7.1 Introduction

The ultrawideband (UWB) ground-penetrating radar (GPR) allows nondestructive control and search of different objects buried in ground. However, it can be also used for characterization of ground. Sometimes, GPR allows detection of water or oil leakages from buried pipelines or determination of soil moisture content in a variety of fields such as hydrology, agronomics, etc. Because interpretation of radar data requires complicated processing, which usually can be made only by experienced operator, the automation of object-detection process and ground-permittivity measurement using GPR images is of great commercial interest. This chapter presents methods to automate the evaluation of ground characteristics by permittivity estimation. The relative permittivity ε is one medium characteristic indicating the physical change of the medium. If we know ε, then it is possible to find the moisture content of soil by definite relations.

Besides well-known and widely used techniques for measurement of soil moisture (gravimetric, neutron, and capacitive sounding, and the “time domain reflectometry”), the GPR can also be used for this purpose [1,2].

Lunt and Huisman showed how to apply two well-known GPR techniques for permittivity measurement. The first technique uses the speed of radio wave propagation [1–4], and the second one estimates the reflection factor from ground layers [5,6]. The first technique has a shortcoming in that it requires interactive (with a GPR operator) data processing, and the second method is only suitable for monostatic stepped-frequency continuous wave radars.

The speed of electromagnetic wave propagation in a medium is V=c/εμ, where c is the velocity of electromagnetic waves in vacuum and μ is the relative permeability. Thus, if μ = 1 (it is true for almost all types of soil), the problem of ε determination consists in the measurement of V.

In order to measure object coordinates, it is enough to find a location of object response on a timescale. Time multiplied by the propagation speed of an electromagnetic wave in a medium gives the distance. Taking into account the respective locations of transmitting and receiving antennas, it is possible to calculate the object coordinates.

The basis of the GPR technique is widely known [1,3,4]. Figure 7.1 schematically represents the subsurface sounding process and the GPR image (“a profile”) obtained by the GPR. The local radar-contrast objects with small dimensions form the hyperbolic curves in a GPR image, as shown in Figure 7.1. The larger is the permittivity, the faster do the hyperbola’s arms go down. An example of the GPR profile with several hyperbolas is shown in Figure 7.2.

7.2 Permittivity Measurement

A well-known interactive technique for measurement of ν works as follows. The operator draws a reference hyperbola over the GPR profile (Figure 7.2) using the data-processing software and superposes it with an object curve [7]. Then, we can calculate ν using the profile parameters.

Now, consider a novel technique that is intended for automatic (without a GPR operator) determination of permittivity if the response from a local object (e.g., a pipe or a cable) takes place in the profile. Actually, the procedure consists in automatic searching for a hyperbolic curve in the GPR profile and then in automatic finding of ground permittivity corresponding to the hyperbolic curve.

A technique based on the Hough transform (HT) [8] provides an effective way for searching contour curves including hyperbola in a binary image [9]. This technique already has wide applications in pattern recognition, automatic object identification, and GPR data processing [10–13]. The theory of HT for fast and precise detection of local objects and for determination of soil properties has been stated in the papers [14,15]. The distinguishing feature of the proposed approach lies not only in the technique for ground permittivity determination, using the known location of a subsurface object, but also in allowing to simultaneously solve the problems of object detection and permittivity determination by means of the method of sequential iterations. The most important feature is that the proposed approach based on the HT algorithm is suitable for both automatic determination of ground permittivity and automatic object detection.

7.3 Use of the Hough Transform for Detection of Hyperbolic Curves

7.4 Dependence of the Form of Hough Space on the Error in Permittivity Value

Consider how the Hough space of a hyperbola changes when the value of ε_c used in calculations, called ε_calc, differs from the actual value of permittivity. The Hough space has been imaged for a test hyperbola similar to that shown in Figure 7.4a when x₀ = 50, y₀ = 70, and at several different values of ε_calc. The actual value of ε_c is 12.

