Vector Tomography Research Articles

Development Of A Spatio-Temporal Acoustic Pressure Measurement Technique Utilizing Background Oriented Schlieren Technique

We developed an image and optical-based technique to measure spatio-temporal acoustic pressure fields using BOS instead of a hydrophone. Fast checker demodulation (FCD) and vector tomography (VT) were employed for displacement detection and axisymmetric tomography, respectively. In experiments, BOS and hydrophones of different diameters measured focused ultrasound, revealing BOS's higher spatial resolution and ability to capture temporal evolution at 4.554 MHz. However, BOS showed systematic errors under amplified pressure. Using synthetic data, we identified that FCD's underestimation issues could be defined by the integrated power spectra of distorted images. Thus, BOS with FCD and VT provides high-resolution measurements but requires criteria to avoid displacement underestimation.

Permittivity tensor imaging: modular label-free imaging of 3D dry mass and 3D orientation at high resolution

The dry mass and the orientation of biomolecules can be imaged without a label by measuring their permittivity tensor (PT), which describes how biomolecules affect the phase and polarization of light. Three-dimensional (3D) imaging of PT has been challenging. We present a label-free computational microscopy technique, PT imaging (PTI), for the 3D measurement of PT. PTI encodes the invisible PT into images using oblique illumination, polarization-sensitive detection and volumetric sampling. PT is decoded from the data with a vectorial imaging model and a multi-channel inverse algorithm, assuming uniaxial symmetry in each voxel. We demonstrate high-resolution imaging of PT of isotropic beads, anisotropic glass targets, mouse brain tissue, infected cells and histology slides. PTI outperforms previous label-free imaging techniques such as vector tomography, ptychography and light-field imaging in resolving the 3D orientation and symmetry of organelles, cells and tissue. We provide open-source software and modular hardware to enable the adoption of the method.

3D imaging of magnetic domains in Nd2Fe14B using scanning hard X-ray nanotomography.

Nanoscale structural and electronic heterogeneities are prevalent in condensed matter physics. Investigating these heterogeneities in 3D has become an important task for understanding material properties. To provide a tool to unravel the connection between nanoscale heterogeneity and macroscopic emergent properties in magnetic materials, scanning transmission X-ray microscopy (STXM) is combined with X-ray magnetic circular dichroism. A vector tomography algorithm has been developed to reconstruct the full 3D magnetic vector field without any prior noise assumptions or knowledge about the sample. Two tomographic scans around the vertical axis are acquired on single-crystalline Nd2Fe14B pillars tilted at two different angles, with 2D STXM projections recorded using a focused 120 nm X-ray beam with left and right circular polarization. Image alignment and iterative registration have been implemented based on the 2D STXM projections for the two tilts. Dichroic projections obtained from difference images are used for the tomographic reconstruction to obtain the 3D magnetization distribution at the nanoscale.

Real space iterative reconstruction for vector tomography (RESIRE-V)

Tomography has had an important impact on the physical, biological, and medical sciences. To date, most tomographic applications have been focused on 3D scalar reconstructions. However, in some crucial applications, vector tomography is required to reconstruct 3D vector fields such as the electric and magnetic fields. Over the years, several vector tomography methods have been developed. Here, we present the mathematical foundation and algorithmic implementation of REal Space Iterative REconstruction for Vector tomography, termed RESIRE-V. RESIRE-V uses multiple tilt series of projections and iterates between the projections and a 3D reconstruction. Each iteration consists of a forward step using the Radon transform and a backward step using its transpose, then updates the object via gradient descent. Incorporating with a 3D support constraint, the algorithm iteratively minimizes an error metric, defined as the difference between the measured and calculated projections. The algorithm can also be used to refine the tilt angles and further improve the 3D reconstruction. To validate RESIRE-V, we first apply it to a simulated data set of the 3D magnetization vector field, consisting of two orthogonal tilt series, each with a missing wedge. Our quantitative analysis shows that the three components of the reconstructed magnetization vector field agree well with the ground-truth counterparts. We then use RESIRE-V to reconstruct the 3D magnetization vector field of a ferromagnetic meta-lattice consisting of three tilt series. Our 3D vector reconstruction reveals the existence of topological magnetic defects with positive and negative charges. We expect that RESIRE-V can be incorporated into different imaging modalities as a general vector tomography method. To make the algorithm accessible to a broad user community, we have made our RESIRE-V MATLAB source codes and the data freely available at https://github.com/minhpham0309/RESIRE-V.

