Eddy-Current Sensors with Asymmetrical Point Spread Function Janusz Gajda and Marek Stencel * Department of Measurement and Electronics, Faculty of Electrical Engineering, Automatics, Computer Science and Biomedical Engineering, AGH University of Science and Technology, al. A. Mickiewicza 30, 30-059 Krakow, Poland; [email protected] * Correspondence: [email protected]; Tel.: +48-12-617-3647 Academic Editors: Changzhi Li, Roberto Gómez-García and José-María Muñoz-Ferreras Received: 11 July 2016; Accepted: 15 September 2016; Published: 4 October 2016

Abstract: This paper concerns a special type of eddy-current sensor in the form of inductive loops. Such sensors are applied in the measuring systems classifying road vehicles. They usually have a rectangular shape with dimensions of 1 × 2 m, and are installed under the surface of the traffic lane.The wide Point Spread Function (PSF) of such sensors causes the information on chassis geometry, contained in the measurement signal, to be strongly averaged. This significantly limits the effectiveness of the vehicle classification. Restoration of the chassis shape, by solving the inverse problem (deconvolution), is also difficult due to the fact that it is ill-conditioned.An original approach to solving this problem is presented in this paper. It is a hardware-based solution and involves the use of inductive loops with an asymmetrical PSF. Laboratory experiments and simulation tests, conducted with models of an inductive loop, confirmed the effectiveness of the proposed solution. In this case, the principle applies that the higher the level of sensor spatial asymmetry, the greater the effectiveness of the deconvolution algorithm. Keywords: point spread function; deconvolution; inverse problem; eddy-current; magnetic signature; spatial scanning

1. Introduction The spatial aperture of eddy-current sensors, used in the measurement of road traffic, causes the blurring of the signal which represents the shape of the object. This causes a distortion in information on the actual shape of the object. The greater the width of the aperture, the stronger its averaging properties. Averaging of the signal, however, results in a loss of information on the details of the geometry of the object studied. In many applications, inductive loops are used to scan the shape of the chassis of vehicles. The measuring signal obtained in this way is known as the magnetic signature of the vehicle, and is commonly used for its classification [1–4]. Loss of information on details of the geometry of vehicles therefore causes a loss of resolution in their classification. The specific design of inductive loops used in the measurement of traffic parameters makes their properties significantly different from those of conventional eddy-current sensors used in non-destructive testing (NDT). The main differences concern: the dimensions of the sensor, which in the case of the inductive loop are in the order of meters, the spatial extent of the generated magnetic field to approx. 0.8 m and a spatial resolution of 0.2 m. Identical however, is the fundamental purpose of the research work conducted with both groups of sensors. It is the increasing effectiveness of the classification of the measured objects. Studies described in this work are the confirmation of this thesis [5]. It proposes the use of PCA (Principal Component Analysis) for increasing the efficiency of subsurface crack measurement. The success, however, is conditioned by the amount of data contained in the raw signal of the sensor. The solution described in this paper allows an increase of the amount Sensors 2016, 16, 1642; doi:10.3390/s16101642

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increasing the efficiency of subsurface crack measurement. The success, however, is conditioned by 2016,of 16,data 1642 contained in the raw signal of the sensor. The solution described in this 2 of 13paper the Sensors amount allows an increase of the amount of data contained in a signal which can then be further processed (e.g.,ofPCA, neural network, wavelet resulting in the(e.g., correct classification of the object. data contained in a signal whichanalysis can thenetc.), be further processed PCA, neural network, wavelet Blurring of the magnetic is caused by spatial averaging properties of the eddy current analysis etc.), resulting in thesignature correct classification of the object. sensor, whose signal signature depends ison the by total metal surfaceproperties of the object affected by the Blurringoutput of the magnetic caused spatial averaging of the eddy current sensor, whose output signal depends on metalThis surface of the object affected by the in alternating alternating magnetic field generated bythe thetotal sensor. phenomenon is illustrated the diagram magnetic generated the sensor. This phenomenon is illustrated the diagram presented in presented in field Figure 1. It is by present not only in the sensors described in above, but also in all types of Figure 1. It is present not only in the sensors described above, but also in all types of sensors used in sensors used in the process of spatial scanning [6]. the process of spatial scanning [6].

point spread function sensor

scanned object

area of sensor operating

Figure 1. Illustration of the spatial aperture of an eddy-current sensor.

Figure 1. Illustration of the spatial aperture of an eddy-current sensor.

