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An Entropy-Based Propagation Speed Estimation Method for Near-Field Subsurface Radar Imaging

EURASIP Journal on Advances in Signal Processing volume 2010, Article number: 636458 (2010) Cite this article

Abstract

During the last forty years, Subsurface Radar (SR) has been used in an increasing number of noninvasive/nondestructive imaging applications, ranging from landmine detection to breast imaging. To properly assess the dimensions and locations of the targets within the scan area, SR data sets have to be reconstructed. This process usually requires the knowledge of the propagation speed in the medium, which is usually obtained by performing an offline measurement from a representative sample of the materials that form the scan region. Nevertheless, in some novel near-field SR scenarios, such as Microwave Wood Inspection (MWI) and Breast Microwave Radar (BMR), the extraction of a representative sample is not an option due to the noninvasive requirements of the application. A novel technique to determine the propagation speed of the medium based on the use of an information theory metric is proposed in this paper. The proposed method uses the Shannon entropy of the reconstructed images as the focal quality metric to generate an estimate of the propagation speed in a given scan region. The performance of the proposed algorithm was assessed using data sets collected from experimental setups that mimic the dielectric contrast found in BMI and MWI scenarios. The proposed method yielded accurate results and exhibited an execution time in the order of seconds.

1. Introduction

Subsurface Radar (SR) is a reliable technology that is currently used for an increasing number of nondestructive inspection applications [1–5]. SR techniques are used to image and detect inclusions present in a given scan region by processing the reflections produced when the area is irradiated using electromagnetic waves. Some advantages of SR technology are the use of nonionizing radiation and a highly automated and/or portable operation [1]. Targets present nonlinear signatures in raw SR data that difficult the proper determination of the correct dimensions and locations of the inclusions inside the scan region [6, 7]. This phenomenon is caused by the different signal travel times along the scan geometry and the wide beam width exhibited by antennas that operate in the Ultra Wide Band (UWB) frequency range. To properly detect and visualize the inclusion responses, SR datasets must be properly reconstructed.

Several reconstruction techniques have been proposed to form SR images [2, 5–8]. These approaches transfer the recorded responses from the spatiotemporal domain where they were collected to the spatial domain where the data will be displayed. Since SR image formation methods use either the time of arrival of the recorded responses or the wavenumber of the radiated waveforms, the wave speed in the propagation medium is required to accurately map the target reflections to their original spatial locations. This value can be obtained from offline measurements using a representative sample of the materials forming the scan area or by using an estimation technique. Any errors in the estimate will cause shifts in the location of the reconstructed responses and the formation of artifacts.

To determine the propagation speed in SR scenarios, a wide variety of estimation techniques have been proposed. These approaches can be divided into two main categories, focal quality measurement techniques and wave modeling approaches. Focal quality measurement techniques reconstruct the collected datasets using different propagation speed values and calculate a focal quality metric that is used to determine a suitable estimate [9–11]. Wave modeling, also called tomographic, techniques perform a minimization process by solving iteratively Maxwell's equations for a set of possible scan scenarios until the difference between the measured data and the analytical solution satisfies a stop criterion [12–15]. Techniques in both categories have been validated on experimental data, yielding accurate results in far-field SR imaging settings.

In the last decade, SR has been used for a series of novel near-field imaging scenarios, such as Breast Microwave Radar (BMR) and Microwave Wood Inspection (MWI). The targets in these applications have sizes in the order of millimetres making necessary the use of large bandwidth waveforms (>5 GHz) to achieve spatial resolution values within this order of magnitude. To the best of the authors' knowledge, only a few propagation speed estimation techniques for this SR imaging setting have been proposed [16–18]. Nevertheless, these methods have some limitations that can potentially limit their use in realistic scenarios. The parametric search proposed in [16] requires a large number of datasets from the scan region to generate accurate estimates. The wave modeling approaches presented in [17, 18] rely on computationally intensive procedures that result in processing times that can range from several minutes to a couple of days [17, 18], resulting in low data throughput rates. Additionally, the method proposed in [17] has limited use when the radiated waveform has a bandwidth over 3 GHz, which is quite common in BMR and MWI scenarios.

