Geostatistical classi cation for remote sensing: an introduction

Transcription

1 Computers & Geosciences 26 (2000) 361±371 Geostatistical classi cation for remote sensing: an introduction P.M. Atkinson a, *, P. Lewis b a Department of Geography, University of Southamptom, High eld, Southampton SO17 1BJ, UK b Department of Geography, University College London, 26 Bedford Way, London, WC1H 0AP, UK Received 15 August 1998; accepted 13 January 1999 Abstract Traditional spectral classi cation of remotely sensed images applied on a pixel-by-pixel basis ignores the potentially useful spatial information between the values of proximate pixels. For some 30 years the spatial information inherent in remotely sensed images has been employed, albeit by a limited number of researchers, to enhance spectral classi cation. This has been achieved primarily by ltering the original imagery to (i) derive texture `wavebands' for subsequent use in classi cation or (ii) smooth the imagery prior to (or after) classi cation. Recently, the variogram has been used to represent formally the spatial dependence in remotely sensed images and used in texture classi cation in place of simple variance lters. However, the variogram has also been employed in soil survey as a smoothing function for unsupervised classi cation. In this review paper, various methods of incorporating spatial information into the classi cation of remotely sensed images are considered. The focus of the paper is on the variogram in classi cation both as a measure of texture and as a guide to choice of smoothing function. In the latter case, the paper focuses on the technique developed for soil survey and considers the modi cation that would be necessary for the remote sensing case Elsevier Science Ltd. All rights reserved. Keywords: Variogram; Classi cation; Geostatistics; Smoothing; Texture 1. Introduction * Corresponding author. Tel.: ; fax: address: pma@soton.ac.uk (P.M. Atkinson). This paper reviews geostatistical techniques for the classi cation of remotely sensed images. Geostatistics is a set of techniques for the analysis of spatial data. All geostatistical techniques are characterised by their dependence on a model of the spatial covariance function or, more frequently, the variogram. Since the techniques described in the main body of this paper depend on the variogram, it is introduced here brie y. For continuous variables, such as re ectance in a given waveband, the experimental semivariance is de ned as half the average squared di erence between values separated by a given lag h, where h is a vector in both distance and direction. Thus, the experimental variogram g n (h) (sometimes referred to as the semivariogram, but more generally abbreviated to variogram) may be obtained from a=1,2,...,p(h) pairs of observations {z n (x a ), z n (x a +h)} de ned on a support n at locations {x, x+h} separated by a xed lag h: /00/$ - see front matter Elsevier Science Ltd. All rights reserved. PII: S (99)00117-X

2 362 P.M. Atkinson, P. Lewis / Computers & Geosciences 26 (2000) 361±371 g n h ˆ 1 2P h X p h aˆ1 z n x a z n x a h Š 2 : Curran (1998) and Woodcock et al. (1988a,b) provide readable introductions to the variogram while Dungan (1998) reviews geostatistical techniques for estimation and simulation in a remote sensing context. Geostatistical techniques that utilise spatial information in classi cation can be split into two distinct groups. In the rst, spatial information is used to provide data on texture. It is implicit in such approaches that texture varies spatially across the image, and particularly between the classes of interest, so that data on texture can be used to inform classi cation. In the second group, spatial information is used to smooth the classi ed image. The rationale for smoothing is that inaccuracies that arise from simple spectral classi- cation applied on a pixel-by-pixel basis can be reduced using the spatial dependence between neighbouring pixels. Proximate pixels are likely to be similar (where the spatial resolution is ne relative to the scale of variation) and this dependence can be formalised (e.g., in a modelled variogram) and utilised to increase classi cation accuracy. The goal is to choose a smoothing function based upon this variogram model. The two approaches (texture and smoothing) to using the variogram in multivariate classi cation are considered separately in the following sections. First, the principles of supervised spectral classi cation are revisited to provide a foundation and context within which to regard geostatistical approaches. 