If the vehicle moves over the sensor along a straight line, a one-dimensional signal is created at If the vehicle moves over the sensor along a straight line, a one-dimensional signal is created at the output of the sensor, which is a function of the time or distance covered by the vehicle, including the output of the sensor, which is a function of the time or distance covered by the vehicle, including averaged information on the shape of the chassis. Both the surface width and length of the vehicle averaged information on the shape of the chassis. Both the surface width and length of the vehicle are averaged. are averaged. There is a large number of studies which describe this process by means of a convolution model [7–9]. There is a which large is number of studies which describe this processasbya Point means of aFunction convolution The function, responsible for the averaging process, is interpreted Spread model [7–9]. function, which the averaging process, (PSF) of theThe sensor. The output signalisofresponsible the sensor is for interpreted as a convolution [10]isofinterpreted the function as a describing shape of(PSF) the object and sensor. the PSF of the output sensor, and is described by Equation (1). Point Spread the Function of the The signal of the sensor is interpreted as a

convolution [10] of the function describing the shape of the object and the PSF of the sensor, and is + w∞ described by Equation (1). y (l ) = h (l − λ) x (λ) dλ (1) − ∞

l hdescribing l xthe geometrical d In which: h(λ) is the PSF, x(λ) isya function shape of the object, and y(l) (1) averaged (spatially blurred) result of the scan. is the output signal of the sensor, which is the A graphic illustration of the convolution operation described by Equation (1), relevant to a In which: h() is the PSF, x() is a function describing the geometrical shape of the object, and one-dimensional case, is presented in Figure 2.

y(l) is the output signal of the sensor, which is the averaged (spatially blurred) result of the scan. A graphic illustration of the convolution operation described by Equation (1), relevant to a one-dimensional case, is presented in Figure 2.

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sensor scanned object

Point Spread Function - PSF

0

function of object shape

signal of sensor response

M

l- Figure 2. One-dimensional object scanning with a sensor with aperture. Figure 2. One-dimensional object scanning with a sensor with aperture.

2. Convolution Matrix 2. Convolution Matrix Taking into account the limited dimensions of the object, Equation (1) looks as follows Equation (2): Taking into account the limited dimensions of the object, Equation (1) looks as follows Equation (2): λ wM y (l ) =

M

h (l − λ) x (λ) dλ

y l λ=0 h l x d

(2)

(2)

0

in which: 0 ≤ λ ≤ λM — the dimension of the object. Following of the distance in which: 0 ≤ λthe ≤ λdigitisation of the λ object. M = ∆l × M, l = ∆l × n, λ = ∆l × m, in which ∆l is the M — the dimension quantum of the the distance, a convolution form is expressed Equation Following digitisation of the distance 𝜆𝑀 = 𝛥𝑙 × by 𝑀, the l = Δl ×n, 𝜆 =(3). 𝛥𝑙 × 𝑚, in which Δl is the quantum of the distance, a convolution form is expressed by the Equation (3). M y (n) M hn−m xm = y = for 0 ≤ m ≤ M (3) n ∑ y∆ln yn m=0hn m xm for 0 ≤ m ≤ M (3)

l

m 0

By applying a matrix form, Equation (3) can be presented as Equation (4) By applying a matrix form, Equation (3) can be presented as Equation (4) h 0 h −1 h −2 · · · h − M y0 y h h h h.M x y 0 h 0 h 1 h 2 . . . .. 0 x0 1 1 0 x h h0 h−11 . y .. y2 1 = 1 1 x1 . . h1 h0 . h2 .. y2 h h h x x2 .. 2 . .. 2 . . 1 . . 0 . . xM . . . . . y M h M · · · · · · · · · h0

yM

or symbolically as Equation (5) or symbolically as Equation (5)

hM

(4) (4)

h0 xM

Y = H · X,

(5)

, y ÷ y , are samples of the output signal, (5) Y ofH X in which h−M .....h0 , h1 ... hM , are spatial samples the PSF, 0 M while x0 ÷h–M xM.....h , are samples of the signal of the object’s shape. in which 0, h1 ... hM, are spatial samples of the PSF, y0 ÷ yM, are samples of the output signal, The size of the H matrix depends the observed signals at the beginning and end while x0 ÷xM, are samples of the signalon of the the length object’sofshape. of theThe sensor, is H equal to the lengthon of the thelength vector of signal X. size and of the matrix depends the observed signals at the beginning and end of the sensor, and is equal to the length of the vector signal X.

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The averaging effect of the sensor depends on the elements of the H matrix. This effect may be eliminated or at least limited by calculating, from Equation (5), the signal of the shape of the object X, assuming that the Y signal and PSF of the sensor used are known. An estimate of the X vector can be obtained from Equation (6). This is a reverse operation to the convolution operation and is known as deconvolution: XOD = H−1 · Y

(6)

in which the H −1 matrix is the inverse matrix of H, XOD is the estimate of the signal of the shape of X. Equation (6) defines the inverse problem, which, due to the singularity of the H matrix or low value of its determinant, is ill-conditioned. This is the cause of substantial numerical errors which distort the solution of this problem. This is why one speaks of an estimate of the signal of the shape and not of its precise calculation. The precision of this estimate depends on the effectiveness of the numerical deconvolution algorithm applied. The form of the H matrix depends on the PSF of the sensor, which in turn depends on its construction. It is always, however, a matrix with a Toeplitz structure [11,12]. In the most general case, it takes the form of Equation (7).

h0

h 1 H= h2 .. . hM

h −1

h −2

h0

h −1

h1 .. . ···

h0 .. . ···

· · · h− M .. .. . . .. .. . . .. .. . . · · · h0

(7)

