Nonseparable wavelet-based cone-beam reconstruction in 3-D rotational angiography

In this paper, we propose a new wavelet-based reconstruction method suited to three-dimensional (3-D) cone-beam (CB) tomography. It is derived from the Feldkamp algorithm and is valid for the same geometrical conditions. The demonstration is done in the framework of nonseparable wavelets and requires ideally radial wavelets. The proposed inversion formula yields to a filtered backprojection algorithm but the filtering step is implemented using quincunx wavelet filters. The proposed algorithm reconstructs slice by slice both the wavelet and approximation coefficients of the 3-D image directly from the CB projection data. The validity of this multiresolution approach is demonstrated on simulations from both mathematical phantoms and 3-D rotational angiography clinical data. The same quality is achieved compared with the standard Feldkamp algorithm, but in addition, the multiresolution decomposition allows to apply directly image processing techniques in the wavelet domain during the inversion process. As an example, a fast low-resolution reconstruction of the 3-D arterial vessels with the progressive addition of details in a region of interest is demonstrated. Other promising applications are the improvement of image quality by denoising techniques and also the reduction of computing time using the space localization of wavelets.     

有限角CT少数投影重建图像技术

叙述了有限角CT少数投影重建图像技术研究所取得的进展：正交全息光路、干涉条纹处理系统、修改的联合代数重建技术、频谱分析图像重建技术和神经网络重建技术。     

3-D RECONSTRUCTION OF SHAPE AND STRUCTURE FOR THE SPACE OBJECT FROM SERIAL CROSS-SECTIONS

A new method for reconstructing a 3-dimensional object from serial cross-sectionsis presented in this paper.The method is based on the principle of sampling and considersevery point in cross-sections as a sampling point and performs the interpolating of nonlinearfunction with these sampling points.Compared with other methods,this method has manyadvantages such as higher precision and fewer requested known sampling points.The result ofreconstruction with this method is an“entity”which involves the exterior shape and interiorconstruction information of the object simultaneously.     

Rotate-and-slant projector for fast LOR-based fully-3-D iterative PET reconstruction

One of the greatest challenges facing iterative fully-3-D positron emission tomography (PET) reconstruction is the issue of long reconstruction times due to the large number of measurements for 3-D mode as compared to 2-D mode. A rotate-and-slant projector has been developed that takes advantage of symmetries in the geometry to compute volumetric projections to multiple oblique sinograms in a computationally efficient manner. It is based upon the 2-D rotation-based projector using the three-pass method of shears, and it conserves the 2-D rotator computations for multiple projections to each oblique sinogram set. The projector is equally applicable to both conventional evenly-spaced projections and unevenly-spaced line-of-response (LOR) data. The LOR-based version models the location and orientation of the individual LORs (i.e., the arc-correction), providing an ordinary Poisson reconstruction framework. The projector was implemented in C with several optimizations for speed, exploiting the vertical symmetry of the oblique projection process, depth compression, and array indexing schemes which maximize serial memory access. The new projector was evaluated and compared to ray-driven and distance-driven projectors using both analytical and experimental phantoms, and fully-3-D iterative reconstructions with each projector were also compared to Fourier rebinning with 2-D iterative reconstruction. In terms of spatial resolution, contrast, and background noise measures, 3-D LOR-based iterative reconstruction with the rotate-and-slant projector performed as well as or better than the other methods. Total processing times, measured on a single cpu Linux workstation, were approximately 10x faster for the rotate-and-slant projector than for the other 3-D projectors studied. The new projector provided four iterations fully-3-D ordered-subsets reconstruction in as little as 15 s--approximately the same time as FORE + 2-D reconstruction. We conclude that the rotate-and-slant projector is a viable option for fully-3-D PET, offering quality statistical reconstruction in times only marginally slower than 2-D or rebinning methods.     

