4.3 Six Scenarios of Synthetic Images

Section 4.2 describes how to simulate subsample targets or mixed-sample targets according to their characteristics. In this section, we will discuss on how to simulate synthetic images with target panels inserted in accordance with certain desired properties.

4.3.1 Panel Simulations

First, the real image scene with reflectance data shown in Figure 1.12(c) is used to simulate panels of interest where the reflectance spectra of five USGS ground-truth mineral spectra: alunite (A), buddingtonite (B), calcite (C), kaolinite (K), and muscovite (M) are used to simulate 25 panels of various sizes that are arranged in a img matrix as shown in Figure 4.2.

Figure 4.2 25 simulated panels.

img

Each row of the five panels in Figure 4.2 is simulated by the same mineral signature and each column of five panels has the same size. Among 25 panels are five img-pure pixel panels, img for img in the first column, five img-pure pixel panels, img for img in the second column, five img-mixed pixel panels, img for img in the third column, five subpixel panels, img for img in the fourth column, and five subpixel panels, img for img in the fifth column. The purpose of introducing five panels in the third column is to conduct a study and analysis on the five mineral signatures with different mixtures in a pixel. Table 4.1 tabulates the mixing details of mineral composition in the 20 panels in the third column, while subpixel panels in the fourth and fifth columns are simulated with their simulated abundance fractions tabulated in Table 4.2, where the background (BKG) is simulated by the sample mean of the real cuprite image scene in Figure 1.12(a).

Table 4.1 Simulated 20 mixed panel pixels in the first column.

Row 1 img img
img img
Row 2 img img
img img
Row 3 img img
img img
Row 4 img img
img img
Row 5 img img
img img

Table 4.2 Abundance fractions of subpixel panels in the fourth and fifth columns.

Row Fourth column Fifth column
1 img img
2 img img
3 img img
4 img img
5 img img

According to Tables 4.1 and 4.2, the simulated synthetic image in Figure 4.2 has a total of 130 panel pixels present in the scene, 80 pure pixel panels in the first column, 20 pure pixel panels in the second column, 20 mixed panel pixels in the third column, five 50%-abundance subtarget panel pixels in the fourth column, and five 25%-abundance subpixel target panel pixels in the fifth column. These 130 pixel panels include a total of 26 spectrally distinct signatures (5 pure mineral signatures in the first and second columns, 20 mixed signatures in the third column 5 along with 1 BKG signature). Figure 4.3 graphically plots the abundance fractions of all these 130 panel pixels where (a-e) indicate the five mineral signatures, A, B,C,K, and M, used to simulate 26 panel pixels in each of five rows in Figure 4.2 respectively.

Figure 4.3 Graphical plots of abundance fractions of 130 panel pixels in Figure 4.2.

img

By virtue of the 25 simulated panels in Figure 4.2, two target insertions, TI and TE, can be designed to be used for experiments conducted in this book.

1. Target implantation (TI): Three scenarios to implant target pixels into the background are simulated in such a manner that the target pixels are inserted into the background while their corresponding background pixels are removed. This type of scenario is mainly designed to simulate pure target pixels for extraction. The utility of TI includes applications such as endmember extraction (Chapters 7–11), mixed pixel detection, classification, and quantification (Chapters 12–18).
2. Target embeddedness (TE): Three scenarios to embed target pixels into the background are simulated in such a manner that the targets are inserted into the background by adding the targets directly to their corresponding background pixels. In other words, target pixels are superimposed over their corresponding background pixels instead of removing their corresponding background pixels to accommodate the target pixels as the way target implantation does. This type of scenario is primarily designed for target detection where a binary hypothesis testing problem is cast for detection (see Chapter 2) with the null hypothesis H0 representing background with additive noise against the alternative hypothesis H1 representing signal (target pixels) plus background (background pixels) with additive noise. It can also be used to test whether or not an endmember extraction algorithm can extract the most purest signatures, which also turn out to be embedded target pixels but do not have 100% purity of signatures. A salient difference between TI and TE is worth being mentioned. Since the three TE scenarios insert targets by adding target pixels to and superimposing over background pixels instead of replacing background pixels as the way the three TI scenarios do for their target insertion, the abundance fraction of the pixel into which a target pixel is embedded is not summed to one. These three TE scenarios violated the abundance sum-to-one constraint (ASC) generally imposed on linear spectral mixture analysis (LSMA), and thus, they cannot be used for quantification as shown in Chang et al. (2010) and Figure 4.14 (Section 4.4.2.2). Nevertheless, they are well suited for detection-in-noise model analysis.

4.3.2 Three Scenarios for Target Implantation (TI)

Three interesting scenarios for TI, Scenario TI1, Scenario TI2, and Scenario TI3, presented in this section, are designed for applications in target extraction such as endmember extraction (see Part II) and target quantification. The 25 panels in Figure 4.2 are used as targets of interest and implanted in a synthetic image scene with size of img pixel vectors in a way that the targets to be implanted replace their corresponding background pixels. Each of these three scenarios is described as follows.

