Spatial-Spectral Dimensionality Reduction of Hyperspectral Imagery with Partial Knowledge of Class Labels

Transcription

1 Spatial-Spectral Dimensionality Reduction of Hyperspectral Imagery with Partial Knowledge of Class Labels Nathan D. Cahill, Selene E. Chew, and Paul S. Wenger Center for Applied and Computational Mathematics, School of Mathematical Sciences, Rochester Institute of Technology, Rochester, NY 4623, USA ABSTRACT Laplacian Eigenmaps (LE) and Schroedinger Eigenmaps (SE) are effective dimensionality reduction algorithms that are capable of integrating both the spatial and spectral information inherent in a hyperspectral image. In this paper, we consider how to extend LE- and SE-based spatial-spectral dimensionality reduction algorithms to situations where partial knowledge of class labels exists, for example, when a subset of pixels has been manually labeled by an expert user. This partial knowledge is incorporated through the use of cluster potentials, turning each underlying algorithm into an instance of SE. Using publicly available data, we show that incorporating this partial knowledge improves the performance of subsequent classification algorithms. Keywords: Dimensionality reduction, Laplacian eigenmaps, Schroedinger eigenmaps, spatial-spectral fusion.. INTRODUCTION Dimensionality reduction algorithms have been used in a variety of hyperspectral imaging analysis applications to provide low-dimensional representations of the image data. Since the original image data cannot be assumed to be linear and may implicitly reside on a nonlinear manifold in a high-dimensional space, it is important that dimensionality reduction algorithms yield representations that preserve the structure of the manifold. A variety of nonlinear approaches to dimensionality reduction have been investigated with respect to applications in hyperspectral imaging, including Local Linear Embedding (LLE), 2 Isometric Feature Mapping (ISOMAP), 3 Kernel Principal Components Analysis (KPCA), 4 Laplacian Eigenmaps (LE), 5 and Schroedinger Eigenmaps (SE). 6 Many nonlinear dimensionality reduction algorithms are capable of integrating or fusing both the spatial and spectral information inherent in hyperspectral imagery to provide low-dimensional representations that can be effectively used as input for clustering, segmentation, and classification of hyperspectral imagery. In this paper, we consider how to extend LE- and SE-based spatial-spectral dimensionality reduction algorithms 7 2 to situations where partial knowledge of class labels exists, for example, when a subset of pixels has been manually labeled by an expert user. This partial knowledge is incorporated through the use of cluster potentials, turning each underlying algorithm into an instance of SE. Using publicly available hyperspectral image data (Indian Pines), we manually identify small subsets of pixels with ground-truth labels to be incorporated as partial knowledge in various spatial-spectral dimensionality reduction algorithms. With the resulting low-dimensional representations (generated both with and without the partial knowledge), we carry out Support Vector Machine (SVM)-based classification to predict class labels for all pixels. Our analysis shows that incorporating partial knowledge of class labels in the low-dimensional representations yields classification results that are competitive with or superior to those based on using lowdimensional representations generated without partial knowledge. The remainder of this paper is organized in the following manner: Section 2 provides mathematical preliminaries describing the LE and SE dimensionality reduction algorithms, and how spatial and spectral information can be fused within these algorithms; Section 3 shows how partial knowledge of class labels can be incorporated into LE- and SE-based spatial-spectral dimensionality reduction; Section 4 describes how the partial knowledge can be propagated in local neighborhoods on the manifold; Section 5 uses the resulting reduced-dimensional representations to perform classification experiments on publicly available data, illustrating improved results when partial knowledge is incorporated; and Section 6 provides conclusions and future work. Send correspondence to Nathan D. Cahill: nathan.cahill@rit.edu

