Two-dimensional Radon Research Articles

On a wavelet-like orthonormal basis and its application to two-dimensional Radon transforms

On a wavelet-like orthonormal basis and its application to two-dimensional Radon transforms

Parallelly Sliced Optimal Transport on Spheres and on the Rotation Group

AbstractSliced optimal transport, which is basically a Radon transform followed by one-dimensional optimal transport, became popular in various applications due to its efficient computation. In this paper, we deal with sliced optimal transport on the sphere $$\mathbb {S}^{d-1}$$ S d - 1 and on the rotation group $$\textrm{SO}(3)$$ SO ( 3 ) . We propose a parallel slicing procedure of the sphere which requires again only optimal transforms on the line. We analyze the properties of the corresponding parallelly sliced optimal transport, which provides in particular a rotationally invariant metric on the spherical probability measures. For $$\textrm{SO}(3)$$ SO ( 3 ) , we introduce a new two-dimensional Radon transform and develop its singular value decomposition. Based on this, we propose a sliced optimal transport on $$\textrm{SO}(3)$$ SO ( 3 ) . As Wasserstein distances were extensively used in barycenter computations, we derive algorithms to compute the barycenters with respect to our new sliced Wasserstein distances and provide synthetic numerical examples on the 2-sphere that demonstrate their behavior for both the free- and fixed-support setting of discrete spherical measures. In terms of computational speed, they outperform the existing methods for semicircular slicing as well as the regularized Wasserstein barycenters.

Carbon nanotube-based multiple source C-arm CT system: feasibility study with prototype system.

To extend the field of view while reducing dimensions of the C-arm, we propose a carbon nanotube (CNT)-based C-arm computed tomography (CT) system with multiple X-ray sources. A prototype system was developed using three CNT X-ray sources, enabling a feasibility study. Geometry calibration and image reconstruction were performed to improve the quality of image acquisition. However, the geometry of the prototype system led to projection truncation for each source and an overlap region of object area covered by each source in the two-dimensional Radon space, necessitating specific corrective measures. We addressed these problems by implementing truncation correction and applying weighting techniques to the overlap region during the image reconstruction phase. Furthermore, to enable image reconstruction with a scan angle less than 360°, we designed a weighting function to solve data redundancy caused by the short scan angle. The accuracy of the geometry calibration method was evaluated via computer simulations. We also quantified the improvements in reconstructed image quality using mean-squared error and structural similarity. Moreover, detector lag correction was applied to address the afterglow observed in the experimental data obtained from the prototype system. Our evaluation of image quality involved comparing reconstructed images obtained with and without incorporating the geometry calibration results and images with and without lag correction. The outcomes of our simulation study and experimental investigation demonstrated the efficacy of our proposed geometry calibration, image reconstruction method, and lag correction in reducing image artifacts.

PVRAR: Point-View Relation Neural Network Embedded with Both Attention Mechanism and Radon Transform for 3D Shape Recognition

Owing to the favorable performance of deep neural networks for 3D shape recognition, an increasing number of researchers are interested in designing novel 3D shape descriptors. However, the relationship between multiple views and point clouds needs to be further elucidated. We propose a multimodal method that combines the features of point clouds and multiple views, i.e., point-view relation neural network embedded with both attention mechanism and Radon transform, to obtain better descriptors. First, a two-dimensional linear Radon transform is performed to investigate linear and color features in multiple views, and the features are used as the input of our network to enable significant distinctions between different views. Moreover, a convolutional block attention module is adopted to enhance the features of point clouds and hence improve the expression ability of feature descriptors. The effectiveness of the proposed method is verified using ModelNet40 and ModelNet10 datasets. Experimental results show that our method can effectively improve the capability of feature extraction and expression as well as achieve state-of-the-art performance on two well-known 3D datasets.

A 2-D Radon Transformation for Enhancing the Detection and Imaging of Embedded Defects in Layered Composite Structures Using Millimeter-Wave System

This paper aims to tackle the challenge of detecting the presence of embedded defects in layered composite structures using imaging techniques including a 2D Radon transformation. The proposed technique aims at enhancing the detection and imaging of embedded defects in layered composite structures that is captured using a millimeterwave (MMW) radar system. A 94 GHz MMW system is used in the near field to detect defects such as disbond and delamination embedded in complex glass fiber composite structures. In order to simplify the detection in glass fibers, measurement data attained using a MMW radar are plotted as images using MATLAB. A stable detection and identification of these defects from the collected images is proposed using a robust two-stage algorithm. The Weiner filter-based deconvolution technique is used for the reflectivity deconstruction in the image and a two-dimensional Radon transformation called back-projection is used to as an additional filtering step enhance the detection of the embedded defects. The proposed method allows defects that are embedded 24mm below the surface of the scanning probe to be detected and identified in terms of location. Our results show potential in the proposed method to be employed in accurately imaging defects in composites that are used in the aerospace and civil infrastructure industries.

