IEEE GEOSCIENCE AND REMOTE SENSING LETTERS, VOL. XX, NO. Y, MONTH Z 2007
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Semi-supervised Image Classification with Laplacian Support Vector Machines Luis G´omez-Chova, Gustavo Camps-Valls, Senior Member, IEEE, Jordi Mu˜noz-Mar´ı, and Javier Calpe, Member, IEEE
Abstract— This paper presents a semi-supervised method for remote sensing image classification based on kernel machines and graph theory. The support vector machine (SVM) is regularized with the un-normalized graph Laplacian, thus leading to the Laplacian SVM. The method is tested in the challenging problems of urban monitoring and cloud screening, in which an adequate exploitation of the wealth of unlabeled samples is of critical. Results obtained using different sensors, and with low number of training samples demonstrate the potential of the proposed Laplacian SVM for remote sensing image classification. Index Terms— Semi-supervised learning, regularization, kernel methods, manifold learning, support vector machines.
I. I NTRODUCTION In remote sensing image classification, we are usually given a reduced set of labeled samples to develop the classifier. Supervised classifiers such as support vector machines (SVMs) [1], [2] excel in using the labeled information, being (regularized) maximum margin classifiers also equipped with an appropriate loss function [3], [4]. These methods, nevertheless, need to be reformulated to exploit the information contained in the wealth of unlabeled samples, which is known as semisupervised classification. In semi-supervised learning (SSL), the algorithm is provided with some available supervised information in addition to the unlabeled data. The framework of semi-supervised learning is very active and has recently attracted a considerable amount of research [5], [6]. Essentially, three different classes of SSL algorithms are encountered in the literature: 1) Generative models involve estimating the conditional density, such as expectation-maximization (EM) algorithms with finite mixture models [7], which have been extensively applied in the context of remotely sensed image classification [8]. 2) Low density separation algorithms maximize the margin for labeled and unlabeled samples simultaneously, such as Transductive SVM (TSVM) [9], which have been recently applied to hyperspectral image classification [10]. 3) Graph-based methods, in which each sample spreads its label information to its neighbors until a global stable state is achieved on the whole dataset [11]. In the last years, TSVM and graph-based methods have captured great attention. However, some specific problems are identified in both of them. In particular, TSVM is sensitive to local minima and requires convergence heuristics by using an (unknown) number of unlabeled samples. Graph-based Manuscript received July 2007; Dept. Enginyeria Electr`onica. Escola T`ecnica Superior d’Enginyeria. Universitat de Val`encia. C/ Dr. Moliner, 50. 46100 Burjassot (Val`encia) Spain. E-mail:
[email protected]
methods are computationally demanding and generally do not yield a final decision function but only prediction labels. In this paper, we present a recently introduced semisupervised framework that incorporates labeled and unlabeled data in any general-purpose learner [12], [13]. We focus on a semi-supervised extension of the SVM, which introduces an additional regularization term on the geometry of both labeled and unlabeled samples by using the graph Laplacian [14], thus leading to the so-called Laplacian SVM (LapSVM). This methodology follows a non-iterative optimization procedure in contrast to most transductive learning methods, and provides out-of-sample predictions in contrast to graph-based approaches. In addition, hard-margin SVM, directed graph methods, label propagation methods and spectral clustering solutions are obtained for particular free parameters. The performance of the LapSVM is illustrated in two challenging problems: the urban classification problem using multispectral (LandSat TM) and radar (ERS2 SAR) data [15], and the cloud screening problem using the MEdium Resolution Imaging Spectrometer (MERIS) instrument on board the ESA ENVIronmental SATellite (ENVISAT) [16]. On the one hand, monitoring urban areas at a regional scale, and even at a global scale, has become an increasingly important topic in the last decades in order to keep track of the loss of natural areas due to urban development. The classification of urban areas is, however, a complex problem, specially when different sensor sources are used, as they