28.4 Computer Simulations Using AVIRIS Data

In order to demonstrate the utility of KFSCSP techniques in spectral estimation, identification, and quantification, computer simulations and real data experiments were conducted for performance evaluation and analysis. For computer simulations, five Airborne Visible InfraRed Imaging Spectrometer (AVIRIS) reflectance data in Figure 1.8 are reproduced in Figure 28.1 for reference. There are five signature vectors, blackbrush, creosote leaves, dry grass, redsoil, and sagebrush, to be used for experiments where these spectra have 158 bands after water bands are removed.

Figure 28.1 Reflectances of creosote leaves, blackbrush, sagebrush, drygrass, and redsoil.

According to Figure 28.1, the signatures of blackbrush, creosote leaves, and sagebrush are close to each other in terms of spectral shape. In particular, the spectral signatures of creosote leaves and sagebrush are very similar. A detailed quantitative analysis of these three signatures can be found in Chang (2003a).

Since KFSCSP techniques are signature vector-based techniques not to be used for image pixel vectors, KFSSE, KFSSI, and KFSSQ are not designed for classification. Therefore, their performance is evaluated by signature vector-based spectral measures such as spectral angle mapper (SAM), spectral information divergence (SID) rather than image classifiers.

28.4.1 KFSSE

To implement KFSSE, the system gain c_l in (28.5) was set to be 1 for all , the standard deviation of the state noise v, σ_v was empirically set to 10³ and the standard deviation of the measurement noise u, σ_u was chosen to achieve signal-to-noise ratio (SNR) = 30 dB, where the SNR was defined as the ratio of half a signature reflectance mean to the standard deviation of noise (Harsanyi and Chang, 1994). It should be noted that throughout our extensive experiments, the results demonstrated that KFSSE was robust to the selection of σ_u and σ_v. More specifically, once the value of σ_v was greater than 10³, the σ_u had limited impact on the performance of KFSSE. Therefore, for KFSSE implemented in this chapter, the values of σ_u and σ_v were fixed at 30 dB and 10³, respectively.

Figure 28.2(a)–(e) shows the five KFSSE-estimated reflectance spectra of the signature vectors given in Figure 28.1. Since the values of the corresponding estimation errors obtained by KFSSE were very small, Figure 28.3(a)–(e) shows the ratios of estimation errors to their corresponding reflectance spectra in Figure 28.1 by the logarithm function to have better visual assessment.

Figure 28.2 KFSSE-estimated spectra of the five reflectance signatures in Figure 28.1.

Figure 28.3 Ratios of the estimation errors to the five reflectance spectra in Figure 28.1 in the logarithm functions.

According to Figure 28.3, KFSSE was very effective in estimating spectral signatures with estimation errors nearly close to zero. Obviously, large estimation errors always occurred at wavelengths where their spectral values had changed drastically. To the contrary, if there was a smooth transition between two wavelengths, the estimation errors were relatively small.

28.4.2 KFSSI

Two scenarios were implemented by KFSSI: one for subpixel target identification and the other for mixed target identification with the target signature vector t to be identified and assumed to be either known or unknown.

28.4.2.1 Subpixel Target Identification by KFSSI

First of all, we simulated a subpixel target implanted in a signature vector. Without loss of generality, we assumed that the subpixel target signature was the creosote leaves (C) and the background signature was the redsoil (R). Three target pixel vectors, t₁, t₂, and t₃ were generated in accordance with the abundance fractions of the subpixel target signature, creosote leaves set to 75%, 50%, and 25% mixed with the redsoil as the background signature to make up 100% abundance, respectively, that is, t₁ = ¾C + ¼R, t₂ = ½ C + ½R, and t₃ = ¼C + ¾R. KFSSI was then implemented to identify C from the three target signature vectors, t₁, t₂, and t₃ via a database Δ = {creosote leaves and redsoil}. The identification was carried out by first setting the values for both σ_u and σ_v, then randomly choosing a matching signature vector from the database Δ to match the unknown subpixel target panel according to (28.8) and (28.9), and finally identifying the unknown subpixel target panel as the one that yielded the minimum LSE. The resulting LSE corresponding to different sizes of subpixel target panels were tabulated in Table 28.1 with and various values of the standard deviation of noise u, σ_u. However, it should be noted that the selection of σ_u was empirically chosen and dependent upon the target signature vector to be used for experiments. The simulated data conducted in this section were designed to demonstrate and illustrate the utility and effectiveness of KFSSI in signature identification under the impact of the parameter σ_u. In doing so, we had experimentally adjusted the value of σ_u in accordance with our simulation data to dictate the impact of subpixel target size on the performance. Figure 28.4(a) and (b) plot the LSEs of creosote leaves and redsoil, respectively, versus the values of σ_u for three sizes of subpixel targets, ¾, ½, and ¼, where the values of σ_u varying from 10 to 1000 with step size set to 10.

