Morphology From Imagery

7/30/2019 Morphology From Imagery

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Environment and P lanning A 1995 , vo lume 27 , pages 7 59 -7 80

Morphology from imagery: detecting and measuringthe density of urban land useT V Mesev, P A LongleyDepartment of Geography, University of Bristol, University Road, Bristol BS8 1SS, EnglandM Batty, Y X i eNational Center for Geographic Information and Analysis, State University of New York,Wilkeson Quad, Buffalo, NY 142 61 -00 23, USAReceived 15 January 199 4; in revised form 20 Ap ril 1994

Abstract. Defining urban morphology in terms of the shape and density of urban land use hashitherto depended upon the informed yet subjective recognition of patterns consistent with spatialtheory. In this pap er we exploit the po tential of urban image analysis from remotely sensed d atato detect, then measure, various elements of urban form and its land use, thus providing a basisfor consistent definition and then ce com parison. First, we introduce methods for classifyingurban areas and individual land uses from remotely sensed images by using conventional maximumlikelihood discriminators which utilize the spectral densities associated with different elements ofthe image. As a benchmark to our classifications, we use smoothed UK Population Census data.From the analysis we then extract various definitions of the urban area and its distinct land useswhich we represent in terms of binary surfaces arrayed on fine grids with resolutions of approximately 20 m an d 30 m. These images form surfaces which reveal both the shape of land useand its density in terms of the amount of urban space filled, and these provide the data forsubsequent density analysis. This analysis is based u pon fractal theory in which densities ofoccupancy at different distances from fixed points are modeled by means of power functions.We illustrate this for land use in Bristol, England, extracted from Landsat TM-5 and SPOT HRVimages and dimensioned from population census data for 1981 and 1991. We provide for thefirst time, not only fractal measurements of the density of different land uses but measures of thetemporal change in these densities.1 M easuring urban morphologyW h a t e v e r a p p r o a c h t o u r b a n a n a l y s i s i s t a k e n , c i t i e s a r e u s u a l l y v i s u a l i z e d a t s o m es t a g e i n t e r m s o f t h e i r g e o m e t r i c f o r m . U r b a n e c o n o m i c s , t r a n s p o r t a t i o n , a n d s o c i a ls t ruc tu re a re p red ica ted in spa t i a l t e rms , and thus the e f fec t s o f such theory a reo f t e n a r t i c u l a t e d t h r o u g h g e o m e t r i c n o t i o n s i n v o l v i n g t h e s h a p e o f u r b a n l a n d u s ea n d t h e m a n n e r i n w h i c h i t s p r e a d s . S h a p e a n d d e n s i t y m e a s u r e t h e w a y i n w h i c hur ba n spac e i s f i ll ed and a r e thus key e lem en ts in th e def in i t ion o f m orp ho log y .Unt i l r ecen t ly , i t has been d i f f i cu l t to l ink the geomet r ic fo rm of r ea l c i t i e s to fo rmalt h e o r i e s e x p l a i n i n g d e n s i t i e s , a n d t h u s u r b a n e c o n o m i c t h e o r y , f o r e x a m p l e , h a sevo lved wi th in h igh ly idea l i zed geomet r ies which bear l i t t l e r e la t ion to r ea l spa t i a lp a t t e r n s .

The deve lopment o f f r ac ta l geomet ry , however , wh ich i s in es sence a geomet ry o fthe i r r egu la r , has c lea r r e levance to spa t i a l sys tems such as c i t i e s . The rud iments o fa f r ac ta l theory o f c i t i e s now ex i s t s which has the po ten t ia l to syn thes ize many ideasf r o m l o c a t i o n t h e o r y w i t h s p a t i a l f o r m ( B a t t y a n d L o n g l e y , 1 9 9 4 ; F r a n k h a u s e r ,1994) . The no t ion tha t c i t i e s a re s e l f - s imi la r in the i r func t ions has been wr i t l a rgein u rban theory fo r over a cen tu ry , and i s man i fes t in t e rms o f r e la t ions such as ther a n k - s i z e r u l e , h i e r a r c h i c a l d i f f e r e n t ia t i o n o f s e r v i c e c e n t e r s a s i n c e n t r a l p l a c et h e o r y , t r a n s p o r t a t i o n h i e r a r c h i e s a n d m o d e s , a n d i n t h e a r e a a n d i m p o r t a n c e o fd i f fe r e n t o r d e r s o f h i n t e r l a n d ( A r l i n g h a u s , 1 9 8 5 ; 1 9 9 3 ) . A l l t h e s e r e l a t i o n s w h i c hf o r m t h e c o r n e r s t o n e s o f u r b a n g e o g r a p h y c a n b e d e s c r i b e d a n d m o d e l e d b y u s i n g


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760 T V Mesev, M Batty, P A Longley, Y Xie

pow er laws which are fractal. W hat this new geom etry is beginning to do is to tie allthese notions explicitly together in a geometry of the irregular, a geometry of thereal world (Mandelbrot, 1983).We take this fractal theory as our starting point and, in this paper, we use itssimplest elements, which involve the measurement of shape and density. Our purpose here, however, is not to extend this theory very far but to concentrate on theequally important problem of detecting the appropriate shape and density of urbanareas and the land uses which occupy them. Hitherto, most characterizations ofurban land use have depended upon some visual inspection of spatial patterns andthe identification of visual distinctions w hich serve to reinforce functional differences. Such classification is often informed by statistical analysis, but methods forcontrolling the consistency of such definitions between different case studies arelacking, and thus comparative analysis is limited.Measurement, of course, prescribes analysis, and it is only as improved data setsbecome avai lable and methods of represent ing urban phenomena become moretransparent that morphological analysis becomes more tractable. In our own previous work (Batty and Longley, 1994; Longley and Batty, 1989a; 1989b) we havebeen aware that the definitions of 'urbanity' adopted for particular administrativepurposes are not usual ly the most appropriate to more general measurement andanalysis. Yet issues of confidentiality and aggregation dictated that socioeconomicdata are rarely available for disaggregate study at finer resolutions.Where models of socioeconomic data have been devised to interpolate continuous surfaces about point information (Martin, 1991), estimates are not accurate atany scale finer than aroun d a 200 m resolution, and there is the co ncern thatmeasurement of urban form in fact represents the assumptions in the modelingprocedure, rather than the underlying distributions of individuals, buildings, andland uses which it is intended to detect. With the development of remotely senseddata, however, the prospect exists for both finer resolution and consistent classification of urban areas and land use across universally available data. This is the taskwe will address here, showing how the data which are derived can be interpreted bymeans of fractal theory.In this paper we are thus concerned with the detection, then measurement, ofurban morphologies from remotely sensed imagery. We will begin with methods forthe generation of distinct land uses from such images, outlining the conventionaltechnique of discrimination based on maximum likelihood estimators applied to thespectral bands which define such images. We augment this method by using smallarea census statistics from the UK Population Census to ground the interpretationswe make in an independent data source, and from the resulting analysis we definediscrete categories of land use for each pixel in the image. The resulting surfacesshow the presence or absence of a particular land use in each cell, and it is thesethat are then measured in terms of their shape and density by means of fractalmethods .Here we use two broad approaches to measuring density. First , we consider theabsolute amount of space within the total available which is filled by a given landuse and, second, we consider the rate at which space is filled with respect todistance from some central point in the city [usually the central business district or(CBD)]. Both these methods yield various parameters, the most significant of whichis the fractal dimension. However, because these methods yield different estimates,we introduce a new method which is based upon elements of each. We also test thesensitivity of these methods to reductions in the size of the images over which theanalysis takes place.


