Principal component analysis with optimum order sample correlation coefficient for image enhancement

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1 International Journal of Remote Sensing Vol. 27, No. 16, 20 August 2006, Principal component analysis with optimum order sample correlation coefficient for image enhancement QIUMING CHENG*{{, LINHAI JING{ and ALIREZA PANAHI{ {Department of Earth and Space and Engineering, Department of Geography, York University, 4700 Keele Street, Toronto, Ontario M3J 1P3, Canada {State Key Laboratory of Geological Processes and Mineral Resources, China University of Geosciences, China (Received 9 June 2005; in final form 26 January 2006 ) Principal component analysis (PCA) has been commonly used and has played an important role in remote sensing for information extraction. However, the ordinary PCA based on second-order covariance or correlation is capable of forming components on the basis of the statistical properties of a majority of pixel values pixel values around mean values. For many applications, principal components should be constructed on the basis of optimum correlation coefficients so that the components can represent low or high values of minority pixels of interest. A new version of the PCA has been proposed on the basis of an optimum order sample correlation coefficient for enhancing the contribution of the image bands including the low or high minority pixel values that can assist in extracting weak information for image classification and pattern recognition. The ordinary PCA becomes the special case of the new version of the PCA introduced in this paper. The new method was validated with a case study of identification of Au/Cu-associated alteration zones from a Landsat Thematic Mapper (TM) image in the Mitchell-Sulphurets district, Canada. 1. Introduction Principal component analysis (PCA) (Richards 1984, Singh and Harrison 1985, Eklundh and Singh 1993) has become a standard statistical approach for image processing for two main reasons: (1) to reduce the number of correlated image bands to form a small number of independent principal components to represent most of the variability carried by the multiple image bands; and (2) to increase the interpretability of the components as combinations of multiple bands. PCA with a correlation matrix (termed standardized PCA) is very useful in the analysis of time series data, where the interest is in the identification of phenomena or signals that propagate over time (Eastman and Fulk 1993, Anyamba and Eastman 1996). PCA with a covariance matrix (as unstandardized PCA) has been commonly used in remote sensing image processing (e.g. Fraser and Green 1987, Chavez and Kwarteng 1989, Ma et al. 1990). The basis of PCA is the correlation (covariance) matrix measuring the interrelationships among multiple image bands (variables). The concepts of PCA and its relevant terminologies can be found in many references (e.g. Davis 2002). When it is used in dealing with spatial data in a geographic information system *Corresponding author. qiuming@yorku.ca International Journal of Remote Sensing ISSN print/issn online # 2006 Taylor & Francis DOI: /

2 3388 Q. Cheng et al. (GIS) and in image processing, a number of improvements and modifications can be applied to the definition of the correlation matrix (Cheng 1999a, 2002). Cheng (2002) defined a high-order correlation coefficient on the basis of multifractal modelling. Based on the definition of this correlation coefficient between a pair of bands, a method for constructing the matrix of an optimum order correlation is proposed in the current paper. According to the recently developed multifractal theory, high-order statistical moments can characterize the statistical behaviour of relatively high/low values whereas low-order moments (second-order moments) have the advantage of characterizing the statistical regulation of majority values, for example values around the mean value, of variables (Cheng 1999a). The matrix of the sample covariance or correlation coefficient with power-law transformation can measure the interrelationship among variables with the minority values of the variables enhanced. It can therefore enhance the contribution of the minority values to depart away from the mean values of the variables in the construction of the principal components. In many applications where weak information is essential for image classification, the covariance of the transformed bands may reduce the influence of the values around the average pixel value and enhance the minority pixel values. For example, in identification of anomalies of soil contamination, mineralization, extreme weather patterns, flooding and landslides, the targeted pixels are the minority in comparison with the normal pixels. The values of these targeted pixels may be high or low in comparison with the background normal pixels. Using the ordinary PCA, the components relevant to these types of objects are often minor components with relatively small magnitudes (small eigenvalues). Therefore, patterns related to these types of components may be easily neglected because of the large uncertainty and strong influence of other irrelevant major components acting as noise. Generally the weak information in image processing is found in two situations: one is due to the pixel values of the targeted entities not being significantly different from the neighbourhood pixels so that the entities are hidden in the neighbourhood pixels, and the second is due to the spatial extent of the targeted entities. These entities cover a relatively small area on the image so that a relatively small number of pixels are available as samples to be used in the statistical analysis. For ordinary statistical methods based on low-order moment statistics (mean, standard, variogram, covariance, etc.), a small number of samples will not produce significant variances. Identifying minor patterns for extracting weak information has attracted increasing attention in remote sensing image processing and pattern recognition. The main objective of the current paper is to optimize the correlation between image bands by applying power transformations with certain orders to the bands prior to calculation of the correlation coefficient or covariance so that this image stretching process can enhance the components associated with the minority pixels. There have been several attempts to enhance the principal components; for example, taking the band ratio to enhance the information of the relevant bands (Frazer and Green 1987) or restricting the performance in subgroups of pixels by masking techniques (e.g. Ma et al. 1990, Cheng 1997, 2000), or choosing specific multispectral bands containing the feature information of targets (e.g. Chavez and Kwarteng 1989, Crosta and Moore 1989). Improving the correlation (covariance) matrix enhances the ability of the PCA to characterize the minority pixel values in defining components. The principle and formulism of the model will be introduced first, followed by a case study of identification of Au Cu mineralization-associated alteration zones using Landsat

