**4. Experiments**

In this section, the proposed PSP-OMPBS is tested on two publicly-available real hyperspectral images. To evaluate the performance of PSP-OMPBS, we conduct three different studies, including BS accuracy analysis, land cover classification, and computational efficiency.

#### *4.1. Hyperspectral Dataset and Experimental Setting*

The first dataset used in the experiments was a real hyperspectral image, which was collected by the reflective optics system imaging spectrometer (ROSIS) optical sensor over an urban area of the University of Pavia. The Pavia image measures 610 × 340, with very high spatial resolution of about 1.3 m per ground pixel. The original data contains 115 spectral bands, with a spectral range from 0.43 to 0.86 μm. After removing 12 noisy bands, the remaining 103 bands were used for the experiments. Figure 4a–c respectively show the image scene of the 50th band, the geometric locations of all target classes, and the corresponding spectral signatures. According to the ground truth in Figure 5b, there are 9 classes in this image scene, consisting of several urban targets, such as vegetation, soil, and roads.

The second dataset used in our experiments is another real hyperspectral image data, Purdue's Indiana Indian Pine test site, which was collected by the airborne visible-infrared imaging spectrometer (AVIRIS) system. It has been extensively studied in the literature and provides a good candidate for those who are interested in algorithm design and analysis. It has a 20 m spatial resolution and a 10 nm spectral resolution in the range of 0.4–2.5 μm with size 145 × 145 pixel vectors, taken from an area of mixed agriculture and forestry in Northwestern Indiana, U.S. It was recorded in June 1992 with 220 bands, among which bands 104–108 and 150–162 were removed, whereas the remaining 202 bands were retained. Figure 5a,b shows the image of band 20 and the ground truth map, respectively. The ground truth map contains 16 crops classes and one background class.

**Figure 4.** ROSIS image scene: University of Pavia. (**a**) Band 80. (**b**) Ground truth map for nine classes. (**c**) Spectral signatures of nine classes.

**Figure 5.** AVIRIS image scene: Purdue Indiana Pine test site. (**a**) Band 20. (**b**) Ground truth map. (**c**) Spectral signatures of 17 classes.

According to the introduction, the development of PSP-BS is not for achieving outstanding BS for a particular data analysis, such as image classification or spectral unmixing, but for the real-time BS monitoring during data transmission. In this case, the experiment is conducted based on self-comparison, instead of comparing with state-of-the-art methods. To evaluate the BS accuracy and efficiency of PSP-OMPBS, we still can adopt OMPBS as the compared method (baseline).

In the following progressive experiments, we set *n* = 100–207,400 for the Pavia data, and *n* = 100–21,025 for the Purdue data. For the *p* value, we set 16 for the Pavia data and 34 for the Purdue data, according to the virtual dimensionality (VD) algorithm [39] with false alarm 0.01. Three PTS methods mentioned in Section 3.4 were adopted for PSP-OMPBS methods. We empirically set *k* as 200 and 100 and *b* as 50 and 25 for the Pavia and Purdue data, respectively. Table 2 lists the parameters used in the experiment.

**Table 2.** The parameters used in the experiments.


#### *4.2. BS Accuracy Analysis*

The accuracy of the instant BS result during data transmission is an index to evaluate the effectiveness of PSP-OMPBS. Theoretically, the BS would gradually converge to the final BS result (i.e., the result of OMPBS performed on the complete image cube) over time. To observe this phenomenon, we use the results of OMPBS implemented on both images, as the ground truths. For instance, Pavia's ground truth is ΩPavia OMPBS(*N*) = {91, 62, 16, 1, 34, 3, 73, 105, 5, 46, 85, 8, 83, 2, 11, <sup>78</sup>}. To evaluate BS correctness, the accuracy index (ACC) is defined by

$$\text{ACC}(n) = \frac{||\Omega\_{\text{PSP}-\text{OMPBS}}(n) \cap \Omega\_{\text{OMPBS}}(n)||\_0}{p}. \tag{15}$$

Base on Equation (15), ACC(*n*) indicates the ratio of the target bands in overall *p*-selected bands, when the number of received pixels is *n*. Higher ACC values mean higher BS correctness. Figure 6a–d plots the ACC curves of OMPBS, PSP-OMPBS, S-PSP-OMPBS, and B-PSP-OMPBS, all implemented on the Pavia data. Several observations can be seen:

1. Comparing Figure 6a with Figure 6b, the tendency of the ACC curves of OMPBS and PSP-OMPBS are nearly the same. This implies that the derivation of Re-OMPBS is correct. In fact, these two curves are still slightly different in some regions. Based on our study, the difference was caused by the numerical errors produced by using the recursive equations derived from Woodbury's Identity.