2. Supervised classi cation revisited In this paper, model-based multivariate supervised (e.g., maximum likelihood) classi cation is the primary interest. However, it may be instructive to consider brie y empirical approaches for univariate classi cation, following Carr's (1996) example Empirical approaches One of the simplest approaches to supervised classi cation is the minimum-distance-to-mean classi er, also referred to as the k-means or (here) c-means classi er. This classi er utilises the Euclidean distances in (spectral) feature space between (i) the pixels to be classi ed and (ii) the class means (obtained from training data). In the univariate case, a training site is selected which belongs to a desired class and the values for all pixels at the site are averaged to obtain the class mean. This procedure is repeated for all other classes of interest. Each pixel in the remainder of the image may then be allocated to the class mean to which it is nearest in 1 univariate feature space. In a simple empirical approach presented by Carr (1996) this allocation is executed through the prior construction of a simple look-up table such that any pixel value will have associated with it the destination class without the need to compare the distances to all class means. This procedure is readily extended to multivariate feature space where the distances to class means may be obtained by Euclidean geometry. A second popular method of supervised classi cation is maximum likelihood (ML) classi cation based on Bayes' Theorem. ML classi cation proceeds by selecting the largest posterior probability rather than the minimum distance. In the univariate case, the simple empirical method for ML classi cation described by Carr (1996) proceeds as for c-means classi cation except that for each training site the complete distribution of values is retained and a histogram computed in place of the mean. For 8-bit remotely sensed imagery, each histogram (per class) has 256 bins. The number of occurrences in each bin relative to the total number of occurrences determines the conditional probability distribution (conditional because it is per class). Once the conditional probability distributions are computed, the simpli ed equation for determining class membership probability, as derived from Bayes' Theorem, is: p c j z ˆ X t p z j c rˆ1 p z j r where p(cvz ) is the conditional probability of having class c given pixel value z, p(zvc ) is the conditional probability that pixel value z belongs to class c and there are r =1,...,t classes. A pixel is then allocated to the class with which it has the highest posterior probability of membership. This empirical approach depends on there being su cient training data to characterise fully the probability distributions for each class. In practice, it is more common to replace the probability distribution with a probability density function with parameters (mean, variance and covariance between wavebands) estimated from the training data. The model-based approach is described in the next section Model-based approaches In remote sensing the observation vectors are commonly treated as random variables characterised by a multivariate Gaussian distribution. A multivariate mean is determined for each class and the Euclidean distance metric used in c-means classi cation is replaced by the Mahalanobis distance, a 2

3 P.M. Atkinson, P. Lewis / Computers & Geosciences 26 (2000) 361± standardized statistical distance from which the a posteriori probability of class membership can be obtained using Bayes' Theorem. The maximum likelihood classi cation is achieved by allocating each pixel to the class with which it has the highest a posteriori probability of membership. It is useful to view multivariate, and in particular ML, classi cation as a series of stages from the initial remotely sensed image to the classi ed image, each of which contains potentially useful spatial information for classi cation purposes (see Fig. 1). For example, start with the remotely sensed image of K wavebands. The information in the image can be viewed in (spectral) feature space (each waveband providing a dimension or axis) instead of geographical space. In this feature space it is possible to compute Euclidean distances from each pixel to each of the class means (obtained from training data) using simple geometry. This could provide an image of t distances to each class mean. Since the distributions of the training data are likely to form ellipsoids in feature space the Euclidean metric is often replaced with the Mahalanobis distance M ci which takes into account the variance±covariance matrix V c associated with a given class c: M ci ˆ z k x i u kc T V 1 c z k x i u kc Š 3 where z k (x i ) is the vector of {k =1,...,K } waveband values at pixel locations x i and u kc is the vector of means in K wavebands for class c. In Fig. 1, the variance±covariance matrices per class are represented graphically by probability contours. It is clear that whereas the pixel to be classi ed is closer to the Vegetation class mean, it is more likely to be classi ed as Tarmac 2 when the variance±covariance matrix is Fig. 1. Stages in maximum likelihood classi cation: (1) original imagery, (2) Mahalanobis distances, (3) probabilities derived from probability density function (pdf), (4) a posteriori probabilities derived from Bayes theorem and (5) nal classi ed image.