Figure 7.6 shows the simulation results as two-dimensional images of the Hough space. The lighter areas in the image correspond to the larger values of accumulators in collecting elements. Histograms of the accumulators in collecting elements along the axis y₀, that is, vertical profiles of accumulators in collecting elements (VPACE), are also presented in Figure 7.6, on the lower line. Corresponding values of ε_calc are given below images. The VPACE is a cross-section of the Hough space with the plane y₀S along the straight line x0=x0′, which stands perpendicular to the plane x₀y₀. Figure 7.7 shows how to form a VPACE.

Focusing: As shown in Figure 7.6d, focusing means the exact matching of the actual value ε_c and ε_calc, where the VPACE contains the peak of the high amplitude at y0′. When ε_calc < ε_c, then S(y₀) = 0 in the range from 0 to y0′. When the depth increases, S(y₀) increases rapidly at first and then decreases as shown in Figure 7.6b. The nearer ε_calc is to ε_c, the faster S(y₀) increases and decreases. When ε_calc > ε_c, then S(y₀) = 0 in the range from y_max to y0′. And when the depth decreases, S(y₀) increases rapidly and then decreases as shown in Figure 7.6f. The dependence of the rate of change of S(y₀) on changes of ε_calc is similar to the above-mentioned dependence. We can observe how when ε_calc changes there are phenomena typical for the focusing.

Therefore, when a specified value of permittivity does not correspond to its actual value, then there is a defocusing in the Hough space. The defocusing pattern depends on the difference between ε_calc and the actual value of ε_c. The most important feature is that the defocusing pattern shows a mismatch direction. If ε_calc < ε_c, S(y₀) mostly equals to zero above the point x0′,y0′, which corresponds to maximal S(y₀) and vice versa. If ε_calc > ε_c, S(y₀) mostly equals to zero below the point x0′,y0′. Thus, from the defocusing pattern, it is obvious whether the ε_calc is more or less than ε_c.

7.5 Algorithm for Accurate Calculation of Permittivity

The algorithm for the adaptive selection of ε has been developed based on the behavior of S(ε_calc). Having specified the initial value of and having constructed the VPACE also define the defocusing direction (upward or downward). Hence, it is obvious what increment of ε_calc (positive or negative) must be applied. Then should be changed according to the increment and VPACE should be reanalyzed. It is necessary to repeat these steps until receiving the maximal value of S at one point and zero values of S at neighboring points. Using this approach, it is possible to find the permittivity.

Owing to the noise and clutter in real radar data, the values of the accumulator are not equal to zero in regions where they are zero in Figure 7.6. Therefore, the automatic focusing procedure with a real profile should be carried out until VPACE becomes symmetrical to the horizontal line passing via the point with a maximal value of S.

The iterative procedure has been carried out by the interval bisection technique according to the algorithm shown in Figure 7.8:

1. The interval of values ε_calc ∈(εcalcmax,εcalcmin) is specified.

2. HT is calculated for the whole profile, the maximal value of S is searched through the whole Hough space in the specified interval of ε_calc, and the part of the profile containing this maximum is chosen for further consideration.

3. HT is calculated for a binary image of the profile section when εcalch∈12(εcalcmax+εcalcmin).

4. The VPACE containing the maximal value of S in the specified profile section is searched.

   

   FIGURE 7.8
   Block diagram of the algorithm for automatic determination of soil permittivity.

5. The VPACE symmetry is estimated, and the interval εcalc∈(εcalcmax,εcalch) is chosen for further consideration for the case shown in Figure 7.6b. The interval εcalc∈(εcalch,εcalcmin) is considered for the case shown in Figure 7.6f.

6. Steps (3) and (4) are repeated until the VPACE curve becomes symmetrical to the straight line y0=y0′ with the maximal S(y₀) or until the increment module of ε_calc specified for the next iteration becomes smaller than a specified accuracy value.

The final value of ε_calc is the “conventional” permittivity ε_c, which determines the actual value of ε based on Equation 7.3.