In the remote sensing community, the Purdue image was considered as a tough image for algorithm evaluation because of its noise and heavily-mixed properties. Figure 7a–d plots the ACC curves of OMPBS, PSP-OMPBS, S-PSP-OMPBS, and B-PSP- OMPBS, implemented on the Purdue data, respectively. Theoretically, the larger *p* and the special properties of the Purdue image, may lead to different consequences. Compared with the Pavia case, we have several interesting findings. Firstly, the oscillation of the four ACC curves is more notable. This is probably due to the noise property of the Purdue image. Secondly, the tendency of OMPBS and PSP-OMPBS curves are the same. This is in accordance with our expectations, because both methods are essentially the same. However, their ACC values are slightly different at some places. The PSP-OMPBS curve of the Purdue image is more unstable and lower at middle region. We think such a phenomenon is caused by the larger setting of *p*. The greater *p* is set, the more numerical errors will be accumulated while selecting later bands at each *n*. Thirdly, and most importantly, we found that the advantages of using sampled sequences (i.e., step and block) is not obvious in the Purdue experiment. We think the strange phenomenon is caused by the homogeneity of spectral profiles of the 16 ground classes. In this case, using any kind of PTS may obtain analogous pixel sets, so that the BS results are similar. From this point of view, in the case of the Purdue data, the OMPBS selects bands based more on spectral variation, instead of spatial/geographical variation. Despite this issue, it is still observed that S-PSP-OMPBS and B-PSP-OMPBS outperformed OMPBS a little bit with regard to ACC stability. The ACC trends of S-PSP-OMPBS and B-PSP-OMPBS did not drop after they reached over 80%, particularly at *n* ∈ [6000,13,000]. In conclusion, the ACC performance of PSP-OMPBS methods varies with the properties of images. Using PSP-OMPBS on the images with lower noise seemed to produce more stable ACC curves. The sampled pixel sequences were more suitable for the images with larger heterogeneity in the spatial domain. The spectral similarity of ground classes was another important factor. Finally, we can find the curve of B-PSP-OMPBS did not end at 100% when *n* = *N* = 21,025. This was caused by the numerical error.

**Figure 6.** Accuracy index (ACC) curves of different methods implemented on ROSIS Pavia dataset: (**a**) OMPBS, (**b**) PSP-OMPBS, (**c**) step sequence PSP-OMPBS (S-PSP-OMPBS), and (**d**) block sequence PSP-OMPBS (B-PSP-OMPBS).

**Figure 7.** *Cont*.

**Figure 7.** ACC curves of different methods implemented on the AVIRIS Purdue dataset: (**a**) OMPBS, (**b**) PSP-OMPBS, (**c**) step sequence PSP-OMPBS (S-PSP-OMPBS), and (**d**) block sequence PSP-OMPBS (B-PSP-OMPBS).

Tables 3 and 4 list the corresponding BS results of the Pavia and Purdue experiments, where *n* is selected by 50% of *N* for each case. The last row shows the BS ground truth. It can be seen that the BS accuracies of PSP-OMPBSs are better than OMPBS at the 50% middle of transmission.

**Table 3.** List of the 16 bands, selected by four different methods, implemented on the Pavia data at *n* = 103,700.


**Table 4.** List of the 34 bands, selected by four different methods, implemented on the Purdue data at *n* = 10,513.



**Table 4.** *Cont.*

#### *4.3. Land Cover Classification*

Image classification is a common procedure to evaluate the effectiveness of a BS approach. For each dataset, we implemented supervised classification using a PSP-OMPBS-selected band <sup>Ω</sup>P(*n*), with a support vector machine (SVM) classifier [40]. The radial basis function (RBF) kernel was adopted for SVM with the selected parameter [*σ*/4, *σ*/2, *σ*, 2 *σ*, 4 *σ*] where r is calculated by the average pairwise distance among training data *σ* = *E* - - - *xi* − *xj* - - - -. The training samples were obtained by randomly selecting 10% of date samples in each class, according to the ground truth maps shown in Figures 4b and 5b. The other 90% of samples were used as the test samples.

For measuring the classification performance, average accuracy (AA), overall accuracy (OA), and Cohen's kappa coefficient were used as the performance metrics. Since PSP-OMPBS provided 207,400 and 21,025 BS results for two datasets respectively, we simply chose the BS results using 20%, 40%, 60%, 80%, and 100% of *N* for the experiment. Three PSP-OMPBS algorithms were considered. The classification result of using full bands was also used for the comparison.

Table 5 lists all the AA, OA, and kappa coefficient values of the SVM results performed on PSP-OMPBS selected bands for both the Pavia and Purdue datasets. We observed that using the selected bands of PSP-OMPBS preserved sufficient spectral information to achieve comparative classification performance, compared to the results of using full bands. The highest accuracies of the AA, OA, and kappa coefficient values reached were 0.907, 0.868, and 0.874, respectively, which is close to the full bands results of 0.914, 0.880, and 0.884, respectively. This simply implies that the bands selected by PSP-OMPBSs are informatively complementary.

