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Pulsed excitation terahertz tomography – multiparametric approach

  • Przemyslaw Lopato EMAIL logo
Published/Copyright: April 18, 2018
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Open Physics
From the journal Open Physics Volume 16 Issue 1

Abstract

This article deals with pulsed excitation terahertz computed tomography (THz CT). Opposite to x-ray CT, where just a single value (pixel) is obtained, in case of pulsed THz CT the time signal is acquired for each position. Recorded waveform can be parametrized - many features carrying various information about examined structure can be calculated. Based on this, multiparametric reconstruction algorithm was proposed: inverse Radon transform based reconstruction is applied for each parameter and then fusion of results is utilized. Performance of the proposed imaging scheme was experimentally verified using dielectric phantoms.

Keywords: electromagnetic waves; terahertz imaging; tomography; reconstruction; artificial neural networks
PACS: 81.70.Tx; 84.40.-x; 81.70.Ex

1 Introduction

Electromagnetic waves in terahertz frequency range (T-rays) enable non-invasive, non-contact and non-ionizing inspection of dielectric materials and semiconductors. T-rays are sensitive to complex permittivity changes, therefore any defect or structure detail causing its noticeable disturbance can be detected. The most common method of defects localization is transmission or reflection imaging based on pulsed terahertz Time Domain Spectroscopy (THz TDS) [1,2,3,4,5].

Tomography is defined as the cross-sectional imaging of an object under test (OUT) by measuring scattered wave (rays). There are several terahertz imaging techniques [6,7,8,9,10]:

  • time of flight reflection tomography (THz ToFRT),

  • digital holography,

  • diffraction tomography (THz DT),

  • tomography with binary lens,

  • computed tomography (THz CT).

Terahertz CT enables analogous to conventional x-ray CT reconstruction of the objects physical properties distribution based on sectional slices (Figure 1). However, in case of terahertz tomography both amplitude and phase data are collected (opposite to intensity of radiation for x-ray CT). The above fact causes that THz CT provides more information about the examined object [1]. For each slice object under test is rotated by angle Θ and shifted by distance r. The terahertz radiation passing through the object is recorded in the measuring plane, where the projection R (Θ, r) is obtained. In Ref. [7] it has been shown that the change in phase of the electromagnetic wave in the measuring plane can be used to determine the distribution of the refractive index n (and consequently permittivity ∊) in the cross-section of the object. Forward problem (projection R (Θ, r) calculation) mathematically is described by Radon transform [7, 9, 11]:

Figure 1 Simplified scheme of THz CT principle of operation
Figure 1

Simplified scheme of THz CT principle of operation

RΘ,r=∫−∞∞∫−∞∞gx,yδr−xcos⁡Θ−ysin⁡Θdxdy(1)

where: g(x,y) – spatial distribution of the selected electromagnetic parameter of the imaged object, e.g. refractive index n(x, y), δ (•) – Dirac delta function, Θ – projection angle, r – ray distance from the axis of rotation.

The inverse problem – the reconstruction of the spatial distribution of the OUT’s electromagnetic parameters – is mathematically described by the inverse Radon transform [9, 11]:

gRx,y=∫0π∫−∞∞RΘ,rδr−xcos⁡Θ−ysinΘdrdΘ(2)

In this paper terahertz computed tomography system with pulsed excitation is shown. Moreover, the reconstruction algorithm based on standard inverse Radon transform (2) and artificial neural network is proposed and verified using tomographic inspections of phantoms made of foamed polystyrene.

2 Measuring system

The proposed tomographic system is based on TRay4000 pulsed terahertz spectroscope of Picometrix (Figure 2). Picosecond order pulses are generated and detected by transmitter and receiver heads using photoconductive antennas (PCA). Terahertz beam focused by HDPE lenses is transmitted through examined object. The OUT was mounted on a (Θ, r, z) rotation/translation stage and scanned in order to obtain a slice image. For each position r and projection angle Θ, a transmission measurement is performed and a time-domain signal s(t) is recorded. Exemplary waveform obtained during tomographic inspection as well as some basic features of THz pulse are shown in Figure 3. Whole system was controlled by a software implemented in Matlab environment.

