Assessing Alternatives for Modeling the Spatial Distribution of Multiple Land-cover Classes at Sub-pixel Scales

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1 Assessing Alternatives for Modeling the Spatial Distribution of Multiple Land-cover Classes at Sub-pixel Scales Yasuyo Makido, Ashton Shortridge, and Joseph P. Messina Abstract We introduce and evaluate three methods for modeling the spatial distribution of multiple land-cover classes at subpixel scales: (a) sequential categorical swapping, (b) simultaneous categorical swapping, and (c) simulated annealing. Method 1, a modification of a binary pixel-swapping algorithm, allocates each class in turn to maximize internal spatial autocorrelation. Method 2 simultaneously examines all pairs of cell-class combinations within a pixel to determine the most appropriate pairs of sub-pixels to swap. Method 3 employs simulated annealing to swap cells. While convergence is relatively slow, Method 3 offers increased flexibility. Each method is applied to a classified Landsat-7 ETM dataset that has been resampled to a spatial resolution of 210 m, and evaluated for accuracy performance and computational efficiency. Introduction Modeling sub-pixel spatial distribution requires knowledge of the expected proportions of land-cover classes for each pixel (e.g., that a given pixel is 40 percent forest and 60 percent water); as estimated by classification methods, including mixture modeling (e.g., Kerdiles and Grondona, 1996), supervised fuzzy c-means classification (e.g., Foody and Cox, 1994), and artificial neural networks (e.g., Kanellopoulos et al., 1992). After classification, the data may be resampled to a finer spatial resolution (e.g., 30 m pixels divided into nine 10 m cells). Finally, values of sub-pixels are assigned a hard class according to the pixel proportions. One key challenge is to identify a plausible spatial distribution of these proportional values. Several alternative algorithms have been proposed for allocating classes to sub-pixels. Atkinson (1997) was first to propose super-resolution mapping using only the output from a soft classification, with the idea to maximize the spatial autocorrelation between neighboring sub-pixels while honoring the original pixel proportions. It was assumed that the land-cover is spatially dependent both within and among pixels. This assumption of maximum class autocorrelation at the target resolution underpins several approaches to super-resolution mapping (Atkinson, 2001 and 2005; Tatem et al., 2001, 2002, and 2003; Verhoeye and De Wulf, 2002; Mertens et al., 2003). Atkinson (2001 and 2005) examined a pixel-swapping optimization algorithm within a geostatistical framework. The algorithm predicted accurately when applied to relatively simple two class simulated and real images. Thornton et al. (2006) tested the algorithm with soft classified imagery containing several classes. The pixel-swapping algorithm along with mathematical morphology, used to suppress error in the sub-pixel scale output, provided moderately accurate results. Atkinson (2004) developed the two-point histogram method, a geostatistical spatial simulated annealing algorithm, which could recreate any target spatial distribution. Foody et al. (2005) employed both two-point histogram and contour-based approach to map the waterline at sub-pixel scales. Muslim et al. (2006) examined both the two-point histogram method and the pixel-swapping method to accurately map the shoreline. These studies showed the considerable potential of super-resolution mapping techniques. Several alternative algorithms have also been proposed for allocating classes of sub-pixels. However, many of these techniques, such as Hopfield Neural Networks (HNN), Genetic Algorithms (GA), and Markov Random Field (MRF) model-based approaches are not easily accessible to the remote sensing practitioner. The objective of this paper is to examine algorithms, which can be easily coded in any scientific computing language without additional data, for modeling the spatial distribution of multiple land-cover classes at sub-pixel scales. Methods This section describes the three methods for super-resolution mapping used in this paper: sequential categorical swapping, simultaneous categorical swapping, and simulated annealing. All three methods employ the notion of attractiveness, which was first introduced in the pixel-swapping optimization algorithm (Atkinson, 2001). Pixel-swapping Algorithm The objective of the pixel-swapping algorithm is to maximize the spatial correlation between neighboring sub-pixels Yasuyo Makido and Joseph P. Messina are with the Department of Geography & Center for Global Change and Earth Observation, Michigan State University, East Lansing, Michigan (makidoya@msu.edu). Ashton Shortridge is with the Department of Geography, Michigan State University, East Lansing, Michigan Photogrammetric Engineering & Remote Sensing Vol. 73, No. 8, August 2007, pp /07/ /$3.00/ American Society for Photogrammetry and Remote Sensing PHOTOGRAMMETRIC ENGINEERING & REMOTE SENSING August

