In this paper, the problem of nonlinear wave tomography is formulated as a coefficient inverse problem for a hyperbolic equation in the time domain. Efficient methods for solving the inverse problems ...of wave tomography for the case of transparent boundary conditions are presented. The algorithms are designed for supercomputers. We prove the Fréchet differentiability theorem for the residual functional and derive an exact expression for the Fréchet derivative in the case of a transparent boundary in the direct and conjugate problems. The expression for the Fréchet derivative of the residual functional remains valid if the experimental data are provided for only a part of the boundary. The effectiveness of the proposed method is illustrated by the numerical solution of a model problem of low-frequency wave tomography. The model problem is tailored to apply to the differential diagnosis of breast cancer.
Abstract
This paper proves the theorem of uniqueness for the solution of a coefficient inverse problem for the wave equation in (with two unknown coefficients: speed of sound and absorption. The ...original nonlinear coefficient inverse problem is reduced to an equivalent system of two uniquely solvable linear integral equations of the first kind with respect to the sound speed and absorption coefficients. Estimates are made, substantiating the multistage method for two unknown coefficients. These estimates show that given sufficiently low frequencies and small inhomogeneities, the residual functional for the nonlinear inverse problem approaches a convex one. This solution method for nonlinear coefficient inverse problems is not linked to the limit approach as frequency tends to zero, but assumes solving the inverse problem using sufficiently low, but not zero, frequencies at the first stage. For small inhomogeneities that are typical, for instance, for medical tasks, carrying out real experiments at such frequencies does not present major difficulties. The capabilities of the method are demonstrated on a model inverse problem with unknown sound speed and absorption coefficients. The method effectively solves the nonlinear problem with parameter values typical for tomographic diagnostics of soft tissues in medicine. A resolution of approximately 2 mm was achieved using an average sounding pulse wavelength of 5 mm.
A method is proposed to compute and synthesize a microrelief to produce a new nano-optical element for forming 3D images with full parallax at the zero order of diffraction. The synthesis of ...nano-optical elements requires the use of multilevel structures. A method is developed for the first time to compute the phase function of such nano-optical elements. Optical security elements that produce the new security feature are synthesized using electron-beam technology. The accuracy of microrelief formation is 10 nm in terms of depth. A sample optical security element is manufactured, which when illuminated by white light, forms a 3D image at the zero order of diffraction. Photos and video of the new 3D visual effect exhibited by real optical elements are presented. The optical elements developed can be replicated using standard equipment employed for manufacturing security holograms. The new optical security feature is easy to control visually, safely protected against counterfeiting, and designed to protect banknotes, documents, ID cards, etc.
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.
We develop efficient iterative methods for solving inverse problems of wave tomography in models incorporating both diffraction effects and attenuation. In the inverse problem the aim is to ...reconstruct the velocity structure and the function that characterizes the distribution of attenuation properties in the object studied. We prove mathematically and rigorously the differentiability of the residual functional in normed spaces, and derive the corresponding formula for the Fréchet derivative. The computation of the Fréchet derivative includes solving both the direct problem with the Neumann boundary condition and the reversed-time conjugate problem. We develop efficient methods for numerical computations where the approximate solution is found using the detector measurements of the wave field and its normal derivative. The wave field derivative values at detector locations are found by solving the exterior boundary value problem with the Dirichlet boundary conditions. We illustrate the efficiency of this approach by applying it to model problems. The algorithms developed are highly parallelizable and designed to be run on supercomputers. Among the most promising medical applications of our results is the development of ultrasonic tomographs for differential diagnosis of breast cancer.
This study focuses on the development of efficient methods for solving inverse problems of ultrasound tomography as coefficient inverse problems for the wave equation. The inverse problem consists in ...finding the unknown wave propagation velocity as a function of coordinates in three-dimensional space. Efficient iterative methods are proposed for solving the inverse problem based on a direct computation of the residual functional. One of the most promising directions of ultrasound tomography is the development of ultrasound tomographs for medical research, and primarily for the differential diagnosis of breast cancer. From a medical viewpoint, diagnostic facilities for the differential cancer diagnosis should have a resolution of 3 mm or better. Because of this requirement, inverse problems of ultrasound tomography have to be solved on dense grids with sizes of up to 1000 × 1000 on cross sections of three-dimensional objects studied. Supercomputers are needed to address such inverse problems in terms of the wave model described by second-order hyperbolic equations. The algorithms developed in this study are easily scalable on supercomputers running up to several tens of thousands of processes. The problem of choosing the initial approximation for iterative algorithms when solving the inverse problem has been studied.