2 2. MATHEMATICAL PRELIMINARIES In this section, we will describe the LE and SE algorithms and we will detail various approaches that have been developed to incorporate or fuse spatial and spectral components of the high-dimensional data in the dimensionality reduction process. We will denote X = {x,...,x k } to be a set of points on a manifold M R n, where n is assumed to be large. Each dimensionality reduction algorithm will identify a set of corresponding points Y = {y,...,y k } in R m, where m n, so that the relationships of points in Y are similar to the relationships of the corresponding points in X. 2. Laplacian Eigenmaps Laplacian Eigenmaps 3 (LE) is a popular graph-based dimensionality reduction algorithm that was introduced by Belkin and Niyogi in 23 and involves the following three steps:. Construct an undirected graph G = (X,E) whose vertices are the points in X and whose edges E are defined based on proximity between vertices. 2. Define a weight W i,j for each edge x i x j in E. 3. Compute the smallest m+ eigenvalues and eigenvectors of the generalized eigenvector problem Lf = λdf, wheredisthediagonalweighteddegreematrixdefinedbyd i,i = j W i,j, andl = D W isthelaplacian matrix. If the resulting eigenvectors f,..., f m, are ordered so that = λ λ λ m, then the points y T, y T 2,..., y T m are defined to be the rows of F = [f f 2 f m ]. One of the great strengths of LE is its flexibility in allowing different ways to define edges and edge weights. Common ways to define edges use ǫ-neighborhoods or (mutual) k-nearest neighbors search in some metric space. ( To define edge ) weights, the heat kernel is a common choice; i.e., the weight W i,j is defined to be exp x i x j 2 /σ if an edge exists between x i and x j or zero otherwise. 2.2 Schroedinger Eigenmaps Schroedinger Eigenmaps 6,4 (SE), a straightforward, yet powerful, generalization of LE, incorporates a potential matrix V that encodes extra information about the data that may be available. The potential matrix includes barrier potentials for points in X that pull the corresponding points in Y towards the origin, and/or cluster potentials for points in X that pull the corresponding points in Y towards each other. Barrier potentials are created by defining V to be a nonnegative diagonal matrix, with V i,i defined to be positive for each of the selected x i s. Cluster potentials are created by defining V to be a weighted sum of nondiagonal matrices V (i,j) that encode individual cluster potentials between x i and x j : V (i,j) k,l =, (k,l) {(i,i),(j,j)}, (k,l) {(i,j),(j,i)}, otherwise. () With a potential matrix defined, SE proceeds in the same manner as LE, but with the generalized eigenvector problem in step 3 replaced by the problem (L+αV)f = λdf, where α is a parameter chosen to relatively weight the contributions of the Laplacian matrix and potential matrix. It is the ability of SE to incorporate cluster potentials that we will exploit in Section 3 to encode partial knowledge of class labels. We will define cluster potentials for pairs of points in X that have the same known class label. 2.3 Spatial-Spectral Fusion The structure of the manifold M is influenced by the spectral (intensity) information at each pixel in an image as well as the spatial relationships between the spectra of neighboring pixels. A variety of techniques have been used to incorporate both spectral and spatial information in LE- and SE-based dimensionality reduction. To summarize these techniques in a common framework, we first consider each manifold point x i to be represented by concatenating the pixel s spectral information x f i with its spatial location x p i ; i.e., xt i = [ x f i T x p i T ] T.