2 while maintaining the overall proportional composition within a pixel. Initially, the pixel-swapping algorithm randomly allocates class codes to sub-pixels. The spatial arrangement of sub-pixel values is changed iteratively based on a distance weighted function (attractiveness, O i ) of each sub-pixel in order to maximize the correlation between neighboring sub-pixels. In this algorithm, the exponential weighting function is used to calculate the attractiveness: where n is the number of neighbors, Z (X j ) is the value of the binary class z at the j th pixel location X j, and ij is a weight predicted as: where h ij is the lag between the pixel location for X i and X j, and a is the range parameter of the exponential model. Within a pixel, the least attractive location currently allocated to a 1 (i.e., a 1 surrounded mainly by 0 s) and the most attractive location currently allocated to a 0 (i.e., a 0 surrounded mainly by 1 s) are stored. If the least attractive value is less than the most attractive value, then the classes are swapped. This procedure is repeated either for a fixed number of iterations or until the optimization algorithm fails to make a change. Makido and Shortridge (2007) explored various weighting functions as alternatives to the exponential model: specifically the Gaussian, IDW, and equally-weighted nearest neighbor functions. The authors suggested that a Nearest Neighbor (NN) function could be used to maximize accuracy instead of using more complex models of spatial structure. In this NN model, attractiveness O i is simply the sum of the values at the nearest neighbors: where n is the number of neighbors, and Z (X j ) is the value of the binary class z at the j th pixel location X j. The NN function is not only simpler and computationally more tractable but also provides equivalent results to using a distance-based weighting function (Makido and Shortridge, 2007). Sequential Categorical Swapping Sequential categorical swapping is a modification of the pixel-swapping algorithm that employs the NN function. The algorithm allocates each class in turn to maximize internal spatial autocorrelation. The algorithm considers the landscape as a binary scheme. Once the first class is allocated, the algorithm uses only the remaining cells to allocate the second class. This ordered allocation procedure continues until the final class is allocated to the few remaining unassigned cell spaces. Thus, the order of the input classes must be specified for this method, with order determined by the degree of spatial contiguity of each class. In this study, Moran s I is used as an index of the spatial autocorrelation for the landscape (Griffith, 1987; Bailey and Gatrell, 1995). For a spatial proximity matrix (W) spatial correlation in attribute values (y i ) is estimated as: I n n O i n l ij Z(X j ) i 1 j 1 n n i 1 j 1 l ij exp h ij a O i n j 1 Z(X j ) w ij (y i y)(y i y) (y i y) 2 i j w ij. (1) (2) (3) (4) The Moran index is positive when nearby areas tend to be similar in attribute, negative when they tend to be more dissimilar than one might expect, and approximately zero when attribute values are arranged randomly and independently in space (Goodchild, 1986). Two input class orders are examined in terms of Moran s I: descending (high I to low I) and ascending (low I to high I) order. Since it is desirable to first allocate the most auto-correlated classes, it is assumed that descending order would achieve higher classification accuracy than the ascending order. Sequential categorical swapping may require prior class information, such as Moran s I. However, in general, Moran s I at the sub-pixel level is not obtainable. Moreover, it is not feasible to estimate the Moran s I of each class at subpixel scales from the coarse spatial resolution image, since the relationship between resolution and Moran s I is unpredictable (Makido and Shortridge, 2007). Simultaneous categorical swapping and simulated annealing introduced in the next section do not require prior class information. Simultaneous Categorical Swapping The second method simultaneously examines all pairs of cell-class combinations within a pixel to determine the most appropriate pair of sub-pixels to swap. Initially, the algorithm allocates classes randomly to all cells in each pixel based on the class proportion. In Figure 1, for example, assume that prior soft classification probabilities indicated that 66 percent of the southwestern cell belonged to class 2, while 33 percent of the southeastern cell belonged to class 2. Therefore, two-thirds of the sub-pixels in the southwestern cell are assigned to class 2, and one-third of the subpixels in the southeastern cell are assigned to class 2. Based on this initial allocation, binary arrays for each class are created as shown in Figure 2. Then, attractiveness O i is calculated for each class (Figure 3). As mentioned before, attractiveness O i is simply the sum of the values of the nearest neighbors. The example below uses the first-nearest neighbors. The center pixel of the northeastern cell is occupied by class 3, and there are four sub-pixels of class 3 (including the center pixel itself) within its nearest neighbors. Thus, the attractiveness of the center pixel is 4. Based on the attractiveness O i value, a decision rule table is created for each pixel (Table 1). Index 1 is the minimum O i value occupied by the class (class a, location x) (Figure 3). Index 2 is the maximum O i value occupied by the other class (class b, location y). Index 3 is the O i value at location y for class b. Index 4 is the O i value at location x Figure 1. Initial random allocation. 936 August 2007 PHOTOGRAMMETRIC ENGINEERING & REMOTE SENSING

