- FUZZY CLUSTERING ALGORITHMS ON LANDSAT IMAGES FOR DETECTION OF WASTE AREAS: A COMPARISON. - For example, they can be used to map changes due to the effects of pollution and environmental degradation over different periods of time. - In this paper we will present a comparison of fuzzy clustering algorithms for the segmentation of multi-temporal Landsat images. - A relabeling stage is performed after the classification in such a way clusters of different segmentations, but corresponding to the same lithological area, are led to a homogeneous color-map.. - Typical applications concern, e.g., the mapping of changes due to the effects of pollution and environmental degradation over different periods of time, thanks to the high frequency of coverage of the Earth surface by satellites.. - [1]) clustering algorithms can overcome the limits of classi- cal classifiers, such as the need of a priori hypothesis on the data distribution, sequentiality, etc. - In this paper, we shall discuss some relevant clustering algorithms proposed in literature, and then we will compare them with supervised techniques in the segmentation of multi- spectral LANDSAT thematic mapper (TM) images for the detection of waste areas.. - In the comparison we will consider unsupervised classifiers based on Hard C-Means (HCM) [4], Fuzzy C-Means (FCM) [5], Possibilistic C-Means (PCM) [6, 7], and Deterministic Annealing (DA) [8].. - HCM is an efficient approximation of the Maximum Likelihood technique for estimating clusters centers, using { 0, 1 } membership values of patterns to classes. - We notice that HCM is subjected to the problem of confinement to local minima of the objective function during the descent procedure. - Moreover, concerning the specific application, the crisp memberships for pixels to a class is a too strong constraint due to the limited resolution of sensors. - This problem is especially critical for pixels in the border of regions.. - The Deterministic Annealing (DA) is a different fuzzy approach to clustering based on the minimization of a Free Energy which has been demonstrated [9] equivalent to the FCM functional. - The main difference with the FCM concerns the updating of fuzziness control parameter (that here has the meaning of a temperature) during the optimization of the objective function. - It is worth of noting an on-line version of FCM, introducing also a scheduling of the fuzzifier parameter, has been recently proposed with the names of FKCN [10] and FLVQ [2].. - HCM, FCM, DA and FLVQ use the probabilistic constraint that the memberships of a pattern across clusters must sum to 1, therefore the membership of a point in a cluster depends on the membership of the same point in all other classes. - On the contrary, the PCM algorithm is based on the assumption that the membership value of a point in a cluster is absolute and it doesn’t depend on the membership values of the same point in any other cluster.. - It is fundamental to lead clusters, coming from different segmentations, relative to the same kind of geographical area, to a homogeneous color-map.. - In the next Section we will discuss the FCM, PCM and DA algorithms. - In Section 3 we will describe the relabeling algorithm. - 2 Fuzzy Clustering Algorithms. - 2.1 The Fuzzy C-Means Algorithm. - The Fuzzy C-Means (FCM) algorithm proposed by Bezdek [5] aims to find fuzzy partitioning of a given training set, by minimizing a fuzzy generalization of the Least-Squares functional.. - is the set of cluster centers;. - E j (x k ) is a dissimilarity measure (distortion) between the sample x k and the center y j of a specific cluster j. - In this paper we use the Euclidean distance: E j (x k. - The minimization of J m , under the probabilistic constraint P c j=1 u jk = 1, leads to the iteration of the following formulas:. - (3) It is worth noting that choosing m = 1 the Fuzzy C-Means functional J m (Eq. - 1) reduces to the expectation of the global error (which we denote as <. - and the FCM algorithm becomes the classic Hard C-Means algorithm [4].. - 2.2 The Deterministic Annealing Algorithm. - The Deterministic Annealing algorithm is an approach to hierarchical cluster based on the minimization of the objective function depending on the temperature. - The objective function to be minimized is the Free Energy:. - k x k − y j k 2 and the parameter β can be interpreted as the inverse of tem- perature T (β = 1/T ) [8. - u jk = 1 if x k belongs to the cluster j , and u ik = 0 ∀ i 6 = j, i.e., each sample is associated to exactly one cluster (hard limit).. - It is worth noting that, whereas standard clustering algorithms need to specify the num- ber of clusters, the Deterministic Annealing algorithm can start with an over-dimensioned number of clusters. - In order to allow a possibilistic interpretation of the membership function as a degree of typicality, in the Possibilistic C-Means (PCM) the probabilistic constraint is relaxed so that the elements of the fuzzy membership matrix U must simply verify:. - [7], Krishnapuram and Keller presented two versions of the Possibilistic C-Means algorithm. - In this paper we consider the second one.. - This formulation of PCM [7] is based on a modification to the cost function of the HCM:. - the objective function contains two terms, the first one