3 LE-Based Spatial-Spectral Fusion Various references propose different ways of incorporating spatial and spectral information into LE by defining specific ways to determine graph edges and compute their weights: Shi-Malik (SM): 7,8 Edge between x i and x j if x p i xp j < ǫ; edge weights defined by: 2 ) 2 xfi xfj exp ( xp i xp j 2 W i,j = σf 2 σ, (x p 2 i,x j ) E. (2), otherwise Gillis-Bowles (GB): 9 Edge between x i and x j if x p i xp j < ǫ; edge weights defined by: 2 ( ( ) exp cos W i,j = f x i,xf 2 j xpi xpj ) x f i x f σ, (x j p 2 i,x j ) E. (3), otherwise Hou-Zhang-Ye-Zheng (HZYZ): Edge between x i and x j if x i and x j are mutually in the k-nearest neighbors of each other according to the measure: x f i xf j 2 ( ( x p i d(x i,x j ) = exp exp xp 2 )) j ; (4) 2σ 2 f 2σ 2 p edge weights defined by: W i,j = {, (xi,x j ) E, otherwise. (5) Benedetto et al. Fused Metric (BM): Edge between x i and x j if x i and x j are mutually in the k-nearest neighbors of each other according to the measure: d β (x i,x j ) = x f i xf j 2 ( x p 2 i β +( β) xp 2 ) j 2, (6) σ 2 f σ 2 p where β ; edge weights defined by: { ( W (β) exp d i,j = β (x i,x j ) 2), (x i,x j ) E β, otherwise. (7) A slightly different approach involves constructing two graphs: Benedetto et al. Fused Eigenvectors (BE):. Construct graphs G and G so that the sets of edges E and E are defined based on mutual k-nearest neighbors according to the metrics d and d from (6), respectively. 2. Define edge weights for G and G according to (7) with β = and, respectively. 3. Choose m and m so that m +m = m. Compute the smallest m + eigenvalues and eigenvectors L f () = λd f (), and compute the smallest m + eigenvalues and eigenvectors of L f () = λd f (). Assuming each set of eigenvectors is sorted so that [ the eigenvalues are increasing, then the points y T, y2 T,..., ym T are defined to be the rows of F = f () f m () f () f m () ].

4 SE-Based Spatial-Spectral Fusion Cahill et al. 2 proposed the Spatial-Spectral Schroedinger Eigenmaps (SSSE) algorithm, which defines graphs with spectral information and uses cluster potentials to encode spatial proximity. Edges are defined based on proximity between the spectral components of the vertices, and edge weights are defined according to: exp W i,j = 2 xfi xfj ( σf 2 ), (x i,x j ) E, otherwise. (8) A cluster potential matrix V is defined to encode proximity between the spatial components of the vertices: ( k x p V = V (i,j) i γ i,j exp xp 2 ) j, (9) i= x j Nǫ(x p i) where N p ǫ (x i ) is the set of points in X whose spatial components are in an ǫ-neighborhood of the spatial components of x i ; i.e., σ 2 p N p ǫ (x i ) = {x X x i s.t. x p i xp ǫ}, () V (i,j) is defined as in (), and γ i,j can be chosen in a manner that provides greater influence for spatial neighbors having ( nearby spectral components. Cahill et al. 2 proposed two versions of SSSE: SSSE - SSSE with γ i,j = ) ( ( )) exp x f i xf j 2 /σf 2, and SSSE2 - SSSE with γ i,j = exp cos f x i,xf j. With these choices for x f i x f j γ i,j, the coefficients of each V (i,j) in (9) are equivalent to the Shi-Malik edge weights in (2) or the Gillis-Bowles edge weights in (3), respectively. 3. PARTIAL KNOWLEDGE OF CLASS LABELS AS CLUSTER POTENTIALS In many scenarios, an expert user or analyst can be queried to identify a small set of pixels or regions of an image that should have the same class labels. Wagstaff et al. 5 defines this type of partial knowledge of class labels as must-link constraints. Normalized Cuts, 8 a graph-based algorithm for data clustering and image segmentation, solves the same generalized eigenvector problem as LE; it has been generalized to handle must-link constraints, either as hard 6,7 or soft 8 constraints. We extend the idea of Chew and Cahill 8 to show how soft must-link constraints in data clustering have the natural analog of cluster potentials in dimensionality reduction. To formalize this idea for graph-based dimensionality reduction, suppose that each of the ordered pairs in the set M = {(x i,x j ),(x i2,x j2 ),...,(x im,x jm )} represent two graph vertices that should ultimately be given the same class label. By introducing cluster potentials for each of these ordered pairs, we can bias the dimensionality reduction process to yield lower dimensional representations so that the distances y ik y jk, k =,2,...,m, are small. These cluster potentials can be encoded in a matrix M given by: M = η ik,j k V (i k,j k ), () (x ik,x jk ) M where V (i,j) is defined as in (), and η i,j can be chosen to weight each must-link constraint individually, if desired. All of the LE-based dimensionality reduction techniques described in Section 2.3 are based on solving generalized eigenvector problems of the form Lf = λdf. By adding a scalar multiple of the cluster potential matrix M to the graph Laplacian so that the generalized eigenvector problems take the form (L+βM)f = λdf, we can bias the representations toward satisfying the must-link constraints. This modification of the generalized eigenvector problems turns each original LE-based algorithm into an instance of Schroedinger Eigenmaps. The SSSE algorithm in Section 2.3 can also be extended to incorporate partial knowledge in the form of mustlink constraints by changing the generalized eigenvector problem (L+αV)f = λdf to (L+αV+βM)f = λdf.