3 Figure 2. Binary matrix. Figure 3. Attractiveness O i (Grey color indicates the pixel is occupied by the class: (a) Class 1, (b) Class 2, and (c) Class 3. TABLE 1. DECISION RULE TABLE INDEX oi_oa oi_ua oi_ob oi_ub _min _max class class calss : num_class Max for class b. The indices 5 and 6 are calculated using the results from index 1 to 4. The index 5 (index 2-1) indicates how much more class a is attracted to location y than current location x. The index 6 (index 4-3) also indicates how much more class b is attracted to location x than current location y. The index 7 is the sum of indices 5 and 6. Thus, the larger the value of index 7, the greater the swapping attractiveness of this cell pair. Subsequently, a row that shows maximum value at index 7 is selected. One pair of sub-pixels (class a at location x and class b at location y) is swapped. This swapping aims to increase the degree of contiguity for both classes. This procedure is repeated for all pixels. Thus, the simultaneous method does not require any prior information about the classes. Simulated Annealing Simulated annealing is a family of optimization algorithms based on the principle of stochastic relaxation (Aarts and Korst, 1989; Goovaerts, 1997). An initial image is gradually perturbed using pixel swapping so as to match constraints (Goovaerts, 1997). There are different criteria, which can be used to decide whether a given perturbation is accepted or rejected during the optimization process. In this study, the Maximum a Posteriori (MAP) model is used (Goovaerts, 1997), and Moran s I is employed as the objective. Moran s I PHOTOGRAMMETRIC ENGINEERING & REMOTE SENSING August

4 of each class is weighted based on the number of pixels of the class. (Griffith, 1982) Here the MAP model only accepts swaps that increase the local Moran s I value. Prob {Accept i th swapping} 1 if Moran's I (i) Moran's I(i 1) 0 otherwise The computation time would be exceedingly long if the Moran s I value needed to be calculated for each trial, especially for a large extent problem. Therefore, Moran s I will be only recalculated after a certain number of iterations for each decision criteria. The basic steps involved in the algorithm are given below: 1. Let I objective be the target Moran s I value and N max be the maximum allowable number of iterations. 2. Randomly allocate sub-pixels within a pixel based on the class proportions. 3. Calculate Moran s I for the current image (I current ). 4. Repeat the following steps: while (I current I objective ) AND (number of iterations N max ). i. Calculate the attractiveness O i value for each class (oi_array). ii. Repeat the following steps a and b for (1% of N max ) times. a. Randomly pick two sub-pixels within a pixel. b. Swap or un-swap based on the O i value. iii. Calculate Moran s I value for the current image (I current ) (go back to Step 4). At step 4-ii-b, sub-pixel x is currently occupied by class a and sub-pixel y is occupied by class b. The O i value at subpixel x for class a and the O i value at sub-pixel y for class b is added (Non_swap_Oi). The O i value at sub-pixel x for class b and the O i value at sub-pixel y for class a is added (Swap_Oi). If Swap_Oi is larger than Non_swap_Oi, then class a and class b are swapped. One advantage of this algorithm over the other simultaneous and sequential methods is that the users can specify the target Moran s I, which means the algorithm does not always maximize the contiguity of the landscape. However, as previously discussed, knowledge of Moran s I at sub-pixel level is generally not possible to obtain. Therefore, target I can be specified only if we have prior information of the landscape. In this study, the target Moran I is set to 1.0 to maximize the autocorrelation. One disadvantage of the simulated annealing algorithm relates to the fact that the algorithm randomly selects two sites. In contrast, the sequential and simultaneous algorithm deterministically selected two sites most in need of swapping based on the attractiveness index O i. Consequently, O i dependent methods are relatively fast since convergence occurs in far fewer iterations. Case Study The sequential categorical swapping, simultaneous categorical swapping, and simulated annealing algorithms are assessed in terms of their