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Measuring the density of urban land use 761

The data for which we develop these applications are derived from the LandsatTM-5 image taken in Apri l 1984 and from the SPOT HRV image taken in June1988 for the urban area of Bristol, England. From these images, together withsocioeconomic data from the 1981 and 1991 Population Censuses, different typesof land use are extracted, and these form the surfaces which are then subject tofractal analysis. Four distinct components of land use can be identified from the1981 data, eight from the 1991, and this enables some comparison, through fractaltheory, of urban form in terms of individual land-use patterns. Temporal comparisons are also made, with the emphasis on comparative analysis demonstrating howsuch analysis can be made rout ine by use of remotely sensed data. We conclude thepaper with some comments on the potent ial for using remotely sensed data forextensive comparative analysis of urban morphologies, a potential which has notbeen realizable hitherto.2 Extracting land-use patterns from remotely sensed dataStandard methods for classifying remotely sensed data exist within proprietarysoftware image analysis packages, and here we have used the most commonly usedcommercial per-pixel classifier, the maximum likelihood classifier based on Bayesianprinciples. The version we have chosen is implemented in the Imagine Version 8.0.2package from ERDAS Inc., Atlanta, GA. This software computes the likelihood ofa pixel belonging to any class, with the spectral categories of the image representedby Gaussian probability density functions described by associated mean vector andcovariance matrices. Readers who are unfamiliar with these methods and theprocess are referred to any of the standard works, such as Strahler (1980). Themaximum likelihood discriminant function, Fik, is given as

Fik = ^ p IQI"* cxp[-\(Xt -MjC^X; -M k)] , (1)where Fik is the probability of pixel / belonging to class k; n is the number ofspectral bands; Xt is the lxn pixel vector of spectral band values for each pixel /,where / = 1, 2, ..., N; Mk is the lxn mean vector for class k over all bands; andCk is the nXn var iance-covar iance matr ix for class k over all bands. Mk and Ckare based on those subsets of observations of pixels in the image that represent'training samples' that the user identifies as being representative of each class k.Given these parameters, it is possible to compute the statistical probability of agiven pixel value being a member of a particular spectral class assuming equal classprior probabilities. Parametric classifiers such as this take into account not only themarginal properties of the data sets but also their internal relationships through thestandardizat ion on the inverse of the variance - covariance ma trix of each class.This is one of the prime reasons for the great robustness of the technique as wellas for its relative insensitivity to distributional anomalies. If the spectral distributions for each class deviate greatly from the normal , the procedure identifies poorperformances, and if no information about the actual dimension of the group isavailable, areal estimates tend to be highly inaccurate (Maselli et al, 1992). Theproblems encountered with the conventional maximum likelihood classifier havebeen, to a certain extent, circumvented by nonparametric algorithms, such as theone developed by Skidmore and Turner (1988), who documented a 1 4 % - 1 6 %improvement in overall accuracy. Their decision rule lies on the extraction of classprobabilities from the gray-level frequency histograms. In this way, all the information about the class probability distribution is extracted from the analysis of


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training samples, with modifications intended to preserve the areal estimates of eachland-cover type.In our applications, we have avoided the equal probability assumption byincorporating as prior probabilit ies knowledge from outside the spectral domain(Mesev, 1992). Prior probabilities describe how likely a class is to occur in thepopulation of observations. They can simply be seen as estimates of the proportionof the pixels which fall into a particular class. Formally, this is the conditionalprobability, Pk\x it vf> of spectral class k given pixel vector X t and some ancillaryvariable Vj. This forms the basis of a modified decision rule, assigning the ithobservation to that class k which has the highest probability of occurrence, giventhe multispectral dimension vector X t (which has been observed) and ancillaryvariable Vj. The probability in equation (1) can thus be modified as P ik with respectto these priors,^-Trr*-- (2)

wThe numerator shows how the joint prior probabilit ies of the spectral and ancillaryvariable are incorporated into the maximum likelihood estimator, F ik, with thedenominator ensuring that all conditional probabilities sum to 1.Classification of urban images is common within the remote-sensing literature,although the purpose of most such exercises is the differentiation between broadland-use categories rather than the differentiation of land-use types within the urbanmosaic. The process of classifying an image by means of any of the standardtechniques requires the user to identify in advance the number of distinct classesinto which the image is to be partitioned, in this case land-use categories. To thisend, we adapt established methods based upon equation (1) by defining urban areasof the image which show clusters of like spectral values as 'training samples'; theseare used in an iterative process of classification of the image into relevant classesand in the com putation of the app ropria te l ikelihood statistics. T he p rocess is, ofcou rse, intuitive to a de gree b ut it is aided by the va riance - cova riance analysis, and,in this case, we have used socioeconomic data from the Population Censuses toinform the process.To this end, we have used the surface smoothing technique developed byBracken and Martin (1989). When applied to socioeconomic data based on irregular zonal units, this method transforms the data to regular units at finer levels ofspatial disaggregation (see also the app roa che s of Lan gford et al, 19 91 ; Sadler andBarnsley, 1990). In these applications, socioeconomic data recorded in PopulationCensus enumeration district (EDs) are spread to a finer grid. Data in each ED areassociated with their appropriate centroids and are distributed spatially (accordingto assumptions of distance-decay) by centering, in turn, a moving window (kernel)over the cells containing each centroid. T he distance-deca y mo del is then used tocompute the probability of each local (within-window) cell containing a proportionof the co unt. For any cell /, th e variable Vj is allocated as

Vj= I VmWJm, (3)m e Cwhere Vj is the estimated value in the ;th cell of the output surface; Vm is the value ofthe variable assigned to the rath centroid (where C is the total number of centroidsin the model area) and W Jm is a uniqu e w eighting of cell / relative to cen troid m(based on the distance-decay assumptions). This method enables data on irregularsurfaces to be disaggregated to spatial units which reduce the dependence of density