Figure 28.4 Relationship between σ_u and LSE according to different sizes of the subpixel target.

Table 28.1 LSE corresponding to different sizes of subpixel target panels (creosote leaves) according to (28.10) via Δ = {creosote leaves and redsoil}

For example, in order to correctly identify the subpixel panel with ¼ size of a pixel, according to Table 28.1, the σ_u must be greater than 159 compared to the subpixel panel with ¾ size of a pixel which only requires σ_u smaller than 12. Additionally, LSEs resulting from KFSSI was in proportion to the values of σ_u.

As pointed out, the selection of σ_u and σ_v had a significant impact on the performance of KFSSI and must be chosen appropriately in order to achieve acceptable LSEs. According to our exepriments, once σ_v was fixed, the minimum value of σ_u could be set approximately in the same order of magnitude in order to correctly identify the subpixel target as creosote leaves. Table 28.2 is included here to illustrate the relationship between σ_v and σ_u with three different values of σ_v and seven subpixel targets of size specified by (1/8, 2/8, 3/8, 4/8, 5/8, 6/8, 7/8).

Table 28.2 Relationship between σ_v and σ_u with creosote leaves as target signature embedded into redsoil.

In order to provide further insights into KFSSI, similar experiments were also conducted by replacing creosote leaves with blackbrush, sagebrush, and drygrass, with their results tabulated in Tables 28.3–28.5, respectively.

Table 28.3 Relationship between σ_v and σ_u with sagebrush as target signature embedded into redsoil.

Table 28.4 Relationship between σ_v and σ_u with blackbrush as target signature embedded into redsoil.

Table 28.5 Relationship between σ_v and σ_u with drygrass as target signature embedded into redsoil.

According to Tables 28.2–28.5, two factors affected the selection of σ_u. One is target size and the other is the standard deviation of the state noise σ_v. When similar experiments were conducted by replacing the creosote leaves with blackbrush, sagebrush, and drygrass, the changes in σ_u were noticeable as shown in Table 28.2–28.5. Three interesting findings from Tables 28.2–28.5 are intriguing.

Finding 1: When target size was fixed, the value of σ_u was in proportional to the value of σ_v.

Finding 2: The range of σ_u was closely related to the subpixel target signature to be identified. For example, according to the study by Chang (2000, 2003a), the creosote leaves was the most difficult signature to be discriminated, since it was very similar to both sagebrush and blackbrush. Table 28.2 demonstrates this fact by providing this evidence that the range of σ_u must be bounded below from 18 to 123 as the subpixel target size from 1/8 to 5/8 with σ_v fixed at . Interestingly, once the size was greater than 5/8, the range of σ_u was suddenly reversed from bounded below to bounded above. The situation was slightly improved when the subtarget signature vector was sagebrush in Table 28.3, where a sudden change in the range of σ_u occurred at size ½. This phonemonon was further evidenced by the blackbrush in Table 28.4, where the range of σ_u was all bounded below from 10 to 32 that were inversely proprtional to target size with σ_v also fixed at . Finally, since the drygrass was very dissimilar to all the three signature vectors, creosote leaves, sagebrush, and blackbrush, Table 28.5 reflects the fact that the range of σ_u was all bounded above from 4 to 35 that were proprtional to target size when with the σ_v was fixed at , a case that was completely opposite to blackbrush in Table 28.4.

Finding 3: The selection of initial values for σ_u and σ_v was very much dependent on the data to be processed, such as spectral similarity among signature vectors. This must be done on an empirical basis. For example, if signature vectors to be considered were spectrally distinct, the results would be very robust to the selection. On the other hand, if the signature vectors were spectrally similar, the results would be sensitive to how the initial values were selected. In the above experiments, they were determined empirically based on a priori knowledge about the material substance signatures and subpixel size. Nevertheless, from our extensive experiments a general guideline to determine the initial values of σ_u and σ_v may be useful. Without loss of generality, we can assume a mixing signature vector specified by , where s and b can be interpreted as the embedded signature and background, respectively. In this case, the size of subpixel target also has a significant effect on the subpixel target by the background signature. The initial guess for the value of σ_u to properly identify the signature vector s embedded into the signature vector r is closely related to σ_v in a form of , where the target signature vector t is any auxiliary signature vector related to s as specified by (28.8) and (28.9). In regard to the value of σ_v, it can be empirically set to , , , etc. However, we would like to point out that this guideline only serves as a reference and should not take as a criterion for all the cases.