4 364 P.M. Atkinson, P. Lewis / Computers & Geosciences 26 (2000) 361±371 taken into account. This provides a second image of t Mahalanobis distances (Fig. 1). From the Mahalanobis distances it is possible to determine an estimated (Gaussian) probability density function for each class of interest. This can be expressed as in Eq. (4): p z x i jc ˆ 1 2p K=2 j V c j 1=2 exp 1=2M ci Š 4 where p(z(x i )vc ) is the probability density for pixel z(x i ) at location x i as a member of class c (Foody et al., 1992; Thomas et al., 1987). This provides a third image, this time of t probabilities one for each class. To use the above probabilities in maximum likelihood classi cation (and particularly in the presence of non-equal a priori probabilities) these must be converted to a posteriori probabilities using Bayes' Theorem. The a posteriori probability of a pixel z(x i ) belonging to class c, L(cvz(x i )), may be obtained from Bayes' Theorem as before using Eq. (2) which may now be written more fully to include the a priori probabilities (Eq. (5)): L c j z x i ˆ X t Pc p z x i jc rˆ1 P r p z x i jr where P c is the a priori probability of membership of class c. This creates a fourth image of t a posteriori probabilities (Fig. 1). The maximum likelihood classi cation is achieved by allocating pixels to their most likely class of membership. This results in a nal fth image of {c =1,...,t } classes. It is notable that fuzzy approaches to classi cation are equivalent to omitting the nal step, presenting the investigator with the posterior probabilities of membership (or alternatively fuzzy memberships) to each class for each pixel. Fuzzy classi cation has become increasingly popular in remote sensing in recent years as more remote sensing researchers realise the ubiquity of the solution o ered (see, for example, Foody, 2000; Smith et al., 2000). However, the accuracy and utility of fuzzy classi cation can also be increased using spatial information: geostatistical approaches are not restricted to hard classi cation. Each of the ve di erent images described above contains useful information for classi cation purposes and we shall return to this point later in the discussion of this paper. We now turn our attention to the rst of the two approaches for incorporating spatial information into the classi cation procedure, referred to as texture classi cation Texture classi cation There are many possible approaches to texture classi cation (see, for example, Chen et al., 1997a; Haralick et al., 1973). These generally involve the computation of further image layers or `wavebands' of texture information using an image lter. Thus, for example, the variance within a local moving window could be computed and used as an additional image layer to increase the accuracy of multivariate classi cation (see Haack and Bechdol, 2000). A range of techniques (e.g., neural networks) can then be used to perform the classi cation (Chen et al., 1997b; Raghu et al., 1995). Recently, research has focused on the variogram computed within a local window as a measure of texture. Carr and Miranda (1998) compare this measure with more `traditional' co-occurrence-based measures. They found that the texture measure that achieved greatest accuracy depended upon the nature of the data and texture. Variogram approaches provide the focus of this section. In general, two approaches for texture classi cation which utilise the variogram may be distinguished. In the rst, the actual values of the sample variogram at discrete lags are used directly. In the second, the variogram is modelled and the model coe cients are used. These two approaches are reviewed brie y in this section The sample variogram in texture classi cation The use of actual semivariance estimates in texture classi cation was made popular in the eld of remote sensing by Miranda et al. (1992, 1996, 1998), Miranda and Carr (1994) and Carr (1996). In all of these papers variations on the same algorithm were implemented, referred to generally as the semivariogram textural classi er (STC). The basis of the approach is described below. A window of xed size is de ned and is used to extract representative l by m areas for each class for use in training the classi er. Miranda and Carr (1994) used a window of pixels. Within each window a kernel of r by s pixels is de ned and is moved over the window allowing semivariances for lags of up to r 1ors 1 (whichever is the larger) to be estimated per window. Miranda and Carr found that, for their particular study, a kernel of 7 7 pixels allowed su ciently accurate estimation of the semivariance while keeping the risk of the kernel straddling a class boundary small. The mean semivariance (and the standard deviation of semivariance) per lag computed within the 7 7 pixel kernel over the entire image provided the information on which class allocation was based. This information was provided as (r 1, s 1) max `wave-