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on the geographic tessellation used (Openshaw, 1984). In these applications, thesurfaces are in a raster format with a resolution of 200 m x 200 m.The process of generating land-use classes is informed by the surfaces generatedfrom census data in two ways: by directing the training sample process and bydetermining the areal estimates for each class (Mesev, 1993). First, samples ofclasses are ne ed ed for all superv ised image classifications. Th es e are usuallyspectrally homogeneous areas of the image that represent a distinct category.U rb a n -r u ra l distinctions are relatively straightforward. Artificial structures arephysically more solid and smoother than vegetation or soil. As a result, urbanareas tend to reflect higher proportions of their incident energy than their ruralcounterparts .The spectral recognition of housing types is primarily based upon the amount ofbuilding materials per pixel, where, for example, detached housing represents thelowest ratio of materials to nonbuilding materials (vegetation, soil, water, etc).Additional information is essential to define the most appropriate breaks in thesomewhat continuous multispectral data that represent built-up land, and this iswhere the census data are used. The surface model can display areas with highprobabilit ies of occurrence of concentrations of particular census variables. Forexample, ED cells with large relative counts of, say, terraced housing can be used todirect the analyst towards that part of the image for the selection of a trainingsample to be labeled ' terraced housing' . For this to be possible, both the satelli teimage and the surface model need to be in very close spatial agreement. We illustrate this in the empirical work which follows.Second, the deterministic probabilities of census variables are in effect the priorprobabilit ies. As described, they can weight each training sample within the maximu m likelihood classifier. For exam ple, if the census indica tes that terra ce d ho usingrepresents 39% of the residential land cover of a scene, then this should beincorporated into the maximum likelihood algorithm as a prior probability of 0.39.The effect is to preserve the areal approximations of the terraced-housing category.We will show how these various elements are used in the analysis of the Bristolimages when we broach the empirical work introduced below where we indicatehow probabilit ies are transformed to distinct (binary) land-use categories.3 Density and fractal dimensionThere are a multitude of geometric measures of morphology which could be appliedto cities, but, as we noted in the introduction, we concentrate here upon the spreadof cities in terms of their density, using ideas from fractal geometry which relate theconventional theory of urban density to the irregularity and self-similarity of urbanshapes (Batty and Longley, 1994). Urban systems, particularly in the developedworld, display clear patterns of density with respect to their historic development,despite changes in overall densities and massive shifts in population over the lastcentury. Even where development no longer focuses upon this coreedge cities asthey are called in North Americadensity still declines with distance from theoriginal seed of development because of inertia in the built form and the fact thatrent per unit of space is always higher in denser areas.

The traditional model originates from Clark (1951) and is based on the definition of population density p[R) at distance R from the core (or CBD) as a negativeexponential function of that distance R. T h en ,p(R) = Xexp(-XR), (4)


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where X is a friction-of-distance parameter, or elasticity, controlling the spread ordensity. In fact, we argue elsewhere (Batty and Kim, 1992) that this model, althoughoriginally predicated as a better alternative to the Pareto or inverse power lawwhich appears extensively in social physics, is flawed: its parameter is scale dependent whereas urban systems manifest a degree of scale independence in terms of theextent to which development fills the space available. The model adopted here canbe stated as

p(R) = KR~\ (5 )wh ere a is a pa ram ete r related to the spread of the function, and K is a normalizing constant.It is thus p ossible to show, given limits on th e rang e of equ ation (5), that thecumulative populatio n N(R) associated w ith the density p[R) can be modeled as

N(R) = GR2~a, (6 )where G is som e consta nt. Fro m equ ation (6) it is clear that the area A(R) overwhich density is defined w ith resp ect to dista nce from the C BD is given by

A(R) = ZR\ (7 )wh ere Z is som e consta nt. For an area which is enclosed by a perfect circle,Z = jt. Th ere are very strong connections betw een urba n density and urba nallometry based on equations (5)-(7), but the most appealing link between theserelations is through ideas from fractal geometry which seek, through the conceptof fractal dimension, to relate density to the extent to which population fillsspace. The rationale for these links has been developed extensively (Batty andLongley, 199 4; Frankh auser, 199 4; L ongley et al, 1991 ), and in this paper wewill only doc um ent results. In shor t, the pow er law defined by e qu ation (6) isconsistent with systems whose activities are distributed according to the principleof similitude or self-similarity. Such laws thus apply wh atever the scale. In thiscontext, it is easy to show that the density parameter a is related to the fractaldimension, D, that is,

D=2-a,and that the cumulative population relation can be written as

N(R) = GRD, (8)where D is a measure of both the extent and the rate at which space is filled bypopulation with increasing distance from the CBD.

Fractal geometry provides a much deeper insight into density functions than hasbeen available hitherto, in that it provides ways in which the form of developmentcan be linked to its spread and exten t. In this pa pe r we do no t discuss form pe r sebut simply concentrate upon the values of the parameters as measures of the wayspaced is filled and the rate at which this space-filling changes with respect todistance from the CBD. In fact, the parameters D and a both measure more thanone effect. Fir st, as these param eter s are related to the slopes of their res pectivefunctions, they can be used to detect the attenuation effects of distance. At thesame time, the extent to which space is actually filled is also measured by D and a.This interpretation is both important and problematic. It is argued that in urbansystems the fractal dimension of any development should lie between 1 and 2; that


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cities (and their land uses) fill more than the linear extent of the two-dimensionalspace in which they exist (where D = 1) but less than the entire space (D = 2).Notwithstanding the notion that cities may fill some of their 'third dimension', theway we define density as simple occupancy of space means that we constrain thedimension to lie between 1 and 2. However, D and a also measure the attenuationeffects, and it is quite possible for these parameters to take on values outside therang e 1 - 2 if significant chang es in the slopes of their respective functions occu rover the distances used to detect them. Accordingly, we have developed severaltechniques for estimating dimension, the first set of which emphasizes space-filling,the second set, density attenuation.4 Estimating fractal dimensionsThe simplest method, and perhaps the most robust, involves approximating thedimension from the models of occupancy based on various forms of idealizedlattice. Co nsider a square lattice centrally positioned on the CB D, each square ofthe grid being either occupied (developed) or not. Then, if every square wereoccupied (with the implication that D = 2), then the sequence of occupancy wouldbe as follows: first, for 1 unit of spacing (or distanc e), 4 poin ts wou ld be occu pied;for 2 units, 16 po ints occ upie d; for 3 units , 36 po ints oc cup ied; and so on. In thiscase, it is clear that the relation is given by N(R) = 4R2 = {2R)2, and therefore wemight approximate this for a grid which is less than entirely occupied asN(R) = 4RD. Thus, for any value of R, we can compute directly the value of Dor a . A s we argue below, prob ably the most ap pro pria te value for R is the meandistance, R', which, for a discrete distribution of populations n t where / indicatesthe grid location, is given as

R = ~ ^ v >i

Note that R t is the distance from the core to /. Using R f, we calculate the dimension as follows,~N{R'D = i ^7 log10 (9)

This is our first estimate of dimension from which a can be computed as 2 D.Equation (9) depends upon the constant being known, and strictly this constantis not 4 but 2D or, in the case of a continuous system, J I . However a simple translation of this involving density rather than cumulative population removes thisrestriction. The area A[R) of the grid is proportional to its full occupation, that is,A(R) = ZR2. Forming the density directly from equations (7) and (8) gives/x GR Dp W = ^

RD~2 = R~a, (10)where we assume that G ~ Z. Rearranging equation (10) gives a second estimate ofdimension for any value of R. Thus, for the mean value R', D is given by

p 2 + l 2 l ! 0 P ( ^ ( 1 1 )log10i?