3 Optimum PCA for image enhancement 3389 Thematic Mapper (TM) bands 1 to 7 in the Mitchell-Sulphurets mineral district, northwestern British Columbia, Canada. 2. Sample correlation coefficient (covariance) Remotely sensed images represent multiple entities on the ground. These entities often correspond to pixel values following various frequency distributions. The majority of the values represent the main entities and minority values represent anomalous entities. For example, the main values may represent land cover and minority pixel values represent edges or boundaries of patches of land cover types. To characterize the entire range of pixel values including not only majority values around the average but also minority values along the two tails has become of increasing interest with the development of nonlinear theory, particularly multifractal theory (Cheng 1999a). The pixel values of these types of entities can follow different statistical distributions from those of normal entities. While the ordinary central limit theorem in statistics ensures that the majority of values around the mean value follow normal and log-normal distributions, the generalized central limit theorem has been extended so that the values on two tails follow power-law distributions (e.g. Schertzer and Lovejoy 1991, Evertsz and Mandelbrot 1992, Cheng et al. 1994, Agterberg 1995). As more advanced data acquisition techniques have been developed and the resolution of remote sensing images has increased significantly, these high-resolution images reflect not only the major entities but also minor entities. Accordingly, the application of image processing will be used not only for general purposes, such as land use classification, but also for specialized entity identification, such as identifying small targets or anomalous entities with relatively small extents; for example, edges, boundaries, extreme weather patterns, extreme geological and hydrological entities and military objects on the sea surface. From a multifractal point of view, lower-order moment statistics are only capable of characterizing the properties dominated by values around the mean value. The behaviour of high or low values along the two tails needs to be characterized with high-order (both positively and negatively) moment statistics. The statistical regularities of the anomalous or extreme values (both directions) often show nonlinearity with respect to various measuring scales. The definition of the correlation coefficient between transformed bands was proposed to enhance the influence of the anomalous and extreme values of minority pixels on the measurement of correlation (Cheng 1999a, 2002). The definition can be expressed as P A q 1 RA q 1,B q ij {Aq 1 B q 2 ij {Bq 2 2 ð Þ~ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 A q 1 ij {Aq 1 P ð1þ 2 B q 2 ij {Bq 2 where A q 1 and B q 2 are the pixel values of bands A and B raised to the powers of q 1 and q 2, respectively. R(A q1, B q2 ) is the q 1 - and q 2 -order correlation coefficient of A and B, which can be calculated with the ordinary correlation coefficient equation applied to the power-transformed bands A and B. The orders of the power exponent q 1 and q 2 can be any real values including negative values and noninteger values. It has been proved that the effect of applying a qth power transformation to the image values enhances either the influence of high values with positive q (&1) or the influence of low values with negative q (%0). If we let q 1 5q 2 51, the correlation coefficient just becomes the ordinary correlation coefficient. The correlation