5 P.M. Atkinson, P. Lewis / Computers & Geosciences 26 (2000) 361± bands' (features), one for each lag. To these features, the original image wavebands could be added allowing a combined spectral and textural classi cation. Several di erent methods of supervised classi cation could be used to allocate pixels on the basis of the texture information provided by the semivariances at di erent lags. Miranda and Carr (1994) used a parallelepiped classi er because it was simple and computationally e cient. They used the standard deviation to de ne decision boundaries for each class. Where boundaries overlapped a c-means classi er was induced as described earlier. The approach was extended to maximum likelihood classi cation in Carr (1996). Also, Miranda and Carr (1994) and Carr (1996) chose to adopt the simple empirical look-up table classi er described earlier in this paper. The approach could readily be extended to the parametric equivalent, which is also described above. Carr (1996) summarises the earlier work involving the STC and provides Fortran 77 code (two programs, MXTEXT for univariate classi cation and MXMULT for multivariate classi cation) to execute several variations of the STC classi er including empirical c- means and ML classi cation. Lark (1996) applied this approach to aerial photographs, while Chica-Olmo and Abarca-He rnandez (2000) provide an example in this issue Variogram model coe cients in texture classi cation The second approach to using the variogram in texture classi cation involves assuming some kind of form or model for the sample variogram and obtaining the coe cients of the model for use as features in classi cation in place of the semivariances themselves. Ramstein and Ra y (1989) were the rst to automate such an approach in a remote sensing context, while Herzfeld and Higginson (1996) have automated a similar approach for the classi cation of elevation on the mid-atlantic sea oor. The general approach is described brie y below. Ramstein and Ra y (1989) selected a kernel of size pixels for their particular study and used it to estimate a non-linear parameter a' (equal to approximately one third of the e ective range) for every pixel. They suggested that least squares model tting for every pixel would be possible, although prohibitive in terms of computer time. Instead they proposed an approximation of a' based on the exponential form of model given by Eq. (6): g h ˆc 0 c 1 1 exp h=a 0 Š g 0 ˆ0 6 where h is the lag distance and c 0 is the nugget variance, c 1 is the structured variance and a' is the nonlinear parameter: the parameters of the exponential model. Given the theoretical value of the covariance C (equal to c 0 +c 1 ) where the variation is second-order stationary (Eq. (7)): C ˆ lim g h ˆ lim h 4 1 h h mean x2v z x i h z x i 2i ˆ mean x2v z2 x i 7 From Eqs. (6) and (7) one obtains Eq. (8): " # 1 g h a 0 ˆ h= log : 8 mean x2w z2 x i Ramstein and Ra y used the approximation given by Eq. (8) to estimate a' for each pixel in a Landsat Thematic Mapper channel 3 image of the urban area of Strasbourg, France. The resulting univariate feature (one waveband representing a') was then used as a single discriminating variable in a classi cation procedure. Ramstein and Ra y found that the non-linear coe cient provided much better discrimination between land cover classes than other linear coe cients such as C. The utility of this approach depends on the appropriateness of the exponential model for all pixelcentred kernel locations. However, Ramstein and Ra y found that it produced visually appealing results when used to classify land cover in the urban area of Strasbourg. Ramstein and Ra y's research suggests that approaches based solely on semivariance values may miss vital information: approaches based on the range should not be overlooked. Herzfeld (1993) adopted the modelling approach to texture classi cation and developed it speci cally for sea oor classi cation. This amounted to de ning, in the rst instance, several di erent target sea oor classes such as `sediment pond', `inside corner mountains' and `abyssal hill terrain'. The sample variograms obtained for these training areas representing each class were then examined for di erences between classes. The analysis involved both the calculation of anisotropic variograms and residual variograms (from a trend m (h)), also referred to as the centred or detrended variogram. A similar kind of analysis was performed by Wallace et al. (2000) to discriminate between vegetation communities in the Mojave desert. It appeared from Herzfeld (1993) that su cient di erences existed between the variograms of di erent classes to merit automated classi cation. Herzfeld and Higginson (1996) automated a classi cation procedure based on several estimated variogram coe cients which were designed speci cally to exploit the di erences most relevant to sea oor classi cation and, in