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Clearly, the dimensions in equations (9) and (11) can be computed by using valuesof R other than the mean R', but we restrict our usage to the mean here, which hasbeen found in previous work to give the best estimates (Batty and Longley, 1994).Last, from these equations, it is clear that D must lie between 1 and 2.The two methods of estimation defined so far do not take account of anyvariance within the distributions other than throug h the use of the mean. How ever,an obvious and well-used method is to linearize the power laws for density as inequation (5) a n d /o r for the cumu lative po pulatio n relation in eq uation (6) andperform regression to give values for K and a and /or fo r G and D, respectively.These linearized forms are given as follows for the case where discrete densities p tand cumulative populations N t are defined. Th en ,

Pi = l o g 1 0 i ^ - a l o g 1 0 ^ , (12)N t = l o g 1 0 G - > l o g 1 0 i ? / . (13)

From these equations, we can compute the whole range of performance measuresassociated with linear regression, and we will use the square of the correlationcoefficient, r2, as a m easu re of fit. Th is will give us som e idea of the stren gth of therelationships. Also, because equations (12) and (13) are related to one another, theparameter values of a and 2-D will be identical for consistently defined data (seeBatty and Kim, 1992).The regression method of estimation can generate fractal dimensions which areou tside the logical limits associated with space-filling in two dime nsion s. Bec ausethe slope parameters in equations (12) and (13) measure the rate at which densityattenuates and population increases with respect to distance, dimensions greaterthan 2 occur when physical constraints restrict development near the CBD or whendensity profiles show major departures from the norm of monotonic decline fromthe cor e. Values less than 1 can also occu r if reversals from the nor m app ear. Inboth of these cases, the divergence of dimension outside the range 1 < D < 2 isreflected in the value of the intercept constants K and G in equations (12) and (13).However, if these constants are constrained to values which reflect idealized spacefilling, then estimates of a and D will be within the space-filling limits. In this way,a measure of the variance in the distribution of densities and populations can beaccounted for through constrained regression. This is tantamount to replacing theconstant i in equation (12) with a value such as 1, and G in equation (13) with avalue such as 4 or jt. We will use these c ons traints in the em pirical work w hich follows.5 Constructing the land-use surfacesThe two images used to create the surfaces reflecting early-1980s and late-1980sdistributions of urban land use in Bristol are based upon the Landsat TM-5 three-band(blue, red, green) image taken on 26 Ap ril 1 984, and the SP O TH RV , three-band,false-color, 1024 x 102 4 pixel satellite images taken on 18 June 198 8. Th eseimages were judged to be sufficiently different to measure land-use change which wegrounded in socioeconomic data taken from the 1981 and 1991 Population Censuses.Hereafter, we refer to these as 1981 and 1991 distributions, although the usualcaveats apply where data are being synthesized from sets at different points in time.Th e m ethods outlined in section 2 and illustrated in figure 1 were applied. Thisprocedure involves taking each image and identifying relevant land-use categorieswhich are taken from census variables by using the training samples explainedearlier. In figure 2 we show the two elemen ts of this process the 19 84 La nd satand 1988 SPOTHRV images (in terms of the urban-rural contrast in the blue


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wavelength), and examples of the 1981 and 1991 numbers of household surfaces (interms of a threefold gray-scale classification) which are used in the process ofcreating the residential-housing land-use category. Also shown is a schematic mapof the study area, urban Bristol, illustrating the main route network and physicalconstraints on the development, to provide the reader with some sense of the scaleof the problem.After each surface was generated with respect to each land-use category, eachscene was processed by means of the Unix-based ERDAS software, geo-referenced

ERDAS Processing

B

Satellite data

1Imagepreprocessing

1Class training -raximumlikelihood

iayesian decision ru

Thematicclassification

^^

^ i^e

Regressionanalvsis ^~

Census data1SASPAC retrieval system

Selectedcensus data atthe ED level1MENITAB statistical program

Po

information

Prior probabi

Probabilitydistributions1pulation surface moT

Probabilitysurfaces1Fortran program

E R D A Sprobabilitysurfaces

del

ities

ASCII dumpFortran ^program

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Binary matrix1Fortran program1

Fractalmeasurements| Signature

1 ^ U U i l l1 Density

Figure 1. The procedure for generating land-use categories from the remotely sensed data.Note: ED, enumeration district .


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by ground control points with a first-order bilinear interpolation and classified byusing the specially modified maximum likelihood algorithm sketched above insection 2. Nonurban areas were first excluded with a standard spectral density sliceas indicated by figure 2. For the 19 81 /1 98 4 data, four land-use ca tegories w eredefined: first, urban land as defined by the Office of Population Censuses andSurveys (Longley et al, 1992); seco nd, built-u p land , a m or e restrictive definitionthan the first but both of these categories being defined by use of spectral values only;third, residential land; and, fourth, nonresidential land, both defined with respect tothe population count and household size surfaces. These were the only relevantvariables in the 1981 Census which were consistent with the 1991 Census.

The Bristol urban area

Landsat TM-5 image, 26 April 1984 Residential-housing surface, 1981

SPOT HRV image, 18 June 1988 Residential-housing surface, 1991Figure 2. The basic data: the 1984 and 1988 images of Bristol and the 1981 and 1991residential surfaces.


