y = Xβ + ε B. Sub-pixel Classification

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1 Sub-pixel Mapping of Sahelian Wetlands using Multi-temporal SPOT VEGETATION Images Jan Verhoeye and Robert De Wulf Laboratory of Forest Management and Spatial Information Techniques Faculty of Agricultural and Applied Biological Sciences, University of Gent Coupure Links 653, B-9000 Gent, Belgium Abstract - Africa supports some of the world s largest swamps. These wetlands are very extensive and dynamic ecosystems. These characteristics make them suitable objects for study using satellite images with coarse spatial but high temporal resolution, such as SPOT-VEGETATION data. At coarse spatial resolutions pixels inevitably become mixed. For mixed pixels soft classifiers can be used, which assign a pixel to several land cover classes in proportion to the area of the pixel that each class covers. The basic hypothesis of the linear spectral unmixing model is that the value of each pixel can be modeled as a linear combination of the land cover spectra present in the image and their respective fractions. Multiple linear regression techniques can be used to solve these models. The resultant fraction values can be assigned to sub-pixels, based on the assumption of spatial dependence and the application of linear optimisation techniques. I. INTRODUCTION Africa supports some of the world's largest swamps. Some of the most extensive of these occur within the Sahelian zone. During the last 40 years the growth of the human population and the associated increased demand for irrigation water and arable land, has put increasing pressure on these wetland ecosystems. In an effort to improve the economic situation of the local populations, large-scale hydro-agricultural projects are being planned, comprising large dams and extensive irrigation schemes. Some of these have already been implemented, causing significant ecological damage and economic losses. These wetlands show two important characteristics: They are very extensive, some exceeding a surface of 5,000 km2 (Denny, 985). These are very dynamic systems. Their vegetation is influenced by two phenomena: the alternation between dry and wet season, and the filling and emptying of the floodplains These characteristics make the wetlands suitable objects for study using satellite images with coarse spatial but high temporal resolution, such as NOAA-AVHRR or the new SPOT-VEGETATION images. II. METHODOLOGY A. Introduction Traditional classification techniques are 'hard' in the sense that a single pixel is assigned to a single land cover class. At coarse resolutions (-km pixel size) pixels inevitably become mixed. For these mixed pixels 'soft' classifiers can be used, which assign a pixel to several land cover classes in proportion to the area of the pixel that each class covers. Several researchers have addressed this sub-pixel mixture problem. Among the most popular techniques for subpixel classification are artificial neural networks (e.g. Kanellopoulos et al. (992)), mixture modelling (e.g. Kerdiles and Grondona (996)) and supervised fuzzy c-means classification (e.g. Foody (994)). These techniques aim at estimating the proportions of specific classes that occur within each pixel. The result is a number of fraction images, one for each land cover class concerned. While this information describes the class composition, it does not provide any indication as to how this is spatially distributed within the pixel. The outcome is thus quite different from the classic classification algorithms, where a single land cover map, containing all classes, is produced. One way to go from fraction images to a classic land cover map would be to use "hardeners", where rules can be formulated to determine which class dominates the pixel. The main disadvantage is that information is lost much in the same way as happens with classic hard classifiers. Atkinson

2 (997) has formulated an idea for an alternative approach. It consists of converting raster data to vector data by threading the vector boundaries through the original image pixels (in stead of between pixels, as classic raster-tovector conversion algorithms would do). This process is called sub-pixel mapping. The key problem is determining where the relative proportions of each class are most likely to occur. B. Sub-pixel Classification The unmixing technique that will be applied is based upon Linear Mixture Modelling. The basic hypothesis is that the image spectra are the result of mixtures of surface materials, shade and clouds, and that each of these components is linearly independent of the other. The mixtures in the image are expressed as linear combinations of their respective spectra in the image. Spectral response of each image pixel in every spectral band can be considered as a linear combination of the response of each component (or endmember) present in the mixture. So, every pixel contains information about the proportion (or fraction) and spectral response of each component. Hence the spectral reflectance r, for every image pixel in any band, can be modelled as follows: r = f * p f k * p k + ε where f stands for fraction, p for pure component reflectance, and ε is an error term. This equation can become a linear system of n equations and k unknowns in two ways: applying the equation to every pixel in a single band (n pixels) applying the equation to every band of a single pixel (n bands) In order to solve this set of equations, the number of unknowns cannot be larger than the number of equations. In case the linear mixture model is applied to every pixel in a single band, this condition is not likely to be limiting, as the number of pixels will normally largely exceed the number of endmembers. In the other case, where the model is applied to every band for a given pixel, this condition is much