Analytical calculation of volumes-of-intersection for iterative, fully 3-D PET reconstruction

Use of iterative algorithms to reconstruct three-dimensional (3-D) positron emission tomography (PET) data requires the computation of the system probability matrix. The pure geometrical contribution can easily be approximated by the length-of-intersection (LOI) between lines-of-response (LOR) and individual voxels. However, more accurate geometrical projectors are desirable. Therefore, we have developed a fast method for the analytical calculation of the 3-D shape and volume of volumes-of-intersection (VOI). This method provides an alternative robust projector with a uniformly continuous sampling of the image space. The enhanced calculation effort is facilitated by using several speedup techniques. Exploiting intrinsic symmetry relations and the sparseness of the system matrix allows to create an efficiently compressed matrix which can be precomputed and completely stored in memory. In addition, a new voxel addressing scheme has been implemented. This scheme avoids time-consuming symmetry transformations of voxel addresses by using an octant-wise symmetrically ordered field of voxels. The above methods have been applied for a fully 3-D, iterative reconstruction of 3-D sinograms recorded with a Siemens/CTI ECAT HR+ PET scanner. A comparison of the performance of the reconstruction using LOI weighting and VOI weighting is presented.     

Motion-compensated and gated cone beam filtered back-projection for 3-D rotational X-ray angiography

This paper presents a method to reconstruct moving objects from cone beam X-ray projections acquired during a single rotational run using a given motion vector field. The method is applicable to voxel driven cone-beam filtered back-projection reconstruction approaches. Here, a formulation based on the algorithm of Feldkamp, Davis, and Kress (FDK) is presented. The motion correction is applied during the back-projection step by shifting the voxel to be reconstructed according to the motion vector field. The method is applied to three-dimensional (3-D) rotational X-ray angiography. Projections from a beating coronary heart phantom are simulated. Motion-compensated reconstructions with varying accuracy of the applied motion field are carried out for a late diastolic heart phase and compared to the reconstruction obtained with the standard FDK-method from projections of the corresponding motion-free model in the same heart phase. Furthermore, gated reconstructions are calculated by weighting the projections according to their cardiac phase without using a motion vector field. Different gating window widths are applied, and the reconstructions are compared. Using the correct motion field with the motion-compensated reconstruction, the image quality of the standard reconstruction from the corresponding motion-free coronary model can almost be recovered. The reconstructed image quality stays acceptable if the accuracy of the motion field sampling points is better than 1 mm. The gated reconstructions with a window width of 15%-20% of the cardiac cycle lead to superior results compared to nearest neighbor gating, especially for histogram based visualization and analysis. The motion-compensated reconstructions provide sharp images of the coronaries far surpassing the image quality of gated reconstructions.     

A knowledge-based approach for 3-D reconstruction and labeling of vascular networks from biplane angiographic projections

An approach to the three-dimensional reconstruction of coronary arteries is presented. The principal objective is to show how modeling of a vascular network, together with algorithmic procedures, can lead to accurate 3-D structure and feature labeling. The labeling problem is stated directly within the 3-D reconstruction framework. The reconstruction ambiguities inherent to biplane techniques are solved by means of a knowledge base, modeling of the object, and heuristic rules. Feasibility in near-real situations has been demonstrated. The critical importance of the object 3-D reference to achieving the data and modeling matching is emphasized, and a way to deal with it is pointed out. The overall system implies an incremental development in methodologies and experiments. All of them have been elaborated and tested independently, and the most appropriate ones have been selected for integration into a modular system. All the stages of the process (calibration, segmentation, reconstruction, and display) are discussed, with the main focus on modeling. Examples of automatic reconstruction from a phantom are provided.     