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For the 1988/1991 data, these first four categories were also defined underidentical criteria, but four others were added because of the availability of censusdata. These were: fifth, detached housing; sixth, semidetached housing; seventh,terraced housing; and, eighth, residential flats. Each of these last four categories isbased on the numbers of detached, semidetached, terraced, and flatted properties ineach 100 m grid squa re of the assoc iated surfaces. T h e final land-us e ca tegoriesfor each of the two years were then compiled simply by selecting the maximumprobability for each pixel in the image. Thus, for each year in question,

f 1 , in land-u se category k, if P ik = maxP /M ,,L ik=\ - (14)I 0 , otherw ise .The four surfaces for 1981 are shown in figure 3 (see over), the eight for 1991 in figure 4. These form the basic data upon which the empirical analysis is accomplished.The density analysis first requires the definition of a central 'seed' site aboutwhich the urban area is deemed to have grown, and in the Bristol case we havetaken the historical center to be the Bristol Bridge. The 1988/1991 SPOTHRVimage was then reduced to a 453 x 453 pixel subsection ce ntered upon th e bridge, and,as the nominal spatial resolution is 20 m, the geographic coverage is 9060 m x 9060 m(82084 km 2). The 1981/1984 Landsat image has a resolution of 30 m, and this wasthen sectioned and scaled to the same area. These requirements are necessary forall subsequent analyses of urban density measurements in this paper. To organizethe data for this analysis, we then counted the numbers of cells of each land use insuccessive rings of ab ou t 13 m w idth, thus giving the variables w e refer to s ubse quently as the 'count' values N t (where / now refers to the individual ring). Thesecounts, when normalized by area of each ring, form the densities p i9 and each ofthese densities is defined for the land uses in question. The cumulative count anddensity profiles are base d o n these definitions. We are now, at last, in a pos ition topresent the empirical analysis.6 Applications to land-use densities in BristolThe first measures of density and dimension are close to the idea of measuring theoccupancy of each land use, and give values between 1 and 2, that is, fractaldimensions which show that each land use fills more than a line across the space(D = 1) but less than the complete plane (D = 2). Equations (9) and (11) wereapplied to the surfaces in figures 3 and 4, yielding the dimensions shown in table 1.There is remarkable consistency in these estimates between land uses at each timeslice. First, all the estimates fall in the range 1.399 < D < 1.837, with the meanTable 1. Space-filling dimensions, based on occupancy class.Land-use category

UrbanBuilt-upResidentialNonresidentialDetached housingSemidetached housingTerraced housingApartments or flats

Equat ion19811.7801.7761.7561.598

(9) of text19911.8171.8081.7771.5681.5561.6041.6631.399

Equat ion19811.7951.7901.7681.598

(11) of text19911.8371.8261.7921.5561.5421.5961.6301.362


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Figure 3. The 1981 binary land-use surfaces.value of D over all land uses, the two time points, and the two equations being1.673. T he the ory of the fractal city sketched out elsewh ere by two of the a utho rs(Batty and Longley, 1994) argues that cities are likely to have fractal dimensionsbetween 1.66 and 1.71, which are those consistent with randomly constraineddiffusion pro ces ses such as D L A (diffusion-limited aggregation) (Vicsek, 1989). T heresults bear out these notions.The order given by the fractal dimensions [Z)(land use)] with respect to all landuses is also consistent between the estimates from the two equations and from thetwo time slices. This order is

D(urban) > D(built-up) > D(residential) > D(terraced) > D{semidetached)> D(nonresident ial) > D(detached) > Departments or f lats) ,and there are no reversals of rank in any of the four columns of table 1. There isperhaps a logic to the absolute values of these dimensions too. Urban, built-up, andresidential land uses clearly occupy more space than any of their disaggregates, andthe order within these three land uses also reflects that 'residential' is a propersubset of 'built-up' , which in turn is a proper subset of 'urban' . In terms of propertytypes, ' terraced' and 'semidetached' are l ikely to be most clustered in the city than'detached' and 'apartments or flats' , and the values generated are similarly consistent. 'Non residential ' is a residual category, the comp leme nt of the residen tialland-use set, and as such there is an obvious relation between each. How ever,although it may be possible to derive one fractal dimension from the other, it is notclear how this might be done in terms of equations (9) or (11) because of the


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Urban Built-up

Residential Nonresidential

' - - * * * a H 5

vv>; !;.*, . -'** / -

" V

' . : , A

r* -'.

Detached housing

. . ... *.rj&vA'::":. $*, # . ' : ,

Semidetached housing

&:-.

Terraced housing Apartments or flats

Figure 4. The 1991 binary land-use surfaces.


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772 T V Mesev , M Bat ty , P A Lo n g ley , Y Xie

definition of the mean distances R r. This is a matter for further research. The lastsubstantive point relates to the change in dimensions between the 1981/1984 andthe 1988/1991 data. There is a small but significant increase in the dimensions ofurban, built-up, and residential land use over this period, with a slight decrease inthat for non residential land use . Th is is consistent with urba n growth in this period butto generate stronger conclusions, m ore and better tem pora l information is required.The two types of estimate just presented do not account for the variance in eachland-use pattern and thus it is impossible to speculate upon the shapes of these patternswith resp ect to dens ity profiles. Th us the ma in bod y of analysis is bas ed o n fittingregression lines to the profiles generated from each surface in terms of the count ofland-use cells in each circular band from the core (the Bristol Bridge), given as N t,and their normalization to densities, p t. These data are then used to estimate theparameters in equations (8) and (5), respectively, from their logarithmic transformsin equations (13) and (12), respectively. The profiles based on the count anddensity for the four land-use categories in 1981 are shown in figures 5 and 6, andthe same profiles for the eight categories for 1991 are shown in figures 7 and 8(see over).Some comment on these profiles is required. There are strong similarit iesbetween the 1981 and 1991 graphs for the same land uses, as might be expectedfrom data which are, in part, only four years different in terms of the remotelysensed images, and this also provides some check on the confidence we have in ourmethods for extracting land uses. But the most significant features pertain to thedensity plots, which reveal the existence of clear physical constraints on development in our example (see figures 6 and 8). These densities fall in the manneranticipated as distance increases away from the core for all land uses in both of thetime periods. For the disaggregate housing land uses this decline is steepest, but at

Urban Built-up1 3 .0 -111.0 -

1 ? 9 . 0 - 7 . 0 - 5.0 -

3.0 -1 0l . U |

13.01 1 . 0 -

g 9 . 0 -& 7 . 0 -1 5 . 0 -

3 . 0 -1 0

i i

Residential

T&s/

1 i

350

I ' U I " I " - i 1 i ' i 1 ' 1

1.0 2.0 3.0 4.0 5.0 6.Clogio^

--

*2

309

3 2 ^ ^

i i i i

Nonresidential

184

I 1.0 2.0 3.0 4.0 5.0 6.0logio^Figure 5. The 1981 count profiles for four land uses defined. Note: num bers within thegraphs are the zone numbers of the data points indicated.


