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A Curvilinear Component Analysis Adapted to the Band Number Reduction in Multispectral Images

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A Curvilinear Component Analysis Adapted to the Band Number Reduction in Multispectral Images
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  A Curvilinear Component Analysis Adapted to the Band Number Reduction in Multispectral Images  Ludovic Journaux, Irène Foucherot, Pierre Gouton  Lab. LE2I UMR-CNRS 5158 - University of Burgundy  BP 47870 - 21078 DIJON Cedex FRANCE  Email: ljourn@u-bourgogne.fr Abstract This paper proposes an optimization of the curvilinear component analysis (CCA) based on the principal component analysis (PCA), in order to apply it to the bands number reduction of LANDSAT multispectral images. We tend to show that the CCA is really a non linear extension of the PCA and that the CCA optimization through the PCA (called CCAinitPCA) allows a reduction of the calculations, providing a result identical to that of CCA. We then estimate the pertinence of the information obtained from PCA, CCA and CCAinitPCA by the comparison of the results of segmentation of the images thus obtained. Keywords: Curvilinear Component Analysis, Principal Component Analysis, LANDSAT 7, Multispectral images, k-means, segmentation Introduction The satellite multispectral images are very useful tools for the landcover analysis [1]. However, the number of data and the redundancy of information contained in these images make them heavy to handle. It's so necessary to reduce the quantity of data while preserving their intrinsic properties. Such reduced images would be sufficient for a structural and informational analysis and would allow to maximize the automatic processing with reasonable run-time. To reduce the number of dimensions of a data set, we can use linear methods such as Principal Component Analysis (PCA), Independent Component Analysis (ICA) or Projection Pursuit (PP) [2]. The problem of these methods is that they only reveal the linear relation between the spectral bands. To exceed the limits of the linear transformations, nonlinear methods of projection such as "Sammon's mapping" or "Multidimensional Scaling" [3] would be a priori advantageous. However, these non-linear methods have prohibitive run-time and convergence problems. A new non-linear method recently appeared called Curvilinear Component Analysis (CCA) [3, 2]. This method is inspired by the auto-adaptative maps of Kohonen (Self Organizing Maps) [4] and frequently interpreted as a nonlinear extension of the PCA [5]. Its principle is to reproduce the topology of the initial data in a n   space to a subspace  p  ()  pn < , without constraining statically the configuration of the topology. It's an automatic autoadaptation to the real shape of the scatter of points in the vectorial space. The mathematic aim of the CCA is to minimize the error function which characterizes the topology between initial and final projection space: 2,() 1()()2  ppnCCAijijijijij  EddFd  ≠ = − ∑ ∑   nij d  and  pij d  : Euclidian distances between the vectors i  x  and     j  x in the respected srcinal space n R and  p R   :[0,1] F   + → R , Decreasing function of  pij d  , allows the local topology to be favored with respect to the global topology In order to extract only the pertinent and useful information in the data set, we estimate the intrinsic dimension of the multidimensional projection by a Local Linear Component Analysis (LPCA). We estimated the intrinsic dimension of the data as 3  p  =  (dimension of the projection subspace). A new approach The optimization of the CCA we propose here for multispectral image processing combines the CCA with the PCA. The PCA determines an optimized linear projection in the vectorial subspace. The result obtained with it corresponds thus to a solution approaching that obtained by the CCA. In our method, we then replace the random initialized matrix traditionally used in the CCA, by the matrix of the first principal components resulting from the PCA previously carried out. We called this method CCAinitPCA. In order to compare PCA, CCA, CCAinitPCA, we have applies them to a set of 40 LANDSAT 7 multispectral images (8 spectral bands). Comparison of resulting images Figure 1 shows a color image extracted from a LANDSAT image and figure 2 the resulting bands after applying PCA, CCA and CCAinitPCA on the satellite image.    Figure 2.Resulting Images bands after the three treatments The inter-band correlation coefficients show three main reports: The CCA is really a nonlinear extension of the PCA. The CCA and the CCAinitPCA give the same results, and the CCAinitPCA confers on the algorithm a greater rapidity of convergence [6] with a significant reduction of calculations and topology conservation. Results of the segmentation In order to study the relevance of the information obtained after the reduction of number of spectral bands, we segmented the resulting images by a K-means unsupervised classification method [7]. The results are illustrated below (figure 3). Figure 3: segmented images obtained by K-means on  (a) srcinal image (8 bands), (b) PCA image, (c) CCA image, (d) CCAinitPCA image The images segmented by unsupervised classification show that the nonlinear methods give better results than the linear ones. The organization of the landscape is preserved. Moreover, unsupervised classification enables us to find under the same label all the zones of the same category. We so observe a relative conservation of the properties of each landscape category. Conclusion In this article, we have presented a new non-linear projection method which combines CCA and PCA. We have applied CCA, PCA and CCAinitPCA on satellite images in order to compare the resulting images. We have shown that CCAinitPCA have more interesting run-time and convergence than CCA and that segmentation of the images resulting from CCAinitPCA and of the srcinal images provide an identical splitting. Moreover, the CCAinitPCA images seem to contain sufficient information to apply a classification on them with relevant results. An analysis of the rate of "good" classification with a supervised classification method would confirm this statement. References 1. Girard, M.-C. and C. Girard, Traitement des données de télédétection . Dunod ed., 1999, Paris, pg. 530. 2. Lennon, M.,  Méthodes d'analyses d'images hyperspectrales , Université de Rennes I, 2002, pg. 359. 3. Demartines, P.,  Analyse de données par réseaux de neurones auto-organisées ., Institut national polytechnique de Grenoble, 1994, pg. 204. 4. Kohonen, T., Self-organizing maps (third edition) , Springer series in information sciences, Berlin, Heidelberg, New York, Vol. 30, 2001, 5. Dreyfus, G. Martinez, J.M., Samuelides, M., Gordon, M.B., Badran, F., Thiria, S., and L . Hérault,  Réseaux de neurones , ed. Eyrolles, 2002, pg. 386. 6. Journaux, L., I. Foucherot, and P. Gouton.  Nonlinear reduction of multispectral images by Curvilinear Component Analysis: Application and optimization . in CSIMTA'04 International Conference , Cherbourg, France, (2004) 7. Duda, R.O., P.E. Hart, and D.G. Stork, Pattern Classification (2nd Edition) , ed. W. Interscience, 2001, pg. 654. Biography Ludovic JOURNAUX is preparing his Ph.D. in Image Processing at the University of Burgundy, France, since 2002. His research interests include image processing, multispectral images, statistical analysis, multiresolution segmentation and classification. Irène FOUCHEROT is an assistant professor at the University of Burgundy since 1996. Her research interests are colour image processing, especially region-based segmentation of multi-spectral images and artificial life. Pierre GOUTON is a professor in image and signal processing. He has worked on passive power components. Since 1993, his main research topic carries on the segmentation of images by linear methods (edge detector) or non-linear methods (mathematical morphology, classification). Member of ISIS (a research group in signal and image processing of the French National Scientific Research Committee) and of the French Color Group. PCA CCA CCAinitPCA   Bands 3Bands 1   Bands 2   (a) (b) (c) (d) Figure 1. Color image
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