Motion correction for coronary stent reconstruction from rotational X-ray projection sequences

This paper presents a new method for 3-D tomographic reconstruction of stent in X-ray cardiac rotational angiography. The method relies on 2-D motion correction from two radiopaque markerballs located on each side of the stent. The two markerballs are on a guidewire and linked to the balloon, which is introduced into the artery. Once the balloon has been inflated, deflated, and the stent deployed, a rotational sequence around the patient is acquired. Under the assumption that the guidewire and the stent have the same 3-D motion during rotational acquisition, we developed an algorithm to correct cardiac stent motion on the 2-D X-ray projection images. The 3-D image of the deployed stent is then reconstructed with the Feldkamp algorithm using all the available projections. Although the correction is an approximation, we show that the intrinsic geometrical error of our method has no visual impact on the reconstruction when the 2-D markerball centers are exactly detected and the markerballs have the same 3-D motion as the stent. Qualitative and quantitative results on simulated sequences under different realistic conditions demonstrate the robustness of the method. Finally, results from animal data acquired on a rotational angiography device are presented.     

Efficient fully 3-D iterative SPECT reconstruction with Monte Carlo-based scatter compensation

Quantitative accuracy of single photon emission computed tomography (SPECT) images is highly dependent on the photon scatter model used for image reconstruction. Monte Carlo simulation (MCS) is the most general method for detailed modeling of scatter, but to date, fully three-dimensional (3-D) MCS-based statistical SPECT reconstruction approaches have not been realized, due to prohibitively long computation times and excessive computer memory requirements. MCS-based reconstruction has previously been restricted to two-dimensional approaches that are vastly inferior to fully 3-D reconstruction. Instead of MCS, scatter calculations based on simplified but less accurate models are sometimes incorporated in fully 3-D SPECT reconstruction algorithms. We developed a computationally efficient fully 3-D MCS-based reconstruction architecture by combining the following methods: 1) a dual matrix ordered subset (DM-OS) reconstruction algorithm to accelerate the reconstruction and avoid massive transition matrix precalculation and storage; 2) a stochastic photon transport calculation in MCS is combined with an analytic detector modeling step to reduce noise in the Monte Carlo (MC)-based reprojection after only a small number of photon histories have been tracked; and 3) the number of photon histories simulated is reduced by an order of magnitude in early iterations, or photon histories calculated in an early iteration are reused. For a 64 x 64 x 64 image array, the reconstruction time required for ten DM-OS iterations is approximately 30 min on a dual processor (AMD 1.4 GHz) PC, in which case the stochastic nature of MCS modeling is found to have a negligible effect on noise in reconstructions. Since MCS can calculate photon transport for any clinically used photon energy and patient attenuation distribution, the proposed methodology is expected to be useful for obtaining highly accurate quantitative SPECT images within clinically acceptable computation times.     

Reconstruction of 3-D objects from cone beam projections

A true three-dimensional reconstruction (TTR) algorithm which is applicable to a cone beam with 4π detection geometry is introduced. The TTR differs from conventional slice by slice 3-D reconstruction methods. Promising areas of application for the TTR algorithm are position and X-ray computerized tomographic image reconstruction.     

Exact reduced-complexity maximum likelihood reconstruction of multiple 3-D objects from unlabeled unoriented 2-D projections and electron microscopy of viruses.

In cryo-electron microscopy, the data is comprised of noisy 2-D projection images of the 3-D electron scattering intensity of the object where the orientation of the projections is unknown. Often, the images show randomly selected objects from a mixture of different types of objects. Objects of different type may be unrelated, e.g., different species of virus, or related, e.g., different conformations of the same species of virus. Due to the low SNR and the 2-D nature of the data, it is challenging to determine the type of the object shown in an individual image. A statistical model and maximum likelihood estimator that computes simultaneous 3-D reconstruction and labels using an expectation maximization algorithm exists but requires extensive computation due to the numerical evaluation of 3-D or 5-D integrations of a square matrix of dimension equal to the number of degrees of freedom in the 3-D reconstruction. By exploiting the geometry of rotations in 3-D, the estimation problem can be transformed so that the inner-most numerical integral has a scalar rather than a matrix integrand. This leads to a dramatic reduction in computation, especially as the number of degrees of freedom in the 3-D reconstruction increases. Numerical examples of the 3-D reconstructions are provided based on synthetic and experimental images where the objects are small spherical viruses.     