Depending on the PSF shape of the sensor, the form of the H matrix can take slightly different forms. For causal systems described for time-domain, this is a lower triangular matrix. For systems described in terms of spatial coordinates, as in the case of an inductive loop, this is in the form of Equation (7). If the PSF is relatively narrow with regard to the range of the input signal X, the H matrix becomes a diagonal. Outside of the band around the main diagonal, the values of the matrix elements are equal to zero. Sensors described by this type of matrix are characterised by high spatial resolution, and causes only small distortions in the signal of the shape of the object. For systems with a relatively wide PSF, this matrix takes the form of a full matrix. Such sensors significantly average the shape signal, blurring geometric details of the object. In the case of spatial systems, H matrices, either band or full, may be either symmetrical or asymmetrical. This symmetry, or its absence, results from the PSF shape, which is in turn associated with the construction of the sensor itself. For sensors with a symmetrical PSF, the H matrix is also symmetrical, while in the opposite case it is asymmetrical. A set of different forms of the H matrix is presented in Table 1. Equation (6) describes the simplest but also least precise deconvolution algorithm, resulting directly from the convolution Equation (5). In practice, other algorithms are used which minimise the influence of distortions that appear in the measuring signals [7,8,13,14]. One of these is the least squares algorithm (LS) [1,15,16], described by Equation (8). −1 XOD = HT H HT · Y

(8)

In order to improve the conditioning of the inverse problem, described by Equation (8), additional regularisation is also often used [17,18]. The necessity of regularization results from the need to minimize the numerical errors of the matrix invertion. The determinant of the symmetric matrix equals zero and inverting of such a matrix is impossible. The proposed method de-symmetrizes the convolution matrix using the sensors with an

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Figure 3. Illustration of a method of PSF (Point Spread Function) asymmetry estimation.

asymmetric PSF. a result, determinant value of the inverted matrix’s increases and the numerical Figure 3. As Illustration of the a method of PSF (Point Spread Function) asymmetry estimation. Forare a symmetrical PSF Wof= a1.0, whereas Wmax = Spread 2.0 for Function) maximum asymmetry, where B = 0, and Figure 3. Illustration method of PSF (Point asymmetry estimation. errors smaller. a symmetrical PSF A is aFor number other than 0. W = 1.0, whereas Wmax = 2.0 for maximum asymmetry, where B = 0, and Table shape and its equivalent convolution matrix H form. a symmetrical PSF 1.0, whereas Wmax = 2.0 for maximum asymmetry, where B = 0, and A is aFor number other than 0. W1.= PSF 1. PSF shape and its equivalent convolution matrix H form. A is a number other Table than 0. Type of System PSFits Shape Convolution Table 1. PSF shape and equivalent convolution matrix H form. Matrix H Form

Type of System PSF Shape Convolution Matrix H Form matrix Table 1. PSF shape and its equivalent convolution h0 h1 H form. h2 0 0 ··· 0 Type of System PSF Shape Convolution Matrix H Form h0 h1 h2 0 0 . . .. 0 h Type of System PSF Shape H Form 0 0 . . 0 1 Convolution h0h hh1 hh2 0Matrix

h10 h01 h12 h2 0 . h1hh10 hhh001 hhh121 h02h2 h0 . . . .. 0 hh02 h1 hh0 h1 h02. . 0 0 h2 21 h1210 h1010 012h1 12 . 0 h02 hh01 h0 hh1 h 2. h0 2 21 10 01 . . 2 0 0 hh2 hh1 hh0 h h2 0 02 h . 0 . 21 10 h 01 h12 .. .0. h. . . h. . . h . . . h h . 2 . .00 01 h h h 1 0 2 1 2 1 0 0 ·· ·0 · ·· 0 h1 h0h 0h2 h 2 h1 h01 h2h h3h h·4· · h h0 h0 h10 h1h 3 0 4 h2 h1 h Mh0 2 M h2 h 3 h4. . hM h h 0 1 1 0 1 2 3. h . 1 h h0 h h1 hh2 hh3 h . . h h210 h101 h012 h hM 4 3 12 2.3 4 h2 hh1h1 hh0h0 hhh1 1 hhh22 h 3. . h4 h 2 1 0 1 2 34 3 2 1 0 1 hh 234 h hh432h hh321h hh210h hh10h1 hh0.12. . h 3 2 0 3 1 1 h h h h h h h h 43h 32h 2h1 h10 0.1. . h 123 h4 h3 2 hh2 1 h h10 h h0 h 2 h2 hhM4. 3 3. . 2. . 1 01 .. . . 4. . . h . . . h . h . h . hh1 h h1 M 4 3 2 1 hh0 · · ·h1h4 hhh2 3 h0h2 h0h1 hh0 h00 hM 4 3 2 1 0 M 0 0 h0 hh−101 hhh0−12 h120 h020 0 · · · h h h 0 0 0 h hh210 hh101 hh012 h120 h2. . . ... 0 1 h0 −2 −1 h321 hh210 hh101 hh012 h012 0 .. h2 hhh1 . 0 h h h h 0 hh3210 hh210−1 hh101−2 hh012 2 32 4 .. h h 0 h hh43 h h h h h . 2 1 0 1 0 h31 2h0 1h−1 0 2 12 3 h3 h1 h0. h hh h04 hh h h 2 2 . 4 3 2 1 0 . h−2 1 h3 h2 h1 h0 h4 hh1 0 h h h h 4. . 3. . 2. . 1 0 .. . . .. . 0. . h4 . h3 . h2 . h1h−1 h 0 0 ··· h h h h h

