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Preprint of IEEE Transactions of Instrumentation and Measurements 59,7,1834-1840, July 2010. DOI: 10.1109/TIM.2009.2030918 “© © 2009 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works.”

Unmixing Low Ratio Endmembers in Hyperspectral Images through Gaussian Synapse ANNs Fernando López Peña, Juan Luís Crespo, and Richard J. Duro * *Grupo Integrado de Ingeniería, Universidade da Coruña Mendizabal s.n., 15403 Ferrol, Spain Email: {jcm,richard,flop}@udc.es Abstract – The application of Gaussian Synapse based Artificial Neural Networks to the detection and unmixing of endmembers within hyperspectral images is the objective of this paper, especially in cases where some of them are mixed in a low ratio as is usually the case in target detection problems. These networks and the training algorithm developed are very efficient in the determination of the abundances of the different endmembers present in the data using a very small training set that can be obtained without any knowledge on the proportions of endmembers present. The validation and test of these networks is carried out through their application to a benchmark set of artificially generated hyperspectral images containing five endmembers with spatially diverse abundances. As a second test, we applied the strategy to a real image and checked its behavior in regions where there were transitions between zones that were labeled differently and compared them to a hypothetical evolution of the spectrum from the endmember in one of the regions to the endmember in the other. A very good correspondence was found.

Keywords – Hyperspectral images, neural networks, Gaussian synapses.

I. INTRODUCTION According to Manolakis et al [1], most algorithms used in hyperspectral applications can be grouped into four categories: target / anomaly detection, change detection, classification, and spectral unmixing. Here we are going to concentrate on the first and last of these categories as particular instances of classification problems, and our objective is going to be to formulate techniques for the case where subpixel information is involved. Obviously, the performance of any target detection approach based on hyperspectral imaging is dependent on the spatial resolution of the imaging sensor. In an ideal case, as indicated by [2], the spatial resolution of a hyperspectral target detection system should be chosen just high enough to ensure that pure

target pixels are present in the image data. In reality, however, in most cases, the spatial resolution of airborne or space based hyperspectral imagers is poor compared to other remote sensing imaging systems. Consequently each pixel of the images thus obtained encompasses areas of tens or hundreds of square meters and often contains information from several types of elements or materials found in this area, leading to a measured spectral signature that results from the mixture of the signatures of the individual elements present in the pixel. The objective of any analysis or classification method for these types of images is to profit from the high spectral resolution in order to decompose the measured spectra into their constituents; that is, to be able to identify what spectrally distinct materials (endmembers) and in what proportions (abundances) are present in each pixel. This is the so-called unmixing process. Two different objectives may direct the process of unmixing spectra: image segmentation or target detection. In the first, and most common, case, the process seeks the differentiation of the main components within a pixel [3], often ignoring those with a very low abundance. In this line, many types of statistical approaches have been proposed [4][5][6]. In the second case, a target must be detected, even if it covers only a very small fraction of the area of the pixel [1], this is what is called low abundance or low ratio endmember detection. In fact, target detection when working with subpixel information can be taken as an instance of the spectral unmixing problem, but with some special characteristics and constraints as indicated by [7]. On one hand, the sparseness of the target class implies a lack of calibration data to train a statistical classifier for target and background classification. On the other hand, the criteria generally used for the design of optimal detectors are not valid because, as the target class appears as a very low proportion of the image, and of the pixels when in subpixel mode, the global error can usually be minimized by accepting the non presence of this target, which defeats the purpose. Therefore, to solve this problem it is important to obtain algorithms that are able to perform a classification with very little training data and that in their operation do not lead to the trivial (and erroneous) solution of the non presence of the low ratio endmember. The unmixing process is commonly carried out by performing endmember extraction and abundance estimation independently of each other and introducing considerable participation of human operators. The approach followed in the work leading to this paper is based on an Artificial Neural Network architecture and a training algorithm that implement an automatic procedure for extracting endmembers and estimating