15/22


a distance of about 3 - 4 km from the co re, the city is cut by river valleys and g orges(see figure 2). Once these, and other (for example, public open space) constraintsare passed, land uses increase in density quite rapidly, soon peaking, then decliningagain in the manne r expected. This is bou nd to lead to poo r estimates of performance in terms of fitting these profiles by means of equation (12). For the cumulative counts shown in figures 5 and 7, these reversals simply manifest themselves interms of breaks in slope, but the implications of fitting equation (13) to these graphsis that fractal dimensions are likely to exceed 2 in several cases.The results of fitting equations (12) and (13) to all land uses from each of thetime slices are show n in table 2. T h e par am eter estimates from each of theseequations are identical, but the performance measures based on r2 differ substantially. For the fractal dimensions, the order is quite different from that associatedwith the occupan cy values in table 1. Th ere is little point in comm enting on theorder, although it is consistent for both time slices, as this order reflects the gradientchanges arising from physical constraints. In fact, only for detached housing andflats is a clear inverse distance decay of densities observed, and this is reflected inthe fact that the fractal dimensions of these profiles are in the correct range andgive high r2 values both for counts and for densities. The performance of the countdistributions is good, as is always the case with cumulative profiles.Examination of the profiles in figures 5-8 reveals immediately the problemswhich are reflected in table 2. Th ere are two ways in which we might generate m ore'acceptable' dimensions from these data. First, we can constrain the regression byfixing the intercept values in advance, for, as we indicated above, this will ensurethat the values fall within the range 1 < D < 2 (subject to a normal error range).Second, we can reduce the range of the plots over which these regressions aremade, thus cutting off extreme segments and omitting outliers. We will develop the

Urb an Built-up

sAtoo

0.0- 1 . 0 -- 2 . 0 -- 3 . 0 -- 4 . 0 -- 5 . 0 -- 6 . 0 -

~~T^v

i f

Residential178 202

^ " ^ - ^ f 35036 62

i 1 1 1 1 1 . r 1

Nonresidential

1.0 2.0 3.0 4.0log io^ 5.0 6.0 3.0 4.0log io^Figure 6. The 1981 density profiles for the four land uses defined. Note: see figure 5.


16/22


constrained regression first, and the results of fitting equations (12) and (13) in thisway are presented in table 3. W ith this m ethod , the fractal d imensions generatedfor the cou nt and the densities now differ. In gene ral, the perform anc e of theestimates improves for the density profiles and worsens slightly for the counts, butthese results are no t significant. W hat do es hap pe n as expe cted is that all the valuesgenerated fall now within the range 1 < D < 2, with the exception of the urbandensity profile for 1 9 9 1 , which yields a D value of 2.01 2 from equ ation (13).

Urban Built-up

Residential Nonresidential

Detached Semidetached

Terraced Apartments or flats

3.0 4.0logio#Figure 7. The 1991 count profiles for the eight land uses defined. Note: see figure 5.


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M e a s u r i n g t h e d e n s i t y o f u r b a n l a n d u s e 7 7 5

This is because the fact that the intercept values which have to be fixed in advanceare always approximations to an unknown value which will ensure the constraints aremet. In fact, it is likely that the intercept values used should be higher, and thus thiswould lower the dimensions a little in all cases, enough to keep all values withinthe range.The order of va lues genera ted in th is way now corresponds much moreclosely to that noted above for dimensions associated with equations (9) and (11).Urban Buil t-up

toO

u.u- 1 . 0 -- 2 . 0 -- 3 . 0 -- 4 . 0 -- 5 . 0 -- 6 . 0 -

Residential

I C T ^ ^ N / " "43

, i i i

1683 2 7 1

i r^

Nonresidential

Detached Semidetached

Terraced Apartments or flats

3.0 4.0log io^ 3.0 4.0log io^

Figure 8. T h e 1 9 9 1 d en s i ty p ro f i l e s fo r t h e e ig h t l an d u ses d e f in ed . N o t e : see f i g u re 5 .


18/22


Table 2. Dimensions, D, from regression of equations (12) and (13) in text, and r2 values forequations (12) and (13) (r$2) and rfa, respectively).Land-use category 1981 data 1991 dataUrbanBuilt-upResidentialNonresidentialDetached housingSemidetached housingTerraced housingApartments or flats

D1.9882.2932.2511.429

ylr(ll)0.0020.3040.2640.802

ylr(13)0.9770.9640.9660.962

D1.8612.1902.1232.1181.5342.3752.6571.407

ylr(ll)0.3350.2740.1790.0600.8730.5290.4900.824

ylr(13)0.9890.9800.9830.9530.9860.9780.9400.964

Table 3. Dimensions, D, obtained from the constrained regression, and r2 values. Note:subscripts (12) and (13) indicate the data applying to equation (12) and equation (13),respectively.Land-use category

UrbanBuilt-upResidentialNonresidentialDetached housingSemidetached housingTerraced housingApartments or flats

1981 (D(U )

1.7411.7091.6911.557

dataylr(12)

0.0330.3020.3250.622

Al3)1.9541.9211.9031.787

ylr(13)

0.9600.8080.9170.831

1991 dataD(ii)

1.8001.7591.7301.4791.4981.5101.5231.313

ylr(U )

0.4830.3470.3740.2670.9410.6910.3560.954

Al3)2.0121.9731.9441.7011.7131.7271.7381.539

ylr(13)

0.9350.8830.8980.7660.9080.7110.6150.949The order for the 1991 data for both the count and density land-use categories interms of D isD(urban) > D(built-up) > D(residential) > D (terr ace d) > D(semidetached)

> D(detached) > D(nonresidential) > Departments or f lats) ,and this is the same as that noted above, with the exception of /^(nonresidential),which moves down a couple of ranks. The 1981 data are consistent with this ordertoo, with the values increasing a little for urban, built-up, and residential land usesfor both count and density over the time period, with the value for nonresidentialland uses falling a little. One interesting difference is that the estimates of dimensions from the density profiles are higher by around 0.2 in value than those fromthe cou nt, but this is likely to be a result of the different interc ept values used . O neissue has become very clear from this analysis and this is that space-filling and theattenuation of densities are different aspects of morphology and that, in futurewo rk, some attentio n shou ld be given to the extent to which different dim ensionscan be dev elope d to reflect this.The second method of refining the dimensions involves cutting down the set ofobservations which constitute the pop ulation density and count profiles. In terms ofa theory of the fractal city, if such cities are generated according to some diffusionabout a seed or core site, as historic cities in the industrial and preindustrial agesclearly were, then it is likely that the profiles observed show departures frominverse distance relations in the vicinity of the core and the periphery. Around thecore, population has been displaced, whereas on the periphery the city is still