This paper proposes a novel technique to accurately determine the propagation speed in near-field SR scenarios. This technique reconstructs a given dataset using different propagation speed values and calculates the Shannon entropy to measure their focal quality. The value used to form the minimum entropy image is then processed to estimate the propagation speed in the scan region. Entropy metrics have been used for airborne radar to estimate the motion parameters of a given target and in SR to eliminate artifacts in reconstructed images arising from a random air-soil interface [19, 20]. The entropy of a radar image is an indicator of its focal quality. As the image is blurred, the uncertainty in the location and dimensions of a target increases. On the other hand, as the focal quality increases, the uncertainty in the position and size of each inclusion decreases. Therefore, the best focal quality is achieved when the entropy of the reconstructed SR image is minimized [21]. The proposed technique exhibits a number of improvements over standard propagation speed estimation methods for near-field imaging, including lower execution time and the ability to generate accurate results using a single data set. This paper is organized as follows. The signal model is described in Section 2. In Section 3 the proposed approach is explained. In Section 4, the performance of the proposed technique is assessed using experimental data sets. Finally, concluding remarks can be found in Section 5.

2. SR Imaging in Homogeneous Media

2.1. Signal Model

Consider a linear scan geometry formed by scan locations in the () plane. The problem domain contains targets over the intervals [] on the axis and [] on the axis and is assumed to have a constant propagation speed . The distance between the scan location and the th target is given by , where () and () are the antenna and the th target coordinates, respectively. In this scan geometry, the antenna element(s) face downwards.

At (), a waveform is radiated, and the reflections from the targets inside the scan region are recorded at the same scan location. The remaining scan locations are inactive during this process. This process is repeated for each scan location. The responses recorded at this scan location can be expressed by

(1)

where is the reflectivity of the th target. Now consider the responses from the th target . The Fourier transform of along the direction is given by

(2)

where , and it is known as the wave number. Equation (2) is known as the spherical phase function of the scan geometry.

Since the targets are located at near-field distances, the differences between travel times at adjacent scan locations are not negligible. These differences lead to the formation of hyperbolic signatures, which make it difficult to properly assess the dimension and location of the targets inside the scan region [1]. To properly visualize the dimensions and locations of these inclusions, the collected data must be reconstructed. One of the most effective image formation algorithms for near-field SR imaging is the frequency-wavenumber migration algorithm [6, 8]. This technique has been used in seismic applications for more than three decades, and it is extensively used in subsurface radar imaging. This method can be summarized as follows. First the Fourier transform of is calculated in the direction yielding

(3)

In order to transfer the data contained in to the rectangular frequency space (), a mapping is performed. The resulting spectrum is given by

(4)

Finally, the reconstructed image, , is obtained by calculating the inverse 2D fast Fourier transform of .

2.2. Propagation Speed Uncertainty Effects

In the majority of the SR scenarios, it is assumed that the medium propagation speed is known a priori. This consideration can have several negative effects on the reconstructed images if there is a difference between the value used in the reconstruction process and the propagation speed of the scan region. Let us denote the propagation speed estimate and the scan region propagation speed . Their corresponding wavenumbers are and , and the wavenumber difference is given by . If is reconstructed using , the mapping function would have the form

(5)

where the value of will introduce a nonlinear error in the frequency mapping process. Depending on the magnitude, two cases can occur. If , then and

(6)

The resulting spectrum has a frequency shift on the axis that decreases as the value increases. Since determines the spatial frequency of the reconstructed data, the mapping error will produce a nonlinear displacement on the axis. Given that the error varies along the axis, the targets in the reconstructed images will have concave signatures. Alternatively, if , then and

(7)

then the error introduced by the mapping process would produce convex signatures in the spatial domain. Although the length of these target signatures will not be as large as they would have been had been left unprocessed (due to the subtraction of the term in the mapping process), the target signatures still present augmented sizes and nonlinear behaviour.