abundances simultaneously. This automatic procedure concentrates on what is relevant and ignores what is not straight from the training set. It was designed and tested on images where the pixels contained mixtures of contributions from different endmembers. Due to their inherent characteristics, ANNs appear as a very suitable and effective alternative to deal with spectral image analysis difficulties [2][8]. The current application is focused on supervised classification. Supervised classification may be defined as the process of determining the distribution of some given classes in an image by using the spectral information derived from a reduced training data set provided by some expert. The main difference between supervised and unsupervised classification methodologies is that supervised classification requires training data as input. In the case of ANNs trained using supervised training algorithms, the main drawback when performing the analysis and classification of hyperspectral remote sensing data is the difficulty in obtaining adequately labelled sample data that can be trusted to the point of using it for quantitative evaluation. The scarcity of ground truth data has been recognized and specific training strategies have been devised to cope with this handicap [9][10]. In fact, the ideal objective would be to obtain a training procedure that produces good classifiers from very small training sets. Usually, in the case of detecting a target, detection itself is the most important objective and, therefore, larger errors in its quantitative determination are allowed. This problem becomes even more pronounced when pixels correspond to combinations of materials, that is, within the spectral unmixing process [2]. If the mixture of different types of materials in a pixel is of rather large pieces, then the mixture of the corresponding spectra can be represented as a linear model. However, if the mixture is of microscopic or small particles in an intimate way, then the spectrum mixture models become more complex and non-linear. In this work we assume a linear mixture model, in which several basic materials (endmembers) are combined according to some abundance coefficients at each image pixel. Taking their spatial distribution, the abundance coefficients may be visualized as abundance images for each one of the materials, which provide a description of the spatial distribution of the material. As stated before, the procedure of unmixing a set of endmember spectra in a given pixel is just the computation of the corresponding abundance coefficients for the pixel. If the endmembers are given, the unmixing procedure is equivalent to the parallel detection of the spectral features represented by the endmembers. This is the approach that was followed here. A new neural network structure based on Gaussian Synapses and a

training algorithm that, for this type of applications, can obtain very good classifiers from small training sets in a very small number of training epochs was used on these multidimensional data sets. This strategy was applied in a previous paper to the classification of pixels, assuming a single material per pixel, and their assignment to their different categories within hyperspectral images without any atmospheric correction, and it has proven to be very efficient in this type of classification [11]. In the present work we extend it to cases where mixtures of materials are a common occurrence aiming to ascertain the quantitative proportion of each endmember present in each pixel. The emphasis is on a pixel by pixel classification without using any neighborhood information in order to test the appropriateness of using Gaussian synapse backpropagation for these tasks. Previous research has proven the capability of this type of networks to handle hyperspectral image segmentation and pointed towards its specific ability to handle the unmixing of endmembers even in cases where they are present in a small proportions or low ratios [12]. It is the objective of this work to test this hypothesis using appropriately constructed benchmark hyperspectral mixture images. As it is better explained later on, these images have been generated by linearly mixing on each pixel the spectra of five different endmembers and then adding some noise, thus simulating highly mixed hyperspectral images having several areas where the abundance of some of the endmembers are in a low ratio.

II. THE NETWORK AND TRAINING ALGORITHM As stated earlier, studies from different researchers prove that ANNs are effective in analyzing and classifying the multidimensional data cubes defining hyperspectral images. These cubes correspond to the two spatial (as in any other type of image) and one spectral (a spectrum per pixel) dimensions present in all of these images. In these ANNs the size of the input layer usually corresponds to the number of features on which the classification is based (i.e. the number of spectral bands used to perform the analysis) while the size of the output layer is often equal to the number of endmembers or output classes. Another possibility is to use one independent ANN with a single output for each endmember. This is the approach followed in the present study, instead of a classification problem, it resembles more a target detection problem and the output provides the abundance of the endmember. In addition, the size of the input layer equals the total number of spectral bands without performing any preprocessing to disregard bands contributing less to the