more likely to be constraining. Thus the spectral dimensions of the image data set will limit the total number of endmembers. In hyperspectral remote sensing this limitation generally is no restriction. In multispectral image analysis however this is a serious drawback. The problem can be addressed by using a multitemporal data set, where the maximum number of endmembers equals the number of spectral bands multiplied by the number of dates for which images are available. The first application has been widely used to calculate pure endmember reflectances, given the endmember fractions, while the second application is typically being used to derive endmember fractions, given the pure reflectances. Using a more general notation, the set of equations can be more conveniently represented as a matrix: y = Xβ + ε In general, y is a (n * ) vector of observations, X is a (n * k) matrix of the levels of the independent variables, β is a (k * ) vector of the regression coefficients and ε is a (n * ) vector of random errors. It can be proven (Montgomery and Runger, 994) that the least square estimate of β is β ˆ = (X X) ' ' X y The common way of expressing classification accuracy in "hard" classifications is by preparing a classification error matrix or contingency table. These matrices compare, on a class by class basis, the relationship between known reference data and the corresponding results of an automated classification. In "soft" classifications the estimated variables (the fractions or proportions of each land cover class) are continuous, ranging from 0 to 00 percent coverage within a pixel. Consequently the above mentioned technique is not suitable for accuracy assessment. Nevertheless it is possible to apply statistical techniques that use a comparable approach and yield similar results. By randomly dividing the known land cover proportions derived from the high resolution images into two data sets, it is possible to use the first half to calibrate the linear model, while keeping the second data set for validation purposes.

3 The validation can be achieved by calculating the sample correlation coefficient R, which is the estimator of the correlation coefficient ρ, between the known land cover proportions from the validation set and the soft classification output. In the case of perfect agreement, the correlation coefficient would be and the linear regression line would have a zero intercept and 45 degrees slope. The pixel unmixing procedure and evaluation module described above have been integrated into a processing chain. It has both high and low-resolution images as input, and fraction images and a measure of accuracy as output. C. Sub-pixel Mapping The key problem of sub-pixel mapping is determining the most likely locations of the fractions of each land cover class within the pixel. Assuming a spatial dependence within and between pixels can solve this. The coarse pixels are divided into smaller units and the land cover is allocated to the smaller cells within the larger pixels, in such a way that spatial dependence is maximised. This amounts to an increase in resolution and consequently an increase in data. However it does not correspond to an equivalent increase of information, as uncertainty will also increase. Furthermore the resulting image will contain much redundant information. The main advantage of applying this new technique is that it will avoid losing important information the way classic classifiers and hardeners do. The present approach is based on the phenomenon of spatial dependence, also commonly referred to as spatial correlation or autocorrelation. It is based on the tendency for spatially proximate observations of a given property to be more alike than more distant observations, a principle central to the field of geostatistics. It is assumed that the land cover is spatially dependent both within and between pixels. Such an assumption is realistic on condition that the intrinsic scale of spatial variation in each land cover class is the same as or greater than the scale of sampling imposed by the image pixels (Atkinson, 997). A simple representation of the problem and a possible solution is given in fig. (adapted from Atkinson, 997). 32% 00% 40% 2% 40% 6% 0% 0% 0% a. b. c. Fig.. A raster grid of 3 by 3 coarse pixels, each discretised into 5 by 5 sub-pixels It shows a raster grid of 3 by 3 coarse pixels, with associated proportions of one land cover class (Fig..a). A single coarse resolution pixel is to be divided into 25 sub-pixels, each corresponding to 4% coverage of the coarse pixel. In order to reach, for example, a fraction of 40 %, 0 sub-pixels have to be selected. One possible arrangement is shown in fig..b. Although the number of sub-pixels assigned to the land cover class (Fig..b) agrees with the indicated proportion (Fig..a), the spatial structure of selection conflicts with the expectations of spatial order or dependence. An alternative solution is presented in fig..c, in which the spatial order in the image is maximised, both within and between the coarse resolution pixels. Sub-pixel mapping can be formulated as a linear optimisation problem, more specifically as a special case of the assignment problem. Suppose the linear pixel unmixing module yields fraction images for NLC land cover classes and the coarse resolution pixels is to be divided into NP sub-pixels. The number of sub-pixels that have to be assigned to land cover class i is NPLC i and has been derived from the fraction images. A measure for spatial dependency PLC ij has been calculated for land cover class i and each sub-pixel j. Each sub-pixel has to be assigned a value or 0 for each land cover class, indicating an assignment to the particular land cover class. The problem now becomes assigning land cover classes to the sub-pixels while maximizing the spatial dependency.

