Shape and orientation classification of objects based on their electromagnetic signatures using convolutional neural networks

The field of machine learning continues its rapid and massive evolution, proposing new variants of existing network models. New families of networks are appearing less frequently, but it is clear that the trend is for these networks to increase in complexity, whatever the field of application and the type of data considered.

As studies [24, 25] have shown, this significant increase in complexity does not bring the same proportional gain in performance. The most classical networks, and the underlying machine learning mechanism based on the optimization power of the frameworks associated with them, provide in most cases very satisfactory results for a reasonable computational cost and memory footprint.

It is worth noting that the larger the networks, the greater the amount of data required to train them, and such data are not always available for scientific problems. In other words, the size of the network must be adapted to the size of the data generated, which here is limited to 12 objects multiplied by the number of observation angles (36 by 73). Increasing the number of observations means increasing simulation times, which are already substantial, and is therefore of no real interest. The reduction of the spatial step to 2 degrees has already been dictated by the need to generate sufficient data per object for our simple CNN architecture in the angular classification part of this work. Thus more complex networks would have need to significantly increase the size of the databases.

More recent network architectures such as ReSNet [26], U-Net [27], or transformers [28] have often proved indispensable for solving complex ML problems such as object detection, language processing (NLP) or multimodal fusion. But on simpler classification problems, such as the one considered in this paper, their use is often unjustified, as they often only allow to gain the few last performance points at the cost of a significant multiplication in complexity. This is illustrated by the example of image classification, which serves as a benchmark for all the models in the literature.

As an example, ConvMLP-S is a hierarchical Convolutional MLP, which is a light-weight, stage-wise, co-design of convolution layers, and MLPs. ConvMLP achieves 76.8% top-1 accuracy on ImageNet-1k with 9 M parameters and 2.4G MACs. In comparison, most recent models as ViT-e reaches 90.9% but at the cost of 3900 M parameters and 5777 GFlops. Even ReSNet-152 with 152 layers and 11.3 billion FLOPs only reaches 78,5%. On a less complex dataset as CIFAR-10, ConvMLP obtained already 98.6% where all ViT models obtained between 99 and 99.5%.

The cost/performance trade-off offered by CNN networks is thus perfectly suited here to the nature of the data and the objectives of our study. Indeed, as we will show in sections 2.6 and 3.1 a convolutional model is already capable of achieving over 90% accuracy on our inverse problem, so not justifying to increase model complexity by several orders of magnitude. Especially since one of our long-term objectives is to aim a full integration of the SEM data capture and processing system on an on-board embedded target, in order to create an autonomous and portable detection system.

For all these reasons, we adopted in this work the LeNet-5 architecture to train both SF datasets [29]. The filter size was changed to be applied to the 1-D input signals. For the SEM data, we applied one convolution layer with 6 filters, followed by one hidden layer with 32 neurons, and a ReLU activation function applied in all layers.

The aim of this study is to classify objects based on a single observation of the SF over an ultra wide frequency range (i.e. ultra wide band (UWB) illumination). The first step is to generate a SF database of all the categories of objects to be classified. The UWB SF from 12 objects with different geometries or materials was computed in the far-field region from 10 MHz to 5 GHz. They were illuminated using a plane wave, as shown in figure 1, where the incident wave vector $\vec{k}_{\textrm{inc}}$ is defined as follows:

$$\begin{equation} \vec{k}_{inc} = \begin{bmatrix} cos\left(\phi_{\textrm{inc}}\right) sin\left(\theta_{\textrm{inc}}\right)\\ sin\left(\phi_{\textrm{inc}}\right) sin\left(\theta_{\textrm{inc}}\right)\\ cos\left(\theta_{\textrm{inc}}\right) \end{bmatrix}. \end{equation} \tag{ 1 }$$

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First, we have five spheres with different permittivity ε_r and conductivity σ where the SF is computed analytically using Mie series algorithm implemented in Matlab [30]. The spheres were illuminated using an x-polarized incident plane wave propagating along z axis. The SF is recovered in a bistatic configuration, where θ varies from 0^∘ to 180^∘ with 5^∘ steps and φ from 0^∘ to 90^∘ with 10^∘ steps. Second, we have seven different perfect electric conductor (PEC) objects, in which the simulations are carried out in a monostatic configuration by using the time domain solver of CST Microwave Studio, which is a 3D electromagnetic simulation software [31] . Hence, they were illuminated using a plane wave with multiple incident directions, chosen according to the symmetry planes of each object.

