A) Introduction

The synthetic RCS database has been developed during the MOSAR project [Reference Barès, Brousseau and Bourdillon2, Reference Brousseau3] with the support of the French Ministry of Defence (DGA – Direction Générale de l'Armement).

The objectives of MOSAR project are to improve knowledge of frequency response of targets in resonance region by measurements, and to test efficiency of recognition methods. These studies led to:

• – Development of a coherent, pulsed, quasi-monostatic, multifrequency, HF–VHF radar using the 20–100 MHz frequency band and horizontal and vertical polarizations.

• – Development and validation of a simulated RCS database using numerical models of aircrafts in the 20–100 MHz frequency band.

• – Development and tests of NCTR algorithms.

B) Description of the synthetic RCS database

To study aircraft RCS, several possibilities exist. One can perform:

• – Anechoic chamber measurements on real aircrafts or scaled models.

• – In-flight measurements with a radar system.

• – Simulations using a computational model.

Anechoic chamber measurements are not well suited to collect data at various angles of a target but they are useful to validate numerical models. To perform in-flight measurements, it is necessary to use a calibrated radar system and to wipe out propagation effects. The simulation of RCS behavior, using a computational model, is a very attractive scheme but the model must be validated.

To be able to use a small computer like a PC, the simulation of RCS has been made with the free Numerical Electromagnetic Code NEC2 that is based on the method of moments [Reference Burke and Poggio4, Reference David, Brousseau and Bourdillon5]. In this case, the aircraft structure is considered as a perfect electric conducting body. An example of wiregrid model is presented in Fig. 1. Currently, the synthetic database is constituted of eight mid-range airplanes: Airbus A320, BAe 146-200, Boeing 727-200, 737-200, 737-300, 747-200, 757-200, and Fokker 100.

Fig. 1. Example of aircraft modeling using a wiregrid model – BAe 146-200.

For each aircraft, RCS has been determined using the following parameters:

• – Frequency band: 20–100 MHz with a 1 MHz frequency step.

• – Azimuth angle:  − 10 to  + 190° with a step of 2°.

• – Elevation angle: 0–90° with a step of 1°.

• – Polarization: HH, HV, VH, VV.

The range profile is estimated from the 61 samples of the frequency response using an inverse fast Fourier transform with a zero padding on 256 points. So, the range step is 0.6 m. Then, a Hamming windowing is applied on the estimated range profile. The synthetic database is finally constituted of around 300 000 range profiles. Figure 2 shows an example of estimated range profile.

Fig. 2. Example of estimated range profile – BAe 146-200 – HH polarization – frequency band: 20–100 MHz.

A) Introduction

Wavelet transforms and clustering algorithms are useful in a variety of applications. Wavelets provide the analyst with an approximation of the signal and a detail of the signal as well. Clustering aims at finding a structure in a collection of unlabeled data.

These methods have powerful efficiencies. Nevertheless, each of them has its own limitations. In this case, it should be interesting to merge them to built multiresolution hierarchical trees.

For a complete description of wavelet analysis and clustering algorithms, the reader should refer to [Reference Mallat6–Reference Bezdek14]. A brief summary of how wavelets and clustering were used is presented below.

B) Multiresolution wavelet representation of RCS database

The discrete wavelet transform of finite sequences analyzes a signal S by decomposing it into approximations A _i and details D _i by quadrature filter systems [Reference Mallat6, Reference Mallat7], where i is the decomposition level.

The approximations and details are, respectively, obtained by a low-pass filter and a high-pass filter. At each level, filtering process is followed by decimation by 2 that decreases the data size. Figure 3 presents the scheme of the filter system. Finally, the approximation and detail signatures at each level are pre-processed from the original signal and placed in the training data set. Figure 4 plots an example of range profile and its wavelet decomposition computed in four levels using the Haar mother wavelet.

Fig. 3. Filter system of the wavelet transform (where S is the original signal, A _i the approximations, D _i the details, and i the decomposition level).

Fig. 4. Example of range profile and its wavelet transform computed in four levels using the Haar mother wavelet.

A previous work [Reference Brousseau15] has shown that there is no statistically significant difference in performance of the classifier when different wavelets are chosen. Thus, in the next sections, only results obtained with the Haar wavelet are presented.

C) Unsupervised clustering of RCS database

Clustering can be considered as the most important unsupervised learning problem. It deals with finding a structure in a collection of unlabeled data. Clustering is the classification of objects according to similarities among them, and organizing of data in groups.

A popular measure to determine this similarity is the Minkowski metric [Reference Duda, Hart and Stork10]:

(1)

where dim is the dimensionality of the data, and P ≥ 1 is a control of the distance growth of patterns. In our case, we have chosen to use the Euclidean distance where P = 2.

Two types of clustering methods can be defined [Reference Zeng and Starzyk9]:

• – Hard clustering techniques where data are set into C specified number of mutually exclusive subsets.

• – Fuzzy clustering techniques where data can be assigned to several clusters simultaneously, with different degrees of membership.

Data membership to a partition is usually defined by an appropriate matrix U whose factors are equal to 0 or 1 in the case of hard clustering method, or a number between 0 and 1 in the case of fuzzy clustering method. For a complete description of these unsupervised clustering algorithms, the reader should refer to [Reference MacQueen8–Reference Bezdek14].

Previous results [Reference Brousseau15] have shown that better performances are obtained with a hard clustering algorithm, like K-means, in NCTR applications. Thus, in the next sections, only results obtained with the K-means hard partitioning method are presented.

The K-means hard partitioning method is simple and popular [Reference Marques de Sà11]. From an N × n dimensional data set, K-means algorithm allocates each data point to one of C clusters to minimize the following objective function:

(2)

where A _i is a set of objects (data points) in the ith cluster and c _i is the mean for those points over cluster i.

