Transformation into Average-Bar and Hill-Pattern

The vehicle signatures can have a large and unequal sample vector size ni. For vehicle length ranging from 5 to 20 meters and speed ranging from 2 to 25 m/s, the detection event duration will range from 0.2 to 10s. With a sampling rate of 128 Hz for each magnetic axis, the resultant sample vector size ranges from 75 to 3840 samples.

Conventionally, the magnitude of the data samples is normalized to 0 mean, magnitude between {-1, 1} [5.28], and re-sampled to a fixed size M (e.g. M = (75+3840)/2 = 1958).

However, this step is computationally expensive, on the order of the vector size (ni) since interpolation processes have to be done between all sample points. The resulting vector of size M is still so large that its transmission would consume too much power. According to the analysis in section 5.2, instead of a very detailed signature with all tiny tips, a smooth pattern of the “peak” and “valley” is already good enough for a reasonable classification.

Therefore, two transformations, called “Average-Bar” and “Hill-Pattern,” are designed to convert the vehicle signatures of variable vector size (ni) into one with fixed vector size (N):

(Eq. 5.3.2.1)

These two transformations are computationally simple enough to be implemented in a sensor node. The resulting small vector size (N) (e.g. 20) makes radio transmission of the transformed data (ti) feasible from the viewpoint of power consumption. This permits the base station to collect and centralize all the transformed signatures for further classification processing. These two transformation processes are discussed in this section. Their

classification performance and experimental results are presented in section 5.4.

5.3.2.1 Average-Bar (AB)

Average-Bar (AB) is a transformation process designed to trim down the vector size (ni) of vehicle signature (xi) into one with small and fixed vector size (N), while retaining enough pattern information in the transformed data (ti) for further classification. The idea is simple.

We take a signature vector of size ni, group the samples into N sub-vectors and replace each of them by its average value. More precisely, the Average-Bar transformation process is:

i, Consider the signature (xi) with time window K(i)

(Eq. 5.3.2.1.1)

ii, Time window K(i) is divided into N sections

(Eq. 5.3.2.1.2)

iii, The transformed data (tAverageBar(i)) with vector size N is given by

(Eq. 5.3.2.1.3)

Whereas a(q) is a smoothed signal of the signature. Fig. 5.3.2.1.1 below shows three examples of the Average-Bar transformation from the vehicle signature of three 2-axle trucks (FWHA class 5). Signatures of 5 to 8 s and sample vector size 640 to 1024 for each magnetic axis were transformed to a fixed vector size of 20.

(a)

(b)

(c)

Fig. 5.3.2.1.1 Three examples of the Average-Bar transformation from vehicle signatures of three 2-axle trucks (FWHA class 5).

iv, Depending on the classification scheme, it can be further normalized to one with magnitude between [-1, 1]

(Eq. 5.3.2.1.4) Fig. 5.3.2.1.2 shows an example of the Average-Bar transformation with magnitude

normalization. Observe that the Average-Bar pattern of the same vehicle traveling at different (constant) speeds is the same. Thus speed information of the vehicle is not necessary for Average-Bar processing. This is a significant advantage over other re-sampling techniques that require speed normalization, in which case two sensors are

needed to obtain speed and so the re-sampling cannot be done locally within a sensor node.

Secondly, as will be seen in section 5.4, we find that N = 10 or N=20 is sufficient for purposes of classification. This is a significant reduction from the original signatures which can have average size of 1,000 samples.

Fig. 5.3.2.1.2 Example of the Average-Bar transformation with normalized magnitude

5.3.2.2 Hill-Pattern

Hill-Pattern (HP) is a transformation process designed to dramatically compress the vehicle signature into a three-valued {+1, 0, -1} signal, while keeping the minimum essential amount of pattern information needed for classification. This numerically simple

transformation packs the signal into a highly compressed one, and makes the classification computationally cheap, power efficient and executable in real time. It is very surprising that although almost all of the information in the original signature is dropped by the Hill-Pattern procedure, the classification results remain at a reasonable level with such a simple algorithm. The construction of the Hill-Pattern process is described next while its

corresponding classification performance is discussed in section 5.4.

Intuitively, the Hill-Pattern process extracts the pattern of “peaks” and “valleys” exhibited by the smoothed vehicle signature (xi). A peak is a local maximum and a valley is a local minimum of the signal xi. Thresholds have to be employed to avoid tiny fluctuations in the signal from contributing to a peak or valley. The precise steps in the Hill-Pattern process are as follows.

i, The rate of change of the signal is first transformed into an intermediate ternary or {+1, 0, -1}-valued signal according to some pre-defined threshold levels

(Eq. 5.3.2.2.1)

ii, An ad-hoc state machine is used to filter out peaks and valleys (local maxima and minima) with small amplitude or short duration. The state diagram is shown in Fig.

5.3.2.2.1. Unless the amplitude and duration of a hill pattern is large enough, the last hill pattern value will be adopted. Therefore, all the small and short hill pattern can be filtered out.

iii, In the example shown in Fig. 5.3.2.2.2, the resultant hill-pattern 2-bit array is [+1 -1 +1 -1].

Fig. 5.3.2.2.1 State diagram of the ad-hoc state machine designed to filter out hill patterns with small amplitude or short duration

Fig. 5.3.2.2.2 Example of the hill pattern transformation process, [+1 -1 +1 -1]

(a)

(b)

(c)

Fig. 5.3.2.2.3 Three examples of the Average-Bar transformation from vehicle signatures of three 2-axle trucks (FWHA class 5).