It has been determined that it is possible to obtain more precise results of ε calculation when the horizontal position (the x₀ coordinate) of the VPACE containing the Hough space maximum is not fixed after the first HT calculation and when a search of global maximum and determination of its new horizontal coordinate x₀ are performed after each calculation cycle (steps 3-5). Thus, the algorithm can self-adjust by knowing the true maximum of the Hough space. When the coordinate x₀ is fixed after the first search and remains invariable until the end of calculations, then the defocusing type could be determined incorrectly after the analysis of the VPACE curve if the VPACE maximum horizontal coordinate is displaced relative to the true position of the hyperbola vertex. This error in the horizontal coordinate may result in specifying an incorrect ε_calc increment and make the algorithm nonconvergent. Finally, it leads to an incorrect or absurd result. This especially occurs when processing a very noisy image or an asymmetric contour of the hyperbola.

This algorithm has been tested using several simulated and experimental GPR profiles. The results are shown in Figure 7.9. It contains gray-scale and binary images of real profiles and the binary images of synthetic data. The first two simulated images were obtained using the geometric tracing of hyperbolas as a binary image. The second two simulated images were obtained using the finite difference time domain (FDTD) software [23] for simulating the problem of diffraction of electromagnetic field pulse by a local object. Figure 7.9 shows exact values of permittivity (for the simulated data): the values determined by superposition of a synthetic hyperbola with the experimental data from the Internet and the values determined using the HT.

A conversion of the gray-scale image to the binary one is done using the 20% threshold as described in Section 7.3 (i.e., only the local maxima with amplitudes greater than 0.2M, where M is the amplitude of the global maximum of the whole profile, are displayed as black pixels) and by thinning of hyperbola contours and clutter elimination. As it is seen from Figure 7.9, the 2% error in permittivity estimation can be achieved for the geometrically generated curves. The error increases to 10% for FDTD-simulated images and to 12% for experimental data because of the presence of clutter in the images. In the case of geometrically generated curves, such an error is conditioned by quantization errors, which typically arise if a continuous image is converted into a digital one consisting of a small amount of pixels. The additional reason for error in the case of FDTD-simulated images used in simulations of bistatic antenna systems distorts the shape of local object response and making it different from hyperbolic.

7.6 Conclusions on Hough Transform Methods

The approach based on the HT provides an automatic (without any operator) and quite precise measurement of the ground permittivity using the GPR technique. It is enough to find a region containing a local radar-contrast object and to specify the range of possible values of permittivity to calculate its exact value.

The algorithm is applicable for automatic GPR data interpretation software, which minimizes the influence of the human factor on data processing. In turn, the data-processing technique for automatic interpretation of GPR results stimulates the development of systems for computerized control and monitoring of ground conditions. After corresponding modification, the proposed algorithm allows automatic and precise search of local objects in the ground.

7.7 Numerical Estimation of the Performance of the Automatic Object-Detection Method

The basic approach remains the same as described for the case of ground permittivity measurement. In the process of GPR-image analysis with HT technique, we obtain coordinates of the maximum of collecting elements as well. It was shown that these coordinates correspond to the objects’ coordinates. Thus, the described algorithm provides both automatic measurement of ground permittivity and automatic calculation of coordinates simultaneously.

Simulated and experimental data tested this method. Section 7.8 shows the relation between the object detection probability, the false alarm probability, and the accuracy of determination of the objects’ coordinates. These metrics allow for evaluating the performance of the automatic interpretation method of GPR images.

7.8 Metrics of Object Detection Method

Sometimes object detection in a radar image cannot be done unambiguously using only one technique. Object discrimination and recognition requires information obtained from different sensors or using various methods. For example, different equipments such as metal detectors, radars, infrared and ultrasound sensors, etc. are used for mine detection in the demining process [19]. Owing to this fact, there is an ensemble of criteria and values for estimating the performance of a method or a system for object classification (a classifier).

When we deal with a so-called binary classifier, which divides a set of detected targets into two classes that include really existing objects (the positive instances) and false alarms (the negative instances), the appointed ensemble of metrics [20] is used for performance measurement.

On the basis of the classification results and of the actual class membership of the objects, the contingency table can be constructed. For the demining problem [21], the contingency table is shown in Table 7.1 and the relative metrics are shown in Table 7.2, but not the absolute values (TP, TN, etc.) used in the data analysis.

