Statistical characterization of scattering delay in synthetic aperture radar imaging

Figure 1.  Plots of $ \left\langle{|I_t|^2}\right\rangle $ and $ \left\langle{|I_s|^2}\right\rangle $ for different values of $ \zeta_{\max} $ and $ \kappa $, see (39) and (27). The dashed lines passing through the origin indicate the ambiguity direction, see (14), (18). For the middle row of plots, the condition $ \kappa \zeta_{\max} \gtrsim 20 $ (see (40)) is satisfied, and the difference in the orientation of the parallelogram-shaped level lines is more apparent than for the top and bottom rows

Figure 2.  Simulated coordinate-delay SAR images with different contrasts. We use $ \kappa = 2.5 $ and $ \zeta_{\max} = 5\pi $, which corresponds to the middle row in Fig. 1. To build each image, we sample $ \zeta_\text{d} $ and $ \psi_\text{d} $ with a step of $ \pi $ each, and for every $ \zeta_\text{d} $ (i.e., each ambiguity line) generate the multivariate circular Gaussian image components according to (36) with the weights $ \sigma^2_ \alpha K_ \alpha $ calculated via (43). The total images are then computed according to the appropriate expression in (42). Simulation of multivariate normal random variables is performed using the $ \text{MATLAB}^© $ function $\mathtt{mvnpdf}$

Figure 3.  Cumulative distribution functions (cdf) for ensembles generated from the s-model and t-model. (a) Examples of cdfs for $ \log \breve p_s $ and $ \log \breve p_t $, see (54). The notations $ \text{cdf}_s[\ldots] $ and $ \text{cdf}_t[\ldots] $ are similar to those in (60). (b) Examples of cdfs for $ l = \log \breve p_t - \log \breve p_s $, see (55) and (60). According to (61) (see also Table 1(a)), we have $ r_t = \text{cdf}_t(0; q) $ and $ r_s = 1- \text{cdf}_s(0; q) $. (c) A zoom-in to the central part of panel (b). The entries $ r'_s $, $ r''_s $, $ r'_t $, and $ r''_t $ from Table 1(b) are determined according to (63) and (65) with $ p = 0.05 $

Figure 4.  Separation between the graphs of $ \text{cdf}_s(\cdot\, ;q) $ and $ \text{cdf}_t(\cdot\, ;q) $ for different values of $ \kappa $ and $ q $. The thick colored vertical bars indicate the percentage of uncertain classifications for the ensembles generated from the s-model (the left set of bars in each plot) and t-model (the right set of bars), see also (65) and Fig. 3(c)

Figure 5.  Dependence of the discrimination quality on $ \zeta_{\max} $, see (39), for two different target contrasts. The left column of plots corresponds to algorithm (56) and Table 1(a), and the right column corresponds to algorithm (57) and Table 1(b). The dashed vertical lines are drawn at $ \kappa \zeta_{\max} = b_\Phi \approx 23 $, cf. (40). The lower and upper parts of the colored panels represent ensembles generated from the s-model and t-model, respectively. The percentage of correct classifications is shown in green (two different shades are used to distinguish between the ensemble types), incorrect in red, and uncertain in yellow (only the right column of plots)

Figure 6.  Dependence of discrimination quality on $ \kappa $, see (27). The notations are the same as in Fig. 5