Representation of Radar Micro-Dopplers Using Customized Dictionaries

2.1. Dictionary learning

Consider the case of I targets. Each target is assumed to be moving in isolation in the propagation environment. The radar data from the target, i, is represented as a vector xiwhich consist of N samples. The training data corresponding to target i consist of M such measurements and hence XiϵN×M. The objective is to represent Xias shown in Eq. (2):

Here, DiϵN×Pand ZiϵP×Mare the dictionary and coefficient matrices corresponding to the target i. The distinction of this dictionary learning process from other data-independent transforms such as Fourier or DCT is that Zimust be a row-wise sparse coefficient matrix with a sparsity value of τ. Therefore, the objective of the dictionary learning algorithm is given as Eq. (3)

It is well known that the l₀—minimization problem stated above is NP-hard [33]. There are therefore, two approaches for solving Eq. (2). One is to implement the l₀—minimization using computationally expensive greedy techniques. Alternately, the l₀—minimization can be reduced to a l₁—minimization technique as shown in Eq. (4)

Here, λis the regularization parameter that trades off between the representational accuracy and the sparsity in the l₁—minimization operation. An iterative alternative minimization approach is used to solve Eq. (4). First, Diis initialized using random columns selected from the training data Xi. Then Ziis determined using Eq. (5) by implementing the iterative soft thresholding algorithm (ISTA) discussed in [34]

Once Ziis determined, Diis estimated using the simple least squares minimization shown in Eq. (6). Each of the P columns of the dictionary has to be normalized to less than unity in order to prevent scale ambiguities

Equations (5) and (6) are iterated until the representation error falls below a pre-defined threshold or when it converges. The process is repeated to learn unique dictionaries for every target. Then, all the dictionaries are concatenated to form an aggregate over-complete dictionary D=D1D2⋯DI. The aggregate dictionary, D∈N×IP, is used for both single channel source separation and for classification of test micro-Dopplers. Both of these are discussed in the succeeding sections.

The dictionaries of multiple targets are learned individually in Eq. (4). An alternate mechanism would be to learn the multiple dictionaries together as suggested by [35]. Here, besides the sparsity penalty in Eq. (3), an additional penalty is introduced to increase the discrimination across multiple target categories. This step increases the computational complexity during the training stages. But since there was no discernible improvement in the performance of the disaggregation and classification algorithms, this scheme is not discussed in this chapter.

2.2. Single channel source separation of multiple radar micro-Dopplers

Now each single test measurement—a vector consisting of N samples, x~∈N×1is considered. In the test scenario, there may be single or multiple targets moving simultaneously in the propagation channel. Therefore, the test signal may be the aggregate of time-domain micro-Dopplers from multiple targets. Using the aggregate dictionary, the test coefficient vector z~1:I∈IP×1is derived using Eq. (7)

Note that the test coefficient vector,z~1:I, is distinct from the columns of the coefficient matrix, Zi, derived earlier from Eq. (5). In z~1:I, the coefficients (z~i) corresponding to each of the I categories of targets are obtained at once as opposed to Eq. (5), where the coefficients corresponding to just the training target category are realized. If one or more targets are present, then the hypothesis is that their corresponding coefficients (z~i∈P×1) extracted from the composite test coefficient vector (z~1:I) will be non-sparse, while the coefficients belonging to the absent targets will be close to zeros. The intuition here is that if the dictionaries are sufficiently discriminative, then they will be able to extract the corresponding target coefficients even when the radar data consists of superposition of returns from multiple targets. Therefore, a target i is determined to be present if the corresponding Diz~iis above a predefined threshold.

2.3. Classification of radar micro-Dopplers

In the previous section, single channel source separation of test data that comprises of radar returns from multiple targets was discussed. Now, the focus is on classification of radar data from an unknown test target category. It is important to note that while the previous problem focused on multiple target detection, the problem in this section focuses on single target classification. In both cases, the aggregate dictionary matrix, D, obtained from concatenation of the individual dictionaries learned from each of the target categories is utilized. For classification purposes, the training features for each target i are derived from the coefficient vector, Z¨1:Ii, obtained using the training data, Xi, and the aggregate dictionary using Eq. (8)

Though the training data matrix is identical, the training feature, Z¨1:Ii, is distinct from the previously derived Zifrom Eq. (5) since Z¨1:Iiis a composite matrix derived from the aggregate D rather than an individual dictionary Di. The the training feature shows the sparsity pattern corresponding to an i^th target within the cluster of coefficients from multiple targets. Any well-established classifier, such as support vector machine or K-Nearest neighbor, etc., can now be trained with multiple columns of the training feature matrix. Each of the columns of Z¨1:Iiis treated as a single training instance for the i^th target and the classifier is similarly trained for all the target classes.

During the test stage, test features, z¨1:I, are extracted from each test measurement, x¨i(from a single target category) using Eq. (9)

The test data are subsequently classified based on the similarity between z¨1:Iand the columns of Z¨1:Iiacross all the Itarget categories. In this chapter, the support vector machine classifier is used. The data were tested on other popular classifiers such as KNN but did not result in any significant difference in the results.