process. It is quite clear that introducing at this stage some preprocessing to reduce the number of inputs might improve the classification results. However, this is not the objective of our work here and we have decided not to consider this in order to test the real capabilities of the networks on the raw data and avoid any bias these preprocessing stages may introduce. Segmenting multi and hyperspectral images consists mainly in a classification process of the pixels in the image in which the correlation of different value intervals for each of the bands participating in the pixel´s spectrum is the discriminant between classes. Consequently, from an Artificial Neural Network (ANN) perspective, we have considered that the best way to allow the network to perform this discrimination is to provide it with mechanisms that allow it to act directly on the values circulating throughout the network, and in particular, through its synapses. This has been achieved by introducing trainable Gaussian functions in each one of the synapses. Following this approach permits each synapse to learn to select the signal intervals that are relevant to the classification being performed in each node and to ignore the rest of the intervals without having to introduce a lot of nodes and connections in the network just to compensate for irrelevant information. Gaussian and other radial functions are in use within the structures of ANNs [14]; but so far they have been used just to serve as basis or activation functions thus, basically, constraining their application to a linear regression scheme through the optimization of the values of the weights. In our scheme the Gaussian functions are in the synapses while leaving the activation functions as sigmoids. As a consequence, the training procedure changes from a simple regression scheme into a more powerful solution by shaping the Gaussian functions automatically and independently for each of the dimensions through the adaptation of their parameters during the training process. The results are not RBF functions, but a sort of skewed RBFs where the width of the Gaussian is different for each dimension. A complete description of this type of networks and their training algorithm can be found in [15]; thus, only a brief description of the training algorithm is included in this section. A comparison on the use of this type of ANNs and other radial based function networks in hyperspectral image classification can be found in [12]. In practice, the architecture of this type of networks is very similar to the classical Multiple Layer Perceptron. In fact, the activation functions of the nodes are simple sigmoids. The only difference is that each synaptic connection

implements a Gaussian function determined by three parameters: its center C, its amplitude A and its variance B:

g (x ) = A * e

B ( x −C )2

(1)

Figure 1 displays a schematic diagram of this network. In order to train this structure, an extension of the backpropagation algorithm, called Gaussian Synapse Backpropagation (GSBP) has been developed [15]. This algorithm works very much like general Backpropagation, by means of the calculation of the gradients of the error with respect to each one of the three parameters of the Gaussian functions determining each weight and modifying them accordingly.

Thus, the outputs of the neurons in the hidden and output layers can be expressed as:

O

k

h

j

⎛ = F ⎜⎜ ∑ h j ⎝ j

A eB jk

(h −C ) ⎞⎟ = F ⎛ 2

jk

j

jk

⎟ ⎠

⎞⎟ ⎜O ⎝ Net k ⎠

⎛ ( − )2 ⎞ ⎞ = F ⎜ ∑ I i Aij eBij I i C ij ⎟ = F ⎛⎜ h ⎟ ⎝ Net j ⎠ ⎝ i ⎠

(2)

Being F the activation function of a neuron (usually a sigmoid), hj the output of hidden neuron j, Ii input i and the sum is carried out over all the inputs to the neuron. We denote as ONet and hNet the values that input the activation function of a node. Therefore, if Tk denotes the desired target for output k of a given training sample and Ok the output actually produced by the network in output neuron k when presented with the inputs associated with this training sample, their difference will constitute the error for neuron k for this sample and the sum of these errors over all of the output neurons gives the total error Etot for a given input sample. Consequently, the gradients of the total error Etot with respect to Ajk, Bjk, and Cjk are obtained for the output neurons as: 2 ∂ E tot = h j (O k − T k ) F´(O Netk ) e B jk (h j − C jk ) ∂ A jk

(3)

2 ∂ E tot = h j (O k − T k ) F´(O Netk ) A jk (h j − C jk ) 2 e B jk (h j − C jk ) ∂ B jk

(4)

2 ∂ E tot = −2 h j A jk B jk (O k − T k ) F´(O Netk )(h j − C jk ) e B jk (h j − C jk ) ∂ C jk

(5)

In the case of the neurons in the hidden layer, if we denote the variation of the error with respect to the net value of the neuron as: Θj =

∂ E tot ∂ E tot ∂ h j ∂ = = E tot F ´(h Netj ) ∂ h Netj ∂ h j ∂ h Netj ∂hj

(6)