Figure 2 Photo of utilized pulsed THz CT system
Figure 2

Photo of utilized pulsed THz CT system

Figure 3 Measured signal s(t) obtained for each position and definitions of basic parameters in the time domain (maxVal, maxPos, minVal and minPos)
Figure 3

Measured signal s(t) obtained for each position and definitions of basic parameters in the time domain (maxVal, maxPos, minVal and minPos)

3 Reconstruction algorithm

Other (than phase shift) parameters of the signal measured in the measurement plane may be used for reconstruction. In the case of pulsed excitation (TDS), the pulse energy, amplitude or time delay introduced by the presence of the object is determined. Time signal acquisition makes it possible to design a very large number of time domain (TD), frequency (FD) and joint time-frequency domain (TFD) parameters. Depending on the parameter used during reconstruction, the resulting image (slice) will have a different character, i.e. other features will be exposed. For example, the use of frequency parameters (the amplitudes of different harmonics) will increase the sensitivity of detection of materials with different frequency dependent refractive indices or absorption coefficients. The use of multiple parameters reconstruction and subsequent fusion of the resulting distributions enables to achieve better results, which will be shown later in this paper. Generally, the multiparametric approach is becoming more popular in non-destructive testing [12].

Based on acquired s(t) time-domain waveform, a set of features ζn was calculated both in time and frequency domain. Each obtained parameter ζn can be utilized in order to form a sinogram (Figure 1). Sinograms acquired by measurements as well as reconstruction results using inverse Radon transform [11] in case of selected parameters ζn of waveform s(t) are shown in Figure 4. Based on observable differences in obtained images, the following conclusion can be drawn – selected parameters carry various information content about spatial distribution of dielectric properties in object under test. Therefore, multiparametric reconstruction algorithm using Radon’s inverse transformation and artificial neural networks in the process of combining information derived from different parameters is proposed. The block diagram of the proposed procedure is shown in Figure 5. The first step is to perform an inspection using the developed tomographic system. Then a preprocessing in form of filtration and deconvolution is carried out. Filtration may be implemented in the form of a one-dimensional median filter applied directly to the measurement signal s(t) or a two-dimensional median filter applied on the sinogram RM(Θ, r). In both cases, instead of median filtration, one- or two-dimensional wavelet denoising may be used. Another task implemented at this stage is the deconvolution of the sinogram RM(Θ, r) – blur limiting operation due to utilization of terahertz Gaussian beam instead of the straight ray beam. The data processed in this way are used in the next step – multi-parametric reconstruction. It is schematically presented in Figure 6.

Figure 4 Sinograms (up) and results of the reconstruction (down) of the object containing the defect for various parameters used in the reconstruction process (maxVal, minVal and minPos)
Figure 4

Sinograms (up) and results of the reconstruction (down) of the object containing the defect for various parameters used in the reconstruction process (maxVal, minVal and minPos)

Figure 5 Scheme of multiple parameters reconstruction algorithm
Figure 5

Scheme of multiple parameters reconstruction algorithm

Figure 6 Scheme of multiple parameters fusion using artificial neural network; gζn (x,y) – reconstruction result based on inverse Radon transform and ζn parameter of waveform s(t)
Figure 6

Scheme of multiple parameters fusion using artificial neural network; gζn (x,y) – reconstruction result based on inverse Radon transform and ζn parameter of waveform s(t)

Twelve two-dimensional distributions of selected parameters ζn (x,y) - obtained by reconstruction using the inverse Radon transform - were scanned by the sliding window as shown in Figure 6. A set of 12 parameters from 27 initial ones was selected based on a preliminary analysis of the signal to noise ratio (SNR). Their definitions are presented in Table 1. Both time domain features (ζ1 – ζ9) and frequency domain (ζ10 – ζ12) were used. Reconstructions obtained using the selected parameters enable identification of both the edges of the examined phantoms and the objects inside the structure, such as artificial or real defects. They contain a variety of information about the OUT. Data from the window that moves over 12 concurrent reconstructions (based on 12 different parameters) is given to the input of a previously trained artificial neural network (ANN). The neural model estimates the value of the final reconstruction OR for the position determined by the center of the sliding window (x, y). Three-layer network was used: 2 hidden layers with sigmoidal activation function and 1 output layer with linear activation function. The neural network was trained using the Levenberg-Marquardt method.