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Measuring the density of urban land use

growing, and thus the inverse relation will be disto rted . It is stan dard pra ctice toexclude o bserva tions in the se a reas , and, in previo us w ork (Batty et al, 1989),dimensions have been refined by using such exclusions. In the case of the Bristolprofiles, this issue is further complicated by the existence of the physical constraintson development because of the river gorges which cut through the city. Accordingly, we have taken the profiles in figures 5 - 8 and simply fitted th e density andcount relations to their various linear segments.The count profiles for 1981 and 1991 shown in figures 5 and 7 have characteristic profiles for all land use involving a single break in slope: most uses accumulateslowly up to the gorge areas of the city and then increase suddenly, thus producingfractal dimensions greater than those generated by the simpler occupancy equations(9) and (11). For some land uses, particularly nonresidential land use, this pattern isreversed, in that the break in slope is from steep to less steep, when the gorges areenc oun tered . In table 4, we hav e selected thr ee coun t profiles for 1 98 1 and four for1991 and have performed the regressions implied by e quation (13) for restrictedpor tions of the curves indicate d in figures 5 and 7 and no ted in term s of zones.Most of these regressions emphasize the steepest parts of these curves and yielddimensions greater than 2, thus illustrating the effect of physical constraints on thedevelopment of the city. Constrained regressions are less meaningful in this context,although these can still be considered relevant for the portion of the curvesconsidered. However, these have been excluded.Table 4. Dimensions, D {13), associated with partial population counts, based on equation (13)in text, and associated r^3) values.Land-use category 1981 data 1991 data

Built-upResidentialNonresidentialTerraced housingApartments or flatsaThis shows the zone numbers which are included in the regressions, where there are 350zones in the complete set. The ranges implied by these numbers are shown in figures 5 and 7.

Reducing the range of density profiles, however, is likely to provide more significant results, for the profiles are distorted in characteristic manner over all land uses.For both 198 1 and 19 91 , as figures 6 and 8 reveal, the curves decline in the usualmanner but at a faster rate than might be anticipated in the absence of constraints.Hence, once the gorge areas have been crossed, densities increase and then declineagain following the more usual pattern. The obvious partit ion is thus into threelinear segments, each of which can be fitted by equation (12). We have not dividedall four land uses in 1981 and all eight in 1991 into these three segments but wehave concentrated upon those curves where breaks in slope are most significant.These breaks are indicated on the profiles in figures 6 and 8, and the associateddimensions and r2 values are presented in table 5 (see over). For both 198 1 and1991 the two segments of these profiles which decline inversely with distance yielddimensions much lower than expected, that is, nearer 1 than 1.7, whereas the positivesegments of these curves give dimensions greater than 2. The model fits, however,are good, with r2 values all greater than 0.8 and most over 0.95. Yet all this analysisserves to do is to dem on stra te that, where physical constra ints have a major effect

zone range 33 2 - 3 0 92 0 - 3 5 02 - 1 8 4

D ( 1 3 )2.7742.5361.518

r(13)0.9780.9770.973

zone range 3

5 3 - 3 1 31 3 - 2 5 41 9 - 3 0 02 - 1 8 8

Al3)2.0932.3103.0341.545

r2r(13)

0.9680.9550.9420.985


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on urban development, it becomes exceptionally difficult to unravel the effect ofdistortions at the core and the periphery which still need to be excluded if the equilibrium fractal dimensions are to be measured. This clearly requires furtherresearch as we begin to augment our research to embrace more thoroughly than anyprevious research the role of constraints on fractal dimension and density.Table 5. Dimensions, D(12), associated with partial population densities based on equation (12)in text, and associated r^ 2) values.Land -use category 1981 data

zone range 3 D(12) r2r{\2)1991 datazone range 3 D{\2) r2r{\2)

Urban 2 0 -7 2 1.083 0.955 13 -6 0 1.309 0.960Urban 73 -1 77 2.816 0.964 16 2- 35 0 1.234 0.952Urban 20 9-3 50 1.065 0.964Built-up 4 -3 6 1.203 0.993 10 -4 2 1.446 0.949Built-up 6 2 -1 7 2 3.531 0.950 16 6- 32 7 1.076 0.959Built-up 207-3 50 1.134 0.951Residential 4 -3 6 1.203 0.993 10 -.4 3 1.431 0.946Residential 62 -1 7 8 3.366 0.955 16 8- 32 7 1.105 0.958Residential 20 2-3 50 1.186 0.940Nonresidential 47 -2 28 0.927 0.990 2 - 1 3 0.452 0.996Nonresidential 10 5- 26 7 0.995 0.961Semidetached 6 -2 6 0.914 0.987Semidetached 27 -1 67 2.826 0.880Semidetached 168-3 11 1.326 0.865Terraced housing 2 -3 6 0.898 0.999Terraced housing 37 -1 70 3.617 0.802Terraced housing 171-3 23 1.012 0.954Apartments or flats 3 8 -2 2 1 1.113 0.9503 This shows the zone numbers which are included in the regressions, where there are 350zones in the complete set. The ranges implied by these numbers are shown in figures 6 and 8.7 Conclusions: future researchThe theory of the fractal city requires extensive empirical testing through thedevelopment of typologies based on different urban morphologies. Using the conceptof fractal dimension and its relation to density, we have made a start upon thisquest, but its successful embodiment depends upon the existence of widely availabledata to define urban form, data which enable consistent comparisons between casestudies. Remotely sensed data fulfill this role. The methods developed here can beapplied routinely, thus generating consistent land-use classifications for differentexamples, and the existence of a universal data set which is continually beingupdated enables many different types of city forms to be explored through time.There is no other data set which has such properties of generality, and althoughcomprehensive digital data exist in the USA (for example, from the TIGER files)this is manually collected in the first instance and cannot be used for the kind oftemporal analysis essential to questions of urban growth. There are problems inclassifying such images into land-use categories, but there is much research on thisfrontier at present within remote sensing, which bodes well for better classifications.There are even possibilities for improving image analysis by means of fractal statistics,which are naturally occurring measures for such images (de Cola, 1993).In this paper, we have been able to extend our analysis to different land usesand to begin some rudimentary analysis of their change over time in terms oftheir morphology. A widely recognized focus within contemporary urban theory