Tomographic Inverse Problems: Mathematical Challenges and Novel Applications

This workshop brought together researchers working on mathematical problems related to tomography, with a particular emphasis on novel applications and associated mathematical challenges. Examples of respective issues represented in the workshop were tomographic imaging with Compton cameras or coupled-physics imaging, resolution and aliasing, vector and tensor field tomography, diffraction tomography, magnetic particle tomography, and limited data, all of which are motivated by the many modern applications. These topics were complemented by novel algorithmic strategies in the solution of tomographic inverse problems, such as stochastic methods and machine learning techniques. Bringing together mathematical and scientific researchers working on these different mathematical problems created a fruitful interchange with novel ideas and strong impact for the future of the field.

Photoacoustic vector tomography for deep haemodynamic imaging.

Imaging deep haemodynamics non-invasively remains a quest. Although optical imaging techniques can be used to measure blood flow, they are generally limited to imaging within ∼1 mm below the skin's surface. Here we show that such optical diffusion limit can be broken through by leveraging the spatial heterogeneity of blood and its photoacoustic contrast. Specifically, successive single-shot wide-field photoacoustic images of blood vessels can be used to visualize the frame-to-frame propagation of blood and to estimate blood flow speed and direction pixel-wise. The method, which we named photoacoustic vector tomography (PAVT), allows for the quantification of haemodynamics in veins more than 5 mm deep, as we show for regions in the hands and arms of healthy volunteers. PAVT may offer advantages for the diagnosis and monitoring of vascular diseases and for the mapping of the function of the circulatory system.

A fast magnetic vector characterization method for quasi two-dimensional systems and heterostructures

The use of magnetic vector tomography/laminography has opened a 3D experimental window to access the magnetization at the nanoscale. These methods exploit the dependence of the magnetic contrast in transmission to recover its 3D configuration. However, hundreds of different angular projections are required leading to large measurement times. Here we present a fast method to dramatically reduce the experiment time specific for quasi two-dimensional magnetic systems. The algorithm uses the Beer-Lambert equation in the framework of X-ray transmission microscopy to obtain the 3D magnetic configuration of the sample. It has been demonstrated in permalloy microstructures, reconstructing the magnetization vector field with a reduced number of angular projections obtaining quantitative results. The throughput of the methodology is × 10–× 100 times faster than conventional magnetic vector tomography, making this characterization method of general interest for the community.

Numerical and experimental study on acoustic tomographic reconstruction of flow field

This paper reports the acoustic vector tomography (AVT) method to reconstruct the airflow as a vector field. In order to increase the efficiency of reconstruction process, proposed radial basis functions approximate the flow velocity components. Tikhonov-Gaussian regularization combined with Generalized Singular Value Decomposition method and modified L-curve are introduced to solve the ill-posed problem and select the optimal regularization parameters. The feasibility and validity of the method are verified by numerical simulation and experiment. The numerical results confirmed the effectiveness of the proposed method with higher accuracy, better anti-noise capability and shorter consuming time compared with other reconstruction methods. Besides, a physical experiment is performed for further testing. We measured the incident flow and vortex field using the proposed acoustic method. The results show that the tendency and data of the acoustical reconstructed flow field are consistent with the strip display and anemometer results. The relative errors of most measurement points are under 15% and our proposed acoustic method is effective in practical velocity field measurement.

Background-oriented schlieren technique with vector tomography for measurement of axisymmetric pressure fields of laser-induced underwater shock waves

This study aims to overcome the problems that existing background-oriented schlieren (BOS) techniques based on computed tomography (CT-BOS) face when measuring pressure fields of laser-induced underwater shock waves. To do this, it proposes a novel BOS technique based on vector tomography (VT-BOS) of an axisymmetric target. The remarkable feature of the proposed technique is the reconstruction of an axisymmetric vector field with nonzero divergence, such as the field of a laser-induced underwater shock wave. This approach is based on an approximate relation between the projection of the axisymmetric vector field and the reconstructed vector field. For comparison, the pressure fields of underwater shock waves are measured with VT-BOS, CT-BOS, and a needle hydrophone. It is found that VT-BOS is significantly better than CT-BOS in terms of better convergence, less dependence on the spatial resolution of the acquired images, and lower computational cost. The proposed technique can be applied not only to fluid dynamical fields, but also to other axisymmetric targets in other areas, such as electromagnetics and thermodynamics.Graphical abstract

Background Oriented Schlieren Technique With Reconstruction Of Three-Dimensional Vector Fields For Pressure Measurement Of Laser Induced Underwater Shock Waves