In both cases, the defocusing caused by propagation speed error can be quantified by using the histogram of the reconstructed image magnitude values. Let us consider the case where . In this case, the histogram would contain a series of components corresponding to the different values. As the wavenumber error increases, the length of the nonlinear signatures grows as well. The defocusing caused the target responses to spread among a larger number of magnitude levels in the image. This will result in an increased number of modes in the histogram compared to the image reconstructed using . Therefore, the image sharpness decreases as the magnitude of increases.

To illustrate this effect, a simulated data set, , was generated using an SR simulator developed by the authors [22]. This data set contained three point scatters located at (0.1, 0.15)m, (0.35, 0.24)m, and (0.8, 0.22)m. The irradiated signal was a Stepped Frequency Continuous Wave (SFCW) with a bandwidth of 11 GHz and a center frequency of 6.5 GHz. The propagation speed in the scan region was  m/s. The unprocessed data set is shown in Figure 1. The image obtained by reconstructing using and its corresponding histogram is shown in Figures 2(a) and 2(b), respectively. To evaluate the effects in the image when , was also reconstructed using propagation speed values of 1.5 and 1.2. The resulting images are shown in Figures 3(a) and 4(a), respectively. The histograms of the reconstructed images are given in Figures 3(b) and 4(b). Notice how the target signatures exhibit a concave shape that becomes more elongated as the wavenumber error increases. The number of modes in the histogram grows as the magnitude of increases as well. It can also be appreciated how the location of the targets is shifted upwards as a result of the wavenumber error.

Figure 1
figure 1

Simulated data set.

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Figure 2
figure 2

(a) Reconstructed data set using . (b) Image histogram.

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Figure 3
figure 3

(a) Reconstructed data set using . (b) Image histogram.

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Figure 4
figure 4

(a) Reconstructed data set using . (b) Image histogram.

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The effects on the reconstruction process when were analyzed by processing using propagation speed values of 0.5 and 0.8. The resulting images are shown in Figures 5(a) and 6(a), respectively. Their corresponding histograms are given in Figures 5(b) and 6(b). Although the signature size in these images is smaller than in the unprocessed data set, they still have a convex shape. Similarly to when was reconstructed using propagation speed values greater than , the spread in the target signatures causes an increase in the image energy levels. This is reflected in the additional modes in Figures 4(b) and 5(b), compared to Figure 2(a).

Figure 5
figure 5

(a) Reconstructed data set using . (b) Image histogram.

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Figure 6
figure 6

(a) Reconstructed data set using . (b) Image histogram.

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2.3. Entropy As a Focal Quality Metric

The focal quality of the image depends on the value of used during the reconstruction process. Therefore, in order to determine the fitness of as an accurate propagation speed estimate, the focal quality of the reconstructed data can be used as a metric. An efficient way of determining the focal quality of a radar image is by calculating its entropy. This metric measures the level of uncertainty in a random variable. Let be a discrete random variable with a probability density function . According to Shannon's definition [23], the entropy of is given by

(8)

Pun [24] defined the entropy of a digital image with intensity levels as

(9)

where are the pixels corresponding to the th intensity level on the image and is the total number of pixels in the image. It can be seen in (9) that the entropy value of an image depends on the pixel intensity distribution. To illustrate the performance of entropy as a focal quality metric, was reconstructed using a set of one hundred different values in the interval ; see Figure 7. The plot of the different entropy values is shown in Figure 8. Note that the minimum entropy value is located at .

Figure 7
figure 7

Entropy values within the interval [].

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Figure 8
figure 8

Dual layer scan scenario sample geometry.

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3. Methodology

3.1. Radar Imaging in a Two-Layer Scenario

Most near-field SR scenarios have a layer formed by air or a homogeneous matching material between the scan geometry and the scan region [20, 25, 26]. This can be modeled as an observation domain composed of two regions, denoted a and , with different propagation speeds, denoted a and , respectively. Since the dielectric properties of are usually known a priori or can be calculated offline, determining is a trivial process. On the other hand, , , where and are the minimum and maximum propagation speed values that are physically feasible for this scan region. Using the signal model illustrated in (1), the recorded signal from a single target in this scenario would have the form

(10)

where and and are the signal travel distance and propagation speed corresponding to the th region, respectively. A diagram for this generic scan geometry can be seen in Figure 8.