induce a highly variable input feature space. On the other hand, the amount of images acquired over the globe every day by Earth Observation (EO) satellites makes inevitable the presence of clouds. Therefore, accurate cloud masking in acquired scenes is of paramount importance to avoid errors on the study of the true ground reflectance, and to permit multitemporal studies, as it is no longer necessary to discard the whole image. However, very few labeled cloud pixels are tipically available, and cloud features change to a great extent depending on the cloud type, thickness, transparency, height, or background. In addition, cloud screening must be carried out before atmospheric correction, being the input data affected by the atmospheric conditions. In machine learning, these problems constitute clear examples of classification in complex manifolds and ill-posed problems, respectively. A manifold is a topological space that is locally Euclidean, but in which the global structure may be more complicated. In both settings, the use of LapSVM is motivated since it permits using the labeled samples, and efficiently exploiting the information contained in the high number of available unlabeled pixels to characterize the marginal distribution of the class of interest. The rest of the paper is outlined as follows. Section II reviews the framework of semi-supervised learning paying attention to the minimization functional and the need for different types of regularization. Section III presents the formulation
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IEEE GEOSCIENCE AND REMOTE SENSING LETTERS, VOL. XX, NO. Y, MONTH Z 2007
of the Laplacian SVM. Section IV shows the experimental results. Finally, Section V concludes and outlines further work. II. S EMI - SUPERVISED L EARNING F RAMEWORK Regularization is necessary to produce smooth decision functions and thus avoiding overfitting to the training data. Since the work of Tikhonov [17], many regularized algorithms have been proposed to control the capacity of the classifier [1], [18]. The regularization framework has been recently extended to the use of unlabeled samples [13] as follows. Notationally, we are given a set of l labeled samples, {xi , yi }li=1 , and a set of u unlabeled samples {x i }l+u i=l+1 , where xi ∈ RN and yi ∈ {−1, +1}. Let us now assume a general-purpose decision function f . The regularized functional to be minimized is defined as: l
1 L= V (xi , yi , f ) + γL f 2H + γM f 2M , l i=1
(1)
where V represents a generic cost function of the committed errors on the labeled samples, γ L controls the complexity of f in the associated Hilbert space H, and γ M controls its complexity in the intrinsic geometry of the marginal data distribution. For example, if the probability distribution is supported on a low-dimensional manifold, f 2M penalizes f along that manifold M. Note that this functional constitutes a general regularization framework that takes into account all the available knowledge. III. L APLACIAN S UPPORT V ECTOR M ACHINES The previous semi-supervised learning framework allows us to develop many different algorithms just by playing around with the loss function, V , and the regularizers, f 2 . In this paper, we focus on the Laplacian SVM formulation, which basically uses a SVM as the learner core and the graph Laplacian for manifold regularization. In the following, we review all the ingredients of the formulation. A. Cost function of the errors The Laplacian SVM uses the same hinge loss function as the traditional SVM: V (xi , yi , f ) = max{0, 1 − yi f (xi )},
(2)
where f represents the decision function implemented by the selected classifier. B. Decision function We use as the decision function f (x ∗ ) = w, φ(x∗ ) + b, where φ(·) is a nonlinear mapping to a higher (possibly infinite) dimensional Hilbert space H, and w and b define a linear regression in that space. By means of the Representer Theorem [1], weights w can be expressed in the dual problem as the expansion over labeled and unlabeled samples w = l+u i=1 αi φ(xi ) = Φα, where Φ = [φ(x1 ), . . . , φ(xl+u )] and α = [α1 , . . . , αl+u ]. Then, the decision function is given by: f (x∗ ) =
l+u
αi K(xi , x∗ ) + b,
(3)
i=1
and K is the kernel matrix formed by kernel functions K(xi , xj ) = φ(xi ), φ(xj ). The key point here is that,
without considering the mapping φ explicitly, a non-linear classifier can be constructed by selecting the proper kernel. Also, the regularization term can be fully expressed in terms of the kernel matrix and the expansion coefficients: f 2H = w2 = (Φα) (Φα) = α Kα.