The gradient of the error with respect to Aij, Bij and Cij may be obtained as: 2 ∂ E tot = I i Θ j e Bij ( I i − C ij ) ∂ Aij 2 ∂ E tot = Θ j I i Aij ( I i − C ij ) 2 e Bij ( I i − C ij ) ∂ Bij

(7) (8)

2 ∂ E tot = −2 Θ j I i Aij Bij ( I i − C ij ) e Bij ( I i − C ij ) ∂ C ij

(9)

Once the gradients have been obtained, the updating of the different parameters every epoch of the training process is carried out in the same way as traditional backpropagation, that is, by modifying the value of the parameter by displacing it in the direction of the largest gradient of the error. As shown above, the algorithm is relatively straightforward and simple to implement. In the next section we will discuss its application to the problem domain we are interested in, that is, endmember unmixing and low ratio endmember detection.

III.

EXPERIMENTS

In this section, we test the presented approach on one synthetic and one real data sets. It is very difficult to obtain benchmarkable results from real hyperspectral images. The main difficulty is related with the labelling of the pixels. In general, most accessible images obtained from airplanes or satellites provide rather sparsely labelled ground truths where most pixels are labelled as a single category. This is a huge problem, as most pixels usually correspond to areas of tens or hundreds of square meters on the ground that contain all kinds of different elements (plants and ground, houses surrounded by grass, etc…) and providing a single value to describe them (grass, house, or whatever) is very misleading and leads to difficulties in the training process. Therefore, it is necessary to provide an algorithm able to discern which are the relevant features characterizing the spectrum of each pixel and which are the correspondences of these features with the different endmembers present in the area the pixel describes. This is the problem we have applied the Gaussian Synapse based Networks to and in order to initially validate them we have made use of a set of synthetically generated images where the proportions of the different elements present in each pixel were known and thus the results from applying this technique could be effectively compared to reality. Afterwards, we have applied the same procedure to spectrally unmix pixels in real images where

the presence of different groundcovers could be inferred and tried to determine the smallest proportions of an element that could be detected using this approach.

The first experiment, based on synthetic images, is designed in order to perform the analysis under controlled conditions while the second one provides an instance to check how the technique performs in real cases. Thus, we have made use of a set of synthetic 64 by 64 pixel and 224 band images generated by Graña and col. [16] containing mixtures of five different endmembers. As indicated above, it is difficult to find real images with the ground truth appropriately labelled for mixtures of materials, and there were no images available to us where this was done for more than two endmembers. In addition, these synthetic images cover a wide variety of mixtures with low abundances of various classes, providing an interesting test set to check how the system performs in classifying these low ratio endmembers. The synthetic hyperspectral images were generated as follows: First, a set of reflectance spectra were selected from the USGS repository (United States Geological Survey. http://www.usgs.gov/) as the ground truth endmembers. Then, a set of abundance images was generated as the simulation of a random Gaussian field with Matern covariance function, which has been recognized as a good model for describing and quantifying soil spatial variability [17]. The images thus generated are renormalized to ensure that they sum up to one at each pixel. Consequently, they can also be interpreted as posterior probabilities. The hyperspectral image pixels are synthesized as a linear combination of the endmembers using the abundance image pixels generated as the combination coefficients. There was a baseline noise level of up to 3% in the quantitative value of the abundances. This experiment assures the most controlled conditions when testing the capability of Gaussian synapse networks not only to recognize patterns, but also to discriminate and identify endmember mixtures. A set of only 11 spectra was created for training, as shown in Figure 2. This set corresponds to 5 spectra of pure endmembers, another spectrum constructed as the arithmetic mean of these five and the last five spectra correspond to the average of every one of the pure spectra with the average spectrum. It is quite significant that only the pure endmembers and just 6 linear combinations of these are used for training as this is one of