4 To construct the mathematical model choice variables x ij are defined so that x ij =, if sub-pixel j is assigned to land cover class i, 0 otherwise. 0 The mathematical model thus becomes: NLC NP Maximize z = xij * PLC Subject to NLC i= j= xij =, j =, 2,..., NP i= NP xij = NPLCi, i =, 2,..., NLC j= ij In order to cast the problem into the traditional linear programming format, the equality constraints have to be converted into pairs of inequalities. Both the function to maximise and the constraints are linear equations, so the model can be solved using linear programming techniques. Several algorithms can be applied, of which the simplex algorithm has been evaluated. III. RESULTS The methodology for sub-pixel classification has been applied both to simulated and real images. Resampling a high resolution image produced the simulated image. This results in an image with a pixel size of km, which is comparable to the pixel size in SPOT-VEGETATION images. The advantage of using simulated images is that the low and high resolution images coincide both temporally or spatially. The results obtained using the simulated data show that a very strong linear relationship exists between the predicted and known fractions, with important clusters near the origin and (00, 00) mark. This is confirmed by the very high coefficients of correlation, varying between 0.94 and Moreover, the intercept of the linear regression lines are close to zero and the slopes close to for all land cover classes (Table ). TABLE : REGRESSION STATISTICS FOR SIMULATED DATA. R Intercept Slope Dryland Water Wetland The methodology has been applied to a set of 22 SPOT-VEGETATION images, dating from August 998 to May 999, and a SPOT-XI image acquired on November 999. Consequently, 88 bands have been used to calculate the fraction images. The coefficients of correlation are somewhat lower than obtained for the simulated data, although still quite high: 0.76 for water, 0.9 for wetland and 0.92 for dryland. The methodology for sub-pixel mapping has sofar only been been applied to simulated data. The simulated data are based on the hard classification a high resolution SPOT-XI image, which is resampled in a way similar to the one described above. The performance of the proposed methodology is evaluated by preparing a confusion matrix, where the reference data is provided by the high resolution classification. A hard classification at 000 m resolution has been produced by assigning each pixel to the class that holds the largest fraction. A classification at 500 m resolution has been obtained by applying the sub-pixel mapping methodology to the same dataset (Fig. 2). TABLE 2: CLASSIFICATION ACCURACY AT 000 M AND 500 M RESOLUTION. Overall accuracy (%) Kappa (%) 000 m resolution

5 500 m resolution Fig. 2. Hard classification result at 20 m (top left), 000 m (top right) and 500 m (bottom) resolution. As measure for spatial dependence the average fraction within a 2 by 2 window comprising neighbouring pixels has been used. The 500 m resolution image gives a better representation of the smaller elements, especially of linear structures such as rivers. The fairly high overall accuracy and Kappa value (Table 2) show that the performance of the sub-pixel mapping methodology is good. IV. DISCUSSION AND CONCLUSIONS The above results prove that methodology for sub-pixel classification is capable of estimating the endmember fractions quite accurately. The main advantage of the presented technique is that mixed pixels are used during the training phase. Other strategies have been described: selection of training sites from large, homogeneous areas (e.g. Atkinson et al., 997). This introduces a degree of randomness as the selection is based upon the experience of the image