5 (a) (b) (c) (d) (e) Figure : Example of incorporating partial knowledge in dimensionality reduction: (a) Indian Pines image (spectral bands 29, 5, 2) with zoomed-in region, (b) ground truth class labels (orange = corn-notill, blue = soybean-mintill), (c) SVM-based class labels after LE dimensionality reduction, with corn-notill region incorrectly labeled as soybean-mintill, (d) orange points manually labeled by an expert user indicating pixels have the same class label, and (e) SVM-class labels after dimensionality reduction that incorporates the partial knowledge indicated in (d), with corn-notill region correctly labeled. Choosing the appropriate weights η ik,j k and β can be tricky. In fact, there is some redundancy, as multiplying each η ik,j k by c and dividing β by c yields exactly the same generalized eigenvector problem for any c. One way to enable a consistent interpretation of β is to define the weights η ik,j k so that tr(m) = tr(l). Hence, if all ordered pairs in M are to be given equal weight (which is reasonable if all ordered pairs should be assigned the same class label), one should define η ik,j k = tr(l)/(2m). β can initially be set to, and it can be increased or decreased logarithmically until the resulting representation appears optimal to the user. More generally, consider the possibility that the ordered pairs in M can be partitioned into l different groups (corresponding to l different class labels), with m ordered pairs assigned to label, m 2 ordered pairs assigned to label 2, etc., and m +m 2 + +m l = m. If the must-link constraints should be equally weighted across class labels, then the weight for each ordered pair corresponding to class label ν should be defined as η ik,j k = tr(l)/(2lm ν ). To illustrate the impact of incorporating partial knowledge in dimensionality reduction, consider the publiclyavailable Indian Pines hyperspectral image, shown rendered in RGB in Figure a. The image has an associated set of ground truth pixels encompassing 6 distinct classes, as shown in Figure b. Performing LE-based dimensionality reduction (with SM graph construction), followed by SVM-based classification yields the predicted class labels shown in Figure c. In the zoomed-in region, we see that a rectangular subregion of orange (cornnotill) pixels has been mislabeled as belonging to the blue class (soybean-mintill). By allowing an expert user to use a paintbrush tool to manually identify a few small sets of pixels across the image (shown highlighted in orange in Figure d) as belonging to the same class, a cluster potential matrix can be defined for these manually labeled pixels. Incorporating this cluster potential matrix into the dimensionality reduction algorithm and then performing SVM-based classification yields the predicted class labels shown in Figure e that correctly identify the corn-notill and soybean-mintill classes in the zoomed-in region. 4. KNOWLEDGE PROPAGATION In some situations, cluster potentials that are generated from small sets of manually provided labels may impact dimensionality reduction in a brittle manner. The cluster potentials may correctly bias dimensionality