the requirements of these types of systems: they must operate with very little training information and what is usually available does not generally provide labelled samples of mixtures, just pure endmembers. Five GSMLP networks were trained to identify the quantitative proportion of the endmembers present in the training set and applied to the whole image, which includes many mixtures of the endmembers in ratios that are very different from those in the training set. The training of the networks was very fast and convergent. A MSE of 0.05 was achieved in just 3 steps of training for each detector; this is an acceptable value for most applications. To test the resulting trained networks, we made use of the abundance images provided with the synthetic source images. An abundance image is a representation of the presence of a given endmember throughout the image where the proportion of the endmember is given as a gray level value. Figure 3 displays the real abundance images for two different hyperspectral cubes with different spatial distributions of the presence of five endmemebers. The figure also shows the abundance images produced by the detectors trained for these categories as mentioned above. In all cases the representations of both the source and detected images look very similar, the distributions of components are preserved and the qualitative performance of the classifier appears to be correct. Even though we could say that the images are evidence of the good behaviour of the nets, in order to really display the appropriateness of the results obtained, which correspond to the proportion of the endmember in the pixels throughout the image, two approaches were followed. Firstly, we present in Figure 4 a comparison of the histogram of the images corresponding to one of the categories. This top histogram represents the real number of pixels having a given abundance of the corresponding endmember. The bottom one provides the same representation but for the abundances provided by the network. The similarity between the histogram corresponding to the original abundance image and that of the one reconstructed by the Gaussian Synapse Network provides a good indication of the power of the GSMLP for extracting spectral proportions from a pixel that is, its ability to unmix endmembers in general. Secondly, we focused on the characteristics of this procedure when operating with low ratio endmembers, that is, with endmembers whose presence in terms of pixel area is very small (below 15%). In order to do this, the original hyperspectral image underwent a filtering process whereby only the pixels that contained a given endmember within an abundance interval (for instance from 5 to 6 % of the total pixel

area) were left for processing and ran the detectors for that endmember over the filtered hyperspectral images to ascertain its ability to detect these endmembers when present in a low ratio. For endmembers present in a high ratio a good performance of the classifier has already been reported [11]. Figure 5 displays the results for this process up to a 15% coverage by the target endmember.. These results show that a GSMLP based segmentation system is able to consistently detect the endmembers when their proportions are as small as 4% of the pixel surface which is quite a good result taking into account that the baseline noise on this image was up to 3%. In fact, the system achieves almost an 80% detection rate for endmember coverage of 4%, reaching 100% detection below 8% coverage. These results compare favourably, for example, to those of [7], who for 3% coverage by the target endmembers only detects 1% of the cases and needs a target as large as 15% for 100% detection. To complement the results from the analysis of controlled synthetic images, and verify how well Gaussian Synapse based networks are adapted to the task of discriminating the presence of those minority elements in the pixels we have tested the network on a very well known real hyperspectral image: The Indian Pines Image [16] that has been previously used in many different studies. This image was obtained from the AVIRIS imaging spectrometer and has a ground resolution of 17x17 meters per pixel. It is mainly made up of two thirds agricultural land and one third forest. The image was taken during the early growth stages of the crops, therefore in the cultivated land the ground coverage of plants is low while most of the surface area corresponding to a pixel is bare soil. From this image we have extracted groups of pixels representing transitions between areas with different labels in the hope that the intermediate pixels contain mixtures of elements from both sides of the transition. As no ground truth was available on the proportions of elements present on these pixels we also generated artificial sets of pixels starting from the same endmembers and simulating the same type of brusque transition so as to provide a validation set for the results obtained from the real image. For that purpose, an arbitrary transition area between two zones was selected on the Indian Pines image, as shown Figure 6. The networks were trained using 5 pixels inside each labeled area and 5 linear combinations of the spectra (80%-20%, 60%-40%, etc…) for each pair of endmembers, for a total of 25 training samples, and then applied to the detection of proportions in the transitions Then, we generated a synthetic set of pixels through the transitions with known proportions and compared them to the real ones obtained from the image in terms of the proportions produced by the

Gaussian Synapse based networks. The evolution of the proportion of endmembers for these two cases along the pixel line can be observed in Figure 7. These results are quite encouraging, as the detection on the real image closely follows that of the synthetic one with well known proportions. The results over the real image, coupled with those obtained using the synthetic images, confirm the adequacy of Gaussian Synapse based networks for the determination of quantitative endmember mixture results after training with very limited numbers of labelled samples for very short training periods.

IV.