Table 1

Parameters utilized for reconstruction

No.Parameters
DefinitionDescription
1.ζ1 = max [s (t)] = maxValmaximum value of s(t) signal (maxVal in Figure 3)
2.ζ2 = td maxtime delay at which the maximum value of the signal occurs (maxPos in Figure 3)
3.ζ3 = min [s (t)] = min Valminimum value of s(t) signal (minVal in Figure 3)
4.ζ4 = td mintime delay at which the minimum value of the signal occurs (minPos in Figure 3)
5.ζ5 = max [s (t)] – min [s (t)]signal peak-to-peak value
6. ζ6=∫t1t2stdtarea under absolute value of signal in time domain
7.ζ7 = mean [s (t)]mean value of the signal
8.ζ8 = med [s (t)]median of sampled signal
9.ζ9 = sid [s (t)]standard deviation of the sampled signal
10.ζ10 = 10log10 |S (f1)|amplitude of selected harmonic f1 = 70GHz
11.ζ11 = 10log10 |S (f3)|amplitude of selected harmonic f2= 250GHz
12.ζ12 = 10log10 |S (f5)|amplitude of selected harmonic f3 = 1.3THz

4 Results

In order to determine the parameters of proposed reconstruction procedure and neural model, the following criterion of reconstructed image evaluation was proposed:

KRwms,NA,kζ=KFNwms,NA,kζ⋅KFPwms,NA,kζ(3)

where: KR – criterion for evaluation of the reconstruction result by the proposed procedure, wms – size of the sliding window,NA – number of neurons in the neural model, kζ – number of ζn parameters used for reconstruction, KFN – sum of false negative indications, KFP – sum of false positive indications.

Such simple criterion can be utilized because of quasi-homogenous character of utilized phantoms. For all considered criteria (KFN, KFP and KR), the smaller the value, the better is the result of the reconstruction. Thus, the minimum value of KR will be sought to determine the parameters of the utilized neural model. In further analysis, the above criterion will be used in a standardized form:

KRN=KRmax(KR)(4)

An experiment was conducted, in which the influence of the number of parameters used in the reconstruction on its quality was investigated. The size of sliding window ranged from 3 to 11 and total number of neurons in ANN ranged from 10 to 60. For each combination of (wms, NA, kζ), 100 networks were trained based on T-shape phantom presented in Figure 7. The validation was performed using several phantoms of different shape. Exemplary results of reconstruction using standard inverse Radon transform and proposed procedure are presented in Figure 8. In case of proposed method, one can observe blur reduction and better defined phantom edges (no oscillation at the air-phantom boundary), compared to standard procedure. This allows a more accurate determination of the size of the examined object and its details. The disadvantage of the proposed algorithm is the need to train a neural model for combining data from various parameters. An influence of number of parameters on reconstruction quality is shown in Figure 9. The order of the features used in this experiment was chosen randomly. One can observe, the reconstruction error (KRN) decreases nonlinearly with increase of kζ. Utilization of 4 parameters causes noticeably smaller KRN than in case of one parameter. Further increase of kζ doesn’t affect the reconstruction error noticeably.

Figure 7 The training set defined by the cross-sectional photo of T-shape phantom (a) and the response of the trained neural model for the training set (b)
Figure 7

The training set defined by the cross-sectional photo of T-shape phantom (a) and the response of the trained neural model for the training set (b)

Figure 8 Exemplary reconstruction results for the phantom with artificial defect: a) using inverse Radon transform based on 1 parameter (signal peak-to-peak value), b) using multi-parametric reconstruction
Figure 8

Exemplary reconstruction results for the phantom with artificial defect: a) using inverse Radon transform based on 1 parameter (signal peak-to-peak value), b) using multi-parametric reconstruction

Figure 9 Criterion for evaluation a trained neural model related to the number of parameters used for training
Figure 9

Criterion for evaluation a trained neural model related to the number of parameters used for training

5 Conclusions

The proposed THz CT system and multiparametric algorithm enable proper reconstruction of the considered objects cross-section (as shown in Figure 8) and cause blur reduction of the resulting image. Moreover, the algorithm enables reduction of reconstruction error in comparison to the case of single parameter based inversion (as shown in Figure 9). That proves correctness of proposed multiparametric approach.