6 (a) (b) (c) (d) (e) Figure 2: Example of the influence of knowledge propagation: (a) Ground-truth class labels for Indian Pines image with zoomed-in regions (orange = corn-notill, light green = grass-pasture); (b) SVM-based class labels after LE dimensionality reduction, with both corn-notill and grass-pasture regions incorrectly labeled; (c) orange and light green points manually labeled by an expert user indicating pixels that have the same class labels; (d) SVM-class labels after dimensionality reduction incorporating the partial knowledge indicated in (c) without knowledge propagation, illustrating the brittle nature of the grass-pasture cluster potential; and (e) SVM-class labels after dimensionality reduction with knowledge propagation, yielding correct class labels for both corn-notill and green-pasture regions. reduction so that the low-dimensional representations of the manually labeled points are close together, but this behavior may not propagate adequately to other points that are nearby in the high dimensional space. Consider the example illustrated in Figure 2. The Indian Pines ground truth class labels are shown again in Figure 2a. We will focus specifically on two subregions: the corn-notill (orange) region in the center of the lower zoomed region, and the rectangular grass-pasture (light green) region in the center of the upper zoomed region. Performing LE-based dimensionality reduction (with GB graph construction), followed by SVM-based classification yields the predicted class labels shown in Figure 2b. Comparing the results to the ground truth, we that as in Figure c, the corn-notill (orange) region is incorrectly labeled as soybean-mintill (blue); in addition, the grass-pasture (light green) region is incorrectly labeled as a mixture of other classes. To try to improve on these results, we manually identify six sets of pixels in the orange corn-notill regions and four sets of pixels in the light green grass-pasture regions, as shown in Figure 2c. Constructing cluster potentials between every pair of manually labeled like-colored pixels, incorporating the cluster potential matrix in dimensionality reduction, and performing SVM-based classification yields class labels shown in Figure 2d. A portion of the corn-notill region that was incorrectly labeled in Figure 2b has now been classified correctly; however, the grass-pasture region has now been correctly labeled only in the pixels that were manually identified. This example illuminates the problem that can occur when the provided manual labels are sparse and two or more clusters of points may be present in the high-dimensional space. If manual labels are provided that indicate a point in one of the high-dimensional clusters should be identified with points in a different high-dimensional cluster, dimensionality reduction may yield a result in which the manually labeled points are pulled together in the low-dimensional representation without pulling together the separate clusters.

7 ThisproblemhaspreviouslybeenidentifiedinthespectralclusteringliteraturebyYuandShi. 6 Theysuggest that for the related clustering problem, constraints on individual points can be smoothed or propagated to nearby points. In dimensionality reduction, this can be achieved by replacing the cluster potential matrix M in () with: M = η ik,j k WD V (i k,j k ) D W, (2) (x ik,x jk ) M where W and D are the weighted adjacency matrix and degree matrix of the graph, respectively. Like the constraint propagation in Yu and Shi, 6 this modification of the cluster potentials encourages points that are close to the manually-labeled points in the high-dimensional space to have low-dimensional representations that are also close. Hence, we refer to the use of (2) as opposed to () in dimensionality reduction as knowledge propagation. Referring back to Figure 2, incorporating cluster potentials with knowledge propagation in dimensionality reduction and performing SVM-based classification yields the class labels shown in Figure 2e. Comparing the results to those in Figure 2d in which knowledge propagation was not used, we see that both the corn-notill and grass-pasture regions have now been classified correctly. 5. CLASSIFICATION EXPERIMENTS In order to determine the effect of incorporating partial knowledge of class labels in LE- and SE-based dimensionality reduction, we explore the publicly-available Indian Pines image and compute various low-dimensional representations without partial knowledge, with partial knowledge, and with partial knowledge and knowledge propagation. Then, using the representations as input, we perform multi-class SVM-based classification. The data, experiments, and results are described in this section. 5. Data The Indian Pines image, shown in Figures and 2, was captured by an AVIRIS spectrometer over the rural Indian Pines test site in Northwestern Indiana, USA. The image contains pixels with spatial resolution of approximately 2 meters per pixel, with 224 spectral bands, 4 of which we have discarded due to noise and water. A set of 249 ground truth pixels associated with 6 classes have been manually identified by an expert analyst for use in training and testing classification algorithms. 5.2 Experimental Setup To compare the various dimensionality reduction algorithms without partial knowledge, with partial knowledge, and with partial knowledge and knowledge propagation, we manually label various subsets of pixels to mimic the type of partial knowledge that could be provided by an expert user. Figure 3 shows larger versions of the original Indian Pines image (rendered in RGB) and the corresponding ground truth class labels. In addition, Figure 3 shows four different sets of partial knowledge to be tested: six sets of pixels in the orange corn-notill regions (Case: C), four sets of pixels in the light green grass-pasture regions (Case: G), the combined sets of orange corn-notill and light green grass-pasture regions (Case: CG), and a set of pixels selected from each of the connected ground-truth regions (Case: All). Using these four cases both with and without knowledge propagation, we compute low-dimensional embeddings using all of the dimensionality reduction algorithms. The low-dimensional embeddings are provided as input to a classifier, through which class labels will be predicted and compared to the ground truth. The classifier we use is based on one-versus-all SVMs with Gaussian RBF kernels as implemented in MATLAB s fitcsvm function. We employ the heuristic procedure available in fitcsvm in order to automatically determine the kernel scaling. We select % of the ground truth pixels from each class for training, and we use the remaining ground truth pixels for testing. From the resulting confusion matrices, we compute per-class accuracy, precision, sensitivity, and specificity, as defined in Table. A few notes about data treatment and parameter choices:

8 Indian Pines Image Case C: corn-notill Case CG: corn + grass Ground Truth Case G: grass-pasture Case All: all classes Figure 3: Indian Pines image (spectral bands 29, 5, 2), ground truth class labels, and manually provided class labels representing different test cases for evaluating dimensionality reduction algorithms. Measure Definition Measure Definition Accuracy Precision TP +TN TP +FP +FN +TN TP TP +FP Sensitivity Specificity TP TP +FN TN FP +TN Table : Definitions of per-class classification performance measures, in terms of true positives (TP), false positives (FP), false negatives (FN), and true negatives (TN).

9 Prior to dimensionality reduction, the spectral components of the data in X are normalized so that (/k) k =. We also assume that components of x p are in units of pixels. 2 i= x f i For all algorithms, we choose the reduced dimension to be n = 5. This value was chosen to ensure that the resulting accuracy of representation is high; i.e., that with other parameters fixed, increasing n does not result in significant changes in accuracy. For algorithms requiring graph construction via k-nearest neighbors (SSSE, HZYZ, BE, BM), we select k = 2. This value was chosen to guarantee that the resulting graphs are connected. For parameter choices in each dimensionality reduction algorithm, we use the values that were reported optimal in Cahill et al.: 2 SM: ǫ = 5, σ f =., σ p = ; GB: ǫ = 5, σ f =.2, σ p = ; HZYZ: σ f =, σ p = ; BE: 8 spatial / 42 spectral eigenvectors, σ f =, σ p = ; BM: σ f =, σ p =, β = 8; SSSE: α = 7.78 tr(l)/tr(v). 5.3 Results Figures 4 illustrate per-class classification performance measures after dimensionality reduction and SVMbased classification. The sequence of figures corresponds to SM, GB, HZYZ, BM, BE, SSSE, and SSSE2 dimensionality reduction methods. In each figure, subplots show accuracy, precision, sensitivity, and specificity measures for each partial-knowledge case (C, G, CG, and All). Within each subplot, three symbols indicate how partial knowledge is used: red circles indicate dimensionality reduction was performed without partial knowledge, blue crosses indicate dimensionality reduction was performed with partial knowledge but without knowledge propagation, and black triangles indicate dimensionality reduction was performed with partial knowledge and with knowledge propagation. A number of observations can be drawn from these figures. For nearly all situations, in cases C, GC, and All, each dimensionality reduction algorithm improves all classification measures in the corn-notill regions when prior knowledge is used. The largest improvements appear to be in SM- and GB-based dimensionality reduction Accuracy Precision Sensitivity Specificity Case C Case G Case CG Case All Figure 4: Per-class classification performance after SM-based dimensionality reduction using: no partial knowledge (red ) partial knowledge without propagation (blue +), and partial knowledge with propagation (black ). Vertical dashed lines indicate classes for which partial knowledge is provided.