6 366 P.M. Atkinson, P. Lewis / Computers & Geosciences 26 (2000) 361±371 particular, several coe cients which exploited class speci c periodicity in the variogram. Herzfeld and Higginson combined linear and non-linear coe cients to form a feature vector for each pixel upon which classi- cation was based. The uses of variogram model coe cients noted above all relate the texture measure empirically to class properties to provide a classi cation. It is worth noting that such measures have also been related both empirically, and using physically-based scene models, to continuous variables such as tree size and density in forested areas. St-Onge and Cavayas (1995), for example, related forest structural parameters empirically to directional variogram properties. Jupp (1997) describes a physically-based model for variance and variograms as a function of viewing and illumination angles based on geometric optics. Any such mapping from variogram characteristics to continuous variables could be applied to classi cation. There are two particular points of note in such work: (i) directional (rather than omnidirectional) variograms are typically used and (ii) physically-based models can describe how the variogram changes as the viewing and illumination angles change Problems with variogram-based texture classi cation The main problem with using the variogram as a measure of texture is that the homogeneous regions of di erent texture within the image must be su ciently large to allow computation of the variogram up to a reasonable number of lags. In many cases, the parcels of interest in the image are too small relative to the spatial resolution of the imagery. Berberoglu et al. (2000) address this issue by computing texture within parcels de ned a priori using vector data. The main problem with variogram model-based approaches to texture classi cation is that automatic tting of (non-linear) models to variograms is unreliable. Thus, the choice of variogram model may be inappropriate for certain regions of the image or for certain classes and the coe cients of the model tted to the local variogram may be misleading. 4. Smoothing the classi cation In the preceding section the variogram was used to provide information on texture for use in classi cation. This is di erent to the common use of the variogram in geostatistics, that is, as a spatial weighting function in kriging (Matheron, 1965, 1971). In kriging, the objective is to smooth or average local values based on the variogram (or other structure function) to estimate optimally an unknown value. In kriging, the variogram is used to determine optimal weights l i to apply to n sample data {z(x i ), i = 1,2,3,...,n ) to form a weighted linear combination z (x 0 ) which estimates some unknown value z(x 0 ), thus: z x 0 ˆXn iˆ1 l i z x i : The advantage and main attraction of kriging is that it estimates optimally by referring to the variogram (model) estimated from the data themselves. It would be attractive also if this advantage could be transferred to multivariate classi cation. Two sets of authors have used the variogram in the kriging sense as a spatial weighting function in unsupervised, multivariate classi- cation (Bourgaullt et al., 1992; Oliver and Webster, 1989). While supervised classi cation is the primary goal, these two papers are reviewed brie y before considering the potential problems of applying the approach to supervised classi cation in remote sensing Oliver and Webster's original idea Some 10 years ago Oliver and Webster (1989) proposed and demonstrated a geostatistical basis for the spatial weighting of multivariate classi cation for application in soil survey. Their approach was particularly suited to sample data provided as a regular lattice or complete cover in the form of a raster array. It would seem, therefore, that their approach might also be suitable for classi cation in remote sensing. Oliver and Webster's (1989) proposal built on the work of others, notably Webster and Burrough (1972) who rst introduced the idea of modifying the dissimilarity matrix by a non-linear function of separating distance in geographical space. However, whereas Webster and Burrough chose non-linear functions (inverse distance square and exponential) arbitrarily, Oliver and Webster's proposal was to use a function obtained from the data themselves. The method is described brie y as follows. The similarity matrix may be constructed for all pairs of observations i and j (pixels) on which K properties (wavebands) have been measured using a similarity coe cient such as Gower's (1971) coe cient (Eq. (10)): S ij ˆ X K kˆ1 1 jz ik z jk j =r k w ijk X K W ijk kˆ where s ij is a measure of the similarity between pixels i and j, z ik is the pixel value at i for class k, r k is a class-