On the other hand, in the Purdue case, the most accurate results for the AA, OA, and kappa coefficient values were 0.736, 0.600, and 0.693, respectively, and were slightly worse than the full bands results of 0.756, 0.586, and 0.716. This might be simply because only using 34 bands is not enough for the Purdue image classification. This issue can be resolved by increasing p, fine-tuning kernel parameters, or use other types of SVM classifiers that utilize spectral–spatial information to improve overall performance. Such a study is beyond the scope of this paper, and thus is not included.

To evaluate classification performance in visual assessment, Figures 8 and 9 plot the classification maps of SVM using the bands selected by PSP-OMPBS, S-PSP-OMPBS, and B-PSP-OMPBS at *n* = 60% of *N* for the Pavia and Purdue data, respectively. The maps using full bands are included for comparison. In the Pavia case, it can be seen that using PSP-OMPBS-selected bands could generate nearly the same classification maps compared to those obtained by using full bands. In the Purdue case, the quality of the PSP-OMPBS-generated maps are a little worse than the full bands map, due to the insufficient number of selected bands.



**Figure 8.** Examples of Pavia classification maps of SVM (RBF) performed on the 16 bands selected by (**a**) PSP-OMPBS, (**b**) S-PSP-OMPBS, (**c**) B-PSP-OMPBS, and (**d**) full 103 bands. For three PSP-OMPBSs, *n* is set at 50% of *N*. The corresponding OA values are 0.86, 0.89, 0.90, and 0.91.

**Figure 9.** Examples of Purdue classification maps of SVM (RBF) performed on the 34 bands selected by (**a**) PSP-OMPBS, (**b**) S-PSP-OMPBS, (**c**) B-PSP-OMPBS, and (**d**) full 202 bands. For three PSP-OMPBSs, *n* is set at 50% of *N*. The corresponding OA values are 0.72, 0.73, 0.73, and 0.76, respectively.

To analyze the classification performance in the progressive manner, Figure 10 plots the kappa OA/AA/kappa coefficient curves of the SVM classification using uniformly selected BS results, performed by PSP-OMPBSs on the *n*-axis with interval 25 for the Pavia dataset. In those figures, the *x*-axis (*n*) indicates band set <sup>Ω</sup>*p*(*n*), and the *y*-axis denotes the corresponding accuracy metrics of the classification. Several observations can be found. First, all the metric curves of PSP-OMPBS and S-PSP-OMPBS are not stable in the beginning *n* ∈ [1,200] because of the poor BS quality that occurs with a low number of received pixels. With too few pixels, PSP-OMPBS could not select correct bands to fulfill the complementary spectral information. This resulted in unstable and lower classification performance. In contrast, the curves of B-PSP-OMPBS could be consistent because the transmitted pixels were uniformly sampled from the image. Second, the overall averaged performances of S-PSP-OMPBS and B-PSP-OMPBS are slightly better than OMP-BS for these three criteria. Third, when *n* is greater than 250, the curves of all three methods tended to be consistent. According to our extended study, all the curves will stay roughly at the same level, with extremely low deviation in the future time *n* ∈ [600,N].

**Figure 10.** SVM classification of using the bands selected by three PSP-OMPBS methods at *n* = [1,600]: (**a**) OA, (**b**) AA, and (**c**) kappa coefficient, for the ROSIS Pavia dataset.

Similarly, Figure 11 shows the OA, AA, and kappa coefficient curves for the Purdue experiment. Similarly, there are some interesting findings. First, the classification performance is low at the beginning *n* ∈ [1,100]. Second, the Purdue image seemed to require more pixels to stabilize the BS quality. We can find that the curves of PSP-OMPBS, S-PSP-OMPBS, and B-PSP-OMPBS could not converge at n ∈ [100,600] in each figure, particular Figure 10b. This is probably because of the heavily mixed and noisy properties of the Purdue image. Each newly incoming pixel may easily disturb the result of band selection. According to our investigation, this phenomenon will last up to *n* = 4000. Finally, the S-PSP-OMPBS and B-PSP-OMPBS did not outperform PSP-OMPBS in the Purdue case. We think this is caused by the homogeneity of spectral profiles of the 16 ground classes. In this case, using a sampled transmission sequence could not significantly improve the collecting of spectral information.

**Figure 11.** SVM classification of using the bands selected by three PSP-OMPBS methods at n = [1,600]: (a) OA, (**b**) AA, and (**c**) kappa coefficient, for AVIRIS Purdue dataset.
