The variational Euler-Elastica model is developed for high-precision terahertz 3D tomographic applications. Our method provides a new approach to mitigate diffraction-limited terahertz reconstructed images with noises and demonstrates the efficient practicality with limited THz datasets

Chen, Ke

Yang, Shang-Hua

Zhang, Yiyao

The variational Euler-Elastica model is developed for high-precision terahertz 3D tomographic applications. Our method provides a new approach to mitigate diffraction-limited terahertz reconstructed images with noises and demonstrates the efficient practicality with limited THz datasets.</jats:p

University of Liverpool Repository

Euler-Elastica Variational Model for PulsedTerahertz 3D ImagingYiyao Zhang1, 2, *, Ke Chen1, and Shang-Hua Yang21 Centre for Mathematical Imaging Techniques, Department of Mathematical Sciences, University of LiverpoolMathematical Sciences Building, Liverpool, L69 7ZL, United Kingdom2 Institute of Electronics Engineering, Department of Electrical Engineering, National Tsing Hua UniversityNo. 101, Section 2, Kuang-Fu Road, Hsinchu, 30013, Taiwan* Yiyao.Zhang@liverpool.ac.uk; yiyaozhang@gapp.nthu.edu.twAbstract: The variational Euler-Elastica model is developed for high-precision terahertz3D tomographic applications. Our method provides a new approach to mitigate diffraction-limited terahertz reconstructed images with noises and demonstrates the efficient practical-ity with limited THz datasets. © 2022 The Author(s)1. IntroductionCredited to the bio-safe penetrability of terahertz (THz) radiation, 3D nondestructive inspection is one of theprimary applications to investigate the interior of visually non-transparent objects. Among different THz 3D imag-ing methodologies (i.e. THz compressed sensing imaging, THz holographic imaging, THz computed tomographicimaging), THz pulsed imaging based on the THz time-domain spectroscopy (THz-TDS) system plays critical rolesin multi-dimensional object information extractions, which is comprised of detailed records by time-resolved THzelectric field signals from the interactions between objects under test and broadband THz wave [1–3]. However,the acquisition time of spatial-temporal-spectral data is increasing exponentially due to the nature of the rasterscanning, and even the computational time is rising if requiring the image with higher resolution. Additionally, 3Dreconstructed quality is strongly influenced by images with diffraction-limited THz beams involving noise sourcesfrom the THz-TDS system. Here, we proposed an Euler-Elastica variational model to address the challenges ofthe efficiency and the quality of undesirable images, paving a new way for high-speed, high-precision THz 3Dimaging applications of nondestructive sensing and inspection.2. Methods and Demonstrations with Quantitative ComparisonsGrounded on the transmission geometry with the schematic diagram of our THz 3D imaging system setup asFig. 1 depicts, the temporal pulse time-resolved signals for each voxel are recorded by the ASynchronous OPticalSampling (ASOPS) THz-TDS system (Menlo TERA ASOPS, Menlo Systems). Without the delay stage assembleby mechanisms in conventional THz-TDS systems, the first acceleration of data collection is profited from twoasynchronous femtosecond Er-doped fibre lasers with the minor variation of input repetition rates in this ASOPSsystem so that the data acquisition time is around 5 ms (whereas it would require minutes or even hours in thetraditional system) for one THz signal trace with 100 ps time span and the temporal resolution of the dataset is0.05 ps (2000 sampling points in each voxel) with about 41.7 dB dynamic range from 0.3 THz to 3 THz [2, 3].Emitter Lens LensObjectLens Lens Detectorfs laser 1 fs laser 2PCMotorisedStageFig. 1: Schematic diagram of THz 3D Imaging system setup by transmission geometry.Similar to the previous work (Fig. 2(b)-(c) and (e)-(f)) [4] and based on the diverse reaction between THzwave and samples (Fig. 2(a)), Fourier transform F is first applied into time-resolved signals f (t) in the timedomain for each voxel, and then the phase Φ(ω) in the frequency domain to reflect the precise depth can becalculated by Φ(ω) = tan−1( ba)where the Fourier amplitude is expressed as F ( f (t)) = F(ω) = a(ω)+ ib(ω) =∫f (t)e−2πiωt dt. As Fig. 2(d) illustrated, here the frequency around 1.0010 THz is chosen based on the intensivelight-matter interactions between THz wave and various ingredients (fat for peanut kernels and fibre for outershell). Subsequently, by inverse Radon transform (IRT), the cross sections (slices) can be derived from the inputsinograms which