The problem that has to be solved while searching for objects is to determine the coordinates of the object and to estimate the errors in coordinate determination. For example, in demining problems [22], if an object has been found within a distance R_halo from the real location of the mine, then it is considered that the mine has been detected and the error estimation is not performed, otherwise a false alarm takes place. Nevertheless, one may need to estimate the accuracy of determination of the coordinates of a subsurface object in some situations.

Correct error estimation can be achieved only when the true object coordinates and true number of objects are known. So the first choice is to simulate the necessary profiles using the FDTD software for solving the problem of diffraction of a pulse by a local object [23]. In addition, we can simply simulate such images with hyperbolic curves using geometric tracing.

The GPR image simulated with FDTD software is shown in Figure 7.10a. It contains reflections from eight metal cylinders located in homogeneous soil. The parameters of the image are as follows: length l = 10 m, time observation interval t = 35 ns, and relative soil permittivity ε = 10. Figure 7.10b shows an example binary GPR image containing 10 hyperbolas obtained by the geometric tracing. Parameters of the profile are as follows: l = 20 m, t = 50 ns, and ε = 10.

7.9 Object Detection in Simulated GPR Images

Let us estimate the probability of accurate detection of the local objects in the binary image according to the above-mentioned criteria of estimation of classifier performance. The term “classifier” means the method of choosing the peaks from the HT space that correspond to the objects really presented in the image. Receiver operating characteristics (ROC) curves are usually used as a means for visualizing and analyzing the classifier performance [20] in machine learning, data mining, medical diagnostics, and also in humanitarian mine clearing. The ROC curve is a two-dimensional graph showing the dependence of correctly classified positive instances (true positive response [TPR]) on incorrectly classified negative instances (false positive response [FPR]).

Let us plot the ROC curves from the results of object detection in the simulated GPR profiles. As the classification result, some classifiers provide a probability or a score, that is, a value representing the membership of an instance to a definite class. As applied to the considered problem, we can use the relative amplitudes h of the peaks in the HT space, which is a ratio of height of the appropriate peak A to the height of the highest peak in the whole Hough space A_max, as such scores: h = A/A_max.

The functioning of the classifier is based on automatic determination of hyperbola vertices described in [15]. It consists in the following:

• Calculating HT for the initial binary image.

• Selecting peaks from the Hough space using the following criterion: the peak’s height must exceed the heights of the eight surrounding elements in the Hough space.

• The selected peaks are sorted descending their amplitudes, and the ROC curve is plotted according to [20].

Figure 7.11 shows the binary profile, the three-dimensional image of HT space, and the ROC curve of Figure 7.10a. The vertical axis in Figure 7.11b represents the values of the Hough space elements.

Table 7.3 contains the list of detected vertices of hyperbolas with their coordinates and the relative amplitudes of peaks corresponding to these vertices in the HT space of the image presented in Figure 7.11a. The objects that are actually present in the image are marked with bold font (here and so on).

Processing the image in Figure 7.9a reveals 15 objects. It follows from Table 7.3 that no false object will be found if the separation threshold goes in the range of h from 0.319 to 0.863, and TPR = 100% while the false alarm probability is equal to zero. If the threshold is less than 0.318, the peaks with heights less than this value will be classified as the actually present objects; at the same time, TPR will remain equal to 100% but false alarms will take place. So, all the peaks of the Hough space corresponding to the test hyperbolas have been positively classified. This fact points at the possibility to detect all objects without false alarms, if we choose the right threshold for class separation.

Consider the results of hyperbola detection (Table 7.4) as applied to the GPR image obtained with FDTD simulation shown in Figure 7.12a. Eleven peaks have been classified as hyperbolas (being false positive instances) in addition to the eight real objects. It is obvious from the results presented in Table 7.3 that TPR = 100% cannot be achieved when FPR = 0, but it is achievable if FPR = 58.3% at the threshold value of 0.5. Thus, it is necessary to increase the threshold in order to decrease the number of false alarms in the presence of a clutter in a binary image (the clutter shows contours and lines that do not belong to hyperbolas); however, it will lead to a decreased TPR.

