Nonlinear dimensionality reduction techniques are becoming increasingly popular in the analysis of hyperspectral images. In this work, some such methods are evaluated as a preliminary stage to the classi cation of hyperspectral images. The list of methods to be studied includes Isomap, Locally Linear Embedding, Laplacian Eigenmaps, Nonlinear Mapping, and also the linear principal component analysis technique. We study the performance of nonlinear methods with both the Euclidean distance and the Spectral angle mapper (SAM) dissimilarity measures. Analyzed methods are evaluated in terms of the classi cation accuracy and runtime. The experiments are carried out using the wellknown hyperspectral scenes.
Hyperspectral images are three-dimensional arrays with two spatial dimensions and one spectral dimension. A pixel in a hyperspectral image can be considered as a vector containing a number (typically, up to a few hundred) of components corresponding to di erent wavelengths.
Being compared to color or gray-scale images, hyperspectral images suggest extended opportunities, for example, to detect materials in a depicted scene or substantially improve the accuracy of classi cation. However, the use of hyperspectral images is accompanied by increased costs for storage, transmission, and processing of such images. For this reason, an important task is to eliminate the redundancy of such images, while maintaining the quality of the solutions to applied problems.
The most widely used solution to the above task consists in the use of dimensionality reduction techniques. As such a technique, in most cases, the principal component analysis is used. Nevertheless, nonlinear dimensionality reduction techniques are becoming increasingly popular in the last years.
In this work, some such methods are evaluated as a preliminary stage to the
classi cation of hyperspectral images. The list of methods to be studied includes
Isomap [
We study the performance of nonlinear methods with both the Euclidean distance and the Spectral angle mapper (SAM) dissimilarity measure as they have been used in hyperspectral image analysis most often. The reason for choosing the above dimensionality reduction methods was the frequency of application of these methods in the analysis of hyperspectral images and the possibility of embedding the SAM measure.
Analyzed methods are evaluated in terms of the classi cation accuracy and runtime. The experiments are carried out on the well-known hyperspectral scenes.
The paper has the following structure. Section 2 is devoted to the brief description of methods used in the paper. Section 3 describes the results of experiments. The paper ends up with the conclusion. 2
In this study, we use the following dimensionality reduction techniques:
{ Principal Component Analysis (PCA) technique [
The main idea of this method consists in the use of geodesic distances instead
of Euclidean distances in classical metric multidimensional scaling (MDS).
Here we use the Landmark Isomap method [
In all the above nonlinear techniques (except PCA), it is assumed that the
Euclidean distance is used as a dissimilarity measure. As we said in the
Introduction, in this paper, we also embed the SAM measure [
In particular, for the Nonlinear Mapping technique, we replace the
calculation of Euclidean distances in hyperspectral space with the calculation of SAM
measures. According to [
Datasets For the reported study, we used several well-known hyperspectral
image scenes [
Both hyperspectral image scenes were acquired using the AVIRIS sensor. The rst scene contains 512 217 pixels, and 224 spectral bands. In our experiments we used the image containing 204 spectral bands, in which some spectral bands were discarded due to a high noise and water absorption. As the Salinas scene contains more than 100 thousand pixels, and it was necessary to perform a lot of runs of nonlinear dimensionality reduction techniques, for our experiments we used regularly sampled test image, which was masked with the provided groundtruth image. The classi ed pixels of the groundtruth image are divided into 16 classes.
Kennedy space center scene contains 512 614 pixels. The version containing 176 spectral bands was used in the experiments. The groundtruth image contains information only on a small amount of pixels, so we applied groundtruth mask, and did not use any sampling. The classi ed pixels of the groundtruth image are divided into 13 classes.
Experimental setup. To perform the experiments we used PCA
implementation provided with Matlab, C++ implementation of Nonlinear Mapping method,
and for LLE, Laplacian Eigenmaps and Isomap, we used Matlab Toolbox for
Dimensionality Reduction [
A laptop based on Intel Core i7-6500U CPU 2.5 GHz, 12 Gb RAM was used to perform experimental studies.
