3.1 Pattern representation in linguistic form
Each input feature Fj be expressed in terms of membership values to each of the three linguistic properties: low, medium, and high. Therefore an n-band-pattern
will be represented as a 3n-dimension vector.
We use the membership function (Pal, 1992) with range [0,1] and r
ÎR
n which is defined as eq. (9) to convert F
j to its three-dimensional form which is given by eq. (8).
where
l is the radius of the
p functuion, c the central point, and || . || the Euclidean norm.
Note that, the membership value decreases when its distance from the central point c(||r-c||) increase. It is maximum (
p=1) when the pattern r lies at the central point c of a category (||r-c||=0).
Let F
jmax and F
jmin are the upper and lower bounds of the feature F
j in all L pattern points. Then the three linguistic property sets are defined as :
where fdenom is the parameter controlling the extent of overlapping. The overlapping structure of the
p functions for the three linguistic properties is shown in Figure 1.
Figure 1: Overlapping structure of the p functions for the three linguistic properties.
3.2 Category membership of output vector
We use the Beta membership function to define the desired output of the network (this allows the desired output, lies in the interval [0,1], not clamped to only state 1 (belong to a category) or state 0 (not belong to a category).
The Beta membership function of the i
th pattern to the k
th category, with range [0,1] are defined as:
Where Z
ik: the weight distance from the i
th input pattern
to the center of the K
th category
b and p: the denominational and the exponential fuzzy generators which control the amount of fuzziness.
M
k: the mean of the mean of the numerical input patterns for the k
th category
F
ij: the value of the j
th component of the i
th pattern point.
Note that when the distance is 0 the membership value is 1 (maximum) and when the distance is infinite the membership value is 0 (minimum).
4. Fuzzy Extension to BP Algorithm
The fuzzy extension to BP algorithm consists of two steps. The first step is the fuzzy logic step. The p membership function (eqs. (8)-(15) are used to convert gray level values of each multi-band pixel into the three linguistic properties: low, medium, and high to the input vector of the network (So that, when we use n-band image, we will have 3n input nodes) and the Beta membership function (eqs. (16)-(17)) are used to define the desired output of the network (this allows the desired output, lies in the interval [0,1] not clamped to only state 1 (belong to a category) or state 0 (not belong to a category)). In the second step, the BP learning algorithm (eqs. (1)-(7)) are used to train the network until eq. (4) is minimized and the network "leans" the input patterns.
5. Experiments Results
A Landsat-TM image, over the area of Chumporn city, Thailand, taken on 10
th March 1998 was used. The data consists of three bands including band 3(0.63-0.69
mm), band 4 (0.76-0.90
mm) and bands 5 (1.55-1.75
mm). The image size is 256*256 pixels. The aim of the classification was to distinguish between the 4 categories: vegetation area, water body, waste land and bare soil. From study area, 4,128 sample patterns are pick up randomly. A subset of 740 patterns in this set are used as training set, and residual 3,388 patterns are used as test set.
| Category | Number of pixels | Conventional method | Proposed method |
| Training | Test | Correct (%) | Correct (%) |
| 1. Vegetation | 272 | 1,243 | 1,185 (95.3) | 1, 159 (93.2) |
| 2. Water body | 198 | 870 | 843 (96.9) | 855 (98.3) |
| 3. Waste land | 126 | 824 | 787 (95.5) | 818 (99.3) |
| 4. Bare soil | 144 | 451 | 434 (96.2) | 442 (98.0) |
| Total | 740 | 3,388 | 3,249 (96.0) | 3,274 (97.2) |
Table 1: Classification results of the proposed method compared with the conventional method.
Figure 2: Band 3 of the tested Landsat- TM image (left), the classified images obtained by the conventional method (middle), and by the proposed method (right).
(light gray: vegetation area, black: water body, dark gray: waste land, white: bare soil)
A four-layer MLP model (the input layer, 2 hidden layers and the output layer) was used in this study. The network consists of 9 input nodes (3 linguistic properties x 3 bands), 4 output nodes (4 categories), and 10 nodes in each hidden layer. The parameter selection is as follows: hand a in eq. (5) are set to 0.01 and 0.9, SSE in eq. (4) is set to 0.003, fdemon in eqs. (12) & (14) is set to 0.8, b and r in eq. (16) are set to 5 and 1 respectively.
A simultation has been performed using MATLAB program running on Pentium pro 200 workstation (64 MB RAM). For a training set of 740 sample patterns, the conventional neural network method used 3 min. On 265 epochs while the proposed method used 8 min, on 3, 766 epochs. After training process, a test set of 3,388 sample patterns was used to evaluate the classification performance of the two methods. The results are shown in Table 1.
From the table, the proposed method seems to work a little better. Its overall correct classification is about 97.2%, against 96.0% for the conventional method. This situation is explained by a better attribution in the proposed fuzzy method case of water body, waste land and bare soil categories, whereas the conventional method make less errors than the proposed method in one case which is vegetation category. The classified images resulted from both methods are also provided in Figure 2.
6. Conclusion
A supervised classification algorithm for multispectral satellite images based on fuzzy logic and neural network has been described. The application of this algorithm to classification of Landsat-TM data has been proposed. The results showed an improvement in classification performance comparing to the conventional neural network algorithm.
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