is the objective function of the HCM, while the second is a regularizing term, forcing the values u jk to be greatest as possible, in order that points with a high degree of typicality with respect to a cluster may have high u jk values, and points not very representative may have low u jk values in all the clusters:. - and the parameter η j depends on the distribution of points in the j-th cluster and is assumed to be proportional to the mean value of the intra-cluster distance.. - If clusters with similar distributions are expected, η j could be set to the same value for each cluster. - In general, it is assumed that η j depends on the average size and on the shape of the j-th cluster.. - A bootstrap clustering algorithm is anyway needed before starting PCM, in order to obtain an initial distribution of prototypes in the feature space and to estimate parameters η j . - In this paper we will use outputs of a FCM in order to estimate η j parameters according to [6]:. - 3 The Relabeling Algorithm. - In order to compare the segmentation results obtained using two different clustering algo- rithms on the same dataset, it is necessary to find a one-to-one mapping between clusters generated by two different algorithms.. - Given a reference classification, obtained by one of the two clustering techniques, the relabeling algorithm calculates a co-occurrence matrix C = [c ij. - where the rows are the labels of regions in the reference segmentation and the columns are the labels of regions in the segmentation to be re-labeled. - The generic element c ij represents the number of points labeled i in the reference. - Table 1: Relabeling Algorithm.. - segmentation and j in the other segmentation. - After the application of the relabeling algorithm we can use homogeneous (consistent) color-maps in the different segmentations.. - The selected geographical area is located between Monte San Michele and Piana di San Marco Vecchio, near Caserta (Italy), and the specific goal was the discrimination and monitoring of caves and wasting areas present in the scene. - In our case we use only six out of the seven available bands (we exclude the thermal infrared sixth band) and we analyzed several combinations of three bands. - In Figures 1 and 2 the set of bands 5, 4 and 1 are depicted respectively for the month of May 1994 and March 1997. - the detection of clusters in the feature space corresponds to a possible segmentation of the input image in agglomerative areas.. - For the HCM and FCM algorithms we fixed the number of clusters to be found to be 8, whereas the Deterministic Annealing algorithm found itself the same number of classes start- ing from an over-dimensioned number (in our case 10 clusters). - Furthermore, the starting point for the PCM algorithm was the FCM output.. - The fuzzifier parameter m in the FCM was chosen to 2, while the other fundamental parameters were set after several trials. - In the PCM algorithm the parameter K (Eq. - In the Deterministic Annealing algorithm the initial value of β (Eq. - 5) was set to 10 − 4 and the scheduling equation was:. - The results of the unsupervised methods were compared to those obtained from the application of the supervised techniques Maximum Likelihood and K-Nearest Neighbour [4].. - 4, the same algorithms are applied to the images dated March 1997. - 5 we show the results generated from the same data set by using the Maximum Likelihood and K -Nearest Neighbour techniques.. - In particular, the fuzzy methods have allowed to identify objects in a more flexible manner, assigning to each pixel degree of membership to the object-classes in the scene.. - Due to these characteristics, the classification results produced by fuzzy methods have allowed to identify a neglected waste site in the geographical area under exam, which was not known before the present study. - Specifically, the waste site is located in the lower-left part of the image and it is evident how it is less wide in the image dated May 1994 with respect to the image dated March 1997.. - In the study reported in this paper we have applied and compared different supervised and unsupervised classification algorithms for the detection of waste areas using LANDSAT TM images.. - It is worth of noting that the 30 meters spatial resolution of the Landsat-TM sensor makes the process of detecting waste areas effective only for medium m 2 ) to large m 2 ) landfills, thus being unusable for small (40-50 m 2 ) ones. - It is however under study the application of the methods presented here to high-resolution images obtained by the bispectral infrared scanner ATL-80 and the panchromatic images sensed by the IKONOS II satellite, where the land resolution is nearly one meter square;. - Figure 5: The Maximum Likelihood (a) and K-Nearest Neighbour (b) classification results over the set of bands 5-4-1 of the Landsat images. - In addition, while spectral knowledge plays an important role in the interpretation of Landsat images, spatial domain knowledge can be efficiently used to adjust image inter- pretation on the basis of the expected relationships (such as contiguity) among different land structures. - Proceedings of the Seventh IFSA World Congress, pp. - Proceedings of the IEEE, vol.86, No
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