Symmetric PSF

Symmetric PSF Symmetric PSF Symmetric PSF

Asymmetric Asymmetric Asymmetric Asymmetric

h2

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In order ordertotoobtain obtain a quantitative description the relation the of PSF a quantitative description of theof relation betweenbetween the degree of degree PSF asymmetry TH matrix determinant, asymmetry and of the Hdeterminant, a factor of PSF asymmetry is introduced and the value ofthe thevalue HT H matrix a factor of PSF asymmetry is introduced in the form in of the form of Equation (9). width is equal to A the Equation (9). Its width is Its equal to the sum of + sum B. of A + B.

W=

2A 2A A+ B B

(9) (9)

B are the the length of theofrelevant sectionssections measured at 10% ofatthe10% maximum in which: which: AAand and B are length the relevant measured of the normalised maximum PSF value, as indicated inindicated Figure 3. in Figure 3. normalised PSF value, as

Figure Figure 3. 3. Illustration Illustration of of aa method method of of PSF PSF (Point (Point Spread Spread Function) Function) asymmetry asymmetry estimation. estimation.

For a symmetrical PSF W = 1.0, whereas Wmax = 2.0 for maximum asymmetry, where B = 0, and A is a number other than 0. Table 1. PSF shape and its equivalent convolution matrix H form.

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For a symmetrical PSF W = 1.0, whereas Wmax = 2.0 for maximum asymmetry, where B = 0, and A 6 of 13 other than 0. 6 of 13

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SimulationStudies Studiesof ofAsymmetrical AsymmetricalPSF PSF 3. Simulation 3. Simulation Studies of Asymmetrical PSF In order to quantitatively study the influence of the PSF asymmetry of the sensor on the quality In order to quantitatively study the influence of the PSF asymmetry of the sensor on the quality of restoration of the object shape signal, simulation studies were conducted. As a PSF model for the of restoration of the object shape signal, simulation studies were conducted. As a PSF model for the sensor, a spline function was used, composed of two normalised Gauss curves with different factor sensor, a spline function was used, composed of two normalised Gauss curves with different factor values. For a defined value of W, the PSF may be both wide, as in the case of a full, asymmetrical H values. For a defined value of W, the PSF may be both wide, as in the case of a full, asymmetrical H matrix; or narrow, as in the the case case of of an an asymmetrical asymmetrical band band matrix. matrix. Examples of a PSF with uniform matrix; or narrow, as in the case of an asymmetrical band matrix. Examples of a PSF with uniform 1.6 and and different different width is presented in Figure 4. asymmetry W == 1.6 asymmetry W = 1.6 and different width is presented in Figure 4.

Figure 4. Examples of two PSF with width of 30 mm and and 90 90 mm mm and identical identical asymmetry factor Figure 4. 4. Examples of of two two PSF PSF with with aaa width width of of 30 30 mm mm factor Figure Examples and 90 mm and and identical asymmetry asymmetry factor values W == 1.6. values W 1.6. values W = 1.6.

The results of simulation studies presented in Figure 5 confirm that the value of the H matrix The results of simulation simulation studies studies presented presented in in Figure Figure 55 confirm confirm that that the the value value of of the the H matrix matrix determinant is strongly dependent on PSF asymmetry. Its value also depends on PSF width. determinant is strongly dependent on PSF asymmetry. asymmetry. Its Its value value also also depends depends on on PSF PSF width. width.

Figure 5. Dependence of the convolution matrix determinant on the asymmetry factor W. Figure Figure 5. 5. Dependence Dependence of of the the convolution convolution matrix matrix determinant determinant on on the the asymmetry asymmetry factor factor W. W.

The simulations conducted were intended to define the relation between the degree of PSF The simulations conducted were intended to define the relation between the degree of PSF The simulations conducted were intended asymmetry and the restoration error Equation (10)to define the relation between the degree of PSF asymmetry and the restoration error Equation (10) asymmetry and the restoration error Equation (10) N

vN X X 22 u N X OD X u1 OD u 1∑ ( XOD − X )2 D N D u 1 N D=u u X 22 N X t 1∑ X 2

(10) (10) (10)

11

in which: X—input shape signal, XOD—estimate of signal X—result of deconvolution. in which: X—input shape signal, XOD—estimate of signal X—result of deconvolution. in which: shape signal, Xfor of signal of deconvolution. OD —estimate In theX—input simulations conducted, the input signal X, aX—result pseudorandom binary signal was used, a In the simulations conducted, for the input signal X, a pseudorandom binary signal was used, a In the simulations conducted, for the input signal X, a pseudorandom binary signal was used, fragment of which is presented in Figure 6. fragment of which is presented in Figure 6. a fragment of which is presented in Figure 6.