By dividing the total travel distance by the signal travel time, the average propagation speed is given by

(11)

or alternatively

(12)

To reconstruct the recorded data using a wavefront reconstruction approach, the stationary point in the following expression must be determined:

(13)

Obtaining a closed form expression for from (13) can be difficult. A feasible approach is to perform the reconstruction process using a constant propagation value estimate, , for the whole scan area. In this case the best focal quality will be achieved for the value that has the smallest error for all the recorded reflections in the data set, which can be expressed as

(14)

By taking the first derivative of the right hand of (14) and equating it to zero, we obtain

(15)

By algebraically manipulating (15), we obtain

(16)

which is equivalent to averaging along the direction. This approach can also be used to determine the value in a multitarget scenario as follows:

(17)

By following a similar approach to the one used in the single-target scenario, the result is

(18)

which can also be written as

(19)

where

(20)

where is the average location of the reflections from the scan region surface, . Note that this estimate takes into account the effects of in the signal travel time.

3.2. Propagation Speed Estimation Algorithm

Based on the previous discussion, we can now formulate a propagation speed estimation method. To detect the surface responses and estimate the average location of the targets in the dataset, the datasets were processed using the approach presented by the authors in [27]. This method uses wavelet multiscale products to eliminate the noise components in the dataset and preserve the target responses. The surface responses are characterized using the method proposed in [28]. The denoised dataset will be reconstructed using a set of feasible propagation speed values, defined as

(21)

The proposed estimation method can be described as follows.

  1. (1)

    Calculate the wavelet multiscale products of the range profile , in the recorded data. The result of this operation is denoted as .

  2. (2)

    Determine the range bin which corresponds to the location the surface.

  3. (3)

    Obtain the denoised range profile, , using the method proposed by the authors in [28].

  4. (4)

    Repeat for .

  5. (5)

    Reconstruct using the th value in the set , yielding .

  6. (6)

    Calculate the discrete probability density function of the energy levels on the reconstructed image.

  7. (7)

    Determine the entropy value of , , using (10).

  8. (8)

    Repeat steps (6) through (8) for each element in .

  9. (9)

    Determine the value, , in which the minimum entropy value is achieved.

  10. (10)

    Next, the image components in are segmented and labelled. Then is estimated using the following operation:

    (22)

    where is the range location of the th target centroid, and the is the number of segmented objects in .

  11. (11)

    The area of is calculated as follows:

    (23)
  12. (12)

    Finally, by algebraically manipulating (22), the value of can be determined using the following operation:

    (24)

    By using the proportion of over the extension of , it is possible to estimate the value of by determining the propagation speed that yields the reconstructed image with the best focal quality. A block diagram of the proposed method is shown in Figure 9.

Figure 9
figure 9

Block diagram of the proposed method.

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3.3. Refraction Effects and Lossy Medium Considerations

Compared to ray tracing approaches, wave front reconstruction methods only consider the phase behavior of the recorded responses to focus the collected data. As shown in [29], the refraction produced as the radiated wavefronts penetrate into will affect the spectral support band, , of the target responses along the scan direction. The spectral support band in a radar system is closely related to the size of its point spread function [7]. In a single medium scenario, the support band size is given by

(25)

where , is the size of the antenna radiation footprint, and is the range extension of the scan region. The refraction caused by the interface between the two mediums will change the emergence angle of the wavefronts [29], affecting the beam width coverage in . By using the approach proposed in [7] and the Huygens-Fresnel principle, the resulting spatial bandwidth is given by

(26)

where , , is the antenna divergence angle, and is the extension of . To satisfy the Nyquist-Shannon criterion along the scan trajectory, the separation between adjacent scan location must satisfy the following rule:

(27)

where is the wavelength corresponding to the maximum frequency component in .

The previous analysis can also be used to be extended to deal with lossy media, by modeling the wavenumber as

(28)

where accounts for the attenuation in the medium. By performing the search process over a 2D search space where and and evaluating the focal quality of the resulting images, an estimate of the attenuation factor in can be obtained. A similar approach was used in [20] to enhance near-field GPR images.