4 3390 Q. Cheng et al. coefficient with mixed negative and positive power orders (q 15 1, q 2 521) is similar to measuring the correlation between bands A and reversed B; the effect is similar to the ratio of bands A to B. Band ratio is common in remote sensing image processing (e.g. Singh and Harrison 1985). It can be seen that the ordinary correlation coefficient and ratio transformation are similar to applying the correlation coefficient with special power transformation. In other words the new correlation coefficient is a more generalized form of the ordinary correlation coefficient and ratio transformation. Whether the ordinary (low-order) or high-order statistics should be used depends on the objectives of individual applications. For some applications, such as identification of alteration zones for mineral exploration or impact zones for environmental assessment from remotely sensed images, the useful information might be carried over by weak or strong signals hidden in the average or background values caused by normal features acting as noise. Use of high-order statistics can enhance the contribution of high or low values in the analysis. The new correlation coefficients of the multiple transformed image bands can be used as the correlation matrix to conduct a PCA. The results will be affected by the orders of the power transformation. In the next section, a method is proposed for determining the optimum orders of powers for the inputting bands for conducting the optimum correlation PCA. 3. Determining the optimum order sample correlation coefficient among a group of images An essential step in the newly proposed PCA is to determine the value of the order of the power transformation for each image. To decide the proper power exponents for calculating the correlation coefficient between each pair of bands, a number of combinations can be tested. The first criterion to determine the order of powers for a pair of bands is the optimum correlation coefficient, the highest correlation coefficient among the combinations of orders of powers applied to each of the two bands. The maximum ranges of power orders for searching for the optimum order are determined either by experience or empirically. It can also be based on the physical meaning of the bands, and the objects to be identified. One procedure that can be used to quantitatively determine the orders of powers for each band can be expressed as follows: (1) for a given pair of image bands, for example B i and B j (i, j 51, 2,, 7), a pair of orders q 1 and q 2 can be determined so that the high-order correlation coefficient R(B q1 i, B q2 j ) reaches the maximum. Such orders q 1 and q 2 can be positive or negative values. (2) The products of these pairs of orders q i q j (i, j51, 2,, 7) form a symmetric matrix (q i q j ) 767. The values of this matrix represent the product of the optimum orders for each pair of bands. To determine the overall approximation of these orders for each of the bands we can use the first eigenvector and eigenvalue to approximate the matrix, as can be seen from the following decomposition of the matrix: q i q j ~l1 Q 1 Q T 1 zl 2Q 2 Q T 2 z:::zl 7Q 7 Q T 7 ð2þ where l i and Q i5 (v 1 v 2 v 7 ) T (i51,, 7) are the ith eigenvalue and eigenvector of matrix (q i q j ). As (q i q j ) is a symmetrical matrix so that l i are real values, Q T i Q j5 1 when i5j, and Q T i Q j5 0 when i? j. The values of the matrix (q i q j ) can be approximated by the largest eigenvalue and its corresponding eigenvector as: q i q j &l1 Q 1 Q T 1 ð3þ

5 Optimum PCA for image enhancement 3391 pffiffiffiffiffiffiffi Therefore, the order of power for the ith image can be approximated with jl 1 jvi. From equation (3) we can see that the product of the power orders determined for the ith and jth bands as l 1 v i v j approximates to the product of the original product q i q j. The sign of v i determines the sign of the power order of the ith band. The sum of the squared power orders is equal to l 1. As the correlation between any band and itself regardless of the power orders applied to it is always perfect, the diagonal values of the matrix (q i q j ) cannot be determined in the same way as other values in the matrix. One way to artificially determine the diagonal values is to take the largest order of the corresponding band in that row: 2 q i q i ~ max j=i q j ð4þ This method will be demonstrated with a case study of identification of alteration zones from one scene of a Landsat TM image from the Mitchell-Sulphurets mineral district, northwestern British Columbia. 4. Identification of alteration zones from the Landsat TM image The study area was chosen from the Mitchell-Sulphurets mineral district, northwestern British Columbia, Canada. The area is noted for extensive alteration zones associated with porphyry copper and molybdenum systems as well as other types of gold and silver mineralization. The area is underlain by Upper Triassic to Jurassic volcanic and sedimentary rocks, primarily from the Hazelton and Stuhini groups (figure 1). The central part of the area is underlain by Cu/Au-associated pyritic altered rocks. Most volcanic and sedimentary rocks in the area are intruded by subvolcanic porphyritic intrusions of diorites, monzonites, syenites and low silica granites. Glaciers, snow and vegetation cover a large portion of the area. More detailed descriptions about the geology and mineralization in the area can be found in Cheng (1994) and Cheng et al. (1994). The data to be used to demonstrate the application of the method introduced here are the seven-band Landsat TM image, received on 9 September 1985 (Rencz et al. 1994), covering the Mitchell-Sulphurets mineral district. This image has been studied for alteration identifications (Ma et al. 1990, Rencz et al. 1994, Cheng 1997, 1999a,b) and for nonlinear modelling and pattern recognition (Cheng 1999a, 2004). The dataset consists of seven TM bands (bands 1 to 7) with 30-m resolution for bands 1 to 5 and 7 (120-m for band 6), each of consists of 496 columns and 777 rows and covers an area of about 350 km 2. TM bands 1, 4 and 7 are shown in figure 2. There are five dominant types of patterns reflecting vegetation covers, lakes, snow and ice, outcropping igneous and sedimentary rocks and Au/Cu-associated alteration zones, respectively. These features generally correspond to distinct patterns in the TM image. However, they may not show sharp boundaries on the image histograms (not shown here). Alteration zones generally show low values on TM bands 1 3 and high values on bands 4, 5 and 7. Because of hydrothermal activity along fractures and contacts of intrusive bodies, the alteration zones show linear trends and occupy a relatively small extent of the area. To enhance the influence of these alteration zones on the principal components (PCs), the highorder coefficient matrix will be applied. To eliminate the influence of snow and ice, pixels indicating snow and ice in the six original TM bands were removed with a mask.