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Figure 6. Arbitrary input signal X used in simulation studies. Figure 6. Arbitrary in simulation simulation studies. studies. Figure 6. Arbitrary input input signal signal X X used used in

The dependence of the restoration error D on the asymmetry factor W for PSF of various widths The dependence of the restoration error D on onFigure the asymmetry factor W for for PSF of various widths was obtained for Algorithm (8) and presented 7. A change in W the asymmetry factorwidths of the The dependence of the restoration error D in the asymmetry factor PSF of various was obtained for Algorithm (8) and presented in Figure 7. A change in the asymmetry factor of the sensor in rangefor ofAlgorithm input signal while maintaining a stable width, a change factor in the of value was obtained (8)X,and presented in Figure 7. APSF change in causes the asymmetry the sensor in range of input signal X, while maintaining a stable PSF width, causes a change in the value of the restoration several degrees of magnitude. Simultaneously, a relation can be sensor in range oferror inputby signal X, while maintaining a stable PSF width, causes a change in observed the value of the restoration error by several degrees of magnitude. Simultaneously, a relation can be observed involving the strong of the restoration error on the PSF width of thecan sensor with a of the restoration errordependence by several degrees of magnitude. Simultaneously, a relation be observed involving the strong factor dependence of the restoration error on as thesensors PSF width ofsmaller the sensor with a constant valueofW. is understandable, PSF widths involvingasymmetry the strong dependence the This restoration error on the PSF width of with the sensor with a constant constant asymmetry factor value W. magnetic This is understandable, as sensors withobject smaller PSFshape widths have better spatialvalue resolution of isthe signature. In such a smaller case, the signal is asymmetry factor W. This understandable, as sensors with PSF widths have better have better spatial resolution of the magnetic signature. In such a case, the object signal shape is weakly averaged. of the magnetic signature. In such a case, the object signal shape is weakly averaged. spatial resolution weakly averaged.

Figure asymmetry factor WW in the fullfull variability range for Figure 7. 7. Dependence Dependenceofofrestoration restorationerror errorDDonon asymmetry factor in the variability range Figure 7.PSF Dependence of restoration error D on asymmetry factor W in the full variability range for widths of the sensor (from 1010mm various for various PSF widths of the sensor (from mmtoto90 90mm). mm).Ideal Idealcase: case: without without distortion distortion in in the the various PSF widths of the sensor (from 10 mm to 90 mm). Ideal case: without distortion in the measuring measuring signal. signal. measuring signal.

In Figure 8, characteristics illustrating the influence of additive random distortions included in In 8, illustrating the of additive random distortions included in In Figure Figure 8, characteristics characteristics the influence influence randomThese distortions includedare in the output signal Y of the sensorillustrating on the restoration error of D,additive are presented. characteristics the output signal Y of the sensor on the restoration error D, are presented. These characteristics are the output Y ofathe sensor onmm the restoration error D,asymmetry are presented. These are relevant to signal PSF with width of 40 and with various factors. As characteristics a measure of the relevant to PSF with aa width of 40 mm and with various asymmetry factors. As aa measure of the relevant to PSF with width of 40 mm and with various asymmetry factors. As measure of the strength of these distortions, their relative value with respect to the maximum input signal X is strength distortions, their relative value withwith respect to thetomaximum input signal Xsignal is applied. strength of ofthese these distortions, their relative value respect maximum X in is applied. The greater the level of distortion, the greater the requiredthe asymmetry ofinput the sensor is, The greater the level of distortion, the greater the required asymmetry of the sensor is, in order to applied. The greater the level of distortion, the greater the required asymmetry of the sensor is, in order to obtain the required value of the restoration error D. obtain the required value of the restoration error D. error D. order to obtain the required value of the restoration

Figure 8. of restoration error D D on 8. Dependence Dependence of restoration error on distortion distortion level level for for sensors sensors with with various various asymmetry asymmetry Figure Figure (W: 8. Dependence ofPSF restoration error D on distortion level for sensors with various asymmetry factors 1.0–1.9) and widths of 40 mm. factors (W: 1.0–1.9) and PSF widths of 40 mm. factors (W: 1.0–1.9) and PSF widths of 40 mm.

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In Figure 9, a characteristic which makes it possible to assess the benefits of using sensors with In Figure 9, a characteristic which makes it possible to assess the benefits of using sensors with asymmetrical PSF is presented. It shows the relative value of the restoration error D(W) of the asymmetrical PSF is presented. It shows the relative value of the restoration error D(W) of the estimate estimate XOD OD, correlated with the value of this error for W = 2.0, that is D(W = 2.0). The characteristic XOD , correlated with the value of this error for W = 2.0, that is D(W = 2.0). The characteristic was was indicated in the presence of additive random distortions included in the signal Y with a relative indicated in the presence of additive random distortions included in the signal Y with a relative standard deviation equal to 0.5%. standard deviation equal to 0.5%.

Figure 9. 9. Value Value of of multiples multiples restoration restoration error function of of the the asymmetry asymmetry factor factor W W in in the the presence presence Figure error as as aa function of additive random distortions. of additive random distortions.