We propose a novel reconstruction of the three-dimensional vector field (VR) of an axisymmetric target to overcome the problems that existing background-oriented schlieren (BOS) techniques based on reconstruction of three-dimensional scalar field (SR-BOS). This approach is based on an approximate relation between the projection of the axisymmetric vector field and the reconstructed vector field. The remarkable feature of the proposed technique is the reconstruction of an axisymmetric vector field with nonzero divergence, such as the field of a laser-induced underwater shock wave. Reconstruction of 3D vector fields, such as conventional vector tomography, is usually used in the field of electromagnetics with vector fields of zero divergence, and its application to fluid phenomena has been di fficult. We have successfully applied this technique to BOS (VR-BOS) and measured the pressure fields of underwater shock waves. Using the experimental data and previous research data, the superiority of VR-BOS was clarified by comparing the measurement results of SR-BOS and VR-BOS, for example, measurement accuracy, in better convergence, less dependence on the spatial resolution of the acquired images, and lower computational cost. In particular, the fact that the measurement accuracy of VR-BOS is almost independent of resolution is expected to expand the maximum pressure values that can be measured by the BOS technique. We also used synthetic data to investigate the effect of noise on VR-BOS. If the synthetic displacement contains noise, the noise is amplified in the vicinity of the axisymmetry axis in the reconstruction distribution calculated by VT. If the distance from the axisymmetry axis is more than 1/4 of the measurement range, the measurement accuracy of the reconstructed distribution can be sufficiently guaranteed. This proposed technique can be applied fluid dynamical fields, as well as other axisymmetric targets with vector reconstruction fields in other areas, such as electromagnetics and material mechanism.

Inversion formulae for ray transforms in vector and tensor tomography

Ray transforms for vector or tensor fields have been studied for a long time, also in connection with Doppler or polarization tomography. In contrast to the classical Radon transform or x-ray transform their known inversion formulae are rather complicated, and represented, for example, using series expansions or singular value decompositions. Recently, applications of the approximate inverse have been studied. In this paper we present a new class of inversion formulae for compactly supported fields similar to those of the x-ray transform. The derivation is based on two observations. First, we apply Derevtsov’s formula relating the ray transforms of vector or tensor fields to a derivative of the x-ray transform of a generating potential, [3]. Second, we apply the calculation of derivatives of a scalar function from its x-ray transform, as it is realized in feature reconstruction, [15, 16]. Generalizations complete the presentation.

Scalar and vector tomography for the weighted transport equation with application to helioseismology

Abstract Motivated by the application to helioseismology, we demonstrate uniqueness and stability for a class of inverse problems of the weighted transport equation. Using 𝐴-analytic functions, this inverse problem is expressed as a Cauchy problem. In this form, we show that, for a finite even trigonometric polynomial weight function, the resulting system is well-conditioned numerically and permits a Carleman-like estimate with boundary terms.

3D magnetic configuration of ferrimagnetic multilayers with competing interactions visualized by soft X-ray vector tomography

Full control of magnetic properties in exchange coupled systems requires a good understanding of 3D magnetic configuration with lateral and in-depth resolution. Here we show results from a soft X-ray tomographic reconstruction which allow determining, solely from the experimental data, a detailed description of the vector magnetic configuration of a ferrimagnetic Gd12Co88/Nd17Co83/Gd24Co76 trilayer with engineered competing anisotropy, exchange and magnetostatic interactions at different depths. The trilayer displays chevron patterns with a distorted closure structure. Near the top Gd24Co76 layer, local exchange springs with out-of-plane magnetization reversal, quasi-domains with ripple-like patterns and magnetic vortices and antivortices across the thickness are observed. The detailed analysis of the magnetic tomogram shows that the effective strength of the exchange spring at the NdCo/GdCo interface can be finely tuned by GdxCo1-x composition and anisotropy (determined by sample fabrication) and in-plane stripe orientation (adjustable), demonstrating the suitability of 3D magnetic visualization techniques in magnetic engineering research.

The Funk‐Hecke type relations and ray transforms for the vector spherical wave functions

The Funk‐Hecke type integral relations for vector spherical wave functions are derived. Applying the obtained Funk‐Hecke integral identities, 3D Fourier transforms of vector spherical wave functions and and are obtained. On the basis of the Central Slice Theorem, the ray transforms for these vector functions are evaluated. This was found to be a useful tool in the scattering and the vector tomography problems.

