3D Radon Research Articles

Анализ трехмерных алгоритмов реконструкции в рентгеновской компьютерной томографии

The article analyzes the existing three-dimensional image reconstruction algorithms in conebeam X-ray computed tomography with a spiral trajectory of the source. The main approaches to the design of the reconstruction algorithms are considered, their brief classification is given. The focus is on approximate Feldcamp-type algorithms and methods of rebinning cone data to parallel geometry. The extended parallel backprojection and advanced single-slice rebinning algorithms, their advantages and disadvantages are described. The authors have considered the exact methods for solving image reconstruction problem for the 3D data set in cone-beam CT. The author describes the additional conditions imposed on the projection data, the computational efficiency of the algorithms, and the quality of the images. The emphasis is on relation between the cone beam and Radon transform and the classical 3D Radon transform inversion. The general scheme for constructing inversion algorithms for cone beam CT is described. The algorithm of cone-beam reconstruction by backprojection and filtering is considered. The advantages and disadvantages of exact reconstruction algorithms are presented.

Sparse 3D Radon Space Rigid Registration of CT Scans: Method and Validation Study.

We present a new method for rigid registration of CT datasets in 3D Radon space based on sparse sampling of scanning projections. The inputs are the two 3D Radon transforms of the CT scans, one densely sampled and the other sparsely sampled (limited number of scan angles/ranges). The output is the rigid transformation that best matches them. The method first finds the best matching between each projection direction vector in the sparse transform and the corresponding direction vector in the dense transform. It then solves a system of linear equations derived from the direction vector pairs (parallel-beam projections) or finds a solution by non-linear optimization (fan-beam and cone-beam projections). Experimental studies show that our method for 3D parallel beam registration outperforms image space registration in terms of convergence range with significantly reduced X-ray dose compared to a full conventional CT scan.

New X-Ray Tomography Method Based on the 3D Radon Transform Compatible with Anisotropic Sources.

In this work, we propose a novel computed tomography (CT) approach for three-dimensional (3D) object reconstruction, based on a generalized tomographic geometry with two-dimensional angular sampling (two angular degrees of freedom). The reconstruction is based on the 3D Radon transform and is compatible with anisotropic beam conditions. This allows isotropic 3D imaging with a source, which can be extended along one direction for increased flux, while high resolution is achieved by a small source size only in the orthogonal direction. This novel scheme for analytical CT is demonstrated by numerical simulations and proof-of-concept experiments. In this way high resolution and coherence along a single direction determines the reconstruction quality of the entire 3D data set, opening up, for example, new opportunities to achieve nanoscale resolution and/or phase contrast with low brilliance sources such as laboratory x-ray or neutron sources.

Advanced methods for calculating Green's function and its derivatives for three-dimensional anisotropic elastic solids

Three advanced methods are presented for constructing and evaluating Green's function and its derivatives for three-dimensional anisotropic elastic solids. The first one is obtained from a line integral followed by Cauchy's residue theorem, and expressed in terms of Stroh's eigenvalues. The second one is obtained by using the explicit solutions of Stroh's eigenvector matrix available in two-dimensional anisotropic elasticity. Whereas the last one is derived from the solution obtained from 2D Radon transform, and expressed in terms of the fundamental elasticity matrix in vertical planes, which makes the derivatives simpler than those obtained from the line integral operated on an oblique plane. The novelty of the first two methods lies in the construction and computation of the first and second derivatives of the Green's function, though the Green's function itself in the first two methods is rather known. In contrast, the third method is quite new for both the Green's function itself and its first and second derivatives. The advantages and disadvantages of these three solution methods are investigated and discussed through their derivations and numerical implementations for isotropic, transversely isotropic, and fully anisotropic elastic solids. Good agreement between the results calculated by these three methods is shown in all numerical examples. Among these three methods, the last one is proved to be the most efficient one.