3-D reconstruction of coronary arterial tree to optimize angiographic visualization

Due to vessel overlap and foreshortening, multiple projections are necessary to adequately evaluate the coronary tree with arteriography. Catheter-based interventions can only be optimally performed when these visualization problems are successfully solved. The traditional method provides multiple selected views in which overlap and foreshortening are subjectively minimized based on two dimensional (2-D) projections. A pair of images acquired from routine angiographic study at arbitrary orientation using a single-plane imaging system were chosen for three-dimensional (3-D) reconstruction. After the arterial segment of interest (e.g., a single coronary stenosis or bifurcation lesion) was selected, a set of gantry angulations minimizing segment foreshortening was calculated. Multiple computer-generated projection images with minimized segment foreshortening were then used to choose views with minimal overlapped vessels relative to the segment of interest. The optimized views could then be utilized to guide subsequent angiographic acquisition and interpretation. Over 800 cases of coronary arterial trees have been reconstructed, in which more than 40 cases were performed in room during cardiac catheterization. The accuracy of 3-D length measurement was confirmed to be within an average root-mean-square (rms) 3.5% error using eight different pairs of angiograms of an intracoronary guidewire of 105-mm length with eight radiopaque markers of 15-mm interdistance. The accuracy of similarity between the additional computer-generated projections versus the actual acquired views was demonstrated with the average rms errors of 3.09 mm and 3.13 mm in 20 LCA and 20 RCA cases, respectively. The projections of the reconstructed patient-specific 3-D coronary tree model can be utilized for planning optimal clinical views: minimal overlap and foreshortening. The assessment of lesion length and diameter narrowing can be optimized in both interventional cases and studies of disease progression and regression.     

Gabor wavelet representation for 3-D object recognition

This paper presents a model-based object recognition approach that uses a Gabor wavelet representation. The key idea is to use magnitude, phase, and frequency measures of the Gabor wavelet representation in an innovative flexible matching approach that can provide robust recognition. The Gabor grid, a topology-preserving map, efficiently encodes both signal energy and structural information of an object in a sparse multiresolution representation. The Gabor grid subsamples the Gabor wavelet decomposition of an object model and is deformed to allow the indexed object model match with similar representation obtained using image data. Flexible matching between the model and the image minimizes a cost function based on local similarity and geometric distortion of the Gabor grid. Grid erosion and repairing is performed whenever a collapsed grid, due to object occlusion, is detected. The results on infrared imagery are presented, where objects undergo rotation, translation, scale, occlusion, and aspect variations under changing environmental conditions.     

Block-iterative methods for image reconstruction from projections

The simultaneous MART algorithm (SMART) and the expectation maximization method for likelihood maximization (EMML) are extended to block-iterative versions, BI-SMART and BI-EMML, that converge to a solution in the feasible case, for any choice of subsets. The BI-EMML reduces to the "ordered subset" EMML of Hudson et al. (1992, 1994) when their "subset balanced" property holds.     

The effect of image distortion on 3-D reconstruction of coronary bypass grafts from angiographic views

Three-dimensional (3-D) reconstructions of coronary bypass grafts performed from X-ray angiographic images may become increasingly important for the investigation of damaging mechanical stresses imposed to these vessels by the cyclic movement of the heart. Contrary to what we had experienced with coronary arteries, appreciable reconstruction artifacts frequently occur with grafts. In order to verify the hypothesis that those are caused by distortions present in the angiographic images (acquired with image intensifiers), we have implemented a grid correction technique in our 3-D reconstruction method and studied its efficiency with phantom experiments. In this article, the nature of the encountered artifacts and the way in which the dewarping correction eliminates them are illustrated by a phantom experiment and by the reconstruction of a real coronary bypass vein graft.     