accuracy and performance using a case study. A Landsat ETM image (path 21, row 30) with a spatial resolution of 30 m acquired on 06 June 2000 of East Lansing, Michigan was used to evaluate the three methods. Figure 4 shows band 4 (0.76 to 0.90 m) of the imagery. A sub-image of 490 pixels by 490 pixels (14.7 km by 14.7 km) was extracted from the original image, representing a variety of landscapes from high density built environment to Figure 4. Landsat ETM image in the near infrared band of East Lansing, Michigan. agricultural land. The sub-image was classified into 20 classes using the ISODATA (Tou and Gonzalez, 1974) unsupervised classification performed using ERDAS Imagine image-processing software. The 20 classes were merged into five land-cover classes by visual interpretation and field survey: high density built environment, low density built environment, vegetation, water, and bare soil (Figure 5). Image classification approaches, such as supervised, unsupervised, and hybrid classification, aim to automatically categorize all pixels in an image into land-cover classes or themes based on their data file values (Lillesand et al., 2004). These spectrally based procedures often result in a salt-and-pepper appearance due to the inherent spectral variability encountered by a classifier when applied on a pixel-by pixel basis. Post classification smoothing is often applied to smooth the classified output (Lillesand et al., 2004). Modal (majority) filtering is one approach for classification smoothing. For example, Atkinson (2005) applied a 7 pixel by 7 pixel modal filter to simulate an input classified image, and Verhoeye and DeWulf (2002) applied a modal filter to the resulting super-resolution maps to eliminate linear artifacts. Since the degree of spatial dependence is considerably affected by the use of modal filters, in addition to the non-filtered original image, various modal filters (3 3, 5 5, and 7 7) were applied to the original classified image. These images are referred to as Non-filter, Mode 3, Mode 5, and Mode 7. Figure 6 shows the results of the 5 5 modal filter (Mode 5). The high density built environment is mainly located in the center of the study site, and agricultural (vegetation) area and bare soils are located in the surrounding areas. The low density built environment is widely distributed across the entire area. Water features, predominantly rivers, run through the central part of the area. The filtered products and the original image serve as the reference data for the case study. The image was spatially degraded by a factor of 7 to a spatial resolution of 210 m by 210 m. The AGGREGATE function in ARCGRID of ARC/INFO was used for this 938 August 2007 PHOTOGRAMMETRIC ENGINEERING & REMOTE SENSING

5 Figure 5. Classified image. classification that would predict the class proportions (Atkinson, 2005). These fraction images for each class are not easy to interpret compared to one classified image, and also they do not provide any indication as to how the classes are spatially distributed within the pixel. The filtered and non-filtered images of 490 pixels by 490 pixels were degraded to 70 pixels by 70 pixels. For all three methods, attractiveness O i is calculated using the equally weighted Nearest Neighbor function. The number of nearest neighbors involved for the computation was 48, which means all first- to third-order neighbors were incorporated. To assess classification accuracy, percent correctly classified (PCC) was used in this study. PCC is calculated by the ratio of the sum of correctly classified sub-pixels in all classes to the sum of the total number of sub-pixels (Congalton and Green, 1999). The reported PCC is the average of 20 trials for the simultaneous and sequential methods. Figure 6. Modal filtered image (Mode 5). degradation process. At this coarser spatial resolution, the contribution of each sub-pixel was summed to obtain a pixel-level proportion for each class (Figure 7). These pixellevel proportions can be considered the output of a soft classification technique. One advantage of this approach is the ability to evaluate the sub-pixel mapping exhaustively, since the reference image is known. Another advantage is the ability to focus on the mapping rather than the soft Results and Discussion Figures 8 and 9 demonstrate visually the effectiveness of subpixel level mapping for the 5 5 modal filter. Figure 8 is the output of the simultaneous method after 30 iterations. For comparison, a possible result of hard classification using reference data is displayed (Figure 9). This image could be regarded a result of hard classification using remotely sensed data with a spatial resolution of 210 m. More detailed shapes can be seen from the output of the simultaneous method than the traditional hard classification image. Some small features are missing from the hard classification image. The hard classification was created using the BLOCKMAJORITY function in ARCGRID of ARC/INFO. This block function was used to control the resampling of a grid from a finer spatial resolution to a coarser one. Here, the function finds the majority value (the value that appears most often) for the 7 7 rectangular neighbor blocks, and cell values within a block are changed to the majority value. Figures 10 through 13 show the relationship between the number of iterations PHOTOGRAMMETRIC ENGINEERING & REMOTE SENSING August