Application of Radon Isotopes to Determine Permeable Zones in Rajabasa Geothermal Field, Indonesia

This study aims to determine permeable zones (i.e. fault or fracture zones) in the Rajabasa geothermal field using radon isotopes. Radon concentration of soil gas was measured by Scintrex Radon detector RDA 200 consisting of 123 measurement station. The measurement site spacing are between 200 m and 350 m. The Radon concentration was classified based on a statistical approach, i.e. low, high and very high. Very high value of Radon concentration, called Radon anomaly, associated with permeable zones. From the Radon anomaly contour map, a 2D Radon migration from the source to the surface was modelled by means of numerical simulation. Finally, the Radon contour map was overlaid with the surface geological map and the resistivity contour map in the area of study. It suggests that the Radon anomaly correlates with the location of permeable zone in the centre part of the study area i.e. mud pool manifestation at the Kunjir and the Cugung (Bulakan) fumarole. Based on numerical simulation, the width of the fault is 2,800 m; velocity of fluid is 0.08 m/s and the Radon concentration is 860,000 Beq/m3. The resistivity contour map results low resistivity value of 0 Wm – 100 Wm indicating the conductive zone coincide with permeable zones based on Radon measurements. Furthermore, the peak of the profile curve of Radon concentration could be an indication of fault systems in this area.

2D and 3D prestack seismic data regularization using an accelerated sparse time-invariant Radon transform

The time-invariant Radon transform (RT) is commonly used to regularize and interpolate sparsely sampled or irregularly acquired prestack seismic data. The sparseness of the Radon model significantly influences the results of regularization. We have developed an effective and efficient method for the regularization and interpolation of 2D as well as 3D prestack seismic data. We used an accelerated sparse time-invariant RT in the mixed frequency-time domain to improve the performance of RT-based seismic data regularization. This 2D sparse RT incorporated the iterative 2D model shrinkage algorithm instead of the traditional iteratively reweighted least-squares (IRLS) algorithm in the time domain, and we computed the forward and inverse RTs in the frequency domain to solve the sparse inverse problem, which dramatically reduced the computational cost while obtaining a high-resolution result. The 2D synthetic and real data examples revealed that our 2D approach can better interpolate beyond aliasing a 2D prestack seismic record that contains a large gap, compared with the least-squares-based RT and the frequency-domain sparse RT methods. To extend the 2D technique to 3D more efficiently, we first formulate the 3D RT as a problem of solving a special matrix equation. Next, we use the iterative 3D model shrinkage algorithm to obtain a high-resolution 3D Radon model. The proposed 3D sparse RT method can be applied in the regularization of 3D prestack gathers, such as in the cable interpolation in a 3D marine survey. We achieved robustness and effectiveness with our 3D approach with successful applications to 3D synthetic and real data.

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.

Reduced-dose patient to baseline CT rigid registration in 3D Radon space.

We present a new method for rigid registration of CT scans in Radon space. The inputs are the two 3D Radon transforms of the CT scans, one densely sampled and the other sparsely sampled. The output s the rigid transformation that best matches them. The algorithm starts by finding the best matching between each direction vector in the sparse transform and the corresponding direction vector in the dense transform. It then solves the system of linear equations derived from the direction vector pairs. Our method can be used to register two CT scans and to register a baseline scan to the patient with reduced-dose scanning without compromising registration accuracy. Our preliminary simulation results on the Shepp-Logan head phantom dataset and a pair of clinical head CT scans indicates that our 3D Radon space rigid registration method performs significantly better than image-based registration for very few scan angles and comparably for densely-sampled scans.