4.3.2.1 Scenario TI1 (Clean Panels Implanted into Clean Background)

This scenario assumes that the image background is clean and simulated by only one BKG signature. The 25 clean panels simulated in Figure 4.2 are then implanted in the background by replacing their corresponding background pixels with the clean panel pixels. The resulting image is a synthetic image shown in Figure 4.4 with clean panels implanted in the clean background image scene.

Figure 4.4 Synthetic image simulated by Scenario TI1.

img

4.3.2.2 Scenario TI2 (Clean Panels Implanted into Noisy Background)

Practically, Scenario TI1 does not exist because of no noise present in the data. Scenario TI2 is more realistic when the noise-free background in Scenario TI1 is replaced with a noisy background image which is corrupted by an additive Gaussian noise to achieve a signal-to-noise ratio (SNR) = 20:1 defined as 50% signature (i.e., reflectance/radiance) divided by the standard deviation of the noise in Harsanyi and Chang (1994). Then clean targets are implanted into such simulated noisy background image. So, the resulting synthetic image has clean targets implanted in a noisy background as shown in Figure 4.5. This scenario simulates a case that true clean targets are indeed present in a noisy image background for target extraction.

Figure 4.5 Synthetic image simulated by Scenario TI2.

img

The synthetic image scene in this scenario is the same as the one in Scenario TI1 except that the image background is not clean, but rather corrupted by an additive Gaussian noise with SNR = 20:1.

4.3.2.3 Scenario TI3 (Gaussian Noise Added to Clean Panels Implanted into Clean Background)

Scenario TI3 is the same as Scenario TI1 except that a Gaussian noise is added to TI1 to achieve an SNR = 20:1. So, in this synthetic image, the clean targets and clean background image are both corrupted by an additive Gaussian noise with SNR = 20:1 as shown in Figure 4.6. It is also similar to Scenario TI2 but the implanted targets are now noise-corrupted compared to the clean targets implanted in Scenario TI2.

Figure 4.6 Synthetic image simulated by Scenario TI3.

img

This scenario simulates a case that the implanted targets are not original true targets and have been contaminated and corrupted by noise. As a consequence of noise corruption, all the pure pixels, mixed pixels, and subpixels are contaminated. So, technically, those panel pixels of 100% purity as endmembers are no longer pure. However, these implanted targets are still considered to be purest and closest to the original targets compared to other pixels present in the image scene. So, they can still be considered as targets of interest. This scenario is designed to test and evaluate how sensitive a target extraction algorithm can be when clean targets are corrupted by noise.

4.3.3 Three Scenarios for Target Embeddedness (TE)

In the previous sections, three scenarios for TI are simulated by inserting the 25 panels into the image scenes by removing background pixels to accommodate these 25 panels for target implantation. As an alternative, this section simulates another type of target insertion, called TE, which inserts the 25 panels into an image scene with a size of img pixel vectors by adding the 25 panels directly to an image scene in such a way that the 25 panels are simply superimposed over their corresponding background pixels. In other words, unlike target implantation in Scenarios TI1, TI2, and TI3 which replaced background pixels with 25 implanted panel pixels, the following three scenarios embed 25 panel pixels by adding panel pixels to background pixels via superimposition. In this case, a pixel which contains an embedded panel pixel also contains a background pixel and thus, its abundance is the sum of abundance of the embedded panel pixel and background pixel. The three counterparts of Scenarios TI1, TI2, and TI3 can also be simulated as TE1, TE2, and TE3, respectively. The design of TE serves different applications such as signal detection (see Chapter 2) and discrimination including subpixel target detection and discrimination, mixed pixel classification, and identification.

4.3.3.1 Scenario TE1 (Clean Panels Embedded in Clean Background)

Scenario TE1 is the same as Scenario TI1 except that the targets implanted into the background are now embedded into the background as shown in Figure 4.7.

Figure 4.7 Synthetic image simulated by Scenario TE1.

img

More specifically, the targets are inserted into the background in such a manner that the targets are added to their corresponding background pixels without removing them similar to Scenario TI1. This scenario simulates an idealistic case for signal detection where two hypotheses represent background versus signal plus background, that is, H0: BKG versus H1: clean signal + BKG. As a result, there are no pure pixels in this particular scenario since pure pixels in the first and second columns are no longer pure due to its inclusion of background pixels. So, technically, there are no pure signatures or endmembers in the image scene, but there are 25 spectrally distinct panel signatures plus one BKG signature.

4.3.3.2 Scenario TE2 (Clean Panels Embedded in Noisy Background)

In analogy with TI2, Scenario TE2 simulates a practical signal detection problem where clean targets are embedded into a noisy background simulated by a background signature with an additive Gaussian noise to achieve an SNR = 20:1. In other words, instead of replacing background pixels with the clean targets as done in Scenario TI2, the targets are actually embedded into and superimposed over clean background pixels. So, in this case, the resulting synthetic image has clean targets embedded into a noisy background as shown in Figure 4.8.