The 12 objects are enumerated as class 0 to 11 as follows:

• Class 0: PEC sphere
• Class 1: lossy sphere ε_r = 4 and σ = 0.5
• Class 2: lossless sphere ε_r = 2
• Class 3: lossless sphere ε_r = 4
• Class 4: lossless sphere ε_r = 9
• Class 5: PEC ring
• Class 6: thin wire
• Class 7: thick cylinder
• Class 8: ovoid
• Class 9: cube
• Class 10: rectangular solid
• Class 11: equilateral pyramid

In this manner, we dispose of raw UWB SF data for these 12 objects. We then apply the SEM to pre-process the raw data and extract the characteristic parameters of each object from its UWB SF.

Pre-processing of SF allows the estimation of CNRs and their associated residues of an object from its complex frequency response. Vector fitting (VF) is a widely used method for fitting a measured or simulated frequency-domain response H(s) using the following rational function [32]:

$$\begin{equation} H\left(s\right) = \sum_{m = 1}^{M} \frac{R_{m}}{s - {a_{m}}} + d +s*e \end{equation} \tag{ 2 }$$

where M is the model order, $s = j\omega$, a_m are the poles, and R_m are the residues. d and e are optional real constants [32]. The poles have the following form: $a_{m} = -\sigma_{m} + j\omega_{m}$, where σ_m is the damping factor, and ω_m is the natural pulsation of the m_th singularity. VF method was selected to determine the CNRs of an object as it showed better robustness to noise and improved accuracy than other SEM techniques (TLS Cauchy method as an example among others) [33].

The first step of the VF algorithm is to estimate CNRs by assuming a set of starting complex poles uniformly distributed over the frequency range. Subsequently, through successive iterations, the algorithm relocates them to the actual poles [34, 35]. If the data are noise-free, two iterations are sufficient to obtain accurate CNRs when the number of starting poles is greater than M. Further iterations are required for convergence in presence of noise. For example, at 10 dB SNR, we applied ten iterations in the VF algorithm. Once the CNRs have been determined, the residues can be computed by solving the corresponding least-squares (LS) problem.

We also computed the quality factor (Q-factor) associated with each pole [16]. This is computed as follows:

$$\begin{equation} Q_{m} = -\frac{2*\pi*f_{m}}{{2*\sigma_{m}}}. \end{equation} \tag{ 3 }$$

The Q-factor allows the description of an object from its CNRs independently of its size. Strong resonating objects, such as thin wires, exhibit resonances with a high Q-factor, whereas weak resonating objects, such as PEC spheres, are characterized by a low Q-factor.

CNNs are among the most commonly used ANN algorithms. They are composed of multiple layers of different types. The most common layers are convolution, pooling, and fully connected layers. The convolution layer is a filter of specific length and width that moves along with the input data. It is composed of kernels or filters applied to extract specific features from an input vector [36].

The pooling layer is a sub-sampling layer that is used to reduce the size of the features computed using the convolution layer. Here, we used the maximum pooling layer that outputs the maximum input value. Fully connected layers are used to process the output features of the last convolution or pooling layer. They are composed of neurons connected to all the neurons in the previous layer. They applied a non-linear activation function to the sum of the products of the inputs related to their weights [36]. Neuron j in layer m performs the calculations indicated in equation (4):

$$\begin{equation} y_{j}^{m} = f\left(\sum^{N_{m-1}-1}_{i = 0}w_{ij}*y_{i}^{m-1}\right) \end{equation} \tag{ 4 }$$

where $y_{j}^{m}$ is the output of neuron j in layer m, N_m is the number of neurons in layer m, w_ij is the weight between the neurons of layer m + 1 and layer m, and $f(.)$ is a non-linear activation function. In this study, we use the rectified linear unit (ReLU) function because of its computational efficiency, enabling the system to converge quickly and prevent over-fitting [37].

To classify an object, we can directly use raw SF data or pre-processed SEM data, as shown in figure 2. We evaluated these approaches to determine the most efficient data type for object classification.

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2.5.1. SF dataset.

