14 different loss functionsH (L1,L2, andL3 are the benchmarks). The results are reproduced from V.
Loss functionH Mean residual errorhσei[m] Image contrast
L1 0.1551 17.9824
formatting was calculated. In the performed tests, 20 different target trajectories were considered. The mean values of these results for 14 different choices ofH are shown in Table 3.1. The benchmarks areL1,L2, andL3. As seen from the results in Table 3.1, the estimation accuracy of the new loss functions (3.1) was as much as 35 percent higher than the accuracy of the benchmarks. Noteworthy, the improved performance was achieved for a very high range resolution (10 cm) and for a significant object rotation (25 degrees during the CPI). The increased estimation accuracy manifested itself as an improvement of about 7 percent in the ISAR image contrast value.
In I, we used a heuristic optimization algorithm to solve the minimization problem. DE was used with a spline polynomial parametrization for the range shifts. The first and second order optimization approaches in IV and V are more sensitive to the energy scale of the signal (which can cause large differences in the values of the different loss functions) and scale differences in the coefficients of the polynomial model. However, as demonstrated in V, the computational burden of the heuristic optimization approach in I is approximately two orders of magnitude higher. The algorithms proposed in IV and V solve the range alignment in a matter of seconds, while the DE approach takes up to a few minutes even with a relatively accurate initial guess.
3.3 Time window optimization
The purpose of time window optimization is to choose an optimal CPI from the available data for the ISAR image reconstruction. This can be achieved by locating a slow-time window during which the object movement is as smooth as possible. To determine this slow-time window an optimization problem with a suitable loss function needs to be defined. The most straightforward way is to use the negative ISAR image contrast, as was done in [43]. In this approach, the ISAR image reconstruction is done by assuming the simple signal model (2.12) and using a one-dimensional Fourier transform. However, this approach has its drawbacks. Because this procedure takes place after range alignment, the residual phase error φe can cause the image to be significantly out of focus. Also,
32 Chapter 3. Optimization framework for imaging non-cooperative moving objects scatterer RCM due to the rotational motion needs to be taken into account when a very high resolution image is the desired outcome.
To deal with the two aforementioned problems, we proposed two modifications for the time window optimization procedure in IV. As in [43], the contrast of the intensity-normalized ISAR image is used as the quality measure to determine the optimal lengthT and location tc for the CPI. To deal with the scatterer RCM, keystone formatting [44,45] is used prior to the image reconstruction. Additionally, autofocus using the PGA algorithm [30,32,118]
is applied to remove the spatially invariant phase errorsφeto obtain a well-focused image.
A drawback of this approach is that the modifications entail an increased computational burden for the time window optimization. The increased computation is countered by suitably modifying the PGA algorithm, using a sub-optimal interpolation in the keystone formatting, and using a local numerical optimization algorithm in solving the minimization problem.
In II, we proposed a method for determining the optimal CPI for keystone formatting without having to reconstruct the ISAR image. The optimization procedure of this approach is based on the same loss functions utilizing the sum envelope of range-compressed signal as in the range alignment problem. In II, time window optimization based on the image contrast was applied after this operation. Thus, the formulation presented in IV combines the two different parts of the algorithm that were used in II, which results in a computationally more efficient algorithm.
Fig. 3.4 illustrates the loss function evaluation process in the time window optimization.
The highlighted motion compensation steps are described next. An ISAR imagesS from a CPI of lengthT located aroundtc is obtained as
sS(tc, T;x, y) =Ft→y
Before evaluating the contrast of the image (3.9), PGA and keystone formatting are applied to remove both spatially invariant and spatially variant defocusing effects. First, the residual phase errorsφeare compensated for by using PGA, because the non-linear part ofφeaffects the result of keystone formatting in an undesirable way. The standard PGA algorithm is used with a simple modification to reduce the need for multiple iterations. Namely, the circular shifting operation of PGA is replaced by a more efficient procedure. The method was originally proposed in [134]. It removes the phase offsets from the phase derivative estimates of different range bins by subtracting the time average of the phase derivative from the estimate in each range bin. This results in a more accurate removal of the offsets than the circular shifting operation, especially if the image (3.9) is severely defocused. This increased accuracy enhances the estimation performance in the first iterations, which speeds up the convergence of the PGA. This causes significant computational savings in the time window optimization, because PGA is used every time the loss function is evaluated. The details of this process are described fully in IV.
Only one iteration of PGA is applied in the time window optimization algorithm to reduce the computational cost. The procedure described above makes sure that we get the best possible result out of this single iteration. Once PGA produces an estimateφbe for the phase errors, the range-compressed signal is phase-corrected according to
ssc(tc, T;x, t) = Π t−tc
T
ss(x, t)e−iφˆe(t). (3.10)
3.3. Time window optimization 33
Figure 3.4: The loss value in the time window optimization is calculated after motion compensation is performed for the time-windowed signal. The proposed motion compensation steps are highlighted.
Table 3.2: Optimal window lengths (T), image contrast values, and relative computational speeds for different motion compensation strategies in the time-window optimization. The first row (COA) represents the benchmark algorithm [43].
Motion compensation T [samples] Image contrast Relative speed
COA 336 77.57 1
COA + Keystone formatting 352 82.55 0.79
PGA 320 90.30 7.5
PGA + Keystone formatting 363 99.89 4.8
Next, keystone formatting (denoted asK) is applied to remove the linear RCM of the scatterers caused by the object rotation as
ssK(tc, T;x, τ) =Fk−1
x→x{Kt→τ{Ssc(kx, t)}}. (3.11) The intensity-normalized intensity ISAR image is obtained by using (3.11) as
Ib(tc, T;x, y) = I(tc, T;x, y) RR∞
−∞I(tc, T;x0, y0)dx0dy0, (3.12) where I(tc, T;x, y) = |sSK(tc, T;x, y)|2. The loss function can be defined in a similar manner as in COA (see (2.15)); the only difference is that now it only depends on two variables (T andtc). The time window optimization is carried out by minimizing this loss function using a black-box local numerical optimization algorithm. Assuming that the PRF is high compared to the time scale in which the motion of the object changes significantly the loss function behaves relatively smoothly. It should be noted that the loss function is not convex, and thus a local optimization approach can lead to a sub-optimal solution.
In the experimental section of IV, the result of the method described above was compared to the maximum contrast time window optimization procedure proposed in [43]. The point of this comparison was to illustrate the effect of the motion compensation strategy included in the loss evaluation. In the benchmark [43], only COA is performed prior to evaluating the image contrast. To ensure that the comparison was done in a meaningful way, a simple coordinate descent optimization algorithm was used in solving the optimization
34 Chapter 3. Optimization framework for imaging non-cooperative moving objects problems. The results of the study are shown in Table 3.2. They show that our strategy of using PGA and keystone formatting is computationally more efficient than the benchmark and it produces the best image contrast value. The inclusion of the keystone formatting increased the contrast by 6 – 10 percent compared to both benchmarks, COA and PGA.
Noteworthy, our strategy of using PGA and keystone formatting was almost five times faster than the benchmark method [43] based on COA alone.