Test Data Sets

tion, and a back plane figure (5.2), which contains the targets and some natural background.

Figure 5.1: The front plane or foliage image used during the simulation. For the purposes of simulation, these images are broken down into three pri-

Figure 5.2: The unobscured back plane image containing the targets used during the simulation.

images. These additional classes have been ignored for expediency and, regardless, the ideas put forward in section 5.3 can be tested without them.

Each class is assigned an average Stokes vector:

Sgrass = h 1 −.05 .05 0 i_T Sf oliage = h 1 .03 .03 0 i_T Starget = h 1 −.46 0 0 i_T (5.24)

Each of the grass and foliage classes have an average degree of polarization of only a few percent while the target, which is made to represent a near specular reflection, is significantly higher.

The average percent pixel obscuration, E[t(u, n)], is 99.9% with a standard deviation of 0.03%. In other words, even though the front plane is “flat” it almost completely obscures the back plane at all times. Likewise, the average percentage of the maximum possible reflection off the backplane, E[r(u, n)] is 50% with a standard deviation of 10%. The purpose of r(u, n) is to partially capture the effects of depth in the obscuration by forcing the backplane to be only partially illuminated with significant overall fluctuation.

The point spread function used to blur the composite Stokes image is shown in figure 5.3 in the same scale as the image. These parts come together under equation (5.20) to form simulated Stokes images in the detector plane.

Figure 5.3: Simulated PSF.

5.4.3 Results. All simulated results are shown with a 100% linear stretch and the MATLAB default color map. Figure 5.4 shows a single simulated image from the Z1 through Z4 channels given by equation (5.18). Though present in the data, the locations of the obscured targets are imperceptible.

True to the predicted eigenvalues for the 4-channel case in equation (5.17); Z1 appears to contain the most information (largest eigenvalue), Z2 contains virtually no information (0 eigenvalue), while Z3 and Z4 both contain some information (but still no indication of the targets). Equation (5.24) shows that the targets are most highly polarized in S1 so, if the targets could be detected, it would likely occur in eigenchannel Z4 (figure 5.4d). Regardless of channel, target detection through the provided obscuration using only a single frame appears unlikely.

Figure 5.5 contains the standard deviation images of channels Z1 through Z4. Each standard deviation image is calculated from 25 separate realizations of an eigenchannel image. Each additional image is generated with a different randomized

(a) Z1 (b) Z2

(c) Z3 (d) Z4

Figure 5.4: A single frame of the simulated eigenchannels.

The derivation in section 5.3 for the 4-channel case predicts that channels Z1 and Z2 should be discarded because their variance is 0 with respect to P , which is assumed to the variable driving fluctuation from image to image. Figures 5.5a and 5.5b, the standard deviation images of Z1 and Z2, appear to generally uphold this prediction with some interesting exceptions.

First, note the foreground grass in figure 5.5a. No portion of the background scene penetrates this portion of the image and, consequently, the variance across this region is very low (the regional variance is low but not zero because of point spread function effects). In other words, since the grass here is impenetrable, it violates the original assumptions made about the obscuration and hence its behavior in the standard deviation image is not predictable by this model.

Second, there is some indication of the right-most target in the Z1 standard deviation image even though this optimization method predicts that there should be

(a) Z1 (b) Z2

(c) Z3 (d) Z4

Figure 5.5: Standard deviation images of the eigenchannels.

none. To explain this apparent incongruity, recall that the distribution over all pos- sible intensities was assumed to be separable from the distribution of possible linear polarization states. This assumption is required by equation (5.10) for formation of the multi-channel covariance matrix but is violated later when the obscured target model is defined. Additionally, variance that occurs purely due to fluctuation in total intensity is ignored throughout this development. Though clearly this assumption is not entirely accurate, it still appears to be a reasonable approximation, especially when the information content in this channel is compared with that in the Z3 and

Z4 channels.

For the same reasons, the Z2 channel and corresponding standard deviation image also contain some information content though none of it is apparently useful. Little further commentary is required to show that, as predicted by this opti- mization method with γ = 0, channels Z and Z clearly contain the most informa-

tion about the obscured targets. Consequently, no target information would have been lost if channels Z1 and Z2 had been discarded completely.

Up to this point, no consideration has been given to the original (i.e. correlated) intensity channels. Figures 5.6 and 5.7 contain a single frame and standard deviation image for the I90 channel, which, for the targets as defined, should contain the most target signal.

Figure 5.6: A single simulated frame from the I90 channel.

Figure 5.7: The I90 standard deviation image.

The I90 standard deviation image does show some evidence of the targets though the quality of the information present is far inferior to that contained in

the Z4 standard deviation image. For brevity’s sake, the other intensity channels are not shown because the results are much the same as the I90 channel.