All simulations were made in the Matlab R2009b software package of Mathworks in the double precision format [19]. 4. Experimental ExperimentalStudies Studies 4.

scanning scanning direction direction

50 50 mm mm

25 25 mm mm

In order to to experimentally experimentally confirm the results of of the the simulation simulation studies, studies, two inductive inductive loop models with withshapes shapesand and dimensions presented in Figure 10 built. were (rectangular: built. (rectangular: turns of models dimensions presented in Figure 10 were turns of coil = 250, coil = 250, resistance 42.3 Ω, inductance = 5.6triangular: mH, triangular: of coil = 270, resistance= =39,2 39.2 Ω, Ω, resistance = 42,3 Ω,=inductance = 5.6 mH, turnsturns of coil = 270, resistance inductance == 4.74 4.74 mH). mH). inductance

50 mm

Figure 10. Rectangular and triangular inductive loop models. Figure 10. Rectangular and triangular inductive loop models.

Figure 11 11 shows The applied applied scanner scanner ensures ensures aa precise of Figure shows the the experiment experiment setup. setup. The precise displacement displacement of eddy-current sensor sensorover overthe thescanned scannedobject objectand andallows allows measurement traveled distance, the eddy-current measurement of of thethe traveled distance, as as well sensor output signal. Maximumlength lengthofofthe thescanned scannedpath pathisis1000 1000 mm, mm, and and attainable well as as thethe sensor output signal. Maximum measurement resolution The method method applied applied by by authors authors for PSF measurement resolution is is 0.1 0.1 mm. mm. The for determining determining the the sensor’s sensor’s PSF employs a pseudorandom binary (PRB) signal generated generated by by the the cut out steel sheet as the input signal exciting the identified ILD (Inductive Loop Detector). If DX means the bit width, the logic state

that means the total geometrical length of a sheet in which the signal was cut out is 5000 mm. The sheet was divided into seven separate sections with an additionally repeated ending sequence of bits from the leading section and initial sequence of bits from the section following the currently scanned portion of the PRB signal, as shown in Figure 3. The tested detector is shifted at a fixed height along consecutive sections. Thus the detector “sees” a continuous PRB signal despite the fact that the Sensors 2016, 16, 1642 9 of 13 experiment setup length is much shorter than the signal geometrical length whereas, due to the large width of the sheet, its solid edges that function only as a structural component are not “seen” by the detector. During the experiment, the sensors have beenIfsupplied with 10 kHz. exciting the identified ILD (Inductive Loop Detector). DX means thethe bitfrequency width, theoflogic stateThe “1” measured signals beenofprocessed using the correlation method. Seekingtothe sensors’ cut PSFout is corresponds to the have presence a steel sheet strip, and the state “0” corresponds an opening determined by the numerical solution of the Winer-Hopf equation. in the steel sheet. The signal length is defined by the number of bits N0.

Figure 11. The scanner scanner used during during the measurement measurement with with an installed eddy-current eddy-current sensor sensor and a steel sheet.

The asymmetrical shape of the triangular detector with of regard to the direction of =scanning During experiments, a signal was used with the number bits N0 = 500 and DX 10 mm, should result in an asymmetrical PSF in this detector. These assumptions are confirmed bysheet the that means the total geometrical length of a sheet in which the signal was cut out is 5000 mm. The characteristics presented in Figure 12. In Figure 12a, the PSF of both detectors is presented. For both was divided into seven separate sections with an additionally repeated ending sequence of bits from of these wereand established experimentally [16]. thethem, leading section initial sequence of bits from the section following the currently scanned portion of the PRB signal, as shown in Figure 3. The tested detector is shifted at a fixed height along consecutive sections. Thus the detector “sees” a continuous PRB signal despite the fact that the experiment setup length is much shorter than the signal geometrical length whereas, due to the large width of the sheet, its solid edges that function only as a structural component are not “seen” by the detector. During the experiment, the sensors have been supplied with the frequency of 10 kHz. The measured signals have been processed using the correlation method. Seeking the sensors’ PSF is determined by the numerical solution of the Winer-Hopf equation. The asymmetrical shape of the triangular detector with regard to the direction of scanning should result in an asymmetrical PSF in this detector. These assumptions are confirmed by the characteristics presented in Figure 12. In Figure 12a, the PSF of both detectors is presented. For both of them, these were established experimentally [16]. (a) (b) The W factor in the case of a triangular detector has a value of W = 1.25, while for the rectangular Figure 12. Experimentally obtained characteristics for inductive loop models (a) PSF for triangular detector that value is W = 1.05. The triangular detector is therefore indeed an asymmetrical detector. (1) and rectangular (2) detector; (b) output signals corresponding to shape signal in the form of PRBS The slight asymmetry appearing in the case of the rectangular detector may have been caused by the (Pseudo Random Binary Signal): 1—Output signal of the triangular detector, 2—Output signal of the uneven placement of the electromagnetic coils or by errors in the PSF identification. The PSF width of rectangular detector, 3—Shape signal—Input signal. both detectors is comparable, for the rectangular detector amounting to A + B = 66.8 mm, and for the triangular detector A + B = 66.2 mm. For both detectors compared, responses to stimulation by the PRBS signal were registered. The output signals obtained from both sensors are very similar. Due to the comparable PSF width of both sensors, they average input signals in a similar way, causing the loss of part of the information regarding the shape of the object measured (the cut steel sheet). The difference between the output signal of the detector and the input signal, measured with regard to error D, is D = 0.59 for the triangular detector, and D = 0.58 for the rectangular detector. In an ideal case, when the PSF of the detector is precisely known and the deconvolution algorithm has been performed perfectly, the restoration input signal should precisely describe the shape of the