The above findings offer a new look of KFSSI in to how to use signature characterization to perform subpixel target identification. The standard deviation of the measurement noise, σ_u, used in KFSSI is a very important parameter for identification and varies with the subpixel target signature to be identified, which makes sense. These interesting findings cannot be observed by any spectral measure as demonstrated in Tables 28.6–28.9, which show the results of the same experiments using SAM and SID, where the SID values are given in parentheses. As shown in these tables, it was impossible for SAM and SID to detect the subpixel target panel if its size was smaller than ½ size of a panel. To the contrary, KFSSI could detect subpixel targets correctly even its size was smaller than ½ size of a pixel as long as σ_u was chosen to be the values tabulated in Tables 28.2–28.5.

Table 28.6 Identification of subpixel panels with creosote leaves as a target signature vector embedded into redsoil by SAM and SID.

Table 28.7 Identification of subpixel panels with sagebrush as a target signature vector embedded into redsoil by SAM and SID.

Table 28.8 Identification of subpixel panels with blackbrush as a target signature vector embedded into redsoil by SAM and SID.

Table 28.9 Identification of subpixel panels with drygrass as target signature vector embedded into redsoil by SAM and SID.

The above experiments demonstrate an important advantage of KFSSI which is that KFSSI could perform well in the subpixel identification, even if the size of a subpixel target was less than ½ of ground sampling distance, that is, pixel resolution, which could not be achieved by any other spectral measure.

28.4.2.2 Mixed Target Identification by KFSSI

In order to make our experiments more interesting and appealing, we further simulated a mixed pixel vector t^mix by equally mixing ¼ blackbrush (s^B), ¼ creosote leaves (s^C), ¼ dry grass (s^D), and ¼ sagebrush (s^S) as follows:

(28.14)

whose spectral signature vector is also shown in Figure 28.1. In this case, no signature vector was preferred to another. KFSSI was used to identify unknown target signature vector t present in the mixed pixel vector t^mix using the database Δ = {drygrass, blackbrush, creosote leaves, sagebrush}. According to the composition of t^mix, all the four signature vectors had equal opportunity to be identified as the target signature vector. The resulting LSEs corresponding to four signature vectors used to match the mixed pixel vector t^mix were shown in Table 28.10.

Table 28.10 LSE between the mixed pixel t^mix and different matching signature vectors according to (28.10)

Interestingly, since both blackbrush and creosote leaves were close to sagebrush by spectral similarity measures, SAM and SID, KFSSI believed that blackbrush and creosote leaves were part of sagebrush with small variations. As a consequence, it identified t^mix as sagebrush. Table 28.10 demonstrates that KFSSI could successfully identify the mixed pixel t^mix as sagebrush as long as and drygrass otherwise. As noted, the values of σ_u and σ_v might be chosen comparably at the same order of magnitude. In this case, σ_v was chosen to be 10, which was of the same order as . If the σ_v in (28.8) was set to 10³, which is similar to the value used in Table 28.1, the LSEs resulting from KFSSI for subpixel target identification were large.

28.4.3 KFSSQ

This section presents experiments to further demonstrate the use of KFSSQ in quantification of subpixel targets and target signature vectors present in mixed pixels. Compared with KFSSI, which is used to identify an unknown target signature vector t, KFSSQ can also be implemented for the target signature vector t, which is either unknown or known a priori. It should also be noted that unlike KFSSI, there is no state noise vector v in the state equation (28.12) implemented by KFSSQ.

28.4.3.1 Subpixel Target Quantification by KFSSQ

Using the same subpixel targets t₁, t₂, and t₃ simulated in Section 28.4.2.1, we implemented KFSSQ to quantify these three creosote leaves-simulated subpixel target panels t₁, t₂, and t₃ of three respective sizes, ¾, ½, and ¼ with two scenarios described in the following examples.

Example 28.1

(Target signature vector t is known)

This example assumed that the target signature t was known to be creosote leaves. In this case, the pair of (28.11) and (28.12) was implemented to quantify three subpixel targets of creosote leaves, t₁, t₂, and t₃, and the results are tabulated in Table 28.5. Since the subpixel target panels only occupied part of a pixel, its size could be interpreted as portion of abundance fraction in terms of percentage. In this case, the three subpixel target panels t₁, t₂, and t₃ with size of ¾, ½, and ¼ could be considered as targets with abundance fractions 75%, 50%, and 25%, respectively, with KFSSQ-estimated abundance fractions given in Table 28.11, where the quantification results were very accurate with appropriate chosen values of σ_u.

Table 28.11 Abundance fractions estimated by KFSSQ.