3.1. Simulation data

Simulation of radar scatterings of still humans has been investigated with full wave electromagnetic techniques as well as the computationally cheaper ray tracing technique at frequencies below X band [36]. The results from the simulations of a uniformly dielectric human body showed that the ray tracing results were comparable to the results from full wave methods. However, both of these methods are computationally not feasible for simulating radar returns from dynamic humans since this requires modeling of multiple human poses. Alternately, the simple primitive-based modeling has proven to be reasonably accurate for modeling human motions [37, 38]. Here, the different body parts on the human are modeled as simple primitives such as the head as a sphere, the torso as a cylinder or ellipsoid, and so forth. The radar cross-sections of these simple shapes (σb) are well characterized for different carrier frequencies (fc) and aspect angles. One or more point scatterers (r→b) are identified for each body part. When the human moves, the time-varying positions of the point scatterers give rise to micro-Dopplers. Then, the radar returns of the human are obtained by the complex sum of the returns from each of the body parts as shown in Eq. (10)

This simplistic model is easy to execute in real time. However, it does not capture the entire physics of the human scatterings. For instance, it does not capture the multiple scatterings of waves between the different body parts or their shadowing effects.

There are three methods that are currently used to describe human motions. The simplest method is to model the swinging motion of the two legs as a double pendulum. A more complete analytical model, known as the Boulic-Thalmann model, was proposed in [37]. This model provides analytical equations to describe different human body parts (arms, legs, hands, and feet) as a function of the human height and relative velocity with respect to height. However, the model is restricted only to a simple human walking motion. More complex and realistic motions, such as crawling, hopping, and running, can be obtained using computer animation data. The radar scattered returns from complex human motions can be therefore obtained by combining animation data with the primitive-based electromagnetic modeling [38]. However, the animation data are obtained through motion capture technologies of a live actor. Therefore, the model cannot be parameterized to obtain varied data for different humans (of different heights or gait patterns) through a single measurement.

In this study, the Thalmann model was used to model human walking motions for multiple human heights (1.5–1.8 m) and velocities (1.5–3.6 m/s). Due to the limitations of the model, a variety of human motions could not be simulated. Instead, just two types of motions were considered—when a human is walking towards the radar and when the human is walking away from the radar. The human moves in the line-of-sight of a monostatic continuous wave radar anywhere from a distance of 2–8 m. The duration of the human motion is 1 s and the sampling frequency of the simulation is 1 kHz. We imparted frequency diversity to the simulation data by varying the carrier frequency across five values—{2.5, 3, 3.5, 4, and 4.5 GHz}. Three hundred and sixty distinct walking motions are simulated both toward and away from radar at each of the carriers. Of these, 80% of the 1800 total simulations—corresponding to four of the five carrier frequencies—were used for training the classification algorithm; and the remaining 20%—from the fifth remaining carrier frequency—were used for testing purposes. The objective here is to test the capability of the dictionary learning algorithm to learn fundamental features from the human micro-Doppler data pertaining to a specific motion despite the variations of the carrier frequency. The dictionaries are learned directly from the raw data without any additional post processing. Therefore, at no stage were any parameters heuristically chosen.

A third category of indoor mover may give rise to significant dynamic clutter—a fan. Hence, a fan was modeled with three blades and multiple point scatterers on each blade. A lot of variety in the experimental data from the fan was generated by changing its speed of rotation (200–400 rpm), the length of the blades (0.2–0.4 m), width of the blades (0.14–0.17 m) and the orientation of the fan with respect to the monostatic radar. The fan micro-Dopplers at five carrier frequencies (1800 distinct simulations) were simulated of which data from four carriers were used for training and the data from the fifth carrier for testing purposes. The results and analyses for dictionary-based classification of simulated human radar data are presented in the following section.

3.2. Measurement data

Next, experimental data were collected in indoor line-of-sight conditions. A monostatic continuous wave radar was set up using a N9926A FieldFox vector network analyzer (VNA) and two linearly polarized broad band horn antennas (HF907). The VNA was configured to make narrowband measurements around a center carrier frequency (Figure 2).

The transmitted power of the VNA was set at +3 dBm. The instrument is highly sensitive with a dynamic range of over 100 dB. The received signal is amplified, in-phase quadrature (IQ) demodulated and digitized inside the instrument and then directly processed in a 2.4GHz Intel Pentium processor. Two sets of experiments were carried out.

The first experiment was carried out to validate the multiple target detection algorithm based on dictionary learning. The center frequency of the VNA was set to 7.5 GHz. Then, human radar data were collected for two types of motions—when a human is walking towards the radar (FH) and when a human is walking away from the radar (BH). Each measurement trace is 2.7 s long with 1000 samples. The low sampling frequency is due to the system constraints of the VNA when it is configured in the narrowband mode. Measurement data were gathered for 40 humans of both genders and of varying heights and gait patterns. The humans move between 1 and 9 m away from the radar in line-of-sight conditions. The third motion class that was considered was of a table fan (TF) with three blades. The table fan was operated at three different rotation rates and was placed at different distances and orientations from the radar. Then, measurement data were gathered with multiple targets moving simultaneously. The cases are: FH + BH, FH + TF, BH + TF, and FH + BH + TF. The objective is to learn dictionaries using training data in the single-target scenario. Then these dictionaries are combined and used to detect targets in multiple target scenarios.