Evaluation. The k-Nearest Neighbor (k-NN) classi er and the Support Vector Machine (SVM) were used in this study. To measure the quality of classi cation we used the overall classi cation accuracy, de ned as the proportion of the correctly classi ed pixels of the test set.
The whole set of ground truth samples was divided into a training (60 percents) and a test (40 percents) subsets in our experiments. The dimensionality of the reduced space ranged from 3 to 30.
Experimental results. The results of the experimental study for the NN classi er are shown in Figure 2. As it can be seen from the gure, the use of the SAM measure was preferable for the Salinas and Kennedy space center hyperspectral scenes, and for almost all the considered nonlinear techniques, as SAM provided a better quality of classi cation compared to Euclidean distance. This observation is also con rmed for two other scenes involved in the experiments.
In all the considered cases, for the NN classi er, the best results were obtained using the Nonlinear Mapping technique. The linear PCA technique provided similar or slightly worse results than the Nonlinear Mapping combined with Euclidean distances. PCA outperformed LLE, LE, and Landmark Isomap methods on Salinas and two other test hyperspectral scenes, except the Kennedy space center scene.
In KSC scene, the linear PCA technique performed much worse than
nonlinear techniques for the dimensionality of the reduced space up to 25. In this
scene, we can also observe that the Nonlinear Mapping technique in
combination with Euclidean distances loses advantages over other nonlinear techniques
based on SAM measures for the dimensionality of the reduced space up to 20.
Thus, on the one hand, we see a signi cant advantage of using the SAM measure
over the Euclidean distance for this scene. On the other hand, we can explain
obtained results by substantially nonlinear properties of this dataset. It noted
earlier that the discrimination of land cover for the KSC scene is di cult due to
the similarity of spectral signatures for certain vegetation types [
The results of the experimental study for the SVM classi er are shown in Figure 3. The experimental results showed that in many considered cases (some results are not shown in the gures) the PCA was a preferable choice. The Nonlinear Mapping performed a bit worse for the SVM classi er. But again, the performance of the PCA technique was drastically reduced on the Kennedy space center scene for the dimensionality of the reduced space up to 25. This indicates the importance of the careful selection of the output dimensionality.
It is worth noting that three graph-based dimensionality reduction techniques, namely Isomap, LLE, and Laplacian Eigenmaps have the mataparameter, which de nes the number of neighbors. In the above experiments, we used the default value equal to 12. We found that these techniques are very sensitive to the choice of this parameter. This was especially evident for the LLE method. In some cases, the classi cation accuracy could be substantially (by some percent) improved over the reported above values by the good choice of the considered metaparameter. But for other cases, the same value could provide worse results. In any case, we were not able to outperform the Nonlinear Mapping by varying this parameter in the reasonable range from 10 to 100 with the step equal to 10. The example dependency of the classi cation accuracy on the metaparameter k is shown in Figure 4.
The same Figure 4 shows the runtime of the considered techniques. The
timing for the PCA technique is not shown as it was about 0.1 sec. that is
negligible compared to the considered nonlinear methods. The run time of Landmark
Isomap, LLE and Laplacian Eigenmaps is less than the run time of the
Nonlinear Mapping technique for the minimum considered value of k, but it raises
fast with the growth of k. So the Nonlinear Mapping becomes faster for k > 30.
Moreover, there are approaches, which allow to speed-up this technique [
While in many cases the combination of the PCA technique with SVM classier provides nice results, for complex hyperspectral scenes containing substantial nonlinear e ects the traditional PCA technique could be a bad choice. In such cases, it is necessary to carefully choose the output dimensionality, and consider the possibility of using the nonlinear dimensionality reduction techniques.
The main drawback of the nonlinear methods is their high computational complexity, which is expressed by their long run time, which exceeds the runtime of the PCA technique by orders of magnitude.
Acknowledgments The reported study was funded by RFBR according to the research project no. 18-07-01312-a.