Patient body motion correction for dynamic cardiac PET-CT by attenuation-emission alignment according to projection consistency conditions.

Patient body motion is known to cause large deviations in the determination of myocardial blood flow (MBF) with errors exceeding 300%. Accurate correction for patient whole-body motion is still a largely unsolved problem in cardiac positron emission tomography (PET) imaging. This study evaluated the efficacy of using Natterer's formulation of the Helgason-Ludwig consistency conditions on the two-dimensional Radon transform to align computed tomography to PET projection data in multiple time frames of a dynamic sequence for the purpose of frame-by-frame correction of rigid whole-body motion. The correction algorithm was evaluated with digital NCAT phantoms using realistic noise added by the analytical simulator. Count rates used in the simulation were derived from clinical patient data. In addition, a proof of concept test using measured data with a cardiac torso phantom was conducted. Motion correction resulted in significant improvement in the accuracy of MBF estimates, especially for high count-rate acquisitions. Maximum errors for 2cm of motion dropped from 325% to 25% and from 250% to 25% using global and regional partial-volume correction, respectively. Median MBF errors dropped from 33% to 4.5% and 27% to 3.8%, respectively. Importantly, the correction algorithm performed equally well to compensate for body motion in both early and late time frames. Cardiac PET-CT data used for attenuation correction (CTAC) alignment using projection consistency conditions was effective for reducing errors in MBF measurements due to simulated patient motion, and can be integrated into the image reconstruction workflow.

Enhancement of Compton camera images reconstructed by inversion of a conical Radon transform

We present a new inversion formula for a weighted conical Radon transform modelling Compton camera data. The formula exploits a large proportion of the acquired events and is easy to implement into fast algorithms. We give for it two equivalent formulations relying on known properties of the two-dimensional Radon transform and we test a semi-iterative algorithm for one of them. From a practical point of view, methods robust to measurement noise and to low number of events are required. We show that adding a constraint on the total variation of the final image strongly improves the results. We illustrate our arguments with Monte-Carlo simulated data in both low and realistic noise configurations.

Single scattering tomography with curved detectors

Classical computed tomography assumes that x-rays propagate without scattering. The data is modeled as the two-dimensional Radon transform of the attenuation coefficient of the scanned object. To reconstruct the image, the Radon transform has to be inverted. Recently, several new imaging methods that use singly scattered rays have been suggested. The data in these cases is modeled by integrals of a function along V-lines (two rays joined at a common vertex). In this paper we study the V-line transform arising in Single Scattering Optical Tomography in the case of a single concave detector. In this setup the direction of one of the rays constituting a V-line depends on the common vertex. We derive an explicit inversion formula which allows the recovery of the absorption coefficient. Our results are illustrated with numerical simulations.

Characteristics of Radon-Concentration Distribution in the Bed of a Building

A mathematical formulation is presented for the boundary problem of stationary twodimensional radon transport in the soil bed of a building. The MAPLE mathematical software package is used to solve the problem. A rule governing variation in the radon-concentration field in the bed of a building is established as a function of its width and depth. A more accurately defined formula is proposed for calculation of the radon load on the embedded portion of the building.

A formalism for anisotropic heat transfer phenomena: Foundations and Green’s functions

A new complex-variable formalism for the analysis of three-dimensional (3D) steady-state heat transfer problems in homogeneous solids with general anisotropic behaviour is proposed in this paper. The derived method is based on the Radon transform, which is used in order to reduce the dimension of the problem to a two-dimensional (2D) Radon space where a solution can be easily handled via a complex-variable method. Subsequently, the 3D solution is obtained by applying the inverse Radon transform. Despite that the main goal of this work is to develop and illustrate the general methodology, the proposed formalism is further applied to derive new Green’s functions as application examples. Contributions include new forms for bimaterial and half-space Green’s functions for line, point, vortex and dipole heat sources. In particular Green’s functions due to heat vortex loop sources for infinite media, half-space and bimaterial systems are presented for the first time. The veracity and computability of the approach are demonstrated with some numerical examples.

