6.5. IMPLEMENTATION ISSUES 151
image-plane Cartesian axes in Figure 6.1), we will follow convention and enumerate the four quadrants as first, second, third, and fourth in a counter-clockwise fashion starting at an angle of 0 .
Table 6.1: Properties of quadrants and quadrant transition schemes. Quadrant Headings Non-Zero Transition Scheme Elements, Z
1 [0 , 90 ] {4, 5, 7, 8}
2 [90 , 180 ] {1, 2, 4, 5}
3 [ 180 , 90 ] {2, 3, 5, 6}
4 [ 90 , 0 ) {5, 6, 8, 9}
For the purpose of this chapter, we shall assume that the aircraft’s post-manoeuvre heading C is known to belong to a specific quadrant. This quadrant assumption enables
a natural description of the heading and transition scheme uncertainty sets P⌫ and ⌅✓,
respectively. In particular, we will use the sets of possible post-manoeuvre headings listed in the second column of Table 6.1 (for each given quadrant) to specify the uncertainty setP⌫ and design our HRC detector (6.16).
Furthermore, our quadrant assumption implies that the post-manoeuvre transition scheme ✓C is a quadrant transition scheme in the sense that it only has four non-zero
transition probabilities (similar quadrant or directional transition schemes were first introduced in [15, Section III.C]). For convenience, we will denote the indices of the non-zero transition scheme elements asZ ⇢ R. The sets Z are given in the final column of Table 6.1 for each quadrant. Importantly, given the quadrant and set of non-zero transition scheme elements Z, we will let the uncertainty set ⌅✓ in the design our TRC
detector (6.31) be
⌅✓ = ✓2 ⇥ : 0.001 ✓i 1 for i 2 Z and ✓i = 0 for i62 Z (6.33)
where we have selected the lower bound 0.001 to avoid numerical issues in finding the transition scheme ˜✓ satisfying (6.30).
Finally, we highlight that the parameter estimation problems that our TGLR (6.22) and TGC (6.29) detectors are required to solve are greatly simplified by our quadrant assumption. Indeed, under our quadrant assumption, the TGLR and TGC detectors are only required to estimate four non-zero elements of the unknown quadrant transition
152 CHAPTER 6. VISION-BASED AIRCRAFT MANOEUVRE DETECTION
scheme ✓C. We next describe the implementation of our TGC detector.
Remark 6.9 Whilst our quadrant assumption allows us to simplify the implementation of our manoeuvre detectors, it is not overly restrictive because target tracking and detec- tion filters for 90 heading ranges are typically implemented in parallel for this application (for example, see [15, 53, 57]). We highlight that we do not use our quadrant assumption to select the transition scheme ✓ given in Section 6.5.1 for our heading-based methods.
6.5.3 Implementation of TGC Detector
In order to implement our proposed TGC manoeuvre detector (6.29) we followed the procedure presented in [15] for selecting a set of test HMM representations . Under our quadrant assumption introduced in Section 6.5.2, we used a set of | | = 4 test HMM representations to estimate the unknown post-manoeuvre quadrant transition scheme ✓C. In particular, we exploited the results of [16] to design from cardinal
(or half-plane) transition schemes matched (in a relative entropy rate sense) to quadrant transition schemes representing aircraft image-plane speeds of between 0.2 and 0.3 pixels per frame (consistent with the aircraft speed assumption introduced in Section 6.5.1). For completeness, in Table 6.2 we have documented the sets of test HMM representations we used to estimate ✓Cwhen we knew it was a first or second quadrant transition scheme. Similarly, Table 6.3 lists the sets of test HMM representations we used to estimate ✓C
when it was known to be a third or fourth quadrant transition scheme. We now describe how these test HMM representations are used in the operation of our TGC detector.
During its operation, our proposed TGC detector (6.29) requires filters of 3 types: 1. A pre-manoeuvre HMM filter (with transition scheme ✓B);
2. A bank of HMM filters (matched to the set of test HMM representations ) that are used to estimate ✓C using our TRER-based estimator (6.26) for each time n in
the window [k w + 1, k 1]; and
3. A bank of post-manoeuvre HMM filters with one filter for each estimated transition scheme ˆ✓k|[n,k] in the window n2 [k w + 1, k 1].