Example 28.2

(Target signature vector t is unknown)

Unlike Example 28.1, the prior knowledge of target signature vector t was not given in this example. In this case, KFSSQ must use (28.5) and (28.6) to first identify the target signature vector t that specified the three subpixel targets t₁, t₂, and t₃ where KFSSE was used for this purpose. The estimated was then used to replace t in (28.11) to produce the abundance vector specified by (28.12) in Table 28.6. As a result, two different values of σ_u were used for measurement noise in KFSSQ: one in (28.5) for KFSSE and the other in (28.13) for KFSSQ. Since both were not correlated, they could be determined independently as tabulated in Table 28.12.

Due to the fact that KFSSE could not correctly estimate the target signature vector t, provided the abundance of subpixel target vector t was below 50%, in which case it was reasonable, the results for the size of subpixel target being ¼ were not included in Table 28.12.

Table 28.12 Abundance fractions estimated by KFSSQ.

28.4.3.2 Mixed Target Quantification by KFSSQ

In this subsection, we used the same mixed pixel vector, t^mix simulated in Section 28.4.2.2 for further experiments. KFSSQ was implemented to quantify the four signatures, blackbrush, creosote leaves, drygrass, and sagebrush, each of which shared a 25% abundance fraction in the pixel vector t^mix. Like Section 28.4.2.3.1, two scenarios were considered.

Example 28.3

(All the four target signature vectors are known)

This example assumed that all the four signatures were known. The pair of (28.11) and (28.12) was implemented for KFSSQ. Table 28.13 tabulates the quantification results of the four signatures with their appropriately chosen values of σ_u. However, it should be noted that the sensitivity of σ_u was generally determined by various applications. For example, when KFSSI and KFSSQ were implemented, the results would be sensitive to the value chosen for σ_u as shown in Tables 28.10 and 28.12. On the other hand, if KFSSE was implemented as an estimator, the results would be relatively robust to the σ_u.

As we can see from this table, KFSSQ-estimated abundance fractions of all the four signatures were very accurate and close to true values. In order to make further comparison, the results produced by the FCLS method were also included and tabulated in Table 28.14, where their results are comparable to those obtained in Table 28.13.

Table 28.13 Abundance fractions of the four signatures embedded into the mixed pixel t^mix estimated by KFSSQ.

Table 28.14 Abundance fractions of the four target signature vectors embedded into t^mix estimated by FCLS.

Example 28.4

(Target signature vector is unknown)

The purpose of including this experiment was to demonstrate that KFSSQ could be implemented even if the target signature vector was not known. The experiment conducted in Section 28.4.2.2 assumed that the exact knowledge of target signature vector, sagebrush, is provided a priori. In this example, this prior knowledge was not given. KFSSQ first used KFSSI to identify the unknown mixed target signature vector t^mix of (28.14) as sagebrush and then used KFSSE to estimate the sagebrush signature vector from t^mix as the desired signature that was used in the follow-up abundance quantifier, KFSSQ, to estimate its abundance fraction, 0.28. Compared with the value of 0.2504 in Table 28.13 and 0.25135 in Table 28.14, it was slightly off the true abundance 0.25, but was still very good. This made sense since the results in Tables 28.13 and 28.14 are obtained by assuming exact knowledge of the sagebrush as opposed to KFSSE-estimated sagebrush used in this example.

As noted further, when KFSSQ was implemented in the above estimation, two cases were considered. If the true signature vector t was used, the value of σ_u was determined by (28.11). On the other hand, if the target signature vector used in KFSSQ was estimated first by KFSSI followed by KFSSE, the value of σ_u in (28.11) was then determined by the value of σ_u used by the two estimators, that is, both (28.5) for KFSSE and (28.7) for KFSSI.

Example 28.5

(Sensitivity of KFSSQ to σ_u)

This experiment was designed to investigate the sensitivity of KFSSQ to the values of σ_u in quantification of subpixel targets. Experiments similar to those in Example 28.1 were conducted by using the values of σ_u given in Table 28.11 for three different target sizes (1/4, 1/2, and 3/4), where three proper ranges of σ_u for three different target sizes (1/4, 1/2, and 3/4) were fluctuated around 7.5, 4.3, and 2.4, respectively, for correct quantification. Figure 28.5(a)–(c) plots the results of KFSSQ for three different target sizes (1/4, 1/2, and 3/4) versus σ_u with step size 0.1.

As shown by the plotted curves in Figure 28.5(a)–(c), the abundance fractions quantified by KFSSQ were inversely proportional to the values of σ_u. The curves in Figure 28.5 indicate that KFSSQ was indeed very sensitive to the value of σ_u, where an appropriate values of σ_u must be carefully selected in order to perform correct quantification for subpixel targets.

Figure 28.5 KFSSQ-estimated abundance fractions versus values of σ_u for different sizes of the subpixel target with step size 0.1.

Finally, three concluding remarks are worth noting.