6 3392 Q. Cheng et al. Figure 1. Simplified geology of the Mitchell-Sulphurets mineral district, northwestern British Columbia (Cheng et al. 1994). In this example, the bands 1 to 5 and 7 were used in the PCA maintain a uniform spatial resolution (30 m). To decide the proper power exponents for calculating the optimum correlation coefficient between each pair of bands, the method introduced in section 3 was used. The optimum pair of orders was calculated at the maximum correlation coefficient between each of the pairs of the six bands. The value ranges of orders were restricted to 3 to 1.9, which were determined to maintain scaling of the moments (a detailed discussion of scaling and multifractal modelling of the TM band can be found in Cheng 1999a). In addition, experiments have shown that this range gives unique optimal pairs of power orders when applied to the six bands. Table 1 shows the partial experimental results on searching the optimum orders of powers applied to bands 2 and 3. The results obtained using other combination of orders are constantly smaller than the results given in table 1. From the values of table 1 it can be seen that the pair 0.3 and 0.6 gives the highest correlation coefficient of Therefore, we can determine that the optimum orders for bands 2 and 3 are 0.3 and 0.6. Similar processes can be applied to all the other pairs of bands and the optimum pairs of orders were determined and are shown in table 2.

7 Optimum PCA for image enhancement 3393 (a) (b) (c) (d) Figure 2. Landsat TM bands 1, 2, 5 and 7 from the Mitchell-Sulphurets mineral district, northwestern British Columbia (Rencz et al. 1994).

8 3394 Q. Cheng et al. Table 1. The experimental results on searching the orders of powers to optimize the sample correlation coefficient between TM bands 2 and 3. q 2 q 1 (TM band 2) (TM band 3) The two values in each of the cells of table 2 represent the optimum orders of the column and row, respectively. From table 2 we can see the optimum order of each band varies depending upon the pair of bands. Therefore, to determine an order for each band such that the order approximates to the overall optimum order of each band, we use the method introduced in section 3. First, the two optimum orders in each cell of table 2 were multiplied by each other to form a product, which is shown in the corresponding cell of table 3. The diagonal values (q i q j ) were determined using equation (4). We can see that table 3 is a symmetrical matrix and the pairs chosen from the first four bands 1 to 4 correspond to the orders q with the same sign, indicating that these four bands are positively correlated. Similarly, the last two bands show the positive correlation with orders in the same sign. However, the pairs from these two groups (for example, one from the group of the first four bands and the other from the second group of the last two bands) show different signs, indicating that these two groups of bands show a negative correlation. To determine the overall optimum orders for the six bands from the data in table 3, the first eigenvalue and eigenvector were calculated and they are: l and Q 1 5(v 1 v 2 v 6 ) T 5( 0.83, 0.47, pffiffiffiffiffiffiffi 0.16, 0.05, 0.20, 0.16) T. Therefore, the orders for the bands are determined as ( jl 1 jv1, pffiffiffiffiffiffiffi jl 1 jv2,:::, pffiffiffiffiffiffiffi jl 1 jv6 )5(22.74, 1.56, 0.53, 0.16, 0.68, 0.51)T. The calculated orders indicate that the power transformation with these orders will stretch the low DN values and compress the high values of the TM bands. These orders were further used to calculate the correlation coefficient matrix using equation (1) for conducting the PCA using the six bands. The results for the correlation coefficients obtained using the new method with the transformed bands Table 2. Optimal orders of power transformation between pairs of bands. Band 1 Band 2 Band 3 Band 4 Band 5 Band 7 Band , , , , , 0.5 Band , , , , , 0.4 Band , , , , , 0.5 Band , , , , , 0.3 Band 5 0.7, , , , , 0.5 Band 7 0.5, , , , , 0.4 2