We highlight that the first type of filter (the pre-manoeuvre filter) is matched to the (known) pre-manoeuvre transition scheme ✓B, and its normalisation factors Nk ✓B are
6.5. IMPLEMENTATION ISSUES 153
Table 6.2: Sets of Test HMM Representations for First And Second Quadrants
Elements Set for First Quadrant Set for Second Quadrant
✓(1) ✓(2) ✓(3) ✓(4) ✓(1) ✓(2) ✓(3) ✓(4) ✓1 0 0 0.007 0.0154 0.007 0.0154 0.007 0.0154 ✓2 0 0 0.038 0.0515 0.153 0.221 0.038 0.0515 ✓3 0 0 0 0 0.007 0.0154 0 0 ✓4 0.038 0.0515 0.153 0.221 0.038 0.0515 0.153 0.221 ✓5 0.757 0.645 0.757 0.645 0.757 0.645 0.757 0.645 ✓6 0.038 0.0515 0 0 0.038 0.0515 0 0 ✓7 0.007 0.0154 0.007 0.0154 0 0 0.007 0.0154 ✓8 0.153 0.221 0.038 0.0515 0 0 0.038 0.0515 ✓9 0.007 0.0154 0 0 0 0 0 0
Table 6.3: Sets of Test HMM Representations for Third and Fourth Quadrants
Elements Set for Third Quadrant Set for Fourth Quadrant
✓(1) ✓(2) ✓(3) ✓(4) ✓(1) ✓(2) ✓(3) ✓(4) ✓1 0.007 0.0154 0 0 0 0 0 0 ✓2 0.153 0.221 0.038 0.0515 0 0 0.038 0.0515 ✓3 0.007 0.0154 0.007 0.0154 0 0 0.007 0.0154 ✓4 0.038 0.0515 0 0 0.038 0.0515 0 0 ✓5 0.757 0.645 0.757 0.645 0.757 0.645 0.757 0.645 ✓6 0.038 0.0515 0.153 0.221 0.038 0.0515 0.153 0.221 ✓7 0 0 0 0 0.007 0.0154 0 0 ✓8 0 0 0.038 0.0515 0.153 0.221 0.038 0.0515 ✓9 0 0 0.007 0.0154 0.007 0.0154 0.007 0.0154
154 CHAPTER 6. VISION-BASED AIRCRAFT MANOEUVRE DETECTION
used to calculate the denominator of the likelihood ratios (6.21) used in the test statistic of the TGC detector.
The second type of filters are used for TRER-based parameter estimation purposes. In our implementation, these filters form a bank of w⇥ | | filters since a filter is run for every HMM in , and every potential manoeuvre time in the window [k w + 1, k]. The bank of filters is “rolling” in the sense that with each time step forward in k, the| | oldest HMM filters are stopped (because their start time is now outside the considered time window n2 [k w + 1, k 1]) and | | new HMM filters (corresponding to manoeuvre time n = k) are initialised with the previous estimate ˆXk 1 ✓B from the pre-manoeuvre
filter. Importantly, at time step k, for each n2 [k w + 1, k 1], an estimated transition scheme ˆ✓k|[n,k] is found by solving the optimisation problem (6.26) using the di↵erence in probabilistic distances (6.27) from the rolling filter bank, and an optimisation routine exploiting the TRER closed form (6.28) (e.g. MATLAB’s fmincon).
Finally, the third type of filters are used to calculate the numerator of the log- likelihood ratios Znk⇣✓ˆk|[n,k]⌘ given by (6.21) with the unknown ✓C replaced by the estimate ˆ✓k|[n,k] for each n2 [k w + 1, k 1]. In particular, at time k, the observations
y[n,k] are reprocessed by w 1 HMM filters (corresponding to each of the determined estimates ˆ✓k|[n,k], for n 2 [k w + 1, k 1]), to produce estimated versions of the
normalisation factors Nn,i
⇣
✓B,Ci ⌘. Given the log-likelihoods for each n 2 [k w + 1, k 1], the test statistic of the TGC detector (6.29) is then trivially calculated by performing a search. We will next describe the computational e↵ort of our proposed TGC detector alongside that of our other transition-based and heading-based aircraft manoeuvre detectors.