The drawback of proposed algorithm is a necessity of training the neural model, thus it is not as widely applicable as transforms based inversions.

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Received: 2017-11-02
Accepted: 2017-11-12
Published Online: 2018-04-18

© 2018 P. Lopato, published by De Gruyter

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

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https://doi.org/10.1515/phys-2018-0018
Keywords for this article
electromagnetic waves; terahertz imaging; tomography; reconstruction; artificial neural networks
Creative Commons
BY-NC-ND 4.0

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  48. Terahertz pulse imaging: A novel denoising method by combing the ant colony algorithm with the compressive sensing
  49. Characteristics of microscopic pore-throat structure of tight oil reservoirs in Sichuan Basin measured by rate-controlled mercury injection
  50. An activity window model for social interaction structure on Twitter
  51. Transient thermal regime trough the constitutive matrix applied to asynchronous electrical machine using the cell method
  52. On the zagreb polynomials of benzenoid systems
  53. Integrability analysis of the partial differential equation describing the classical bond-pricing model of mathematical finance
  54. The Greek parameters of a continuous arithmetic Asian option pricing model via Laplace Adomian decomposition method
  55. Quantifying the global solar radiation received in Pietermaritzburg, KwaZulu-Natal to motivate the consumption of solar technologies
  56. Sturm-Liouville difference equations having Bessel and hydrogen atom potential type
  57. Study on the response characteristics of oil wells after deep profile control in low permeability fractured reservoirs
  58. Depiction and analysis of a modified theta shaped double negative metamaterial for satellite application
  59. An attempt to geometrize electromagnetism
  60. Structure of traveling wave solutions for some nonlinear models via modified mathematical method
  61. Thermo-convective instability in a rotating ferromagnetic fluid layer with temperature modulation
  62. Construction of new solitary wave solutions of generalized Zakharov-Kuznetsov-Benjamin-Bona-Mahony and simplified modified form of Camassa-Holm equations
  63. Effect of magnetic field and heat source on Upper-convected-maxwell fluid in a porous channel
  64. Physical cues of biomaterials guide stem cell fate of differentiation: The effect of elasticity of cell culture biomaterials
  65. Shooting method analysis in wire coating withdrawing from a bath of Oldroyd 8-constant fluid with temperature dependent viscosity
  66. Rank correlation between centrality metrics in complex networks: an empirical study
  67. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering
  68. Modeling of electric and heat processes in spot resistance welding of cross-wire steel bars
  69. Dynamic characteristics of triaxial active control magnetic bearing with asymmetric structure
  70. Design optimization of an axial-field eddy-current magnetic coupling based on magneto-thermal analytical model
  71. Thermal constitutive matrix applied to asynchronous electrical machine using the cell method
  72. Temperature distribution around thin electroconductive layers created on composite textile substrates
  73. Model of the multipolar engine with decreased cogging torque by asymmetrical distribution of the magnets
  74. Analysis of spatial thermal field in a magnetic bearing
  75. Use of the mathematical model of the ignition system to analyze the spark discharge, including the destruction of spark plug electrodes
  76. Assessment of short/long term electric field strength measurements for a pilot district
  77. Simulation study and experimental results for detection and classification of the transient capacitor inrush current using discrete wavelet transform and artificial intelligence
  78. Magnetic transmission gear finite element simulation with iron pole hysteresis
  79. Pulsed excitation terahertz tomography – multiparametric approach
  80. Low and high frequency model of three phase transformer by frequency response analysis measurement
  81. Multivariable polynomial fitting of controlled single-phase nonlinear load of input current total harmonic distortion
  82. Optimal design of a for middle-low-speed maglev trains
  83. Eddy current modeling in linear and nonlinear multifilamentary composite materials
  84. The visual attention saliency map for movie retrospection
  85. AC/DC current ratio in a current superimposition variable flux reluctance machine
  86. Influence of material uncertainties on the RLC parameters of wound inductors modeled using the finite element method
