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504 IEEE GEOSCIENCE AND REMOTE SENSING LETTERS, VOL. 4, NO. 4, OCTOBER 2007 II. MFLDA Let the total scatter matrix S T be defined as S T = n∑ (r i − µ)(r i − µ) T (4) i=1 and it can be related with S W and S B by [1] S T = S W + S B . (5) So the maximization of (3) is equivalent to maximizing q ′ = wT S B w w T S T w . (6) Following the same idea of FLDA, the solution will be the eigenvectors of the generalized eigenproblem: S B w = λS T w. When the only available information is the class signatures {s 1 , s 2 ,...,s p }, they can be treated as class means, i.e., M = [µ 1 µ 2 ···µ p ] ≈ [s 1 s 2 ···s p ].TheS B in (2) becomes Ŝ B = p∑ (s j − ˆµ)(s j − ˆµ) T (7) j=1 where ˆµ is the mean of class signatures, i.e., (1/p) ∑ p i=1 s i = ˆµ. S T in (4) can be replaced by the data covariance matrix Σ, i.e., Ŝ T =Σ= N∑ (r i − ˜µ)(r i − ˜µ) T (8) i=1 where ˜µ is the sample mean of the entire data set with N pixels, i.e., (1/N ) ∑ N i=1 r i = ˜µ. Then, the solution is the eigenvectors of the generalized eigenproblem: ŜBw = λΣw or Σ −1 Ŝ B . Regardless of the actual classes present in the data, replacing S T with Σ represents an extreme case, which means all the pixels are separated into the classes they belong to and selected as samples. Using ŜB as S B represents another extreme case, which means there is only one sample in each class. So the discrepancy incurred comes from two factors: only one sample (i.e., class signature) for each of the p classes is used to estimate S B , and all the pixels are used to estimate S T with the implicit assumption that pixels are put into all the existing classes including unknown background classes (i.e., the actual number of classes p T may be greater than p). In the experiments, it will be shown that the term Σ −1 is very effective in background suppression. Since the rank of ŜB is the same as S B , which is (p − 1), the dimensionality of the MFLDA-transformed data is (p − 1) as that of FLDA. After the data are projected onto this (p − 1)- dimensional space, an algorithm is needed for some tasks, such as classification or detection. A less powerful distance-based classifier such as the Spectral Angle Mapper (SAM) can be applied. Or, a more powerful filter, such as target constrained interference minimized filter (TCIMF), may be used [6]. III. RELATIONSHIP BETWEEN LDA-BASED APPROACHES A. Relationship Between FLDA and CFLDA The CFLDA in [5] imposed a constraint to align the class centers along with different directions [4], i.e., w T l µ j = δ lj , for 1 ≤ l; j ≤ p. (9) This also means that the jth transform vector w j is for the jth class. So the CFLDA-transformed data are actually classification maps. It can be derived that when the constraint was satisfied, w T S B w was a constant. Thus, the constrained problem would be to minimize w T S W w in (3) while satisfying the constraint in (9). Using the Lagrange multiplier approach, it was shown that the desired transform matrix W including all the p transform vectors is W CFLDA = S −1 W M ( M T S −1 W M) −1 . (10) Obviously, the implementation of CFLDA requires the knowledge of the training samples of each class to compute S W . B. Relationship Between CFLDA, CLDA, and MFLDA Following the same idea of FLDA in maximizing the class separability, the CLDA in [2] and [3] imposed the same constraint that different classes were aligned along different directions as in (9). To make the constrained problem easier to solve, it employed the ratio of within-class and between-class distances instead of the Raleigh quotient [4]. It was proved that the transformed within-class distance is a constant when the constraint in (9) was satisfied. It also used the data covariance matrix Σ to substitute S T as in MFLDA. It was proved that the transform matrix W is equivalent to [3] W CLDA =Σ −1 M(M T Σ −1 M) −1 . (11) Equation (11) is similar to (10) except that S W is replaced with Σ. Therefore, CLDA does not require the training samples in each class and it needs the class signatures only. Similar to CFLDA, CLDA was designed for classification, so the classification maps were obtained right after the transform. C. Use of Σ and S W