6 Figure 7. Input images for five classes (Mode 5): (a) High density built environment, (b) Low density built environment, (c) Vegetation, (d) Water, and (e) Bare soil. Figure 8. Output of simultaneous method (after 30 iterations) (Mode 5). and overall accuracy for each algorithm. The horizontal axis shows the number of iterations, and the vertical axis shows PCC as classification accuracy. For simulated annealing, PCC increases with the number of iterations and levels off. The number of iterations for the simultaneous method was 30. For the sequential method, the number of iterations for each 940 August 2007 PHOTOGRAMMETRIC ENGINEERING & REMOTE SENSING

7 Figure 9. Example hard classification output (Mode 5). Figure 10. Overall accuracy of the methods (Non-filter). Figure 11. Overall accuracy of the methods (Mode 3). class was 30, and there are four classes to be allocated (the fifth class is allocated to the remaining cells). Therefore, a total of 120 iterations were used for the sequential method. These numbers were derived empirically. However, it would also be possible to stop the iteration when the algorithm fails to increase the accuracy. For interpretive purposes, horizontal lines depict the accuracy of the simultaneous, sequential, and hard classification. The results suggested that all three methods increased classification accuracy over the hard classification for all modal-filtered images. For the non-filtered image, the subpixel classification methods do not reach to the same accuracy as hard classification. As discussed before, the super-resolution techniques used in this study tended to increase the spatial dependence of the image and, thus, these techniques are better suited for highly autocorrelated images. Therefore, we can assume that the resulting accuracy will be higher for an image which has high Moran s I. Table 2 shows the Moran s I value for each image and the resulting accuracy using the simultaneous method. As Moran s I increases, PCC and its difference between the simultaneous and hard classification also increased. For non-filtered images, the simultaneous method failed to increase accuracy compared to hard classification. For the sequential method, descending (high I to low I) input order shows higher classification accuracy than ascending (low I to high I) input order (Table 3) for all images. Using a two-sample student s t-test assuming PHOTOGRAMMETRIC ENGINEERING & REMOTE SENSING August

8 Figure 12. Overall accuracy of the methods (Mode 5). Figure 13. Overall accuracy of the methods (Mode 7). TABLE 2. ACCURACY COMPARISON FOR SIMULTANEOUS METHOD Accuracy (PCC(%)) Hard Moran s I for Simultaneous Classification Difference Fine Image Non-filter Mode Mode Mode TABLE 3. ACCURACY COMPARISON FOR SEQUENTIAL METHOD Accuracy (PCC(%)) T-test High to Low Low to High Difference Probability Non-filter E-28 Mode E-31 Mode E-05 Mode E-17 unequal variance for the 20 samples, the probabilities for all images are less than 0.05, which indicates two population means are not equal at a 95 percent confidence level. Thus, the order of input classes affected the classification accuracy for this study area. The class input order should start with the class with the largest Moran s I to the one with the smallest Moran s I. In any case, the sequential method achieves lower overall accuracy than any other tested method. Simultaneous categorical swapping and simulated annealing show similar maximum accuracy. However, the simultaneous method needed only 30 iterations to reach the highest accuracy, while simulated annealing needed nearly eight million iterations to reach the same accuracy for mode filtered images. For the simultaneous (and sequential) algorithm, all pixels were visited per iteration. For simulated annealing, the algorithm visited only one pixel per iteration. There are 4,900 pixels in the study area, and therefore, 30 iterations for the simultaneous method could be reframed as 147,000 iterations. However, there was still a substantial difference in the number of the iterations between the two methods. As mentioned before, this was due to the fact that the simulated annealing randomly selected two sites. In contrast, the simultaneous algorithm deterministically selected two sites most in need of swapping based on attractiveness O i. Thus, there was no unnecessary swapping with the simultaneous method. Consequently, computation time for the simulated annealing was much longer than for the simultaneous method. For the simultaneous method, the time taken for 30 iterations (cell 7 to 1) was about 20 seconds on a Pentium 4 computer, versus approximately 10 minutes