Thus, c _i are called the cluster centers (centroids) and are defined as

(3)

where N _i is the number of objects in A _i.

D) Multiresolution hierarchical tree representation of RCS database

These previous techniques are very useful in many applications. These methods give powerful efficiencies but each of them has its own limitations [Reference Brousseau15, Reference Brousseau16].

Application of wavelets representation to NCTR application slightly decreases recognition search time but with a low degradation of false identification probability. At the opposite, use of clustering algorithm gives a very low decrease of recognition search time but with an important degradation of false identification probability. A way to improve these techniques is their association in a multiresolution hierarchical tree.

A) Introduction

To test efficiency of database compression using multiresolution hierarchical trees, many criteria can be used:

• – Probability of false classification (Pfc) as a function of signal-to-noise ratio (SNR).

• – Minimum SNR to obtain a Pfc smaller than 1%.

• – Search computational time (Sct) for a fixed SNR.

B) Probability of false classification

Probability of false classification Pfc is defined for M target classes as

(7)

where m _i is the number of classification error and n _i the number of element in the class i.

The nearest-neighbor algorithm [Reference Cover and Hart17] is used to recognize the target. It is a simple algorithm and is useful to test the efficiency of database compression using multiresolution hierarchical trees. The distance used to find the nearest neighbor is the Euclidean distance d _T^k,r,s between the RCS magnitudes A _j:

(8)

where j is the sample index, N the length of the signal, A _j^T the magnitude of measured RCS, and A _j^k,r,s a database element (aircraft k, azimuth angle r, elevation angle s).

Then, minimal distances to each aircraft are computed and the nearest neighbor k _T for the measure T is extracted like

(9)

C) Signal-to-noise ratio

To see the effect of random noise, zero-mean white Gaussian noise has been added to the signal. The signal-to-noise ratio (SNR) is defined as

(10)

where s _j is the sampled signal, N the length of the signal, and σ ² the variance of Gaussian noise.

D) Search computational time

In computing, to estimate the “search computational time” (Sct), a standard parameter is the number of MFLOPs (Million FLoating point Operations) with which comparison between efficiencies of different processing algorithms can be easily performed.

A) Introduction

Different multiresolution hierarchical trees have been designed from different beginning decomposition levels, from 1 to 4. An example of tree built from the decomposition level 4, using Haar wavelet and K-means hard partitioning algorithm, is shown in Fig. 5.

Fig. 5. Example of multiresolution hierarchical tree built from the level decomposition 4, using Haar wavelet and K-means clustering algorithm.

This tree has 21 final clusters, an average distortion of 0.56, and a partition entropy of 2.9. In this figure, the clusters are designated by C _l,j, where l is the cluster number at resolution j. The number in each circle is the percentage of data in the cluster.

Total distortion and entropy are presented in Figs 6 and 7. A decrease of total distortion and an increase of entropy as a function of the number of clusters are observed which confirms the validity of the tree designing method.

Fig. 6. Estimation of total distortion as a function of the number of clusters.

Fig. 7. Estimation of total distortion as a function of entropy.

In Figs 8 and 9, examples of range profiles contained in some clusters are shown. We can see that range profiles are associated according to aspect.

Fig. 8. Example of range profiles in each cluster at decomposition level 4.

Fig. 9. Example of range profiles in each cluster at decomposition level 3.

Figure 10 presents an estimation of Pfc as a function of SNR for different multiresolution hierarchical trees designed from different beginning decomposition levels, from 1 to 4. This figure reveals that higher is the beginning decomposition level, lower is the Pfc.

Fig. 10. Example of probability of false classification Pfc as a function of SNR for the original set and the multiresolution hierarchical trees designed from different beginning decomposition levels, from 1 to 4, using a Haar wavelet and K-means algorithm.

Figures 11 and 12 show, respectively, the variation of minimum SNR to obtain a Pfc smaller than 1%, and the search computational time Sct for a fixed SNR as a function of the beginning decomposition level used to design the multiresolution hierarchical tree. We observe a degradation of the minimum SNR to have a Pfc < 1% of 8 dB, but Sct is divided by a factor of 13.

Fig. 11. Minimum SNR to obtain a Pfc smaller than 1% as a function of decomposition level for the multiresolution hierarchical trees, K-means algorithm (C = 50), and Haar wavelet decomposition.

Fig. 12. Search computational time Sct as a function of the beginning decomposition level for the multiresolution hierarchical trees, K-means algorithm (C = 50), and Haar wavelet decomposition.

Thus, multiresolution hierarchical trees are a good solution to compress high-resolution data of radar targets. Nevertheless, other methods exist such as the use of wavelet decomposition or the unsupervised data clustering [Reference Brousseau15, Reference Brousseau16]. It must be interesting to compare these techniques as a function of probability of false classification and computational time of search.

Finally, Figs 11 and 12 compare the efficiencies of different techniques (multiresolution hierarchical trees, K-means clustering algorithm, multiresolution Haar wavelet decomposition). The lowest Sct is obtained for clustering algorithm but with the most important degradation of minimum SNR to obtain a Pfc smaller than 1%. Use of approximation signals of wavelet decomposition to NCTR application makes it possible to obtain the weakest SNR to have a Pfc smaller than 1%, in particular for the first (finer) decomposition levels (1 and 2). Use of multiresolution hierarchical trees, designed from the coarser decomposition levels (3 and 4), is a good compromise between the data clustering and the wavelet decomposition, because better performances are obtained for the minimum SNR to obtain a Pfc smaller than 1%, with a similar search computational time.