2.5.2. SEM dataset.

Each dataset was divided into 80% for training and 20% for validation (tables 1 and 2). The remaining 20% was composed of random observation angles that were not present in the training set, where each class had an equal number of samples. The results were averaged over 10 runs. We then test noisy data, which are data affected by an additive white Gaussian noise (AWGN) with different levels of SNR. Next, we present the results of the generalization ability of the CNN for noisy data from unseen object (i.e. according to its size). The object sizes were either larger than those included in the training datasets (${\gt}$15 cm) or smaller (fewer than 5 resonances for the SEM data) as reported in table 3. Indeed, decreasing the object size means that the first five natural frequencies are shifted upward within the frequency band.

2.6.1. Training phase.

We adopted the LeNet-5 architecture to train both SF datasets [29]. The filter size was changed to be applied to the 1-D input signals. For the SEM data, we applied one convolution layer with 6 filters, followed by one hidden layer with 32 neurons, and a ReLU function applied in all layers.

This was performed over 200 epochs for the SEM data and 800 epochs for the SF data, with a batch size of 100. The output layer was composed of 12 neurons, using the softmax activation function to compute the output probabilities. The learning rate was updated using the Adam optimizer [38].

Figure 3 shows the training and validation accuracies as a function of the number of epochs for the three datasets. We can see that the training curve converges quickly to an accuracy of 100% for SEM data from 125 epochs, while presenting much fewer fluctuations than both SF data curves, because the SEM data are easier to learn.

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Table 4 lists the execution times recorded during the CNN training phase. We observed that the training runtime using the SEM data was much faster than that using raw data. The SEM pre-treatment using VF accounts for 1 s when applied to all observation angles of an object. This means that even when considering the VF time in the overall procedure, the computation time remains favorable to the SEM data.

Table 4. Time consumption of the CNN trained using the different datasets.

Database	FD	TD	SEM
Time (s)	950	658	84

2.6.2. Test phase.

First, we present the mean accuracy results (% of data points classified correctly) when testing the 20% of the remaining samples of each dataset. In this case, we note that the CNN trained using FD and TD data did not classify all classes accurately, and the mean accuracy did not exceed 94%. In contrast, when trained with SEM data, the accuracy reached 99.8% where it miss-classified only some samples from class 1 (lossy sphere) with class 11 (pyramid), as seen from the confusion matrix in figure 4.

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In the following subsections, we kept the datasets unchanged.

2.6.3. Noisy data.

Now, we test the classifiers' ability to handle noisy data. Several AWGN levels are added to the SF responses, where the SNR values vary from 65 dB, which is one of the highest values that can be obtained, to 10 dB, corresponding to an unfavorable noise condition. First, we test noisy data on object sizes that have already been seen during training (15 cm) where the five first resonant frequencies are present. VF was applied to extract the CNRs and residues directly from the noisy signals.

The results in figure 5 indicate that the performance of the CNN trained using normalized SEM data was better than that of the SF data, with a gain of approximately 20% for SEM at high noise levels (10 dB SNR). CNNs can classify noisy responses processed with SEM because they can take advantage of the extra information provided by the original structure of the SEM dataset proposed in this study (sparse data, Q-factors, and residues). They can easily identify the first NRF, its associated Q-factor, and residues that are barely affected by noise.

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These results show that the first objective of this study was to achieve high robustness to noise in classification based on SEM data.

2.6.4. Generalization to noisy data from unseen objects.

The larger dimension is chosen to be 30 cm while the smaller one is chosen for each object so that it is smaller than those included in the constructed dataset (but with at least one CNR in the frequency band). The results in figures 6 and 7 confirm that the SEM data can distinguish the object classes correctly regardless of the size from their noisy responses, as opposed to SF data. Moreover, the SEM dataset only requires knowledge of the SF of one size for each object, in contrast to the FD or TD datasets, which are made up of five sizes for each object. This is due to the use of Q-factors and NRFs, which makes it possible to accurately distinguish objects of different sizes not included in the training set. In addition, sparse SEM data allow the maintenance of high classification rates that can attain 80% when there is only a single CNR and noisy data.

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These results prove that a CNN trained using SEM data can efficiently distinguish the classes of different objects of any size from the knowledge of a single object size even at low SNR levels.