4. Results

In order to test the proposed method, a SFCW radar system was used. The system consists of a 360B Wiltron Network Analyzer and an AEL H Horn Antenna which has a length of 19 cm. A bandwidth of 11 GHz (1–12 GHz) was used in all the experiments. The system was characterized by recording the antenna responses inside an anechoic chamber. This reference signal was subtracted from the experiment data in order to eliminate distortions introduced by the components of the system. The data acquisition setup was surrounded by absorbing material in order to reduce undesirable environment reflections. The data was reconstructed using a 3 GHz PC with 1 GB RAM.

The proposed estimation algorithm was tested using experimental data acquired from a 3 × 1 × 13 m rectangular deposit filled with dry sand. The walls of this deposit were covered with electromagnetic wave absorbing material to eliminate their responses. The targets were buried within a region of 20 cms beneath the sand surface. Different distances between the antenna and the sand surface were used in each experiment to assess the effect of the air layer in the estimation method. In the first three experiments the box was filled with silica sand which has a propagation speed of  m/s [30]. The dielectric contrast between the scan region layers in this scenario is similar to the one present in BMI and MWI [31, 32] scenarios. The propagation speeds of the materials used in the experimental setups are shown in Table 1. The values of materials commonly found in BMI and MWI scan scenarios are summarized in Table 2. To demonstrate the robustness of the proposed approach, the search process is performed over the interval [1 × 108 m/s, 3 × 108 m/s] which is significantly larger than the range of values reported in the literature for dry sand ([1.37 × 108 m/s, 2.12 × 108 m/s]) [33]. A total of 200 equidistant values were defined in the search interval. In the experimental setup, the antenna was mounted on a horizontal rail that was 1.2 meters above the bottom of the box. The antenna motion was controlled by a stepper motor that was connected to a custom control interface controlled by a 2 GHz PC with 1 GB RAM. In all the experiments, the step size in the direction was 1 cm. The limit of the near-field region, , on this imaging system is given by

(29)

where is the largest dimension of the antenna at its phase center. Since the maximum distance between the antenna and the sandbox bottom is 1.2 m, the targets in all the experiments were at near-field distances.

Table 1 Propagation speed values of the materials used in the experimental setups.
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Table 2 Propagation speed values of materials found in BMI and MWI scan scenarios.
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The first experimental data set is shown in Figure 10(a). In this experiment, two aluminum pipes with a diameter of 3 cm and two steel pieces with a length of 2 cm and a thickness of 5 mm were used. The average separation between the antenna and the sand surface was 10 cms. Figure 10(b) shows the energy of the denoised data. Note that the target signatures are easier to visualize in this image. The clutter in the image corresponds to stationary waves caused by multiple reflections between the surface and the antenna. Nevertheless, the magnitude of these responses is less than half of the magnitude of the target signatures. Figure 10(c) shows the resulting entropy values for the images formed using the values in the search interval. For this experiment, the minimum value is located at  m/s. The difference between this value and is caused by higher propagation speed of the air layer (3 × 108 m/s). From the mathematical model of described in (22), an increase in the value of will result in an increased . Substituting the values of , , and in (24) yields a value of  m/s, which has a 12% error compared to . The reconstructed image using is shown in Figure 10(d).

Figure 10
figure 10

(a) First experimental data set. (b) Denoised data set. (c) Entropy values for the interval [1 × 108 m/s, 3 × 108 m/s]. (d) Dataset reconstructed using .

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Figure 11(a) shows data collected from the second experimental setup. In this case, the targets were two aluminum pipes with a diameter of 1 cm and a steel plate with a length of 7 cms and a thickness of 1 cm. The average separation between the antenna and the sand surface was 7 cm. It can be seen that the sand surface in this experiment is closer to the antenna, which according to the modeling performed in Section 3 will result in a lower composite propagation speed estimate. Figure 11(b) shows the corresponding denoised image. The entropy values for the search interval are shown in Figure 11(c). The minimum entropy value was located at 2.04 × 108 m/s, and the corresponding propagation speed estimate was 1.63 × 108 m/s. Similarly to the last dataset, the dataset was reconstructed using . The resulting image is shown in Figure 12(d). Notice increased focal quality of these images compared to the previous dataset. These results are consistent with the simulations presented in Section 3. As the error between propagation speed the medium and the estimate decreases, the focal quality of the image improves.