9 Optimum PCA for image enhancement 3395 Table 3. Products of optimal orders of power transformation between pairs of bands. Band 1 Band 2 Band 3 Band 4 Band 5 Band 7 Band Band Band Band Band Band Table 4. Ordinary correlation matrix calculated for the original bands. Band Table 5. Optimum correlation matrix of the transformed data. Band and the ordinary method with the original bands are shown in tables 4 and 5, respectively. Table 5 is provided for comparison purposes. Comparing the results given in tables 4 and 5, we can see that the correlation coefficients calculated using the transformed bands are consistently stronger than those obtained with the original bands; in particular, the correlation coefficients between bands 1 to 2 and bands 5 to 7 have improved significantly. These correlation coefficient matrixes in tables 4 and 5 were further used for calculating the PCs. The six PCs were calculated and their component variance (eigenvalues) are shown in tables 6 and 7, and the loadings of bands on each components are shown in tables 8 and 9, respectively. Comparing the component variances in tables 6 and 7, we can see that the variance of component PC1 as well as the last two PCs 5 and 6 are enhanced by the power transformation. The results in tables 8 and 9 show that the general trends of loadings of bands on the first two components in these two tables are similar, implying that the physical meanings of the PCs are similar, although the signs of the loadings of 5 and 7 are opposite in these two tables due to the negative orders of power transformation applied to bands 1 to 4 from bands 5 and 7. However, the loadings of each band on the last few components show significant differences. These can be confirmed by showing them on scatter plots and the correlation coefficients

10 3396 Q. Cheng et al. Table 6. Component variance of the original data. Component Variance PC PC PC PC PC PC Table 7. Component variance of the transformed data. Component Variance PC PC PC PC PC PC Table 8. Loadings of bands on the components calculated from the original data components. Band PCA 1 PCA 2 PCA 3 PCA 4 PCA Table 9. Loadings of bands on the components calculated from the transformed data. Loading PC1 PC2 PC3 PC4 PC5 Band Band Band Band Band Band calculated between each of the component pairs. The results are shown in figure 3, which gives correlation coefficients between each pair of components calculated with or without power transformations as 20.99, 20.91, 0.90, and 0.60, respectively. As the loadings on PC1 in table 9 are around 0.4 and the loadings for the first four bands are positive and those for the last two bands are negative, and also the power transformation orders for the first four bands are negative and for the last two

11 Optimum PCA for image enhancement 3397 (a) (b) (c) (d) (e) Figure 3. Scatter plots showing the correlations between principal components calculated from the original data (x-axis) and power-transformed data (y-axis): (a) PC1, (b) PC2, (c) PC3, (d) PC4 and (e) PC5. Colour scale coded according to the number of pixels. bands are positive, PC1 in fact represents the overall combination of all the inputting bands with enhancement of the low-value tail of each original band and compression of the high-value tail. It shows an overall albedo of the transformed bands. PC3 is dominated by bands 4, 5 and 7, indicating the spectral contrast between the NIR band 4 and SWIR bands 5 and 7; it mainly shows the outcropped rocks and alterations. PC4 is dominated by bands 1 to 3 and 5; it shows the contrast of band 1 with negative loading and bands 2, 3 and 5 with positive loadings. Considering the opposite transformation orders, PC4 mainly shows a contrast between the two groups of bands: bands 1 and 5 vs. bands 2 and 3. This component

12 3398 Q. Cheng et al. (a) (b) (c) Figure 4. Scores on PCs 1, 3 and 4 obtained using the new PCA method. The loadings on the PCs are given in table 9.