Approximate truncation robust computed tomography—ATRACT

We present an approximate truncation robust algorithm to compute tomographic images (ATRACT). This algorithm targets at reconstructing volumetric images from cone-beam projections in scenarios where these projections are highly truncated in each dimension. It thus facilitates reconstructions of small subvolumes of interest, without involving prior knowledge about the object. Our method is readily applicable to medical C-arm imaging, where it may contribute to new clinical workflows together with a considerable reduction of x-ray dose. We give a detailed derivation of ATRACT that starts from the conventional Feldkamp filtered-backprojection algorithm and that involves, as one component, a novel original formula for the inversion of the two-dimensional Radon transform. Discretization and numerical implementation are discussed and reconstruction results from both, simulated projections and first clinical data sets are presented.

Characterization of the supersonic flowing microwave discharge using two dimensional plasma tomography

A tomographic numerical method based on the two-dimensional Radon formula for a cylindrical cavity has been employed for obtaining spatial distributions of the argon excited levels. The spectroscopy measurements were taken at different positions and directions to observe populations of excited species in the plasmoid region and the corresponding excitation temperatures. Excited argon states are concentrated near the tube walls, thus, confirming the assumption that the post discharge plasma is dominantly sustained by travelling surface wave. An automated optical measurement system has been developed for reconstruction of local plasma parameters of the plasmoid structure formed in an argon supersonic flowing microwave discharge. The system carries out angle and distance measurements using a rotating, flat mirror, as well as two high precision stepper motors operated by a microcontroller-based system and several sensors for precise feedback control.

Investigation of discrete imaging models and iterative image reconstruction in differential X-ray phase-contrast tomography.

Differential X-ray phase-contrast tomography (DPCT) refers to a class of promising methods for reconstructing the X-ray refractive index distribution of materials that present weak X-ray absorption contrast. The tomographic projection data in DPCT, from which an estimate of the refractive index distribution is reconstructed, correspond to one-dimensional (1D) derivatives of the two-dimensional (2D) Radon transform of the refractive index distribution. There is an important need for the development of iterative image reconstruction methods for DPCT that can yield useful images from few-view projection data, thereby mitigating the long data-acquisition times and large radiation doses associated with use of analytic reconstruction methods. In this work, we analyze the numerical and statistical properties of two classes of discrete imaging models that form the basis for iterative image reconstruction in DPCT. We also investigate the use of one of the models with a modern image reconstruction algorithm for performing few-view image reconstruction of a tissue specimen.

Calculation of scattering patterns from phase-shifting objects using the Radon transform

In the case of neutron (and X-ray) scattering by objects that are about 105times larger than the wavelength, the objects can be considered as (inhomogeneous) phase-shifting media. In contrast with small-angle scattering, the scattering patterns from phase-shifting objects are calculated by the superposition of coherent partial waves that penetrate the object. In order to determine the scattering patterns from large complicated objects, it is proposed to use the two-dimensional Radon transform of the objects and Fraunhofer diffraction. This approach is much easier than using the small-angle scattering treatment, as is shown in this paper.

On the v-line radon transform and its imaging applications.

Radon transforms defined on smooth curves are well known and extensively studied in the literature. In this paper, we consider a Radon transform defined on a discontinuous curve formed by a pair of half-lines forming the vertical letter V. If the classical two-dimensional Radon transform has served as a work horse for tomographic transmission and/or emission imaging, we show that this V-line Radon transform is the backbone of scattered radiation imaging in two dimensions. We establish its analytic inverse formula as well as a corresponding filtered back projection reconstruction procedure. These theoretical results allow the reconstruction of two-dimensional images from Compton scattered radiation collected on a one-dimensional collimated camera. We illustrate the working principles of this imaging modality by presenting numerical simulation results.

Detection of Rossby waves in multi-parameters in multi-mission satellite observations and HYCOM simulations in the Indian Ocean

Rossby waves are difficult to detect with in situ methods. However, as we show in this paper, they can be clearly identified in multi-parameters in multi-mission satellite observations of sea surface height (SSH), sea surface temperature (SST) and ocean color observations of chlorophyll- a (chl- a), as well as 1/12° global HYbrid Coordinate Ocean Model (HYCOM) simulations of SSH, SST and sea surface salinity (SSS) in the Indian Ocean. While the surface structure of Rossby waves can be elucidated from comparisons of the signal in different sea surface parameters, models are needed to gain direct information about how these waves affect the ocean at depth. The first three baroclinic modes of the Rossby waves are inferred from the Fast Fourier Transform (FFT), and two-dimensional Radon Transform (2D RT). At many latitudes the first and second baroclinic mode Rossby wave phase speeds from satellite observations and model parameters are identified. Wavelet transforms of these multi-parameters from satellite observations and model simulations help to discriminate between the annual and semi-annual signal of these Rossby waves. This comprehensive study reveals that the surface signature of Rossby waves in SSS anomalies is likely to be between 0.05 and 0.3 psu in the South Indian Ocean.