To identify the angular sector, the space surrounding each object is split into multiple angular sectors to locate the receiving antenna. This is performed according to the symmetries in the geometry of each object. For example, a thin wire oriented along z-axis has a symmetry plane in the azimuth plane; therefore, we only consider the upper half space, as shown in figure 9. Additionally, it has rotational symmetry along z-axis; therefore, the mono-static SF does not vary in the azimuth plane.

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Using a CNN, we determined the angular sector containing the scattered wave vector, which is equivalent to determine the location of the receiving antenna. We present the results obtained for two specific classes of objects: a thin wire, which is a highly resonating smooth object, and a rectangular solid, which is a weakly resonating object with edges.

3.1.1. Dataset construction.

For each object, datasets were created separately because they had their own geometric symmetries. The SFs simulated in 2.2 are used to construct the datasets. To consider the polarization of the incident wave, we include data derived from various rotational angles α that vary from 0^∘ to 90^∘ with a 10^∘ step. This allows the diversification of the polarization of the incident wave and, consequently, the SF. Thus, we can detect the angular sector for various rotations of the object or the incident wave.

First, for the FD and TD datasets, the lengths of the input vectors were the same as those in section 2.5.1. Thus, for FD data, we include the amplitude of both E_θ and E_φ field components that have 500 frequency points, whereas for TD data, we include the first 10 ns of the signal for both field components.

Second, for the SEMs dataset, the input vector is of length 10 and comprises two channels representing the amplitude of residues related to both field components, E_θ and E_φ respectively. The natural frequencies and Q-factor are eliminated because they do not vary with the locations of the antennas and the polarization of the field. Additionally, the dataset was completed with sparse data related to objects of small size, for the same reasons as those mentioned in section 2.5.2. In fact, using residues over raw SF data presents some merits.

• First, there are significantly fewer data points in the input vector. The residue dataset is 100 and 20 times smaller than the SF datasets in the frequency and time domains, respectively.
• Second, the amplitude of the residues is independent of the size of the object, which aids in the generalization process.

Indeed, the natural frequencies at which the residues are derived are inversely proportional to the size of the object; that is, for each object, the residues are independent of their size because they are computed at the same electrical length. Consequently, they are unique to each object but informative about the orientation of that object.

The thin wire is divided into nine sectors making nine classes as follows:

• sector 0: 2 $\unicode{x2A7D} \theta \unicode{x2A7D}$ 10;
• sector 1: 12 $\unicode{x2A7D} \theta\unicode{x2A7D}$ 20;
• sector 2: 22 $\unicode{x2A7D} \theta\unicode{x2A7D}$ 30;
• sector 3: 32 $\unicode{x2A7D} \theta\unicode{x2A7D}$ 40;
• sector 4: 42 $\unicode{x2A7D} \theta\unicode{x2A7D}$ 50;
• sector 5: 52 $\unicode{x2A7D}\theta\unicode{x2A7D}$ 60;
• sector 6: 62 $\unicode{x2A7D}\theta\unicode{x2A7D}$ 70;
• sector 7: 72 $\unicode{x2A7D}\theta\unicode{x2A7D}$ 80;
• sector 8: 82 $\unicode{x2A7D}\theta\unicode{x2A7D}$ 90.

The rectangular solid is a polyhedral object with edges. We tested two angular sector configurations as shown in figure 10. A preliminary study showed that the classification performance was lower when the configuration shown in figure 10(a) was used. This becomes some of the miss-classified samples belonging to the data related to the edge area of this object (sectors 2 to 5). However, when using the configuration shown in figure 10(b), where sectors 4 and 5 covered the edge area, the classification accuracy within this area increased by 8%. Therefore, we show the study realized when this object is divided into ten sectors, as shown in figure 10(b),