Generalized V-line transforms in 2D vector tomography

We study the inverse problem of recovering a vector field in from a set of new generalized V-line transforms in three different ways. First, we introduce the longitudinal and transverse V-line transforms for vector fields in . We then give an explicit characterisation of their respective kernels and show that they are complements of each other. We prove invertibility of each transform modulo their kernels and combine them to reconstruct explicitly the full vector field. In the second method, we combine the longitudinal and transverse V-line transforms with their corresponding first moment transforms and recover the full vector field from either pair. We show that the available data in each of these setups can be used to derive the signed V-line transform of both scalar component of the vector field, and use the known inversion of the latter. The final major result of this paper is the derivation of an exact closed form formula for reconstruction of the full vector field in from its star transform with weights. We solve this problem by relating the star transform of the vector field to the ordinary Radon transform of the scalar components of the field.

Regularization with metric double integrals for vector tomography

Abstract We present a family of non-local variational regularization methods for solving tomographic problems, where the solutions are functions with range in a closed subset of the Euclidean space, for example if the solution only attains values in an embedded sub-manifold. Recently, in [R. Ciak, M. Melching and O. Scherzer, Regularization with metric double integrals of functions with values in a set of vectors, J. Math. Imaging Vision 61 2019, 6, 824–848], such regularization methods have been investigated analytically and their efficiency has been tested for basic imaging tasks such as denoising and inpainting. In this paper we investigate solving complex vector tomography problems with non-local variational methods both analytically and numerically.

Reconstruction in the cone-beam vector tomography with two sources

In this paper, we consider the inverse problem for the cone beam transform of vector fields. This transform maps a vector field to its line integrals along rays coming out from points on a given source trajectory. We proved that the solenoidal part of the field can be reconstructed, if the trajectory satisfies the Kirillov–Tuy condition of order 2. It means that every plane, intersecting the support of the field, intersects the trajectory twice. An exact inversion formula is presented. A reconstruction algorithm, based on the obtained formula is developed. Numerical experiments confirm reliability of the algorithm.

Numerical solvers based on the method of approximate inverse for 2D vector and 2-tensor tomography problems

A problem of reconstruction of 2D vector or symmetric 2-tensor fields by their known ray transforms is considered. Two numerical approaches based on the method of approximate inverse are suggested for solving the problem. The first method allows to recover components of a vector or tensor field, and the second reconstructs its potentials in the sense of feature reconstruction, where the observation operator assigns to a field its potential. Numerical simulations show good results of reconstruction of the sought-for fields or their solenoidal or potential parts from its ray transforms.

An improved exact inversion formula for solenoidal fields in cone beam vector tomography

In this paper we present an improved inversion formula for the 3D cone beam transform of vector fields supported in the unit ball which is exact for solenoidal fields. It is well known that only the solenoidal part of a vector field can be determined from the longitudinal ray transform of a vector field in cone beam geometry. The inversion formula, as it was developed in Katsevich and Schuster (2013 An exact inversion formula for cone beam vector tomography Inverse Problems 29 065013), consists of two parts. The first part is of the filtered backprojection type, whereas the second part is a costly 4D integration and very inefficient. In this article we tackle this second term and obtain an improved formula, which is easy to implement and saves one order of integration. We also show that the first part contains all information about the curl of the field, whereas the second part has information about the boundary values. More precisely, the second part vanishes if the solenoidal part of the original field is tangential at the boundary. A number of numerical tests presented in the paper confirm the theoretical results and the exactness of the formula. Also, we obtain an inversion algorithm that works for general convex domains.

Vector tomography for reconstructing electric fields with non-zero divergence in bounded domains

In vector tomography (VT), the aim is to reconstruct an unknown multi-dimensional vector field using line integral data. In the case of a 2-dimensional VT, two types of line integral data are usually required. These data correspond to integration of the parallel and perpendicular projection of the vector field along the integration lines and are called the longitudinal and transverse measurements, respectively. In most cases, however, the transverse measurements cannot be physically acquired. Therefore, the VT methods are typically used to reconstruct divergence-free (or source-free) velocity and flow fields that can be reconstructed solely from the longitudinal measurements. In this paper, we show how vector fields with non-zero divergence in a bounded domain can also be reconstructed from the longitudinal measurements without the need of explicitly evaluating the transverse measurements. To the best of our knowledge, VT has not previously been used for this purpose. In particular, we study low-frequency, time-harmonic electric fields generated by dipole sources in convex bounded domains which arise, for example, in electroencephalography (EEG) source imaging. We explain in detail the theoretical background, the derivation of the electric field inverse problem and the numerical approximation of the line integrals. We show that fields with non-zero divergence can be reconstructed from the longitudinal measurements with the help of two sparsity constraints that are constructed from the transverse measurements and the vector Laplace operator. As a comparison to EEG source imaging, we note that VT does not require mathematical modeling of the sources. By numerical simulations, we show that the pattern of the electric field can be correctly estimated using VT and the location of the source activity can be determined accurately from the reconstructed magnitudes of the field.