10 Accuracy Precision Sensitivity Specificity Case C Case G Case CG Case All Figure 5: Per-class classification performance after GB-based dimensionality reduction using: no partial knowledge (red ) partial knowledge without propagation (blue +), and partial knowledge with propagation (black ). Vertical dashed lines indicate classes for which partial knowledge is provided. Accuracy Precision Sensitivity Specificity Case C Case G Case CG Case All Figure 6: Per-class classification performance after HZYZ-based dimensionality reduction using: no partial knowledge (red ) partial knowledge without propagation (blue +), and partial knowledge with propagation (black ). Vertical dashed lines indicate classes for which partial knowledge is provided.

11 Accuracy Precision Sensitivity Specificity 8 8 Case C Case G Case CG Case All Figure 7: Per-class classification performance after BM-based dimensionality reduction using: no partial knowledge (red ) partial knowledge without propagation (blue +), and partial knowledge with propagation (black ). Vertical dashed lines indicate classes for which partial knowledge is provided. Case C Case G Case CG Case All Accuracy Precision Sensitivity Specificity Figure 8: Per-class classification performance after BE-based dimensionality reduction using: no partial knowledge (red ) partial knowledge without propagation (blue +), and partial knowledge with propagation (black ). Vertical dashed lines indicate classes for which partial knowledge is provided.

12 Accuracy Precision Sensitivity Specificity Case C Case G Case CG Case All Figure 9: Per-class classification performance after SSSE-based dimensionality reduction using: no partial knowledge (red ) partial knowledge without propagation (blue +), and partial knowledge with propagation (black ). Vertical dashed lines indicate classes for which partial knowledge is provided. Accuracy Precision Sensitivity Specificity Case C Case G Case CG Case All Figure : Per-class classification performance after SSSE2-based dimensionality reduction using: no partial knowledge (red ) partial knowledge without propagation (blue +), and partial knowledge with propagation (black ). Vertical dashed lines indicate classes for which partial knowledge is provided.

13 algorithms. The few situations where there do not appear to be improvements when prior knowledge is used occur when knowledge is not propagated. For the cases G, GC, and All, classification performance in the grass-pasture regions is already quite good without the use of prior knowledge in dimensionality reduction. Incorporating prior knowledge can minimally improve various performance measures when knowledge propagation is used. However, if knowledge propagation is not used, results can be problematic. Note in particular the SSSE and SSSE2 dimensionality reduction algorithms for case G when partial knowledge is used without propagation: all classification performance measures are worse than when partial knowledge is not used at all. In these cases, the pixel regions that were labeled as grass-pasture were all classified incorrectly into a common class. When the partial knowledge is used with propagation, the pixel regions become classified correctly. Also of interest are how partial knowledge of specific classes impacts classification performance of other classes. When the corn-notill regions are provided as partial knowledge with propagation in SM-, GB-, BM-, and BE-based dimensionality reduction, classification performance is also improved in the soybean-mintill regions (class ). This observation confirms the results obtained visually in Figure e. 6. CONCLUSION In this paper, we showed how the dimensionality reduction algorithms Laplacian Eigenmaps and Schroedinger Eigenmaps can be extended to incorporate partial knowledge of class labels, such as pixels or regions that have been manually labeled by an expert user. Using cluster potentials to encode the partial knowledge turns each algorithm into an instance of Schroedinger Eigenmaps. With publicly available data, we illustrated that incorporating this partial knowledge yields low-dimensional representations of the data that can improve the performance of subsequent SVM-based classification. APPENDIX Prototype implementations of the algorithms presented in this paper are available for download at MATLAB Central ( under File ID #589. ACKNOWLEDGEMENTS The authors would like to thank Prof. Landgrebe (Purdue University, USA) for providing the Indian Pines data. Selene Chew was supported in part by Rochester Institute of Technology s 24 Honors Summer Undergraduate Research Fellowship program. REFERENCES [] Prasad, S. and Bruce, L., Limitations of principal components analysis for hyperspectral target recognition, IEEE Geoscience and Remote Sensing Letters 5, (Oct 28). [2] Kim, D. and Finkel, L., Hyperspectral image processing using locally linear embedding, in [Proc. IEEE EMBS Conference on Neural Engineering], (March 23). [3] Bachmann, C., Ainsworth, T., and Fusina, R., Exploiting manifold geometry in hyperspectral imagery, IEEE Transactions on Geoscience and Remote Sensing 43, (March 25). [4] Fauvel, M., Chanussot, J., and Benediktsson, J., Kernel principal component analysis for the classification of hyperspectral remote sensing data of urban areas, EURASIP Journal on Advances in Signal Processing 29(78394), 4 (29). [5] Halevy, A., Extensions of Laplacian Eigenmaps for Manifold Learning, PhD thesis, University of Maryland, College Park (2). [6] Benedetto, J., Czaja, W., Dobrosotskaya, J., Doster, T., Duke, K., and Gillis, D., Semi-supervised learning of heterogeneous data in remote sensing imagery, in [Proc. SPIE Independent Component Analyses, Compressive Sampling, Wavelets, Neural Net, Biosystems, and Nanoengineering X], 844, 2 (June 22).