7 P.M. Atkinson, P. Lewis / Computers & Geosciences 26 (2000) 361± speci c constant and w ijk is a weight. This matrix may then be converted to dissimilarity by (Eq. (11)): d ij ˆ 2 1 s ij Š 1=2 : 11 The objective is to modify this dissimilarity matrix to take account of both the geographical proximity between pixels and the form of spatial variation. This may be achieved by multiplying the dissimilarity by a function of geographical distance (Eq. (12)): d ij ˆ d ij f x i x j : 12 If the function were the exponential model (Eq. (6)) then the modi cation would take the following form (Eq. (13)): d ij ˆ d ij c 1 1 exp h ij =a 0 c 0 Š d ij : c 0 c 1 c 0 c 1 13 The modi ed dissimilarity matrix may be used in unsupervised classi cation whether that be based on hierarchical clustering or non-hierarchical dynamic clustering (Oliver and Webster, 1989). Oliver and Webster applied their method to three small data sets on soil properties to demonstrate its utility as a tool in classi cation, particularly for soil management purposes. The unsupervised algorithm chosen was a form of non-hierarchical dynamic clustering that operates on orthogonal principal coordinates rather than directly on the original dissimilarity matrix. Because of the need to extract a single spatial weighting function from the multivariate feature space the modelled variogram of the rst (or rst few) principal component(s) was used in the modi cation (Eqs. (12) and (13)). This seems to undermine somewhat the unbiasedness desired for such an approach. Further ambiguity was introduced by the suggestion of the authors that the range a of the model could be varied to achieve di erent amounts of smoothing as desired. For example, a larger range (which yields smoother results) may be more appropriate for management purposes. Again, this falls somewhat short of the objective of an unbiased and even optimal solution as is the case with kriging Modi cations to the method Bourgaullt et al. (1992) proposed to replace the variogram of the rst (few) principal component(s) with the multivariate variogram (Eq. (14)) and the multivariate covariogram (or covariance function) (Eq. (15)) de ned as follows: 2G h ˆE Z x Z x h M Z x Z x h T Š 14 K h ˆE Z x m M Z x h m T Š 15 where, Z(x) is a row vector of p second-order stationary random functions, m=e[z(x)], and M is a p by p positive de nite symmetric matrix used as metric in the calculation of (dis)similarities. Bourgaullt et al. also based the dissimilarities matrix on the Mahalanobis distance (Eq. (3)). Unfortunately, despite o ering what would appear to be a less ambiguous method for selecting a variogram model, Bourgaullt et al. (1992) introduced a further ambiguity in that the multivariate variogram and covariogram, despite being computed from common data, produced di erent classi cation results. Although not discussed by Bourgaullt et al., the above can be attributed to what might be termed the `local' and `global' operation of the two measures. Essentially, modi cation of the dissimilarity by the (multivariate) variogram does not modify the contribution of points that lie beyond the range of spatial correlation. In the method of Bourgaullt et al., a mean spatially modi ed dissimilarity is calculated: if all points considered in the calculation of this were further than the range from a candidate point then the weighted dissimilarity would be the same as the unweighted value. If a mean spatially-modi ed similarity measure was used, however, with the spatial modi cation performed using a multivariate co-variogram which tends to zero beyond the range then only points within the range would contribute positively to the estimate of the mean. These measures are non-linearly related, so no linear transformation of modi ed similarity to modi ed dissimilarity will produce equivalent results using the two methods in the general case. The former is `global' in the sense that it includes contributions from all points while the latter is `local' in that only points within the range contribute. Further, in the `global' case, the e ect of spatial modi cation will depend on the image extent considered, as it will vary according to the proportion of observations which lie within the range Alternative smoothing approaches Many alternative approaches to smoothing in supervised classi cation have been employed. These range from simple low-pass ltering of remotely sensed imagery to more intricate processing (e.g., see the graphtheoretic approach adopted by Barr and Barnsley, 2000). Some of these alternative approaches are discussed in this section. An important alternative to the spatial weighting of multivariate classi cation based on the variogram is the Gibbs sampler. Based on Bayes' Theorem, it allows one to incorporate information in neighbouring pixels into the spectral classi cation procedure. The Gibbs