(4)
C. Manifold regularization The geometry of the data is modeled with a graph in which nodes represent both labeled and unlabeled samples connected by weights Wij [5], [14]. Regularizing the graph follows from the smoothness (or manifold) assumption and intuitively is equivalent to penalize the “rapid changes” of the classification function evaluated between close samples in the graph: f 2M =
l+u 1 Wij (f (xi ) − f (xj ))2 = f Lf , (l + u)2 i,j=1
(5)
where L = D − W is the graph Laplacian, D is the diagonal l+u degree matrix of W given by D ii = j=1 Wij , and f = [f (x1 ), . . . , f (xl+u )] = Kα, where we have deliberately dropped the bias term b. D. Formulation By plugging (2)-(5) into (1), we obtain the regularized function to be minimized: l 1 γM min ξi + γL α Kα + α KLKα (6) ξi ∈Rl l i=1 (l + u)2 α∈Rl+u
subject to: l+u yi αj K(xi , xj ) + b ≥ 1 − ξi , i = 1, . . . , l
(7)
j=1
ξi ≥ 0
i = 1, . . . , l
(8)
where ξi are slack variables to deal with committed errors in the labeled samples. Introducing restrictions (7)-(8) into the primal functional (6) through Lagrange multipliers, β i and ηi , and taking derivatives w.r.t. b and ξ i , we obtain: 2γM 1 min KLK α 2γL K + 2α α,β (l + u)2 l −α KJ Yβ + i=1 βi , (9) where J = [I 0] is an l × (l + u) matrix with I as the l × l identity matrix (the first l points are labeled) and Y = diag(y1 , . . . , yl ). Taking derivatives again w.r.t. α, we obtain the solution [13]: −1 γM α = 2γL I + 2 LK J Yβ ∗ (10) (l + u)2 Now, substituting again (10) into the dual functional (9), we obtain the following quadratic programming problem to be solved: l 1 ∗ β = max βi − β Qβ (11) β 2 i=1
GOMEZ-CHOVA ET AL.: SEMI-SUPERVISED IMAGE CLASSIFICATION WITH LAPLACIAN SVM
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l
βi yi = 0 and 0 ≤ βi ≤ 1l , i = 1, . . . , l, where −1 γM Q = YJK 2γL I + 2 LK J Y (12) (l + u)2
subject to
i=1
Therefore, the basic steps for obtaining the weigths α i for the solution in (3) are: (i) build the weight matrix W and compute the graph Laplacian L = D − W, (ii) compute the kernel matrix K, (iii) fix regularization parameters γ L and γM , and (iv) compute α using (10) after solving the problem (11). E. Relation with other classifiers The Laplacian SVM is intimately related to other unsupervised and semi-supervised classifiers. This is because the method incorporates both the concepts of kernels and graphs in the same classifier, thus having connections with transduction, clustering, graph-based and label propagation methods. The minimizing functional used in the standard TSVM considers a different regularization parameter for labeled and unlabeled samples, which is the case in the proposed framework (cf. Eq. (1)). Also, LapSVM is directly connected with the softmargin SVM (γM = 0), the hard margin SVM (γ L → 0, γM = 0), the graph-based regularization method (γ L → 0, γM > 0), the label-propagation regularization method (γ L → 0, γM → 0, γM γL ), and spectral clustering (γ M = 1). In conclusion, by optimizing parameters γ L and γM over a wide enough range, the LapSVM theoretically outperforms the aforementioned classifiers. See [13] for deeper details and theoretical comparison. IV. E XPERIMENTAL RESULTS This section presents the experimental results of the proposed method in two challenging scenarios: urban monitoring and cloud screening. These two examples are well-suited because efficient exploitation of unlabeled samples becomes strictly necessary to attain satisfactory results. A. Model development and experimental setup We used both the linear kernel, K(x i , xj ) = xi , xj , and the Radial Basis Function (RBF) kernel, K(x i , xj ) = exp −xi − xj 2 /2σ 2 , where σ ∈ R+ is the kernel width
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for both SVM and LapSVM. The graph Laplacian, L, consisted of (l + u) nodes connected using k = 6 nearest neighbors, and compute the edge weights W ij using the Euclidean distance among samples. For all experiments, we generated training and validation sets consisting of l = 400 labeled samples (200 samples per class). The semi-supervised LapSVM used u = 400 unlabeled (randomly selected) samples from the analyzed images. We focus on the ill-posed scenario and varied the rate of both labeled and unlabeled samples independently, {2, 5, 10, 20, 50, 100} %. All classifiers are compared using the overall accuracy, OA[%], and the estimated kappa statistic, κ, as a measure of robustness in the classification. Free parameters γL and γM were varied in steps of one decade in the range [10 −4 , 104 ], and the Gaussian width was tuned in the range σ = {10 −3 , . . . , 10} for the RBF kernel. The selection of the best subset of free parameters was done by cross-validation. Figure 1 shows the kappa statistic as a function of the regularization parameters γ L and γM obtained by a LapSVM in the validation set for an illustrative example. This figure