Figure 4.8 Synthetic image simulated by Scenario TE2.

img

This scenario simulates a case for target detection where two hypotheses represent noisy background against signal plus noisy background, that is, H0: BKG + noise versus H1: clean signal + BKG + noise. It is tricky to simulate this scenario since the noisy background pixels are replaced by their corresponding clean background pixels, while SNR must be retained at the given level.

4.3.3.3 Scenario TE3 (Gaussian Noise Added to Clean Panels Embedded in Background)

Scenario TE3 is the same as Scenario TE1 except that a Gaussian noise is added to Scenario TE1 to achieve SNR = 20:1 as shown in Figure 4.9.

Figure 4.9 Synthetic image simulated by Scenario TE3.

img

It is also similar to Scenario TE2 with only difference that the embedded targets are noise corrupted compared to the clean targets embedded in Scenario TE2. More specifically, in Scenario TE3 the clean targets and clean background image are both corrupted by an additive Gaussian noise with SNR = 20:1. So, the 25 spectrally distinct panel signatures are corrupted by noise and BKG signature. This scenario simulates a case that two hypotheses represent noisy background against noisy signal with noisy background, i.e., H0: noisy BKG versus H1: noisy signal + noisy BKG.

Finally, several remarks on the above designed six synthetic images are noteworthy:

1. Most panel pixels in Figures 4.4–4.9 are visible. This may lead to a belief that these six scenarios are not useful or appropriate for experiments. The truth is that what we see from images is generally not what we will expect. Specifically, what we see is only qualitative and not quantitative, a task that a computer algorithm can do well while human being cannot. Although all target panels either implanted or embedded in synthetic images are clearly visible, it does not mean that an algorithm can detect these target panels well. This is exactly what we need from these scenarios to show that “can an algorithm accomplish what human eyes cannot do or do better?” Unfortunately, on many occasions visual assessment may even mislead conclusions. This phenomenon will be demonstrated in the following experiments where human eye inspection can only provide qualitative assessment but not quantitative measure. One such example was demonstrated in Chang and Wang (2008) and Figure 4.14 (Section 4.4.2.2), where a fully constrained least-squares (FCLS) method (Heinz and Chang, 2001; Chang, 2003a) could not estimate abundance fractions of any target panel in the second to fifth rows in Scenario TE2 even when the embedded target panels are clean and known precisely a priori. However, if we examine Scenario TE2 shown in Figure 4.8 closely, all the 130 embedded target panels are clearly visible by visual inspection. Why was FCLS unable to estimate abundance fractions of the embedded target panels correctly? The simple reason of why FCLS failed in Scenario TE2 is not that FCLS was ineffective but rather that the simulated embedded target panels in Scenario TE2 do not satisfy ASC imposed by FCLS. To resolve this dilemma, the nonnegativity constraint least squares (NCLS) method (Chang and Heinz, 2000; Chang, 2003a) was implemented. The NCLS-estimated abundance fractions of all 130 target panels in the TE2 scenario turned out to be very accurate as demonstrated by Chang and Wang (2008) and Figure 4.14 (Section 4.4.2.2). This simple example shows how unreliable human visual inspection can be. It also further explains why the three TE scenarios, which are simulated for various signal detection models involving pure, mixed, and subpixel targets, can be used to evaluate effectiveness of signal detection techniques such as NCLS (Chang and Heinz, 2001), while the three TI scenarios, which contain various simulated pure, mixed, and subpixel targets, can be used to evaluate effectiveness of endmember extraction algorithms such as FCLS used for accurate abundance fraction estimation (Heinz and Chang, 2001). The above six scenarios are precisely designed for these purposes.
2. The TI and TE scenarios introduced above bridge a gap between computer simulations such as a Mote Carlo method and real images and provide basic understanding of real images under complete controllable environments via a set of designed parameters. They can be further used to simulate more sophisticated scenarios such as two or more BKG signatures or different noise distributions or some examples in Chapter 18.
3. It should be noted that since different spectral bands have different signal energies, in order for each spectral band to achieve the same level of SNR defined as 50% signature (i.e., reflectance/radiance) divided by the standard deviation of the noise in Harsanyi and Chang (1994), zero-mean Gaussian noises with different variances are used and added to different bands for this purpose.
4. In hyperspectral imagery noise is generally non-Gaussian. This is mainly due to the fact that many unknown subtle substances such as clutters and interferers uncovered by hyperspectral imaging sensors are actually interference and not noise, in which case these unwanted interferers should be considered as structure noise instead of random noise. If all such unknown substances are removed in the image data, which is the case in these six scenarios, it leaves only random noise. Under this circumstance, the Gaussian noise is the most appropriate assumption, which is exactly the case assumed in signal processing and communications. In light of this interpretation, it is reasonable to simulate Gaussian for Scenarios TI and TE, because the simulated image background is clean.
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