . ,S, . . , U',V):(. _ . , . . .,Pv)

A(. . . S, . . . , U, V ) . ( . . . , . . . Pv) + A(. . ,S ' , . . . , U', V ) : [ . . ., . . .,PJ A( . . . S ,..., U,V) : ( ..., . Pv) + A( . . . , S', ..., U ' , V ) : ( _ . . , . . . , PQ) A(. . . , . . . , Pv) + A(. . . , . . . ,Pv)

musr add to rhe uncertainty in the resulting dara set.

D ' # D v S'# S v T # T - I/ ' > I/

6 Accumulation of Lineage. All transformarions add ro the lineage of the resulring data set, b"# I'. lMore precisely, b"= I/$ 'I-'.

7 Logical-Physical Independence. Transformarions thar change rhe view do nor necessarily have ro change rhe way the dara is physically stored; there are no hard requiremenrs abour, say, using rasrer srrucrures for images or vector srrucrures for fearures:

C' f c * P' # P.

The above characrerizarion of behaviour offers some consrraints on the mappings beween data models, and provides a rational and consisrenr framework by which ro manage and communicate rhe effects of rransformations. Such expressions may be also be used as a guideline ro assess rhe implicit assumprions that are somerimes made when combining essentially disparare data together, simply because it happens ro share rhe same geomerry. As such they form a viable basis for describing the effecrs of data inregrarion, interoperabiliry, and dara exchange on the various absrracr properties of a geographic data ser.

Conclusions

This paper describes some underlying models of geographical space, specifically characterizing the rransformarion of data sets from one model ro

Table 3. Trans formarions beween internal dara structures.

anorher. Transformation expressions are developed and form a useful basis by which to understand and describe data set behaviour. T h e geographic models are nor presented as drfinirive. since orhers are possible (indeed not all may agree with rhe descriprions given). However, rhe key ro data inregrarion, from wharever model, is ro understand rhe behaviour o f a dara ser as it is rransfornied from one model to another and ro this end the work presented appears useful.

T h e operarors used for rhe addition and union of spacial, temporal, uncerrainry, and lineage properries require a more formal and low level definition so rhar rhe actual effects of rransformarions can be calculared, and this is the subject of currenr research by the aurhor. Further work is required ro examine the potenrial of auromared reasoning for data inregrarion. Whilsr implemenrarion is srraighrfonvard, it raises some complex issues from the perspecrive of interface design and user interaction. Specifically, if inregration derails are hidden from rhe user then aurornarion may serve to reduce the user's ability to correctly inrerprer results rather than improve it. With all such systems, one must build inrerfaces ro exactly suir rhe differing needs of a wide user-base, so several modes of inreraction will be necessary.

T h e goal of model-level inregrarion of dara in rhe geosciences has as yet ro be satisfied. Until it is, we run the subsranrial risk of misundersranding the dara we use and incorrectly applying it in the analyses we perform. I r is nor impossible to inregrare acriviries char logically occur in different diira models, but i r does presenr some big chdlsnges in

150

Sp..ifr;ng he frorrskwmafians withh and betweera geographic dartz models

the way in which we design systems and in rhe meaning which we ascribe to dara. T h e modelling capabilities of esrablished CIS are guilty of gross simplification of the geographical environment. Whilst they may be adequare for a large range of problems, they do not provide a sufficiently rich semanric basis from which ro proceed.

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Documents

Specifying the transformations within and between geographic data models