clearly shows that the best classification results are obtained with γL > γM /(u + l)2, which suggests a preference for the regularization of the classifier (supervised information) than for the regularization of the geometry of the marginal data distribution (unsupervised information). B. Urban monitoring 1) Data description: The image used in this first experiment was collected in the Urban Expansion Monitoring (UrbEx) ESA-ESRIN DUP project [19]. Results from UrbEx project were used to perform the analysis of the selected test site and for validation purposes as well 1 . The considered test site was Naples (Italy), where images from ERS2 SAR and Landsat TM sensors were acquired in 1999. An external Digital Elevation Model (DEM) and a reference land cover map provided by the Italian Institute of Statistics (ISTAT) were also available. The ERS2 SAR 35-day interferometric pairs were selected with perpendicular baselines between 20-150m in order to obtain the interferometric coherence from each complex SAR image pair. The available features were initially labeled as: L1-L7 for LandSat bands; In1-In2 for the SAR backscattering intensities (0–35 days); and Co for the coherence. Since these features come from different sensors, the first step was to perform a specific processing and conditioning of optical and SAR data, and to co-register all images. After pre-processing, features were stacked at a pixel level. For full details, see [15]. 2) Model comparison: Figure 2[top] shows the validation results for the SVM and LapSVM with both the linear and RBF kernels. Several conclusions can be obtained from this figure. First, LapSVM classifiers produce better classification results than SVM in all cases (note that SVM is a particular case of the Laplacian SVM for γ M = 0). This gain is specially noticeable when a low number of labeled samples is available (unlabeled samples help estimating the geometry of the manifold) and for the RBF kernel only. These differences are lower with the linear kernel. When the number of labeled samples is high, differences among methods are numerically very similar, but the proposed method produces better results still in this situation. On the right plot, the κ surface for the 1 For further details, visit: http://dup.esrin.esa.int/ionia/ projects/summaryp30.asp
IEEE GEOSCIENCE AND REMOTE SENSING LETTERS, VOL. XX, NO. Y, MONTH Z 2007
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Fig. 2. Results for the urban classification (top row) and the cloud screening (bottom row) problems. Overall Accuracy, OA[%], (left) and Kappa statistic, κ, (middle) over the validation set as a function of the rate of labeled training samples used to build the LapSVM and SVM with an RBF and linear kernel. Kappa statistic surface (right) over the validation set for the best RBF-LapSVM classifier as a function of the rate of both labeled and unlabeled training samples.
LapSVM highlights the importance of the labeled information in this problem. 3) Visual inspection: The best LapSVM classifier obtained before was used to classify the whole scene, which consists of a scene containing 200×200 pixels with urban and non-urban labeled samples. The classification map is shown in Fig. 3. Excellent classification accuracy is obtained, and uniform classification covers can be observed, even with so few labeled training samples.
which are part of the data acquired in the framework of the SPARC 2003 and 2004 ESA campaigns (ESA-SPARC Project, contract ESTEC-18307/04/NL/FF). These two images were acquired on July 14th of two consecutive years (2003-07-14 and 2004-07-14). For our experiments, we used as input 13 spectral bands (MERIS bands 11 and 15 were removed since they are affected by atmospheric absorptions) and 6 physicallyinspired features extracted from the 15 MERIS spectral bands in previous works [20], [21]: cloud brightness and whiteness in the visible (VIS) and near-infrared (NIR) spectral ranges, along with atmospheric oxygen and water vapour absorption features. 2) Model comparison: Figure 2[bottom] shows the validation results for the considered methods. Again, LapSVM models produce better classification results than SVM in all cases. In fact, LapSVM performs especially better than the standard SVM when a low number of labeled samples is available and unlabeled samples help estimating the geometry of the manifold. Third, the linear kernel seems to be more appropriate for the presented cloud screening application if the number of labeled samples is low, while the difference between the linear and the RBF kernels decreases as the number of labeled samples increases. It is well-known that simple (linear) decision functions are more appropriate in extremely ill-posed situations. Finally, the κ surface for the LapSVM in Figure 2[bottom right] confirms, in general terms, the importance of the labeled information in this problem.