TABLE 7.3
Object Detection Results for Figure 7.11a

TABLE 7.4
Object Detection Results for Figure 7.12a

So, in the first example, the classifier has marked all true positive instances as positive ones correctly, though several false alarms appeared. In the second example, some problems with the classification occurred due to the presence of a clutter and due to nonideal shape of the hyperbolic curves.

Consider a coordinate system associated with pixels of image. Comparison between coordinates of the detected peaks and coordinates of the hyperbola vertices shows the following:

• The x coordinates of all vertices in the first example have been determined with zero error. So, there is a theoretical possibility of determining the horizontal coordinates of the object precisely.

• Absolute error in the determination of the y coordinate of the hyperbola vertices did not exceed three pixels (Δd = 0.28 m according to the parameters of the GPR profile).

7.10 Object Detection in Experimental GPR Images

Consider the classification of the objects in the experimental GPR images. Unfortunately, hyperbolic curves of ideal shape cannot be obtained during the GPR survey for reasons such as the inhomogeneity of the soil structure, the variability of relative permittivity, the roughness of ground surface, the bistatic antenna system of GPR that leads to distortion of the hyperbolic shape of response for objects of shallow depth [24], etc. Moreover, because of interference produced by several closely located subsurface objects, even an experienced GPR operator has difficulty interpreting such an image. So automatic processing of such GPR images is accompanied by detection of more false objects than in simulated images.

Two examples shown in Figures 7.13 and 7.14 have been taken from Internet. They were processed with the discussed algorithm. Binary images have been obtained by mapping the maxima of each signal composing the profile after the preprocessing of the halftone image. The classifier has selected only the peaks with the relative amplitude h ≥ 0.5.

Example 1 (Figure 7.13): Five objects at different depths, that is, two pipes situated side by side (1, 2), the cable (3), the plastic pipe (4), and the metal pipe (5) are considered [25]. Table 7.5 shows the object detection results for Figure 7.13. The peaks corresponding to the hyperbolas sometimes have less height than the peaks that are formed by other points of the image owing to clutter in the binary image. Moreover, since the amplitude of reflected pulses from the buried objects has a specific form (e.g., Figure 7.3), one object is represented by several peaks in the reflected pulse and therefore by several hyperbolas in a binary image. However, only the upper hyperbola corresponds to the first maximum (or minimum) of the pulse reflected by the object and to the true location of the object.

So, if there are several hyperbolas located one under another, it has been assumed that the upper hyperbola will be the true positive case. At the object No. 6 is the false alarm. The object No. 10 that is situated higher than the object No. 6 is the correct detection despite the fact that it has less relative amplitude than No. 6.

The ROC curve and Table 7.5 show that if the separation threshold is equal to 0.54, then all objects will be detected (TPR =100 %) and that the false alarm probability will be equal to 91.6%, whereas if the threshold is equal to 0.668, only 60% of all objects (three from five) will be detected with FPR = 0.

Example 2 (Figure 7.14): Five underground fuel storage tanks located at the same depth (the objects 1-5). Unlike the previous example, some objects correspond only to one clearly visible hyperbola in the binary image. There is no ambiguity regarding which peak conforms to each of the objects in the Hough space. One can see that only two peaks do not conform to the real objects among the first six peaks listed in Table 7.6. If the threshold equals 0.59, the detection probability will be 100% and the false alarm probability will be equal to 33.3%, which is less than that in the previous example.

TABLE 7.5
Object Detection Results for Figure 7.13a

TABLE 7.6
Object Detection Results for Figure 7.14a

7.11 Conclusion

The examples of simulated and experimental GPR images show the possibility of automatically detecting subsurface objects in a GPR image and finding their coordinates accurately. The method developed based on the HT allows this. It has been demonstrated that 100% object detection probability is achievable, and at the same time, the false alarm probability is minimal. It means that optimal classification with relation to the balance between TPR and FPR is possible. Further, it is necessary to optimize the criteria for choosing the optimal threshold for separating the peaks in the Hough space. It requires more complete statistical data and enough experimental GPR images.