Low-Frequency Ultrasonic Tomography: Mathematical Methods and Experimental Results

Goncharsky, A. V.; Romanov, S. Yu.; Seryozhnikov, Sergey Y.

doi:10.3103/s0027134919010090

Cited by 26 publications

(18 citation statements)

References 24 publications

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“…To solve these inverse problems, the conventional approach is optimizationbased methods; see e.g. [2,8,[10][11][12]. However, it is well known that these methods may suffer from multiple local minima and ravines, and their convergence analysis is also unknown in many cases.…”

Section: Introductionmentioning

confidence: 99%

Convergence of a series associated with the convexification method for coefficient inverse problems

Klibanov

Nguyen

2020

Journal of Inverse and Ill-Posed Problems

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This paper is concerned with the convergence of a series associated with a certain version of the convexification method. That version has been recently developed by the research group of the first author for solving coefficient inverse problems. The convexification method aims to construct a globally convex Tikhonov-like functional with a Carleman weight function in it. In the previous works, the construction of the strictly convex weighted Tikhonov-like functional assumes a truncated Fourier series (i.e. a finite series instead of an infinite one) for a function generated by the total wave field. In this paper, we prove a convergence property for this truncated Fourier series approximation. More precisely, we show that the residual of the approximate PDE obtained by using the truncated Fourier series tends to zero in L^{2} as the truncation index in the truncated Fourier series tends to infinity. The proof relies on a convergence result in the H^{1}-norm for a sequence of L^{2}-orthogonal projections on finite-dimensional subspaces spanned by elements of a special Fourier basis. However, due to the ill-posed nature of coefficient inverse problems, we cannot prove that the solution of that approximate PDE, which results from the minimization of that Tikhonov-like functional, converges to the correct solution.

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Section: Introductionmentioning

confidence: 99%

Convergence of a series associated with the convexification method for coefficient inverse problems

Klibanov

Nguyen

2020

Journal of Inverse and Ill-Posed Problems

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“…In variant 4, we present a DOE where the observer sees four symbols «A», «B», «C», and «D», which move across the area of the optical element when it is illuminated by a point source. Similar to variants 1 and 2, area The location map of two nonintersecting areas G (1) and G (2) in the area of optical element G. (d) The location map of three nonintersecting areas G (1) , G (2) , and G (3) in the area of optical element G. (e) The location map of four nonintersecting areas G (1) , G (2) , G (3) , and G (4) in the area of optical element G. (f) The location map of subareas G ′ ij in elementary areas G ij . (g) Complementary colour image seen at diffraction angles greater than 60°.…”

Section: Resultsmentioning

confidence: 99%

“…(h) Plots of the function h (i) (x, y) for variants 1-3. G is subdivided into nonintersecting areas G (1) , G (2) , G (3) , and G (4) , as shown in Fig. 2e.…”

Section: Resultsmentioning

confidence: 99%

“…Over the past 30 years, the technology for the mass replication of optical security elements has experienced much development, allowing a single original security element to be replicated hundreds of millions of times, thereby substantially reducing the cost of these copies in large projects. This is an outstanding achievement that has provided the technical conditions to be met by modern equipment to allow security elements to be replicated as numerous copies, with the microreliefs of the copied elements reproducing very precisely that of the original 3 , 4 .…”

Section: Introductionmentioning

confidence: 99%

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Nanooptical elements for visual verification

Goncharsky

Melnik²,

Durlevich

2021

Sci Rep

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This paper focuses on the development of flat diffractive optical elements (DOEs) for protecting banknotes, documents, plastic cards, and securities against counterfeiting. A DOE is a flat diffractive element whose microrelief, when illuminated by white light, forms a visual image consisting of several symbols (digits or letters), which move across the optical element when tilted. The images formed by these elements are asymmetric with respect to the zero order. To form these images, the microrelief of a DOE must itself be asymmetric. The microrelief has a depth of ~ 0.3 microns and is shaped with an accuracy of ~ 10–15 nm using electron-beam lithography. The DOEs developed in this work are securely protected against counterfeiting and can be replicated hundreds of millions of times using standard equipment meant for the mass production of relief holograms.

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“…If this frequency is too high for the computations to be practicable then reducing the highest frequency, at the loss of the achievable image resolution, may be a compromise it is necessary to make. Goncharsky et al [36,37,35,39,38,40] follow this idea and consider using low-frequency transducers. They further combine with a "2.5"D sliceby-slice reconstruction approach.…”

Section: Full Waveform Inversionmentioning

confidence: 99%

High Resolution 3D Ultrasonic Breast Imaging by Time-Domain Full Waveform Inversion

Lucka,

Pérez-Liva,

Treeby

et al. 2021

Preprint

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Ultrasound tomography (UST) scanners allow quantitative images of the human breast's acoustic properties to be derived with potential applications in screening, diagnosis and therapy planning. Time domain full waveform inversion (TD-FWI) is a promising UST image formation technique that fits the parameter fields of a wave physics model by gradient-based optimization. For high resolution 3D UST, it holds three key challenges: Firstly, its central building block, the computation of the gradient for a single US measurement, has a restrictively large memory footprint. Secondly, this building block needs to be computed for each of the 10 3 − 10 4 measurements, resulting in a massive parallel computation usually performed on large computational clusters for days. Lastly, the structure of the underlying optimization problem may result in slow progression of the solver and convergence to a local minimum. In this work, we design and evaluate a comprehensive computational strategy to overcome these challenges: Firstly, we introduce a novel gradient computation based on time reversal that dramatically reduces the memory footprint at the expense of one additional wave simulation per source. Secondly, we break the dependence on the number of measurements by using source encoding (SE) to compute stochastic gradient estimates. Also we describe a more accurate, TD-specific SE technique with a finer variance control and use a state-of-the-art stochastic LBFGS method. Lastly, we design an efficient TD multi-grid scheme together with preconditioning to speed up the convergence while avoiding local minima. All components are evaluated in extensive numerical proof-of-concept studies simulating a bowl-shaped 3D UST breast scanner prototype. Finally, we demonstrate that their combination allows us to obtain an accurate 442 × 442 × 222 voxel image with a resolution of 0.5 mm using Matlab on a single GPU within 24 hours.

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Low-Frequency Ultrasonic Tomography: Mathematical Methods and Experimental Results

Cited by 26 publications

References 24 publications

Convergence of a series associated with the convexification method for coefficient inverse problems

Convergence of a series associated with the convexification method for coefficient inverse problems

Nanooptical elements for visual verification

High Resolution 3D Ultrasonic Breast Imaging by Time-Domain Full Waveform Inversion

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