We develop efficient methods for solving inverse problems of ultrasound tomography in models with attenuation. We treat the inverse problem as a coefficient inverse problem for unknown ...coordinate-dependent functions that characterize both the speed cross section and the coefficients of the wave equation describing attenuation in the diagnosed region. We derive exact formulas for the gradient of the residual functional in models with attenuation, and develop efficient algorithms for minimizing the gradient of the residual by solving the conjugate problem. These algorithms are easy to parallelize when implemented on supercomputers, allowing the computation time to be reduced by a factor of several hundred compared to a PC. The numerical analysis of model problems shows that it is possible to reconstruct not only the speed cross section, but also the properties of the attenuating medium. We investigate the choice of the initial approximation for iterative algorithms used to solve inverse problems. The algorithms considered are primarily meant for the development of ultrasound tomographs for differential diagnosis of breast cancer.
A method is proposed to compute and synthesize the microrelief of a diffractive optical element to produce a new visual security feature - the vertical 3D/3D switch effect. The security feature ...consists in the alternation of two 3D color images when the diffractive element is tilted up/down. Optical security elements that produce the new security feature are synthesized using electron-beam technology. Sample optical security elements are manufactured that produce 3D to 3D visual switch effect when illuminated by white light. Photos and video records of the vertical 3D/3D switch effect of real optical elements are presented. The optical elements developed can be replicated using standard equipment employed for manufacturing security holograms. The new optical security feature is easy to control visually, safely protected against counterfeit, and designed to protect banknotes, documents, ID cards, etc.
We demonstrate a new security feature for visual control of the authenticity of optical security features - the change of the images when the optical element is turned by 180 degrees ("switch-180°"). ...The diffractive optical element has an asymmetric microrelief structure resulting from the asymmetry of the scattering pattern. The phase function of the diffractive optical element is computed in terms of Fresnel's scalar wave model. We developed efficient algorithms for computing the structure of flat optical elements to produce the switch effect. A sample of flat optical element for the "switch-180" effect has been developed using electron-beam lithography. The effectiveness of the development is illustrated by the photos and the video captured from a real sample. The visual "switch-180°" effect is easy to control allowing secure anti-counterfeit protection of the optical security feature developed. The new security feature is already used to protect IDs and excise stamps.
•The velocity is experimentally reconstructed using ultrasonic tomography.•The velocity reconstruction is formulated as a nonlinear coefficient inverse problem.•A two-stage iterative method is used ...for velocity reconstruction.
The possibility of reconstructing the velocity structure of inspected objects with a high spatial resolution and high sensitivity in ultrasonic tomographic nondestructive testing within the framework of a wave model has been demonstrated in a real experiment. In this study, a scheme of a tomographic experiment with rotation is proposed, which provides sounding of an inspected object from multiple sides. A tomographic scheme of the experiment with linear antenna transducer arrays operating at a frequency of ∼5 MHz was used. The experiment was conducted on dedicated samples including inserts with different sound propagation velocities. It was shown that the velocity structure and the boundaries of inserts can be reconstructed in the transmission and the reflection schemes. The reconstruction of the velocity structure was formulated as a nonlinear coefficient inverse problem for a scalar wave equation. Efficient iterative methods for its solution on a supercomputer were developed using direct formulas to compute the gradient of the residual functional between the computed and experimentally measured wave field at the detectors. Nonlinearity of the inverse problem of ultrasound tomography leads to multiple local minima of the residual functional. A two-stage iterative method was used for velocity reconstruction. The transmission scheme enables a spatial resolution of approximately 1 mm with a velocity contrast of 2%.
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