FREQUENCY DOMAIN RECONSTRUCTION FOR PHOTO- AND THERMOACOUSTIC TOMOGRAPHY WITH LINE DETECTORS

This paper is concerned with a version of photoacoustic tomography, that uses line shaped detectors (instead of point-like ones) for the recording of acoustic data. The three-dimensional image reconstruction problem is reduced to a series of two-dimensional ones. First, the initial data of the two-dimensional wave equation is recovered from boundary data, and second, the classical two-dimensional Radon transform is inverted. We discuss uniqueness and stability of reconstruction, and compare frequency domain reconstruction formulas for various geometries.

Remote sensing observations of the coherent and non-coherent ring structures in the vicinity of Lesser Antilles

Abstract. The North Brazil Current Rings (NBCR) penetration into the Caribbean Sea is being investigated by employing a merged altimeter-derived sea height anomaly (TOPEX/Poseidon, Jason-1 and ERS-1, 2), the ocean surface color data (SeaWiFS) and Global Drifter Program information. Four strategies are being applied to process the data: (1) calculations of the Okubo-Weiss parameter for NBCR identification, (2) longitude-time plots (also known as Hovmöller diagrams), (3) two-dimensional Radon transforms and (4) two-dimensional Fourier transforms. A twofold NBCR structure has been detected in the region under investigation. The results have shown that NBC rings mainly propagate into the Caribbean Sea along two principal pathways (near 12° N and 17° N) in the ring translation corridor. Thus, rings following the southern pathway in the fall-winter period can enter through very shallow southern straits as non-coherent structures. A different behavior is observed near the northern pathway (~17° N), where NBC rings are thought to have a coherent structure during their squeezing into the eastern Caribbean, i.e. conserving the principal characteristics of the incident rings. We attribute this difference in the rings' behavior to the vertical scales of the rings and to the bottom topography features in the vicinity of the Lesser Antilles.

The Mathematical Foundations of 3D Compton Scatter Emission Imaging

The mathematical principles of tomographic imaging using detected (unscattered) X- or gamma-rays are based on the two-dimensional Radon transform and many of its variants. In this paper, we show that two new generalizations, called conical Radon transforms, are related to three-dimensional imaging processes based on detected Compton scattered radiation. The first class of conical Radon transform has been introduced recently to support imaging principles of collimated detector systems. The second class is new and is closely related to the Compton camera imaging principles and invertible under special conditions. As they are poised to play a major role in future designs of biomedical imaging systems, we present an account of their most important properties which may be relevant for active researchers in the field.

View-independent reconstruction algorithms for cone beam CT with general saddle trajectory

In Yang et al (2006 Phys. Med. Biol. 51 1157–72), an exact filtered backprojection (FBP) reconstruction algorithm was proposed for cone beam tomography with saddle trajectory based on the seminal works of Pack and Noo (2005a Inverse Problems 21 1105–20; 2005b 8th Int. Meeting on Fully 3D Reconstruction in Radiology and Nuclear Medicine (Salt Lake City) ed F Noo, H Kudo and L G Zeng pp 287–90). However, the artefacts due to discretization and/or sampling errors in the reconstructed images by this method were still visible, especially when the pitch is large. In this paper, two view-independent (VI) algorithms, which are similar to the FDK-type algorithms (Feldkamp et al 1984 J. Opt. Soc. Am. A 1 612–19), are proposed for planar detector geometry. The first VI algorithm involves two filtered projections and a small additional term (two-dimensional (2D) Radon transform term). One of the filtered projections is obtained by ramp filtering (as in the FDK algorithm for circular trajectory) and the other one is obtained by Hilbert transform. The 2D Radon transform term is just like the term which was first derived by Hu (1996 Scanning 18 572–81) for a circular trajectory. The second VI algorithm involves only one filtered projection term, which is obtained by differentiation followed by Hilbert transform and the 2D Radon transform term. Both algorithms involve only one backprojection step with a weighting factor as in the FDK algorithm. The simulation studies show that the pixel values of the reconstructed images by the VI algorithms are more accurate than those by the original view differencing (VD) algorithm, the streak artefacts are also reduced, and their computational times are comparable to that of the original VD algorithm. We also generalize the concept of saddle trajectory and the corresponding reconstruction algorithm. The generalized algorithm is also theoretically exact, has a shift-invariant FBP structure, and does not depend on the concept of π-line.