for the simulated annealing method and eight million iterations. One advantage of simulated annealing over the other simultaneous and sequential methods is that the users can specify the target I instead of maximize it. In this study, target Moran s I was set to 1.0 for simulated annealing, since the objective of the other methods was to maximize the spatial contiguity. Since the Moran s I for the Non-filter image was relatively low (0.66), the Non-filter image was used to examine simulated annealing with target Moran s I The algorithm stopped at roughly 1.4 million iterations when the current Moran s I value exceeded the target Moran s I value. The average PCC for 20 trials using the same target Moran s I was 68.8 percent (Figure 10), which was still less than the maximum PCC for simulated annealing (70 percent). Although the output of simulated annealing had nearly the same Moran s I value as the reference data, the accuracy was still low. Output from the simulated annealing method failed to generate a similar spatial distribution to the reference data. This could have been caused by a limitation of Moran s I that very different spatial scaling properties can have identical Moran s I values. Conclusions Sub-pixel mapping uses the output of a soft classification and transforms it into a hard classification at the sub-pixel scale. The results are easier to interpret and more accurate without using any extra data. The three algorithms presented here can be coded easily in any scientific computing language, and can be used for modeling the spatial distribution of multiple land-cover classes. The sequential method achieved lower overall accuracy than any other tested method. Unlike the sequential method, the simultaneous method and simulated annealing do not require the order of the input classes to be defined. Although both simultaneous and simulated annealing resulted in similar accuracies, the numbers of iterations to reach the maximum accuracy were notably different; 30 iterations for the simultaneous method 942 August 2007 PHOTOGRAMMETRIC ENGINEERING & REMOTE SENSING

9 and 8 million iterations for simulated annealing. Therefore, for this study area, the simultaneous method can be considered as the optimum method in terms of accuracy performance and computation time. This study also examined how the degree of contiguity of the landscape affected sub-pixel mapping. All results suggest, as the spatial dependency of the landscape increased, that the accuracy of the three techniques increased. None of the tested algorithms worked for the non-filtered image because some classes have landscape variation that is much finer than the pixel size. In this study, Moran s I was used as an index of the spatial contiguity of the landscape. In addition to the Moran s I, alternative ways to characterize the landscape should be tested. Various landscape indices, such as mean patch size and patch size standard deviation, can be used. These indices are used as quantitative measures of spatial pattern in heterogeneous landscapes (Cardille and Turner, 2002). Several researchers employ the variogram to capture the spatial pattern of the landscape using sub-pixel mapping algorithms with Hopfield Neural Networks (Tatem et al., 2002), linear optimization techniques (Verhoeye and DeWulf, 2002), and sequential indicator simulation (Boucher and Kyriakidis, 2006). The twopoint histogram is also used geostatistical spatial simulated annealing (Atkinson, 2004; Foody et al., 2005; Muslim et al., 2006). The employment of such alternative measurement tools may provide markedly different behaviors than the results for Moran s I identified in this work. Additional research is necessary to test the techniques for differently structured landscapes. The techniques proposed here should be applicable to imagery from any remote sensing system as long as the basic assumptions about spatial dependence are fulfilled and more broadly in any area of GIS research where data are spatially aggregated. References Aarts, E., and J. Korst, Simulated Annealing and Boltzmann Machines: A Stochastic Approach to Combinatorial Optimization and Neural Computing, John Wiley & Sons, Inc., New York. Atkinson, P.M., Mapping sub-pixel boundaries from remote sensed images, Innovations in GIS 4 (Z. Kemp, editor), London, Taylor & Francis, pp Atkinson, P.M., Super-resolution target mapping from softclassified remotely sensed imagery, Online Proceedings of the 6 th International Conference on GeoComputation, September 2001, Brisbane, Australia, URL: (last date accessed: 20 May 2007). Akinson, P.M., Super-resolution land-cover classification using the two-point histogram, GeoENV IV: Geostatistics for Environmental Applications (X. Sánchez-Vila, J. Carrera, and J. Gómez-Hernández, editors), Dordrecht, Kluwer, pp Atkinson, P.M., Sub-pixel target mapping from soft-classified, remotely sensed imagery, Photogrammetric Engineering & Remote Sensing, 71(7): Bailey, T.C., and A.C. 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