are assembled from the projections. To reconstruct a high resolution surface u∗ε based on lowresolution and noisy input surface defined by the interior uinε and the exterior uexε with 1 and 0 indicated respectivelyby the profile function, our proposed variational Euler-Elastica-based model is to minimise the energyu∗ε = Eε(u) =∫Ω(ε2|∇u|2 + 1εW (u))dΩ+12ε∫Ω(ε △u− 1εW ′(u))2dΩ, (1)which is Γ-convergence approximation to functional energies involving total variations and mean curvatures [5]Γ(L1(Ω))− limε→0Eε(u) =∫ 10√2W (s)ds∫∂Ω(1Ω + |H∂Ω|2)dH 2, (2)subject to the linear obstacle restriction uinε ≤ u ≤ uexε ∈ Ω ∈ R3 and the diffuse interface width parameter ε withthe double-well potential W (u) = 12 u2(1− u)2 and its first derivative W ′(u) = u(u− 1)(2u− 1). Next, numericalsolution for the Euler-Lagrange equation of (1) is by the projected gradient descent method with a semi-implicittime-stepping scheme in time τt where fast Fourier transform will be utilised to speed up the matrix inversionud+1 =(ID − ετt △+τtε△2)−1(ud − τtεW ′(ud+1)+τtε△W ′(ud+1)−τtεW ′′(ud+1)△ud+1 + τtε3W ′(ud+1)W ′′(ud+1)).(3)Accordingly, to quantify the smoothness of results by Willmore-based formulation and Euler-Elastica-basedformulation, the benchmark is devised by computing the standard deviation of Gaussian curvatures (GC) σGC :=σ(κG) and mean curvatures (MC) σMC := σ(κ̄) from the viewpoint of discrete geometry (more rudiments in [6])κG(vi) =2π −NFiR∑k=1θikAiRand κ̄(vi) =12AiR∑j∈RNVi(cotαi j + cotβi j)(v j −vi) (4)for all vertices in surfaces represented by triangular meshes where AiR stands for the appropriately chosen areafrom the patch within R-ring neighbouring vertices RNVi around the vertex vi, θik denotes the angle of the kth faceat the vertex vi, NFiR is the total number of faces in the set FiR around this vertex vi, as well as αi j and βi j are twoangles opposite to the sharing edge in the two triangles. Remark that the lower value of each computing curvaturesindicates the better smoothness concerning the variation of curvatures (as illustrations in Fig. 2(f)-(g)).Eventually, our model with the fast numerical approximation provides a smooth reconstruction (Fig. 2(g)) fromonly 1/3 input slices with noise in the low resolution after the selection of crucial features (Fig. 2(e)) not onlyto improve the quality of 3D reconstruction in the super-resolution but also to reduce scanning time as well asprocessing time by the efficient algorithm under the low resolution. In other applications like X-ray and magneticresonance imaging (MRI), our proposed framework can be readily adapted and applied.(a)123Maximum Intensity (Time Domain)102(b) (c)1.0010 THz-6-22Phase (Frequency Domain)103(d) (e) (f) (g)Fig. 2: Illustrations of (a) the unveiled peanut kernels after scanning by Fig. 1; (b) representative projectionof the maximum intensity in the time domain; (c) reconstruction result of peanuts from (b); (d) representativeprojection of unwrapped phase around 1.0010 THz in the frequency domain; (e) 46 input cross sections (slices)extracted from (d) by IRT; (f) reconstruction result of peanut kernels from (e) by Willmore model [4] with highercomputational time (more iterations) and lower smoothness; (g) final reconstruction result of kernels from (e) bynew proposed Euler-Elastica model with lower computational time (less iterations) and higher smoothness.References1. H. Guerboukha, K. Nallappan, and M. Skorobogatiy, “Toward real-time terahertz imaging,” Adv. Opt. Photon, 10,843–938 (2018).2. W.-T. Su, Y.-C. Hung, P. Yu, C.-W. Lin and S.-H. Yang, “Physics-guided Terahertz Computational Imaging,” IEEESignal Processing Magazine, 40(1) (2023).3. Y.-C. Hung, T.-H. Chao, P. Yu, and S.-H. Yang, “Terahertz spatio-temporal deep learning computed tomography,” Opt.Express 30, 22523-22537 (2022).4. Y. Zhang, K. Chen and S.-H. Yang, “Fast Terahertz 3D Super-Resolution Surface Reconstruction by Variational Modelfrom Limited Low-Resolution Sampling,” 2022 47th International Conference on Infrared, Millimeter and TerahertzWaves (IRMMW-THz), pp. 1-2 (2022).5. M. Röger and R. Schätzle, “On a Modified Conjecture of De Giorgi,” Mathematische Zeitschrift, 254(4), 675-714(2006).6. M. Meyer, M. Desbrun, P. Schröder and A. H. Barr, “Discrete Differential-Geometry Operators for Triangulated 2-Manifolds,” in Visualization and Mathematics III, Springer Berlin Heidelberg, 35-57 (2003).

Euler-Elastica Variational Model for Pulsed Terahertz 3D Imaging

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Euler-Elastica Variational Model for Pulsed Terahertz 3D Imaging

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