Arabic Word Recognition Based on 3D Radon and Multiwavelet Neural Network

Arabic Word Recognition Based on 3D Radon and Multiwavelet Neural Network

Non-iterative partial view 3D ultrasound to CT registration in ultrasound-guided computer-assisted orthopedic surgery

In image-guided orthopedic surgery, rigid registration of intra-operative ultrasound (US) to a pre-operative plan, developed using computed tomography (CT) scans, is an important step for providing real time surgical guidance. The ability to perform this registration accurately, automatically, and in real time is critical for enabling more effective image guidance and anatomic restoration in a number of orthopedic procedures. Several surface-based and intensity-based registration methods have been proposed before to align the US and CT data. Although relatively successful results were reported, both methods require accurate segmentation or localization of the bone surface in US data, which is a challenging task. Furthermore, typically, only partial views of the three-dimensional (3D) bone anatomy are visible in US data, and registration would only converge if a good estimation of the initial alignment between the US and CT datasets is known. We propose a 3D rigid CT to US registration method based on the alignment of local phase bone image projections. The registration is achieved by transforming the local phase bone features, calculated using 3D Log-Gabor filters, to a projection space obtained using 3D Radon transform. Validation experiments show the capability of the method in registering partial view US volumes to full view CT volume. Feasibility experiments, carried out on a phantom and ten volunteer subjects, show an average surface registration, in the region where the US scans were acquired, of 0.42 and 0.78 mm, respectively. The proposed US to CT registration method is fully automatic, non-iterative and requires no initial alignment between the two registering datasets.

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.

Multiple attenuation in crustal-scale imaging: examples from the TAIGER marine reflection data set

During summer of 2009, multi-channel marine seismic reflection data and wide-angle refraction data were acquired as part of the joint NSF and Taiwanese-funded TAIGER program with the goal of understanding the dynamics of arc-continent collision in Taiwan. One of the principle difficulties of crustal-scale imaging with marine reflection data such as these is the prevalent multiple contamination that obscures many of the deep crustal targets. Without effective treatment of multiples, many of the objectives of the TAIGER active source program may not be achieved. We present three profiles, one from each acquisition leg, that demonstrate the effectiveness of 2D surface-related multiple elimination (SRME) and radon filtering in attenuating much of this unwanted energy in broad ranges of water depths, seafloor topographies and lithologies. Two profiles from south of Taiwan image 3–4 km of sedimentary strata overlying moderately extended continental crust along the Eurasia continental shelf and a 5–6 km thick sedimentary section overlying thin crust consisting of faulted blocks and volcanic bodies along the continental slope. Our multiple attenuation efforts also reveal a seaward-dipping normal fault that penetrates into the upper mantle and separates thick crust of the continental shelf from thin crust of the continental slope. A profile from east of Taiwan reveals thin ocean crust of the Philippine Sea plate subducting beneath the Ryukyu trench that may be traced beneath the accretionary prism and Ryukyu forearc. These profiles demonstrate the success of our imaging strategy in the range of imaging environments spanned by the TAIGER marine reflection seismic data.

Image Reconstruction Using Modified Hybrid Transform

In this paper, an algorithm for reconstruction of a completely lost blocks using ModifiedHybrid Transform. The algorithms examined in this paper do not require a DC estimationmethod or interpolation. The reconstruction achieved using matrix manipulation based onModified Hybrid transform. Also adopted in this paper smart matrix (Detection Matrix) to detectthe missing blocks for the purpose of rebuilding it. We further asses the performance of theModified Hybrid Transform in lost block reconstruction application. Also this paper discussesthe effect of using multiwavelet and 3D Radon in lost block reconstruction.

Analytical properties of time-of-flight PET data

We investigate the analytical properties of time-of-flight (TOF) positron emission tomography (PET) sinograms, where the data are modeled as line integrals weighted by a spatially invariant TOF kernel. First, we investigate the Fourier transform properties of 2D TOF data and extend the ‘bow-tie’ property of the 2D Radon transform to the time-of-flight case. Second, we describe a new exact Fourier rebinning method, TOF-FOREX, based on the Fourier transform in the time-of-flight variable. We then combine TOF-FOREX rebinning with a direct extension of the projection slice theorem to TOF data, to perform fast 3D TOF PET image reconstruction. Finally, we illustrate these properties using simulated data.