Figure 11.1642 The scanner used during the measurement with an installed eddy-current sensor and 10 a of 13 Sensors 2016, 16, steel sheet.

object, thus allowing it shape to be perfectly recognised. In reality, conditionsofisscanning fulfilled, The asymmetrical of the triangular detector withneither regardoftothese the direction and the result of deconvolution, only to a greater or lesser extent, reflects the input signal. It is an should result in an asymmetrical PSF in this detector. These assumptions are confirmed only by the 2016, 16, 1642 10 of 13 estimate. The sensor’s construction methods undertaken used are to For improve characteristics presented in Figure 12. In Figure 12a, the and PSF algorithms of both detectors is intended presented. both the precision of this solution. of them, were in established The these W factor the case experimentally of a triangular[16]. detector has a value of W = 1.25, while for the rectangular detector that value is W = 1.05. The triangular detector is therefore indeed an asymmetrical detector. The slight asymmetry appearing in the case of the rectangular detector may have been caused by the uneven placement of the electromagnetic coils or by errors in the PSF identification. The PSF width of both detectors is comparable, for the rectangular detector amounting to A + B = 66.8 mm, and for the triangular detector A + B = 66.2 mm. For both detectors compared, responses to stimulation by the PRBS signal were registered. The output signals obtained from both sensors are very similar. Due to the comparable PSF width of both sensors, they average input signals in a similar way, causing the loss of part of the information regarding the shape of the object measured (the cut steel sheet). The difference between the output signal of the detector and the input signal, measured with regard to error D, is D = 0.59 for the triangular detector, and(a) D = 0.58 for the rectangular detector. (b) In an ideal case, when the PSF of the detector is precisely known and the deconvolution Figure 12. obtained characteristics for loop (a) PSF Figurehas 12. Experimentally Experimentally obtained characteristics for inductive inductive loop models models (a)precisely PSF for for triangular triangular algorithm been performed perfectly, the restoration input signal should describe the (1) and rectangular (2) detector; (b) output signals corresponding to shape signal in the form of PRBS (1) and rectangular (2) detector; (b) output signals corresponding to shape signal in the form PRBS shape of the object, thus allowing it to be perfectly recognised. In reality, neither of theseofconditions (Pseudo Random Binary Signal): 1—Output signal of the triangular detector, 2—Output signal of the (Pseudo Random Binary Signal): 1—Output signal of the triangular detector, 2—Output signal of is fulfilled, and the result of deconvolution, only to a greater or lesser extent, reflects the inputthe signal. rectangular detector, 3—Shape signal—Input signal. rectangular detector, 3—Shape signal—Input signal. It is only an estimate. The sensor’s construction methods undertaken and algorithms used are intended to improve the precision of this solution. deconvolution results obtained for thefor triangular detector and for theand rectangular In Figure Figure13, 13,thethe deconvolution results obtained the triangular detector for the detector using the LS algorithm are presented. In the case of the triangular detector, the restoration rectangular detector using the LS algorithm are presented. In the case of the triangular detector, the signal includes considerably more details more and more precisely describes the shape of the object restoration signal includes considerably details and more precisely describes thescanned shape of the when compared with the result of with the rectangular The restoration error,The accepted as a measure scanned object when compared the result detector. of the rectangular detector. restoration error, of the difference between resultthe XOD and the shaperesult of theXobject, D =shape 0.32 for accepted as a measure of the the deconvolution difference between deconvolution OD andisthe of the triangular = 0.52 fordetector, the rectangular object, is Ddetector, = 0.32 forand theDtriangular and D =detector. 0.52 for the rectangular detector.

Figure 13. Deconvolution Deconvolutionresult result using LS (Least Square) algorithm. the triangular Figure 13. using the the LS (Least Square) algorithm. 1—For 1—For the triangular detector, detector, 2—For the rectangular detector, 3—Shape of scanned object. 2—For the rectangular detector, 3—Shape of scanned object.

A comparison of experimental and simulation results allows Figure 14. It shows a fragment of A comparison of experimental and simulation results allows Figure 14. It shows a fragment of the the characteristics shown in Figure 7 and two points determined experimentally. These points characteristics shown in Figure 7 and two points determined experimentally. These points correspond correspond to the two selected values of the PSF asymmetry factor W i.e., W = 1.05 and W = 1.25. to the two selected values of the PSF asymmetry factor W i.e., W = 1.05 and W = 1.25. The differences between the results of the experiment and the simulation results may be due to several reasons. The PSF used in the experiment is not described by a function and its values were obtained in the identification process. For this reason, they are certainly subject to errors. In the simulation tests, PSF was modeled by compound shape of the two parts of the Gaussian curve. This shape differs from the shape of the PSF of the experiment, although the respective values of the factor W are the same. The simulation study was realized using a perfect digital convolution.