8 368 P.M. Atkinson, P. Lewis / Computers & Geosciences 26 (2000) 361±371 sampler updates iteratively the predicted class of a pixel (chosen at random) conditional upon the previous values of all other pixels (and particularly neighbouring pixels). The iteration is stopped when the sequence converges to a (su ciently) stable solution. In some sense then, this solution may be regarded as optimal (SchroÈ der et al., 2000). However, it is only optimal in that the t achieved using the Gibbs sampler attains maximum pseudo likelihood for the given choice of spatial weighting function and kernel size (Augustin et al., 1996). In most instances, it is necessary to experiment with di erent choices for the weighting function and kernel size to search for a generally optimal solution. van der Meer (1996, 1999) used indicator kriging (Goovaerts, 1997) applied to multivariate data to obtain a classi cation for all pixels in a remotely sensed image. The approach involved de ning indicator variables for each feature (waveband) in an image and obtaining variograms for each indicator. These variograms were then used in block indicator kriging to estimate the average value of each indicator for a block or area of pixels centred on the pixel to be classi ed. This amounts to smoothing of the traditional classi er. However, importantly the spatial information (weighting function) incorporated into the classi cation via the variogram is derived from the form of spatial variation in the variable itself as is desired. The ambiguity in this approach would appear to come from the initial selection of indicator cut-o s and the size of blocks (i.e., the amount of smoothing). Further, extrapolation at the tails of the distribution can alter the results substantially. The above indicator approach is similar to an approach known as regionalised classi cation (Bohling, 1997; Harf and Davis, 1990; Moline and Bahr, 1995). Regionalised classi cation is, in fact, nothing more than the interpolation to unobserved sites of the inputs to or the outputs from some traditional classi er applied to sparse data. However, papers on regionalised classi cation have been useful for pointing out that di erent stages in the classi cation process can be interpolated (or in the present case smoothed). For example, one could interpolate the feature vectors and then proceed with classi cation at all sites. Alternatively, one could interpolate the Mahalanobis distances, or the probabilities for use in ML classi cation. This choice, which is entirely general, represents an important decision for investigators wishing to devise strategies for incorporating spatial information into a classi cation. Following on from the above theme, Palubinskas et al. (1995) applied a smoothing algorithm to the fuzzy (for example, a posteriori Bayesian probabilities) outputs from the classi cation of a remotely sensed image. The algorithm was modi ed locally to account for di erences in class homogeneity. There is much overlap between such an approach and those discussed above. There are many algorithms available for segmenting remotely sensed images (Haralick and Shapiro, 1985). While a review of this literature is beyond the scope of the present paper, it is worth pointing out that segmentation is fundamentally di erent to traditional spectral classi cation because spatial contiguity is an explicit goal of segmentation whereas it is only implicit in classi cation. There are many approaches to segmentation including those based on edge detection (most often based on some high-pass lter) and those based on region-growing (based on the growth of homogenous regions conditional upon similarities between the pixel to be merged and previously merged pixels). There exist many examples of segmentation applied to remotely sensed imagery (Janssen and Molenaar, 1995; Khodja and Mengue, 1996; Lemoigne and Tilton, 1995; Lobo et al., 1996; Ryherd and Woodcock, 1996). Segmentation routines involving region-growing algorithms may be based on the similarities between variograms or the coe cients of variogram models (Lloyd and Atkinson, 1998) resulting in segmentation based on variogram texture. 5. Discussion: spatial weighting for remote sensing classi cation 5.1. Selecting an appropriate space One of the rst decisions facing the analyst wishing to use a spatial weighting in smoothing a classi er for remote sensing is in which space should the spatial (geographical) weighting be determined and applied? The work on regionalised classi cation of Harf and Davis (1990), Moline and Bahr (1995) and Bohling (1997) (among others) illustrates clearly that there are many possible spaces in which the smoothing can take place. These include (see Fig. 1): 1. multivariate feature space (for example, the actual values in the wavebands), 2. Euclidean distance-to-means in multivariate feature space, 3. Mahalanobis distance-to-means in multivariate feature space, 4. probabilities (per class), 5. a posteriori probabilities (per pixel) and 6. the classi ed image. The spatial weighting function could be estimated in any one of these spaces and used to aid classi cation.