Indian Ocean Rossby waves detected in HYCOM sea surface salinity

Rossby waves have been well identified in satellite derived sea surface height (SSH), sea surface temperature (SST) and ocean color observations. Studies of Rossby waves have yet to include sea surface salinity (SSS) as a parameter, largely because presently available in situ measurements of salinity lack sufficient spatial and temporal coverage, and as of now no methods are available for measuring salinity from a satellite. In this paper, we demonstrate that Rossby waves can be observed in SSS in the Indian Ocean by using simulations of the 1/12° global Hybrid Coordinate Ocean Model (HYCOM). HYCOM results compared favorably to SSS data provided by Argo floats and the World Ocean Atlas 2005 (WOA05) in selected grid boxes in the Indian Ocean. Hovmöller diagrams of HYCOM SSS anomalies and gradient show the distinct westward propagating signature of Rossby waves, with a steeper slope in longitude/time plots further from the equator. The propagation speeds, calculated from a 2D Radon Transform are comparable with new theoretical speeds for Rossby waves. Annual westward propagating signals in the SSS simulations at most of the latitudes in the Indian Ocean coincide with previous studies. We hope that future studies of Rossby waves in SSS using model results and eventually satellite measurements of salinity data will allow a better understanding of Rossby wave dynamics.

ON THE EQUIVALENCE OF STABLE AND UNSTABLE FORMS OF THE INVERSE CIRCULAR HARMONIC TRANSFORM SOLUTION FOR THE RADON TRANSFORM

We investigate the inversion of the 2D Radon transform with the technique based on the Circular Harmonic Transform (CHT) using Chebyshev/Zernike polynomials. This polynomial pair is often described as a solution of the unstable or causal form of the CHT inverse. The primary object of this work is to show analytically and explicitly that the polynomial pair is formally a solution to the stable form of the CHT inverse. Hence it is the first solution of its type to be simultaneously a solution to both forms of the inverse. The stable/unstable dichotomy is not fundamental to the CHT inverse. We validate this claim with numerical experiments and image data. We describe a potential application of the algorithm to MRI imaging with nonlinear gradients.

Detector Blurring and Detector Sensitivity Compensation for a Spinning Slat Collimator

This paper extends the well-known frequency-distance relation to a rotating-spinning parallel-slat collimator SPECT system, where the projection data are in the form of weighted planar integrals and the detection sensitivity function is inversely proportional to the distance between the point source and the detector. The three-dimensional (3D) Radon inversion formula is used in image reconstruction. The 3D backprojection is implemented as Marr's two-step two-dimensional (2D) backprojectors. The frequency-distance relation is applied in the second step of the 2D backprojection. Both the collimator blurring and distance dependent sensitivity effect are compensated for

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.

Application of the trial-and-error method for solving the inverse radon problem for ITER neutron tomography

A method for solving the inverse 2D Radon problem for a single temporal cross section as applied to the two-aspect-angle ITER neutron tomographic detection system is described. This method allows reduction of the calculation time owing to the formation of the reconstructed image described by models that are as simple as possible in the case of a single plasma filament (a spatial density distribution of the plateau, hill, crater, and banana type). These models are reduced to the approximation of the solution in the form of a superposition of two elliptic Gaussian curves (or the sum of an arbitrary number of circular Gaussian curves).

An FDK-like cone-beam SPECT reconstruction algorithm for non-uniform attenuated projections acquired using a circular trajectory

In this paper, Novikov's inversion formula of the attenuated two-dimensional (2D) Radon transform is applied to the reconstruction of attenuated fan-beam projections acquired with equal detector spacing and of attenuated cone-beam projections acquired with a flat planar detector and circular trajectory. The derivation of the fan-beam algorithm is obtained by transformation from parallel-beam coordinates to fan-beam coordinates. The cone-beam reconstruction algorithm is an extension of the fan-beam reconstruction algorithm using Feldkamp–Davis–Kress's (FDK) method. Computer simulations indicate that the algorithm is efficient and is accurate in reconstructing slices close to the central slice of the cone-beam orbit plane. When the attenuation map is set to zero the implementation is equivalent to the FDK method. Reconstructed images are also shown for noise corrupted projections.