CONCLUSIONS

The capabilities of Gaussian Synapse Based Artificial Neural Networks for unmixing hyperspectral images data cubes have been shown in cases where some of the known endmembers are present with abundance fractions lower than 15% in a pixel. These networks, trained with the GSBP algorithm, are able to obtain good abundance estimations in the spectral unmixing problem with the additional advantage of requiring very small training sets, which, in the test presented here, are just made up of the pure endmember spectra and 6 linear combinations of these. Additionally, this type of approach is a good option in the case of trying to detect endmembers that correspond to very small proportions of the pixel area. In the experiments carried out, endmembers which covered as little as 4% of the area of the pixel were mostly detected and those covering 8% were always detected. In fact in this work we have thoroughly tested this approach on a set of benchmark hyperspectral images and confirmed the results on the well known Indian Pines image. The results indicate that the combination of this strategy with an unsupervised method for the extraction of the endmembers would be a very promising path to follow in order to produce a system that can automatically obtain a segmentation of an unknown hyperspectral cube with unknown endmembers.

ACKNOWLEDGEMENTS This work was funded by the MCYT of Spain through project VEM2003-20088-C04-01 and Xunta de Galicia through project 07MDS035166PR. REFERENCES [1] D. Manolakis, C. Siracusa and G. Shaw, “Hyperspectral subpixel target detection using the linear mixing model,” IEEE Trans. Geoscience and Remote Sensing, vol. 39, pp. 1392-1409, 2001.

[2] Skauli, T.; Kåsen, I. The effect of spatial resolution on hyperspectral target detection performance, Electro-Optical and Infrared Systems: Technology and Applications II. Edited by Driggers, Ronald G.; Huckridge, David A. Proceedings of the SPIE, Volume 5987, pp. 264-270 (2005). [3] N. Keshava and J. F. Mustard, “Spectral unmixing,” IEEE Signal Processing Magazine, pp. 44-57, Jan. 2002. [4] B. Thai and G. Healey, “Invariant subpixel material detection in hyperspectral imagery,” IEEE Trans. Geoscience and Remote Sensing, vol. 40, pp. 599-608, Mar. 2002 [5] Shah, C.A.; Varshney, P.K.A higher order statistical approach to spectral unmixing of remote sensing imagery Geoscience and Remote Sensing Symposium, 2004. IGARSS apos;04. Proceedings. 2004 IEEE International Volume 2, Issue , 20-24 Sept. 2004 Page(s): 1065 - 1068 [6] S. Tompkins, J.F. Mustard, C.M. Pieters, and D.W. Forsyth, “Optimization of Endmembers for Spectral Mixture Analysis,” Remote Sens. Environ. 59 (3), 1997, pp. 472–489. [7] Ge, Y., N. H. Younan, R. L. King. 2005. Adaptive Subspace Target Detection in Hyperspectral Imagery. International Symposium on Remote Sensing of Environment. [8] J. Ghosh. “Adaptive and neural methods for image segmentation”. In Al Bovik, editor, Handbook of Image and Video Processing, chapter 4.10, pages 401--414. Academic Press, 2000. [9] S. Tadjudin, and D. Landgrebe. “Covariance Estimation with Limited Training Samples”, IEEE Trans. Geos. Rem. Sensing, 37(4) ,(1999) 2113- 2118, [10] S. Tadjudin, and D. Landgrebe. “Robust parameter estimation for mixture model”, IEEE Trans. Geos. Rem. Sensing, 38(1): ,(2000) 439 [11] J. L. Crespo, R. J. Duro, and F. López Peña, “Gaussian Synapse ANNs in Multi and Hyperspectral Image Data Analysis”. IEEE Transactions on Instrumentation and Measurement, Vol. 52, No. 3. June 2003. 724-732. [12] A. Prieto, F. Bellas, R.J. Duro, and F. Lopez-Peña. A Comparison of Gaussian Based ANNs for the Classification of Multidimensional Hyperspectral Signals. In J. Cabestany, A. Prieto, and D.F. Sandoval (Eds.): IWANN 2005, LNCS 3512, (2005). pp. 829 – 836