9 P.M. Atkinson, P. Lewis / Computers & Geosciences 26 (2000) 361± A geostatistical basis for supervised classi cation? For several reasons Oliver and Webster's (1989) approach without modi cation is unlikely to be of much utility for the classi cation of remotely sensed imagery. First, in remote sensing unsupervised classi cation is used less frequently than supervised classi cation and for the latter there is no dissimilarity matrix. The equivalent to the dissimilarity matrix, the distances between the pixel values and the class means expressed in terms of some distance metric in feature space, do not have a spatial dimension (they are not expressed as a function of lag). Each class mean is de ned over the whole image, not for a single pixel located in geographical space. These distances are, therefore, less readily modi ed by a function de ned in geographical space and the variogram cannot be employed in the same way as for unsupervised classi cation. The most straightforward approach would be to use the (multivariate) variogram as a smoothing function directly on the images of distances-to-classmeans (Fig. 1). However, the simple application of the variogram as a smoothing function (that is, convolution of the images of distances with a kernel-based weighting function) is unlikely to result in optimal use of the spatial correlation in the image. The Gibbs sampler discussed above is attractive in this regard because of the lack of an explicit structure function. Second, the data sets must be small for the algorithm to be e ective (for example, one of the data sets analysed by Oliver and Webster consisted of 6 14 cells) and in remote sensing the data sets are typically large (commonly in excess of pixels). Despite the problems discussed above, supervised classi cation does have the advantage of removing the need to compare distances between all pixels and all other pixels (involving nearly 500,000,000,000 comparisons for an image of pixels, which would be prohibitive in most cases). Therefore, small data sets are not a prerequisite for supervised classi cation. Finally, the variogram is de ned as a parameter of a RF model which is stationary in the squared di erences between pairs of locations separated by a given lag h (referred to as the intrinsic hypothesis). For many remotely sensed images the entire scene is not readily modelled in such a way. In particular, for landscapes a ected by human activity (for example, agricultural elds, forest stands, urban areas and so on) the objective of classifying an entire image using a single variogram is ultimately awed because the stationary RF model is unjusti ed. Where a stationary variogram model cannot reasonably be considered, but rather the model should be allowed to vary smoothly across the image, the objective should be to adopt (probably non-parametric) approaches which do not depend on a stationary variogram. It should be noted that the Gibbs sampler does not necessarily help in this regard. The Gibbs sampler is a generally applicable tting procedure most useful for Markov chains. It does not necessarily allow for non-stationarity although it can be used to t models which do so to some extent (for example, Brunsdon, 2000). In most implementations, the result of the Gibbs sampler will be a maximum pseudolikelihood t for a selected smoothing function and kernel size xed over the entire image. A potential solution is provided by a perparcel classi er that utilises a priori knowledge in the form of digital vector boundaries to segment the region of interest into distinct parcels prior to classi cation. Within each parcel the spatial weighting could be applied independently. The per-parcel classi er would also limit naturally the number of data to a tolerable value. 6. Conclusions Most techniques that use the variogram to classify remotely sensed images have their shortcomings. For example, texture classi ers based on the variogram work only where the homogenous regions of each class in the image are su ciently large, homogenous and di erent texturally between classes. Even then there is no guarantee that the extra data on texture will yield useful information (above that provided by the original imagery). Approaches that use the variogram for smoothing in classi cation may be divided into two groups: those based on simple ltering of the image (at any stage in the classi cation process) and the method of Oliver and Webster (1989) in which the variogram is used to modify the dissimilarities in unsupervised classi cation. The former approaches do not deliver the optimality associated with geostatistical techniques such as kriging. The latter approach holds much promise, but problems will need to be overcome if it is to be applied in remote sensing. Acknowledgements This paper was written while PMA was on leave at the School of Mathematics, University of Wales, Cardi. The authors thank Professor Giles Foody for useful information relating to this paper and are grateful to the referees for their comments.

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