Figure 11
figure 11

(a) Second experimental data set. (b) Denoised data set. (c) Entropy values for the interval [1 × 108 m/s, 3 × 108 m/s]. (d) Dataset reconstructed using .

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Figure 12
figure 12

(a) Third experimental data set. (b) Denoised data set. (c) Entropy values for the interval [1 × 108 m/s, 3 × 108 m/s]. (d) Dataset reconstructed using .

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The recorded data from a third experimental setup is shown in Figure 12(a). In this setup, the same targets than in the previous experiment were used. The steel plate target was moved 7 cm deeper to observe the effect on the propagation speed estimate. The average separation between the sand surface and the antenna was 7 cm. The denoised data can be seen in Figure 12(b). The entropy values for the search interval are shown in Figure 12(c). The minimum entropy value was located at 2.17 × 108 m/s, and the corresponding propagation speed estimate was 1.743 × 108 m/s. The reconstructed image using is shown in Figure 12(d).

In order to test the proposed method in a different propagation medium, four targets were buried in desert sand ( m/s) [34]. In this case, the targets were aluminum pipes with a diameter of 1 cm. The average distance between the antenna and the sand surface was 6 cm. The recorded data set is shown in Figure 13(a). The result of the denoising process can be seen in Figure 13(b). The search was conducted also in the interval [1 × 108 m/s, 3 × 108 m/s]. The calculated entropy values are displayed in Figure 13(c). The location of the minimum value (2.3 × 108 m/s) was shifted towards the right. This is consistent with the model presented in Section 3, as the wave propagates faster in this medium than in silica sand. The estimated value of for this data set was 1.9 × 108 m/s. The reconstructed image using is shown in Figure 13(d).

Figure 13
figure 13

(a) Fourth experimental data set. (b) Denoised data set. (c) Entropy values for the interval [1 × 108 m/s, 3 × 108 m/s]. (d) Dataset reconstructed using .

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The performance of the proposed technique was compared to the algorithm described in [16] which is based on the use of the Hough Transform (HT). This technique offers a good balance between execution time and estimation accuracy. The results of this comparison are shown in Table 3. The HT technique has a higher error and a larger execution time than the proposed approach.

Table 3 Estimation errors and execution times of the proposed method and the HT-based estimation technique for each experimental data set.
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5. Conclusions

A novel technique for propagation speed estimation in near-field SR scenarios is presented in this paper. The proposed algorithm focuses the data using initial estimates of the propagation speed on the media followed by the calculation of the focal quality of the reconstructed images using Shannon's entropy as a metric. A clutter removal process is performed on the data in order to allow a more accurate estimation. A search process is performed on the resulting entropy measurements in order to find the propagation speed value associated with the minimum entropy value. The proposed method yielded accurate propagation speed estimates (with an error less that 13%) and has an execution time in the order of seconds. Finally, the proposed algorithm exhibits both lower execution times and estimation errors compared to current noninvasive estimation techniques based on the use of the HT.

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Authors and Affiliations

  1. Department of Medical Physics, CancerCare Manitoba, Winnipeg, MB, Canada

    Daniel Flores-Tapia

  2. Department of Physics and Astronomy, University of Manitoba, Winnipeg, MB, Canada

    Stephen Pistorius

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  1. Daniel Flores-Tapia
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  2. Stephen Pistorius
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Correspondence to Daniel Flores-Tapia.

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Flores-Tapia, D., Pistorius, S. An Entropy-Based Propagation Speed Estimation Method for Near-Field Subsurface Radar Imaging. EURASIP J. Adv. Signal Process. 2010, 636458 (2010). https://doi.org/10.1155/2010/636458

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  • DOI: https://doi.org/10.1155/2010/636458

Keywords

  • Reconstructed Image
  • Propagation Speed
  • Radar Image
  • Sand Surface
  • Minimum Entropy