[13] E. Merényi, T. B. Minor, J. V. Taranik, and W. H. Farrand, “Quantitative Comparison of Neural Network and Conventional Classifiers for Hyperspectral Imagery”. Summaries of the Sixth Annual JPL Airborne Earth Science Workshop, Pasadena, CA, March 4-8, 1996, Vol. 1: AVIRIS Workshop, Ed. R.O. GreeN (1996). [14] N. B. Karayiannis. “Reformulated radial basis neural networks trained by gradient descent,” IEEE Transactions on Neural Networks, vol. 10 no. 3, (1999). 657 –671. [15] R. J. Duro, J. L. Crespo, and J. Santos. “Training Higher Order Gaussian Synapses”. LNCS, Vol. 1606 Springer-Verlag, Berlín (1999) 537-545. [16] M. Graña, B. Raducanu, P. Sussner, and G. Ritter. “On Endmember Detection in Hyperspectral Images with Morphological Associative Memories”. Presented at IBERAMIA 2002, Sevilla, Spain. (2002) 526-535. [17] B. Minasny and A. B. McBratney. “The Matérn function as a general model for soil variograms”. Geoderma. Volume 128, Issues 3-4, (2005), 192-207 [18] D. Landgrebe. “Indian Pines AVIRIS Hyperspectral Reflectance Data: 92av3c”, 1992. Available at http://makalu.jpl.nasa.gov/

FIGURE CAPTIONS Fig. 1. Schematic representation of the type of network used in this work. The spectral components of each pixel are input to the network and the outputs define the presence of a given endmember within this pixel. Fig. 2. The 11 element set used for training the detectors: The five spectra of the synthetic endmembers (left), the mean spectrum (top right), five combination spectra (right).

Fig. 3. Comparison of the original abundance maps for two synthetic hyperspectral images with a distribution of five endmembers mixed in different pixels and those obtained by the GSMLP after analyzing the two hyperspectral cubes. Fig.4. Top: Original Abundance Histogram for C2. Bottom: GSMLP Generated Abundance Histogram. Fig. 5 Percentage of the pixels of the image with the presence of low ratio endmembers for which these are appropriately detected as a function of the coverage of the pixel by the endmember. Fig. 6. The two transition zones on the Indian Pines image where the experiments are performed. Fig. 7. Behavior of the detectors of two given endmembers through a brusque transition for a series of pixels obtained from the Indian Pines image (top) and for the synthetically generated pixels (bottom).

FIGURES

hNet1

Y

h1

X

ONet1 I1

Spectrum

I2

hNet2

h2

I3

ONet2

hNet3

O1

h3

Fig. 1. Schematic representation of the type of network used in this work. The spectral components of each pixel are input to the network and the outputs define the presence of a given endmember within this pixel.

O2

Fig. 2. The 11 element set used for training the detectors: The five spectra of the synthetic endmembers (left), the mean spectrum (top right), five combination spectra (right).

SYNTHETIC SAMPLE IMAGE 1: Original abundance maps for the 5 components:

Abundance maps obtained by the GSMLP system for the five components:

SYNTHETIC SAMPLE IMAGE 2: Original abundance maps for the 5 components:

Abundance maps obtained by the GSMLP system for the five components:

Fig. 3. Comparison of the original abundance maps for two synthetic hyperspectral images with a distribution of five endmembers mixed in different pixels and those obtained by the GSMLP after analyzing the two hyperspectral images.

Normalized Number of Pixels

Fig.4. Top: Original Abundance Histogram for C2. Bottom: GSMLP Generated Abundance Histogram. We can see that, although they are not identical, the number of levels is very similar in both cases. This provides an indication of the generalization ability in terms of abundance level of a net trained with a very limited number of combinations.

0%

50% Abundance

100%

Fig.4. Top: Original Abundance Histogram for C2. Bottom: GNet Generated Abundance Histogram.

Fig. 5 Percentage of the pixels of the image with the presence of low ratio endmembers for which these are appropriately detected as a function of the coverage of the pixel by the endmember.

Fig. 6. The two transition zones on the Indian Pines image where the experiments are performed.

Fig. 7. Behavior of the detectors of two given endmembers through a brusque transition for a series of pixels obtained from the Indian Pines image (top) and for the synthetically generated pixels (botton).