21/22


concerns the evolution of cities, and, in this regard, we see remotely sensed imageryas the database upon which we can truly develop the dynamics of the city in fractalterms (Dendrinos, 1992; Longley and Batty, 1993). For changes in fractal dimension through time, fractal theory is well worked out in terms of growth dynamics(Vicsek, 1989) but there is much more work to be done on how the morphologiesof individual land uses coalesce to produce more aggregate morphologies of urbandevelopment.Here we have produced fractal dimensions for urban development and then forthe land uses which compose it . As yet, we have no formal theory as to how thefractal dimensions of individual land uses 'add up' to those of the whole. Forexample, the four housing types defined from the 1991 data form when aggregatedthe residential class. The question is 'how do the individual dimensions D(land use),where land use = terraced, semidetached, detached, and apartments or flats, combine to produce /^resident ial)? ' There is some formal theory to be developed herewhich involves an extension of fractal theory per se and thus might be applicable toany problem of a form in which the parts add up to the whole. This is a questionwe intend to address in future work.Finally, the example of Bristol we have chosen is one which illustrates howproblematic the analysis becomes when several physical constraints distort thedevelopment of the city. So far, in our more general research program concerningthe analysis of urban form by use of fractals, we have concentrated upon citieswhich have been developed in relatively unconstrained situations, or we have dealtwith cities where physical constraints have been well defined because of naturalfeatures such as rivers and estuaries which rule out development completely (Battyand Longley, 1994). The Bristol example we have dealt with here extends ourapproach considerably, thus raising two issues. First , new measures are requiredwhich distinguish between the space-filling function of the fractal dimension and itsreflection of the attenuation of densities. This suggests that D might be used purelyfor space-filling as implied by equations (9) and (11), and that a new measure of theeffect of attenuation be composed from D ( = 2-a) and K in equation (12), say.Calling D in equation (12) D*, then we might construct indices which measure thecombined effect of D* and K relative to some idealized space-filling norm, or wemight take differences between D and D*. All of these are for the future, but workin this area has implications for fractal theory beyond the immediate domain ofurban systems. The development of constrained regression might be improved too.The selection of predetermined constants is problematic; the lattice-based examplewhich we have treated here as the ideal type is in i tself an approximation, and morethought is needed concerning this.

Nevertheless, what we have shown in this paper is that morphology can beextracted from remotely sensed data and be measured by calling upon new ideaswhich link fractal theory with urban density. The ultimate quest in this work is, ofcourse, the development of a classification of morphologies based on a deeperunderstanding of how urban processes lead to cities of different shapes and densities. There are many policy implications stemming form such work, particularlyinvolving accessibility and energy use as well as questions of social and economicsegregation. It is our view that w e are at the beginning of developing useful ide as inthis domain. The existence of remotely sensed imagery and its classification interms of urban land use, as we have developed here, will speed progress towards thegoal of understanding how cities fill the space available to them and of howdistortions to this process through policy intervention might affect issues of spatialefficiency and equity.


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ReferencesArlinghaus S L, 19 85 , "Fractals tak e a ce ntral place " Geografiska Annaler B 6 7 8 3 - 8 8Arlinghaus S L, 19 93 , "Central place fractals: theoretical geography in an urb an setting", inFractals in Geography Eds N S-N Lam, L de Cola (Prentice-Hall, Englewood Cliffs, NJ)p p 2 1 3 - 2 2 7Batty M, Kim K S, 19 92 , "Form follows function: reformulating urb an po pu lation densityfunctions" Urban Studies 2 9 1 0 4 3 - 1 0 7 0Batty M, Longley P A , 1994 Fractal Cities: A Geometry of Form and Function (AcademicPress , London)Batty M, Longley P A, Fotheringham S, 1989 , "Urb an grow th and form: scaling, fractalgeometry, and diffusion-limited aggregation" Environment and Planning A 2 1 1 4 4 7 - 1 4 7 2Bracken I, Martin D J, 1989, "The generation of spatial population distributions fromcensus centroid data" Environment and Planning A 2 1 5 3 7 - 5 4 3Clark C, 1 95 1, "Urban p opulation densities" Journal of the Royal Statistical Society (Series A)1 1 4 4 9 0 - 4 9 6de Cola L , 1 99 3, "Multifractals in image processing and proce ss imagining", in Fractals inGeography Eds N S-N Lam, L de Cola (Prentice-Hall , Englewood Cliffs , NJ) pp 213-227Dendrinos D S, 1992 The Evolution of Cities (Routledge, London)Frankhauser P, 1994 La Fractalite des Structures Urbaines (Anthropos, Paris)Langford M, M aguire D J, Unwin D J, 19 91 , "Th e areal interpolation p roblem in estimatingpopulation using remote sensing in a GIS framework", in Handling Geographical InformationEds IMasser , MBlakemore (Longman , Har low, Essex) pp 55-77Longley P, Batty M, 1 989a , "Fractal measure men t and cartograp hic line ge neralisation"Computers and Geosciences 1 5 1 6 7 - 1 8 3

Longley P, Batty M, 1 989 b, "On the fractal m easurem ent of geographical bo und aries "Geographical Analysis 2 1 4 7 - 6 7Longley P A , Batty M, 19 93 , "Speculation on fractal g eometry in spatial dynamics", inNonlinear Evolution of Spatial Economic Systems Eds P Nijkamp, A Reggiani (Springer,Ber l in ) pp 203-222Longley P, Batty M, Shepherd J, 1 99 1, "Th e size, shape and dimension of urban settlements"Transactions of the Institute of British Geographers: New Series 1 6 7 5 - 9 4Longley P A , Batty M, Shephe rd J, Sadler G, 19 92, "Do green belts change the shap e ofurban areas? A preliminary analysis of the settlement geography of South East England"Regional Studies 2 6 4 3 7 - 4 5 2Mandelbrot B B, 1983 The Fractal Geometry of Nature (W H Freeman, San Francisco, CA )Martin D J, 1991 Geographic Information Systems and their Socioeconomic Applications(Routledge, London)Maselli F C, Conese L , Petkov L, Resti R, 199 2, "Inclusion of prio r probabilities derivedfrom a nonparametric process in the maximum likelihood classifier" PhotogrammetricEngineering and Remote Sensing 5 8 2 0 1 - 2 0 7Mesev T V , 1 992 , "Integration between rem otely sensed images and popu lation surfacemodels", paper presented to the Regional Science Association, University of Dundee,Scotland, 15-18 September; copy available from authorMesev T V , 19 93 , "Population prio r probabilities in urban image classification", pap erpresented at the Annual Meeting of the Association of American Geographers,

Atlanta, GA , 6 - 1 0 April ; copy available from authorOpenshaw S, 1984 Concepts and Techniques in Modern Geography, 38: The Modifiable ArealUnit Problem (Geo Books, Norwich)Sadler G J, Barnsley M F, 1990, "Use of population density data to improve classificationaccuracies in remotely-sensed images of urban areas", working report 22, South EastRegional Research Laboratory, Birkbeck College, LondonSkidmore A K, Turner B J, 1988, "Forest mapping accuracies are improved using asupervised nonparametric classifier with SPOT data" Photogrammetric Engineering andRemote Sensing 5 4 1 4 1 5 - 1 4 2 1Strahler A H, 1980, "The use of prior probabilities in maximum likelihood classificationof remotely sensed data" Remote Sensing of Environment 1 0 1 3 5 - 1 6 3Vicsek T, 1989 Fractal Growth Phenomena (World Scientific, Singapore)

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Morphology From Imagery