• sector 0: 0$\unicode{x2A7D}\phi\unicode{x2A7D}$21; 0$\unicode{x2A7D}\theta\unicode{x2A7D}$21
• sector 2: 0$\unicode{x2A7D}\phi\unicode{x2A7D}$21; 24$\unicode{x2A7D}\theta\unicode{x2A7D}$33
• sector 4: 0$\unicode{x2A7D}\phi\unicode{x2A7D}$21; 36$\unicode{x2A7D}\theta\unicode{x2A7D}$51
• sector 6: 0$\unicode{x2A7D}\phi\unicode{x2A7D}$21; 54$\unicode{x2A7D}\theta\unicode{x2A7D}$69
• sector 8: 0$\unicode{x2A7D}\phi\unicode{x2A7D}$21; 72$\unicode{x2A7D}\theta\unicode{x2A7D}$90
• sector 1: 24$\unicode{x2A7D} \phi \unicode{x2A7D}$45; 0$\unicode{x2A7D} \theta \unicode{x2A7D}$21
• sector 3: 24$\unicode{x2A7D} \phi \unicode{x2A7D}$45; 24$\unicode{x2A7D} \theta \unicode{x2A7D}$33
• sector 5: 24$\unicode{x2A7D} \phi \unicode{x2A7D}$45; 36$\unicode{x2A7D} \theta \unicode{x2A7D}$51
• sector 7: 24$\unicode{x2A7D} \phi \unicode{x2A7D}$45; 54$\unicode{x2A7D} \theta \unicode{x2A7D}$69
• sector 9: 24$\unicode{x2A7D} \phi \unicode{x2A7D}$45; 72$\unicode{x2A7D} \theta \unicode{x2A7D}$90.

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Note that the steps in θ and φ are a trade-off between the field distribution (which is strongly linked to the shape of the object), frequency, and desired angular accuracy. This value can be set according to these parameters.

3.1.2. Results & discussion.

We divided each dataset into 80% for training and 20% for testing, and the results of the angular sector classification were averaged over 10 runs. We then tested the generalization ability for noisy data and object sizes that were not included in the training sets.

3.1.2.1. Training phase.

The LeNet-5 architecture was adopted for the SF datasets, whereas for the residue dataset, we used one convolution layer with 12 filters, followed by two hidden layers with 32 neurons each. The number of epochs was 256 for the residues and 512 for the SF data with a batch size of 100. The output layer had 9 and 10 neurons for the thin wire and rectangular solid, respectively.

3.1.2.2. Test phase.

For the thin wire, the accuracy obtained using residues as input data was 97.8% whereas for both SF data, we obtained 96%. These results show that it is possible to detect the illuminated angular sector in one of the nine sectors of a thin wire, using either residues or raw data.

For the rectangular solid, the accuracy obtained using residues was 90% whereas for both SF data, we obtained an accuracy of 95% . We obtained a slightly better performance when using raw data in this case, which might be due to the discontinuities in the shape of this object, making residue computation difficult. Additionally, an important part of the miss-classified samples belonged to data with only one or two resonances in the frequency band.

The classification results of the observation angle using the SEM data and raw data were good and almost identical during the test phase. At this stage, the main benefit of the SEM method is the simplicity of building the dataset, which requires only one object dimension in the learning phase and very few parameters which reduces computational cost.

3.1.2.3. Noisy data.

After the test phase, we proceed to the test of noisy data of the 15 cm object size for which the noiseless response is already included in the training datasets. This was tested using multiple SNR levels to corrupt the data.

• Thin wire: figure 11 shows the accuracy results when using the three datasets. We see that the accuracy when using residues or raw data is almost similar, with only 2% difference at 10 dB SNR. This is because the thin wire is a strong resonating object, so the extraction of CNRs in a noisy environment is not very difficult; therefore, the calculation of residues is quite precise even at low SNRs.
• Rectangular solid: figure 12 shows the accuracy results when using the three datasets and highlights a weakness of the proposed method: residues show higher sensitivity to noise than raw data because we have significantly less data in the input vector. However, most of the miss-classified samples, when using residues, belong to data having only one or two resonances in the frequency band (small object sizes) and to data that exist in the borders of each sector (small error).

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3.1.2.4. Generalization to noisy data from unseen object sizes.

The large dimension chosen was 30 cm for both objects, and the small dimension, where there was only a single resonance, was of 5 cm for the thin wire and 3 cm for the rectangular solid.

• Thin wire: figure 13 shows the confusion matrices for larger and smaller thin wires at 10 dB SNR when using residues. The classification accuracy is high for large objects, where it reaches 80% at 10 dB SNR. For a small thin wire, the accuracy was low (62%), which is normal because there is only a single CNR in the frequency band. The results when the raw data were used were low and did not exceed 30%. These results are summarized in table 5.
• Rectangular solid: the accuracy obtained at 10 dB SNR when using residues of the 30 cm length object is 62%. However, for the 3 cm length object, the accuracy decreased to 52% as it was more difficult to classify from only residues associated with the first CNR. In addition, with raw data (FD or TD data), we obtained very low performance that did not exceed 30%. These results are summarized in table 6. The normalized confusion matrices showing the classification of each class using residues at 10 dB SNR are presented in figure 14. It can be noticed that the confusion mostly arises in the sectors that are adjacent to each other.