The next experiment that was conducted was again based on the same set up. However, this time only single-target scenarios were considered. Instead, the carrier frequency of the measurement data was varied across five values—{2.5, 3, 3.5, 4, and 4.5} GHz. A variety of human motions were considered—two humans walking simultaneously before the radar (TH), human standing still but boxing with his arms (HB) and a human walking while holding a stick (HHS). The last case that was considered was of the rotating table fan (TF). The challenge in this experiment is to learn dictionaries and training features from measurement data corresponding to four carrier frequencies, while testing the classifier with data from a fifth distinct carrier frequency. This experiment was specifically chosen since the human micro-Doppler shows a lot of variations due to the carrier frequency and the Dopplers are directly proportional to the frequency. The table fan was selected for both types of experiments, since it is one of the key contributors of dynamic clutter in indoor environment.

4.1. Results from multiple target detection using disaggregation of radar data

First, the measured data that are gathered for multiple target detection are considered. As mentioned in Section 3.2, three target classes—human walking toward the radar (FH), away from the radar (BH), and table fan (TF) are considered. The dictionaries are learned from single target data and use them to detect the presence of multiple simultaneous movers in three scenarios—single-target scenario, two-target scenario, and three-target scenario. The true detection and false alarm percentage for each of these cases is summarized in Table 1. The results show that for a single-target scenario, the true detection is very high (above 93%) in all the three cases. The true detection of the fan is slightly poorer than that of the humans because of greater probability of aliasing arising with the fan micro-Dopplers due to the limited sampling frequency of the radar measurements. This also gives rise to the slightly higher false alarm rate of the fan when compared to the humans. Next, three two-target scenarios are considered. In each of the scenarios, the algorithm correctly detects the presence of two targets in more than 80% of the cases by disaggregating their micro-Dopplers. The false alarm rate though is high especially from the table fan due to the aliasing. In the three-target scenario, the algorithm correctly detects the presence of all three targets in more than 90% of the cases. This sort of multiple target detection cannot be carried out using basic data independent transforms and this result demonstrates the usefulness of representing micro-Dopplers with unique data-dependent dictionaries.

4.2. Results of single target classification using frequency diverse radar micro-Dopplers

Now the objective is to correctly classify that target class when the propagation channel consists of only one type of target class. The challenge here is to classify test data where system conditions (carrier frequency) during test deviate significantly from the training conditions. Therefore, during the training stage, the dictionaries for each target class are learned from diverse frequency data (from four carrier frequencies) and the test data consists of micro-Dopplers from a fifth carrier frequency that was previously not used for training.

The following five carrier frequencies—{2.5, 3, 3.5, 4, and 4.5} GHz are considered. In fold 1 through fold 5, the test frequencies are 2.5, 3,…,4.5 GHz, respectively, and the training data for each fold are obtained from the corresponding complimentary set from the total set of carrier frequencies. The performance of the algorithm for both simulation and measurement data are studied. In the simulation set up, three target classes are considered—a human walking towards the radar (FH), a human walking away from the radar (BH), and a fan (TF). The classification results obtained from the fivefold classification are presented in Table 2. The results show very high classification accuracy (close to 100%) for all the cases. This shows that the algorithm is capable of learning unique dictionaries corresponding to each human motion from frequency diverse training data. Again, this type of classification would not be possible with other data-independent transforms which rely on heuristic parameter selection.

The simulation data are highly sanitized, since they are not affected by real world conditions such as noise, interference, and radar system limitations. Therefore, in the next study, the performance of the algorithm on real world data is studied. Four types of motions—two humans walking simultaneously in the propagation channel (forming a single class of motion—TH), a single human standing still and boxing his arms (HB), a human walking with a stick (HHS), and a table fan are considered. Five-fold classification is performed on the target data and the results are presented in the following two tables. Table 3 shows the confusion matrix obtained for a single fold (fold 4). The column headers show the class labels that the classification algorithm assigns to the test data. The true class labels are shown as row headers. The results show that TH, HB, and TF are classified with 100% accuracy, while the HHS is confused with TH and HB. There are two reasons for this confusion. When two humans are walking together, there are times when the scattered signal from one human may be weaker than the other human (due to shadowing or due to different distances of the two humans from the radar). As a result, there are similarities at times between TH and HHS. Second, when the human is boxing, the micro-Dopplers occur at both positive and negative frequencies due to motions of the arms toward and away from the radar. This can be confused with the HHS, especially from the back swing of the legs, stick, and arms. The micro-Dopplers of both these motions could resemble each other especially across different carrier frequencies.

Table 4 shows the classification results across all the fivefolds. The results show a fairly good classification performance (above 85%) for all of the cases. The confusion of the table fan with the human motions can again be attributed to the limited sampling frequency of the measurement data.