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In fact, the convolution is only an approximation of the process performed during the scan of anFigure object.13.Experimental such the as the identification PSF and1—For scanning of the object, Deconvolutionstudies, result using LS (Least Square) of algorithm. the triangular are subject to2—For the errors resultingdetector, from disturbances contained in the measuring signals. All this detector, the rectangular 3—Shape of scanned object. causes the observed difference in the obtained results, although, in the opinion of the authors, they are A comparison of experimental simulation results allows 14. It shows fragment of acceptable considering the complex and phenomenon occurring duringFigure the scanning of thea metal object thean characteristics Figure and the twomodel, pointsbased determined experimentally. These points by eddy-current shown sensor. in Please note7 that on convolution operations, is only an correspond to the twophenomena selected values of thebetween PSF asymmetry factor i.e., W = 1.05 and W = 1.25. approximation of the occurring the sensor andW the object.

Figure 14. 14. The comparison simulation results results with with the the experiment. experiment. Figure

Both sensors used in experimental studies, in spite of various geometrical shapes, have very similar metrological characteristics. The PSF width of both detectors is comparable, which indicates that both the sensors have a similar averaging ability to that of the measurement signal. This is confirmed by the similar error value (10), used to evaluate the distance between the measured signal and the output signal of the sensor. On this basis, it is justified to claim that the observed improvement in the efficiency of the deconvolution algorithm is due to PSF asymmetry. This thesis is also confirmed by the results of the simulation tests. The impact of the asymmetry factor W of the sensor’s PSF on the accuracy of the signal restoration, in the sense of error D, was investigated. In this study, however, any particular geometric shape of the sensor was not assumed. Experimental tests have been carried out for the triangular sensor, for which the PSF asymmetry factor is equal W = 1.25. Tests for sensors with higher values of the asymmetry factor W were performed by means of computer simulation. Their results, shown in Figure 7, confirm the positive impact of the PSF asymmetry on the effectiveness of deconvolution and accuracy of the measuring signal restoration. Performing experimental tests with the sensors having a required degree of PSF asymmetry is however conditioned by the ability of designing such sensors. Currently, such a method is not known and its preparation requires further long-term experimental and simulation studies. The authors plan to undertake such research. 5. Conclusions This paper presents the results of studies on an inductive loop used in systems classifying vehicles. A commonly used and widely accepted spatial convolution model for the classification of vehicle chassis was used in the study. One of the ways of increasing the spatial resolution of the magnetic signature obtained and the effectiveness of the classification process is a deconvolution operation. In many publications, numerical algorithms are presented to decrease the estimate error of the object shape signal. In this paper, a new approach leading to an improvement in the quality of the restoration signal obtained from an eddy-current sensor, is presented. This was achieved by shaping the sensor PSF. In the simulation studies presented, it is shown that even a slight PSF asymmetry results in a considerable decrease in the restoration error of the input signal. The results of experimental studies confirmed the assumptions of the simulations. Two inductive loop models were used in the experiments, with a symmetrical and asymmetrical shape with regard to the scanning direction. As a

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result of the application of an LS algorithm, a significantly smaller restoration error was obtained for the asymmetrical sensor. In light of these results, it appears justifiable to search for effective methods of designing eddy-current sensors with the PSF asymmetry. Continuing research on the issue described in this paper, authors intend to develop a method allowing the design of eddy-current sensors with desired PSF asymmetry. The use of such sensors, in the form of an inductive loop, in the measurement systems of road traffic parameters, will allow a considerable increase in the information contained in the measuring signals, generated by the vehicles that pass over the sensor. The practical use of such sensors, in the authors’ opinion, will allow the efficient detection of the axles of trucks and cars, the estimation of the location of the axles relative to the characteristic points in the geometry of the vehicle (e.g., front and rear bumpers), and will allow the measurement of distances between axles, as well as detecting the presence of a trailer. As a consequence, the authors expect to significantly improve the effectiveness of vehicle classification using systems equipped only with inductive loops. Effectiveness could be significantly better than this, which is achieved by using the best systems, equipped with axle detectors, while the costs of such a system will be many times reduced. Acknowledgments: The paper was financed by the statutory fund of the Department of Measurements and Electronics AGH UST. Author Contributions: Janusz Gajda and Marek Stencel conceived and designed the experiments and analyzed the simulation and experimental results. Janusz Gajda suggested the use of induction loop sensors having asymmetric PSF and identified the PSF of both sensors used in experiments. Marek Stencel conducted the simulation tests, described the convolution model and different forms of convolution matrix. Conflicts of Interest: The authors declare no conflict of interest. The founding sponsors had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, and in the decision to publish the results.

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