14 [7] Shi, J. and Malik, J., Normalized cuts and image segmentation, in [Proc. IEEE Conference on Computer Vision and Pattern Recognition], (997). [8] Shi, J. and Malik, J., Normalized cuts and image segmentation, IEEE Transactions on Pattern Analysis and Machine Intelligence 22(8), (2). [9] Gillis, D. B. and Bowles, J. H., Hyperspectral image segmentation using spatial-spectral graphs, in [Proc. SPIE Algorithms and Technologies for Multispectral, Hyperspectral, and Ultraspectral Imagery XVIII], 839Q, (22). [] Hou, B., Zhang, X., Ye, Q., and Zheng, Y., A novel method for hyperspectral image classification based on Laplacian eigenmap pixels distribution-flow, IEEE Journal of Selected Topics in Applied Earth Observations and Remote Sensing 6(3), (23). [] Benedetto, J., Czaja, W., Dobrosotskaya, J., Doster, T., Duke, K., and Gillis, D., Integration of heterogeneous data for classification in hyperspectral satellite imagery, in [Proc. SPIE Algorithms and Technologies for Multispectral, Hyperspectral, and Ultraspectral Imagery XVIII], 83927, 2 (June 22). [2] Cahill, N. D., Czaja, W., and Messinger, D. W., Schroedinger eigenmaps with nondiagonal potentials for spatial-spectral clustering of hyperspectral imagery, in [Proc. SPIE Algorithms and Technologies for Multispectral, Hyperspectral, and Ultraspectral Imagery XX], 9884, 3 (June 24). [3] Belkin, M. and Niyogi, P., Laplacian eigenmaps for dimensionality reduction and data representation, Neural Computation 5, (June 23). [4] Czaja, W. and Ehler, M., Schroedinger eigenmaps for the analysis of biomedical data, IEEE Transactions on Pattern Analysis and Machine Intelligence 35, (May 23). [5] Wagstaff, K., Cardie, C., Rogers, S., and Schrödl, S., Constrained k-means clustering with background knowledge, in [Proceedings of the Eighteenth International Conference on Machine Learning], ICML, (2). [6] Yu, S. X. and Shi, J., Segmentation given partial grouping constraints, IEEE Trans. Pattern Analysis and Machine Intelligence 26, (Feb 24). [7] Eriksson, A., Olsson, C., and Kahl, F., Normalized cuts revisited: A reformulation for segmentation with linear grouping constraints, Journal of Mathematical Imaging and Vision 39(), 45 6 (2). [8] Chew, S. E. and Cahill, N. D., Normalized cuts with soft must-link constraints for image segmentation and clustering, in [Proc. IEEE Western New York Image and Signal Processing Workshop], 6 (November 24).