Fig. 3. LandSat TM RGB color composite (top left), SAR intensity map (top right), true map (bottom left) and classification map (bottom right) of the urban scene, indicating ‘urban’ (gray), ‘non-urban’ (white) and ‘unknown’ (black) classes.
C. Cloud screening 1) Data description: Experiments were carried out using two MERIS Level 1b (L1b) images taken over Barrax (Spain),
3) Visual inspection: We used the best LapSVM to classify the whole scenes, which consist of MERIS L1b images of 1153 × 1153 pixels (reduced to around 500000 useful pixels after the projection in Lat/Lon coordinates). The classification map for the 2003 and 2004 images are shown in Fig. 4. In this figure, we use as ground truth a cloud classification made by an operator following the methodology described in [20], [21]. Excellent classification accuracy of 95.51% and 96.48% are obtained for the 2003 and 2004 images, respectively. The value of κ is 0.78 for both images, and reflects the
GOMEZ-CHOVA ET AL.: SEMI-SUPERVISED IMAGE CLASSIFICATION WITH LAPLACIAN SVM
misclassification of a significant number of false cloud pixels while all cloud pixels are correctly classified, suggesting that LapSVM benefits from the inclusion of unlabeled samples and obtains a reliable estimation of the marginal cloud data distribution. The committed errors correspond to high altitude locations and bright bare soil covers which are not wellrepresented in the small, randomly selected, training samples dataset. True Cloud
False Land
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Fig. 4. RGB color composite and classification maps for the analyzed MERIS multispectral images using the best LapSVM classifiers.
V. C ONCLUSIONS A semi-supervised method has been presented for the classification of urban areas and the identification of clouds. This method brings together the ideas of spectral graph theory, manifold learning, and kernel-based algorithms in a coherent and natural way to incorporate geometric structure in a kernelbased regularization framework. The solution of the LapSVM constitutes a convex optimization problem and results in a natural out-of-sample extension from the labeled and unlabeled training samples to novel examples, thus solving the problems of previously proposed methods. Results showed an increase on the classification accuracy provided by the LapSVM respect to the standard SVM both with linear or RBF kernels, suggesting that considered problems hold a complex manifold. This work has also revealed the potential of this classification method in remote sensing image classification, when reduced training sets are available. In particular, it has accurately identified urban areas in multi-sensor imagery and located cloud covers in MERIS multispectral images. In both classification problems, it becomes very difficult to obtain a representative training set for all possible situations, thus motivating the introduction of a semi-supervised method exploiting the unlabeled data. The main problem encountered is related to the computational cost, since a huge matrix consisting of labeled and unlabeled samples must be inverted. Note, however that in this method is not necessary to incorporate all unlabeled samples in the image so the computational load is easily scalable. However, we feel that smart sampling strategies to select the most informative unlabeled samples could yield improve perfomance, something to be considered in the future.