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Table 5. Comparison of classification accuracy for unseen thin wire sizes @10 dB SNR.

Size	Included in database	30 cm (not included)	5 cm (not included)
Raw data	$85\%$	${\lt}30\%$	${\lt}30\%$
SEM data	$82\%$	80%	62%

Table 6. Comparison of classification accuracy for unseen rectangular solid sizes @10 dB SNR.

Size	Included in database	30 cm (not included)	3 cm (not included)
Raw data	$76\%$	${\lt}30\%$	${\lt}30\%$
SEM data	$65\%$	62%	52%

By comparing the classification results obtained using both objects, we can observe that the thin wire has better performace. This is because of to the strong resonating aspect of this object, making the extraction of resonances and the residues computation more accurate even in a noisy environment than for a weak resonating object. Both objects evaluated in this section were chosen because they are representative of all the other simulated objects, which can be divided into two categories: objects that do not present sharp edges (thick cylinder, ovoid, etc) having similar performance as the thin wire, and the remaining objects that perform more like a rectangular solid where the edges strongly scatter the field, which is detrimental in a monostatic configuration, making the estimation of residues using SEM more delicate.

The advantages of using residues are the compactness of the dataset and ease of construction because the residues are independent of the object size (i.e. we can use the SF from a single object size). Moreover, the performance obtained in terms of generalization to other object sizes is excellent and largely surpasses that obtained from the SF, even when training with many object sizes. Finally, we note good robustness to noise owing to the sparse dataset, which maintains a very high performance considering the small number of parameters of the input vector (20 to 100 times more compact).

Once the receiving antenna is localized (i.e. the angular sector is identified), the next information to be retrieved is the polarization of the incident wave, which indicates the rotation of the antenna with respect to the object. This is equivalent to determining object orientation according to the antenna coordinate system. A rotated incident wave or a rotated object means that the scattered wave will be expanded in both θ and φ field components. The angle α used to define the orientation of the linear polarization of the incident wave can be determined according to figure 8 as follows:

$$\begin{equation} \textrm{tan}\left(\alpha\right) = \frac{|Res_{\theta_{\textrm{meas}}}|}{|Res_{\phi_{\textrm{meas}}}|} \end{equation} \tag{ 5 }$$

$$\begin{equation} \alpha = \textrm{tan}^{-1} \left(\frac{|Res_{\theta_{\textrm{meas}}}|}{|Res_{\phi_{\textrm{meas}}}|}\right) \end{equation} \tag{ 6 }$$

where $Res_{\theta_{\textrm{meas}}}$ and $Res_{\phi_{\textrm{meas}}}$ are the residues of the field components in the plane perpendicular to the direction of $\vec{k}_i$.

To estimate the α angle using equation (6), we considered only the residues associated with the first resonant frequency in our calculations. This is due to the fact that, in case of noisy signals, those residues are the least affected by noise, unlike residues related to poles of higher order.

As an example, we present a more complex object formed by merging two PEC objects: a thick cylinder and a hemisphere. The cylinder has a length of 15 cm and the hemispherical has a radius of 7.5 cm (figure 15).

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To verify whether we can compute α accurately to determine the orientation of this object, we rotate it by 45^∘ around y axis and recover the SF at multiple incident angles, where θ varies from 0^∘ to 180^∘ in 5^∘ steps. Additionally, we tested this approach with noiseless and noisy data at 30 dB and 10 dB SNR. VF was then applied to the raw data to extract the residues from the E_θ and E_φ field components. Equation (6) was used to compute the orientation angle of the object. For the noiseless case, we obtained the exact value with α = 45^∘ for all the observation angles. When testing with noisy signals, we noticed that noise affects both field polarizations, and there exists a small error in the computation of α which increases when the SNR value is low. Figure 16 shows the estimated angle α for various observation angles and noise levels.

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Thus, we can easily determine the incident wave orientation with respect to the object, or inversely. Hence, we developed a simple and complete scheme that allows the determination of the orientation of the object.