13 Optimum PCA for image enhancement 3399 Figure 5. Colour composite of PCs 1, 3 and 4 (figure 4) as red, green and blue, respectively. may reflect the intensity of the Fe concentration caused by pyrite alterations. The spatial patterns of scores on the three components (PCs 1, 3 and 4) are shown in figure 4, respectively. A false colour composite of PCs 1, 3 and 4 as red, green and blue is shown in figure 5. Comparing the results in figure 5 with the geological map (figure 1) and alteration information from field photographic interpretation (Cheng and Li 2002) available in the area, it can be seen that the white patterns shown in figure 5 mainly represent the alteration zones. This highlights not only the main alteration zones in the area but also the intensive alterations with NNE SSW orientation probably corresponding to the more intensive potassium alteration zone with gold/copper mineralization. 5. Conclusions The optimum order sample correlation coefficient index introduced in this paper applies a power transformation to the image bands prior to calculating the correlation coefficient between two image bands. Two optimum orders of power

14 3400 Q. Cheng et al. transformation can be determined to maximize the correlation coefficient between the two image bands. The optimum coefficient may be able to enhance the contribution of low/high values of minority pixels due to the image stretching by applying the power transformation. The method proposed here was further used to determine the overall optimum power orders for a group of image bands. Transformed bands with these orders of powers were used to form the sample correlation coefficient matrix from which the PCA was further applied to construct the principal components. The case study of identification of mineralizationassociated alteration zones in the Mitchell-Sulphurets area has demonstrated that the new method, proposed as a more generalized method in comparison with the ordinary PCA or ratio transformation, can enhance the weak anomalies related to minority pixels. The method is expected to become a powerful common tool for general image enhancement. Other types of stretching methods such as exponential and logarithmic methods may also be applied to the bands prior to conducting the PCA. The effects of these types of stretching should be investigated in future studies. Acknowledgements Two anonymous reviewers are thanked for their critical review of this paper and for constructive comments on providing the comparison of the results obtained using the new method and the ordinary method. This research was partially supported by an NSERC individual discovery grant (ERC-OGP ), a Chinese 973 Project (G ) and a Chinese 863 Research Project (2002AA135090) awarded to Q.C. References AGTERBERG, F.P., 1995, Power law versus lognormal models in mineral exploration. In Computer Applications in the Mineral Industry, Proceedings of the Third Canadian Conference on Computer Applications in the Mineral Industry, H.S. Mitri (Ed) October 1995, Montreal, Canada (Montreal, Quebec: CIM), pp ANYAMBA, A. and EASTMAN, J.R., 1996, Interannual variability of NDVI over Africa and its relation to El Niño/Southern Oscillation. International Journal of Remote Sensing, 17, pp CHAVEZ, P.S. JR. and KWARTENG, A.Y., 1989, Extracting spectral contrast in Landsat Thematic Mapper image data using selective principal component analysis. Photogrammetric Engineering and Remote Sensing, 65, pp CHENG, Q., 1994, GIS-based methods for mineral resources assessment: Mitchell-Sulphurets area, Canada. PhD dissertation, University of Ottawa. CHENG, Q., 1997, Fractal/multifractal modeling and spatial analysis. In Proceedings of the International Association for Mathematical Geology (IAMG) Conference, 1, September, 1997, Barcelona, Spain (Barcelona: CIMNE), pp CHENG, Q., 1999a, Multifractality and spatial statistics. Computers and Geosciences, 25, pp CHENG, Q., 1999b, The gliding box method for multifractal modelling. Computers and Geosciences, 25, pp CHENG, Q., 2000, GeoData Analysis System (GeoDAS) for mineral exploration: User s Guide and Exercise Manual, Material for the training workshop on GeoDAS held at York University, 1 3 November. Available online at: CHENG, Q., 2002, New versions of principal component analysis for image enhancement and classification. In Geoscience and Remote Sensing Symposium, IGARSS 02. IEEE International, June 2002, Toronto, Ontario, 6, pp CHENG, Q., 2004, A new model for quantifying anisotropic scale invariance and for decomposition of mixing patterns. Mathematical Geology, 36, pp

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