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ACKNOWLEDGMENTS This paper has been partially supported by the Spanish Ministry for Education and Science under project DATASAT ESP2005-07724-C05-03. The authors wish to thank ESA for the availability of the images acquired in the framework of the ESA-SPARC Project ESTEC-18307/04/NL/FF. R EFERENCES [1] B. Sch¨olkopf and A. Smola, Learning with Kernels – Support Vector Machines, Regularization, Optimization and Beyond. MIT Press Series, 2002. ´ [2] G. Camps-Valls, J. L. Rojo-Alvarez, and M. Mart´ınez-Ram´on, Eds., Kernel Methods in Bioengineering, Signal and Image Processing. Hershey, PA (USA): Idea Group Publishing, Jan 2007. [3] J. Shawe-Taylor and N. Cristianini, Kernel Methods for Pattern Analysis. Cambridge University Press, 2004. [4] G. Camps-Valls and L. Bruzzone, “Kernel-based methods for hyperspectral image classification,” IEEE Transactions on Geoscience and Remote Sensing, vol. 43, no. 6, pp. 1351–1362, June 2005. [5] O. Chapelle, B. Sch¨olkopf, and A. Zien, Semi-Supervised Learning, 1st ed. Cambridge, Massachusetts and London, England: MIT Press, 2006. [6] X. Zhu, “Semi-supervised learning literature survey,” Computer Sciences, University of Wisconsin-Madison, USA, Tech. Rep. 1530, 2005, online document: http://www.cs.wisc.edu/∼jerryzhu/pub/ssl survey.pdf. Last modified on September 7, 2006. [7] N. M. Dempster, A. P. Laird and D. B. Rubin, “Maximum likelihood from incomplete data via the EM algorithm,” Journal of the Royal Statistical Society, Series B, vol. 39, no. 1, pp. 1–38, Jan 1977. [8] Q. Jackson and D. Landgrebe, “An adaptive classifier design for highdimensional data analysis with a limited training data set,” IEEE Transactions on Geoscience and Remote Sensing, pp. 2664–2679, Dec. 2001. [9] V. N. Vapnik, Statistical Learning Theory. New York: John Wiley & Sons, 1998. [10] L. Bruzzone, M. Chi, and M. Marconcini, “A novel transductive SVM for the semisupervised classification of remote-sensing images,” IEEE Transactions on Geoscience and Remote Sensing, vol. 44, no. 11, pp. 3363–3373, 2006. [11] G. Camps-Valls, T. Bandos, and D. Zhou, “Semi-supervised graph-based hyperspectral image classification,” IEEE Transactions on Geoscience and Remote Sensing, 2007, accepted, in press. [12] M. Belkin and P. Niyogi, “Semi-supervised learning on Riemannian manifolds,” Machine Learning, Special Issue on Clustering, vol. 56, pp. 209–239, 2004. [13] M. Belkin, P. Niyogi, and V. Sindhwani, “Manifold regularization: A geometric framework for learning from labeled and unlabeled examples,” Journal of Machine Learning Research, vol. 7, pp. 2399–2434, 2006. [14] M. I. Jordan, Learning in Graphical Models, 1st ed. Cambridge, Massachusetts and London, England: MIT Press, 1999. [15] L. Gomez-Chova, D. Fern´andez-Prieto, J. Calpe, E. Soria, J. VilaFranc´es, and G. Camps-Valls, “Urban monitoring using multitemporal SAR and multispectral data,” Pattern Recognition Letters, Special Issue on “Pattern Recognition in Remote Sensing”, vol. 27, no. 4, pp. 234– 243, 2006. [16] M. Rast and J. Bezy, “The ESA Medium Resolution Imaging Spectrometer MERIS a review of the instrument and its mission,” International Journal of Remote Sensing, vol. 20, no. 9, pp. 1681–1702, June 1999. [17] A. N. Tikhonov, “Regularization of incorrectly posed problems,” Sov. Math. Dokl., vol. 4, pp. 1624–1627, Jan 1963. [18] T. Evgeniou, M. Pontil, and T. Poggio, “Regularization networks and support vector machines,” Advances in Computational Mathematics, vol. 13, no. 1, pp. 1–50, 2000. [19] P. Castracane, F. Iavaronc, S. Mica, E. Sottile, C. Vignola, O. Arino, M. Cataldo, D. Fernandez-Prieto, G. Guidotti, A. Masullo, and I. Pratesi, “Monitoring urban sprawl and its trends with EO data. UrbEx, a prototype national service from a WWF-ESA joint effort,” in 2nd GRSS/ISPRS Joint Workshop on Remote Sensing and Data Fusion over Urban Areas, 2003, pp. 245–248. [20] L. Gomez-Chova, G. Camps-Valls, J. Amoros-Lopez, L. Guanter, L. Alonso, J. Calpe, and J. Moreno, “New cloud detection algorithm for multispectral and hyperspectral images: Application to ENVISAT/MERIS and PROBA/CHRIS sensors,” in IEEE International Geoscience and Remote Sensing Symposium, IGARSS’2006, Denver, Colorado, USA, July 2006. [21] L. G´omez-Chova, G. Camps-Valls, J. Calpe, L. Guanter, and J. Moreno, “Cloud screening algorithm for ENVISAT/MERIS multispectral images,” IEEE Transactions on Geoscience and Remote Sensing, vol. 45, accepted for publication, in press.