Both CFLDA and CLDA apply the constraint in (9), resulting in the similar operators in (10) and (11) with the difference that CLDA uses Σ while CFLDA uses S W . So CLDA does not require the training samples, which is the same as in MFLDA. There is another benefit of using Σ. As mentioned earlier, the true number of classes present in an image scene p T is greater than p due to the difficulty of exhausting all the present classes, in particular, those background classes. In the ideal case when all the pixels in an image scene are put into the p T classes, S T =Σ. Therefore, using Σ in LDA-based approaches represents the best situation for S T , which means
DU: MODIFIED FISHER’S LINEAR DISCRIMINANT ANALYSIS FOR HYPERSPECTRAL IMAGERY 505 Fig. 1. (a) HYDICE image scene with 30 panels. (b) Spatial locations of the 30 panels that were provided by ground truth. all the classes can be well separated without knowing these class information. This is particularly important to suppress the background classes for better extraction of the foreground classes. Σ −1 represents the data whitening term, which has the power to suppress the unknown background classes [7]. Therefore, in general it is reasonable and desirable to use Σ to replace S W or S T in the practical implementation of LDA. IV. EXPERIMENTS A. HYDICE The HYperspectral Digital Imagery Collection Experiment (HYDICE) image scene shown in Fig. 1 includes 30 panels arranged in a 10 × 3 matrix [3]. The three panels in the same row, i.e., p ia , p ib , p ic , were made from the same material of sizes 3 m × 3m,2m× 2m,and1m× 1 m, respectively, which can be considered as one class, P i for 1 ≤ i ≤ 10. The pixel-level ground truth map in Fig. 1(b) shows the precise locations of pure panel pixels. These panel classes have very close signatures for differentiation. A pure pixel from each leftmost panel (3 m × 3m)was used as the corresponding class signature. Fig. 2(a) shows the classification result using SAM on the original data, where the panels could not be classified. Here, the minimum angle was displayed in white, the maximum angle in black, and others in shades between white and black. Fig. 2(b) is the result using TCIMF, which is a more powerful filter, on the original image, where the panels were well detected and separated. Fig. 2(c) shows the SAM classification result using the 9-D MFLDA-transformed data, where the panels were correctly classified. This demonstrates that MFLDA successfully separated the ten panel classes when performing dimension reduction, allowing a less powerful classifier, such as SAM, to correctly classify these classes, which is impossible when using the original 169-D data. To quantify the performance of MFLDA and compare it with that of TCIMF using the original data, each classification map was normalized to [0 1] and converted to a binary map with a threshold η. The binary classification maps were compared with the pure panel pixels provided as ground truth. The accurately detected panel pixels were counted as N D and false alarm as N F . To comprehensively evaluate the performance, the similar concept of receiver operating characteristic (ROC) curve was adopted here [8]. As η was changed from 0.1 to 0.9, an ROC curve could be estimated. For the 28-pixel test set, the resulting ROCs with the averaged probability of false alarm (Pf) and Fig. 2. Comparison between the classification results using the original and MFLDA-transformed data. (a) SAM (soft) classification result on the original data. (b) TCIMF (soft) classification result on the original data. (c) SAM (soft) classification result on the MFLDA-transformed data. probability of detection (Pd) are shown in Fig. 3. The larger the area under a curve, the better the performance [9]. We can see that SAM on the MFLDA-transformed data slightly outperforms TCIMF on the original data. To further compare the results in Fig. 2, Table I lists the largest number of detected pixels in the test set (N D ) when no false alarm exists (N F =0). This happened when η =0.6 for the MFLDA with SAM and η =0.4 for TCIMF. By applying MFLDA followed by SAM, 21 out of 28 panel pixels in the
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