analyst. Furthermore, the number of pixels used for training will probably be much less than used here. compilation of ground or laboratory reflectance measurements, with information of spectral and temporal characteristics of the land cover classes (e.g. Ouaidrari et al., 996). Although this may be the ideal method, it implies considerable investments (laboratory equipment, etc.). selection of pure pixels based on principal components (e.g. Cross et al., 99). These pixels lie at the extremes of the distribution of pixel signatures in the principal component feature space. The extremes are revealed by plotting a scattergram and are used to determine the corresponding pixels in image space. This

6 again introduces a measure of randomness and it may not always be easy to link these pixels to land cover classes. Compared to these other techniques, the present one is simple, cheap and objective. The results of the sub-pixel mapping exercise indicate that the technique can be useful to increase the resolution while keeping the classification accuracy high. Further improvements that can be envisaged are: iterative application of the optimisation algorithm, as the measure for spatial dependency can be expected to have changed after the first run representing the spatial dependence by other measures. Functions, such as the variogram, can be calculated, quantifying both the amount and the scales of spatial variation in the property of interest allowing a limited amount of spill over of fractions into neighbouring pixels. This should be proportional to the overlap between pixels as indicated by the point spread function of the sensor (Atkinson, 997) other linear optimisation algorithms that may handle the assignment problem more efficiently (Kolman B. and R.E. Beck, 995). ACKOWLEDGEMENTS Funding for the project SP0202 was provided by the Belgian Federal Office for Scientific, Technical and Cultural Affairs (OSTC). The SPOT-VEGETATION images were made available within the framework of the VEGETATION Preparatory Program. BIBLIOGRAPHY Atkinson P.M., 997. Mapping sub-pixel boundaries from remotely sensed images, in: Z. Kemp (Ed.), Innovations in GIS 4, pp Atkinson P. M., M. E. J. Cutler, H. Lewis, 997. Mapping sub-pixel proportional land cover with AVHRR imagery. International Journal of Remote Sensing, vol. 8, no.4, pp Cross A. M., J. J. Settle. N. A. Drake, R. T. M. Paivinen, 99. Sub-pixel measurement of tropical forest cover using AVHRR data. International Journal of Remote Sensing, vol. 2, no. 5, pp Denny P., 985. The ecology and management of African wetland vegetation. Geobotany 6. (Dordrecht: Dr W. Junk Publishers). Foody G. M., D. P. Cox, 994. Sub-pixel land cover composition estimation using a linear mixture model and fuzzy membership functions. International Journal of Remote Sensing, vol. 5, no. 3, pp Kanellopoulos I., A. Varfis, G.G. Wilkinson & Megier, 992. Land cover discrimination in SPOT HRV imagery using an artificial neural network: a 20 class experiment. International Journal of Remote Sensing, vol. 3, no.5, pp Kerdiles H., M. O. Grondona, 996. NOAA-AVHRR NDVI decomposition and subpixel classification using linear mixing in the Argentinean Pampa. International Journal of Remote Sensing, vol. 6, no. 7, pp Kolman B. and R.E. Beck, 995. Elementary Linear Programming with Applications. Second edition. Computer Science and Scientific Computing. Academic Press. Montgomery D. C., G. C. Runger, 994. Applied Statistics and Probability for Engineers. (New York: John Wiley & Sons). Ouaidrari H., A. B gu, J. Imbernons, J. M. D'Herbes, 996. Extraction of the pure spectral response of the landscape components in NOAA-AVHRR mixed pixels - application to the HAPEX-Sahel degree square. International Journal of Remote Sensing, vol. 7, no. 2, pp