  87. Cogging force reduction in linear tubular flux switching permanent-magnet machines
  88. Modeling hysteresis curves of La(FeCoSi)13 compound near the transition point with the GRUCAD model
  89. Electro-magneto-hydrodynamic lubrication
  90. 3-D Electromagnetic field analysis of wireless power transfer system using K computer
  91. Simplified simulation technique of rotating, induction heated, calender rolls for study of temperature field control
  92. Design, fabrication and testing of electroadhesive interdigital electrodes
  93. A method to reduce partial discharges in motor windings fed by PWM inverter
  94. Reluctance network lumped mechanical & thermal models for the modeling and predesign of concentrated flux synchronous machine
  95. Special Issue Applications of Nonlinear Dynamics
  96. Study on dynamic characteristics of silo-stock-foundation interaction system under seismic load
  97. Microblog topic evolution computing based on LDA algorithm
  98. Modeling the creep damage effect on the creep crack growth behavior of rotor steel
  99. Neighborhood condition for all fractional (g, f, n′, m)-critical deleted graphs
  100. Chinese open information extraction based on DBMCSS in the field of national information resources
  101. 10.1515/phys-2018-0079
  102. CPW-fed circularly-polarized antenna array with high front-to-back ratio and low-profile
  103. Intelligent Monitoring Network Construction based on the utilization of the Internet of things (IoT) in the Metallurgical Coking Process
  104. Temperature detection technology of power equipment based on Fiber Bragg Grating
  105. Research on a rotational speed control strategy of the mandrel in a rotary steering system
  106. Dynamic load balancing algorithm for large data flow in distributed complex networks
  107. Super-structured photonic crystal fiber Bragg grating biosensor image model based on sparse matrix
  108. Fractal-based techniques for physiological time series: An updated approach
  109. Analysis of the Imaging Characteristics of the KB and KBA X-ray Microscopes at Non-coaxial Grazing Incidence
  110. Application of modified culture Kalman filter in bearing fault diagnosis
  111. Exact solutions and conservation laws for the modified equal width-Burgers equation
  112. On topological properties of block shift and hierarchical hypercube networks
  113. Elastic properties and plane acoustic velocity of cubic Sr2CaMoO6 and Sr2CaWO6 from first-principles calculations
  114. A note on the transmission feasibility problem in networks
  115. Ontology learning algorithm using weak functions
  116. Diagnosis of the power frequency vacuum arc shape based on 2D-PIV
  117. Parametric simulation analysis and reliability of escalator truss
  118. A new algorithm for real economy benefit evaluation based on big data analysis
  119. Synergy analysis of agricultural economic cycle fluctuation based on ant colony algorithm
  120. Multi-level encryption algorithm for user-related information across social networks
  121. Multi-target tracking algorithm in intelligent transportation based on wireless sensor network
  122. Fast recognition method of moving video images based on BP neural networks
  123. Compressed sensing image restoration algorithm based on improved SURF operator
  124. Design of load optimal control algorithm for smart grid based on demand response in different scenarios
  125. Face recognition method based on GA-BP neural network algorithm
  126. Optimal path selection algorithm for mobile beacons in sensor network under non-dense distribution
  127. Localization and recognition algorithm for fuzzy anomaly data in big data networks
  128. Urban road traffic flow control under incidental congestion as a function of accident duration
  129. Optimization design of reconfiguration algorithm for high voltage power distribution network based on ant colony algorithm
  130. Feasibility simulation of aseismic structure design for long-span bridges
  131. Construction of renewable energy supply chain model based on LCA
  132. The tribological properties study of carbon fabric/ epoxy composites reinforced by nano-TiO2 and MWNTs
  133. A text-Image feature mapping algorithm based on transfer learning
  134. Fast recognition algorithm for static traffic sign information
  135. Topical Issue: Clean Energy: Materials, Processes and Energy Generation
  136. An investigation of the melting process of RT-35 filled circular thermal energy storage system
  137. Numerical analysis on the dynamic response of a plate-and-frame membrane humidifier for PEMFC vehicles under various operating conditions
  138. Energy converting layers for thin-film flexible photovoltaic structures
  139. Effect of convection heat transfer on thermal energy storage unit
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