3D reconstruction from projections for stochastic objects(stochastic tomography)

The problem of 3D reconstruction from projections is studied for the case when the object under investigation is described by a random function and different projections correspond to its different realisations. A uniqueness theorem is obtained for the case when a stochastic object has a finite number of conformational states     

Towards 3-D scene reconstruction from broadcast video

Three-dimensional (3-D) scene reconstruction from broadcast video is a challenging problem with many potential applications, such as 3-D TV, free-view TV, augmented reality or three-dimensionalization of two-dimensional (2-D) media archives. In this paper, a flexible and effective system capable of efficiently reconstructing 3-D scenes from broadcast video is proposed, with the assumption that there is relative motion between camera and scene/objects. The system requires no a priori information and input, other than the video sequence itself, and capable of estimating the internal and external camera parameters and performing a 3-D motion-based segmentation, as well as computing a dense depth field. The system also serves as a showcase to present some novel approaches for moving object segmentation, sparse and dense reconstruction problems. According to the simulations for both synthetic and real data, the system achieves a promising performance for typical TV content, indicating that it is a significant step towards the 3-D reconstruction of scenes from broadcast video.     

Assessment of diffuse coronary artery disease by quantitative analysis of coronary morphology based upon 3-D reconstruction from biplane angiograms

Quantitative evaluations on coronary vessel systems are of increasing importance in cardiovascular diagnosis, therapy planning, and surgical verification. Whereas local evaluations, such as stenosis analysis, are already available with sufficient accuracy, global evaluations of vessel segments or vessel subsystems are not yet common. Especially for the diagnosis of diffuse coronary artery diseases, the authors combined a 3D reconstruction system operating on biplane angiograms with a length/volume calculation. The 3D reconstruction results in a 3D model of the coronary vessel system, consisting of the vessel skeleton and a discrete number of contours. To obtain an utmost accurate model, the authors focussed on exact geometry determination. Several algorithms for calculating missing geometric parameters and correcting remaining geometry errors were implemented and verified. The length/volume evaluation can be performed either on single vessel segments, on a set of segments, or on subtrees. A volume model based on generalized elliptical conic sections is created for the selected segments. Volumes and lengths (measured along the vessel course) of those elements are summed up. In this way, the morphological parameters of a vessel subsystem can be set in relation to the parameters of the proximal segment supplying it. These relations allow objective assessments of diffuse coronary artery diseases.     

Interior-point methodology for 3-D PET reconstruction

Interior-point methods have been successfully applied to a wide variety of linear and nonlinear programming applications. This paper presents a class of algorithms, based on path-following interior-point methodology, for performing regularized maximum-likelihood (ML) reconstructions on three-dimensional (3-D) emission tomography data. The algorithms solve a sequence of subproblems that converge to the regularized maximum likelihood solution from the interior of the feasible region (the nonnegative orthant). We propose two methods, a primal method which updates only the primal image variables and a primal-dual method which simultaneously updates the primal variables and the Lagrange multipliers. A parallel implementation permits the interior-point methods to scale to very large reconstruction problems. Termination is based on well-defined convergence measures, namely, the Karush-Kuhn-Tucker first-order necessary conditions for optimality. We demonstrate the rapid convergence of the path-following interior-point methods using both data from a small animal scanner and Monte Carlo simulated data. The proposed methods can readily be applied to solve the regularized, weighted least squares reconstruction problem.     

三维电大目标散射求解的多层快速多极子方法

为进一步提高对电大尺寸目标散射求解的能力,详细研究了多层快速多极子方法.重点设计了用于多层快速多极子方法的各种优化方法包括Morton编号、转移因子修正内插技术与外向波重复存储策略.对于未知量数目为N的三维电磁散射,数值实验显示多层快速多极子方法具有O(NlogN)量级的计算量、O(N)量级的存储量,特别适合求解三维电大尺寸目标的电磁散射.利用该方法在单机(内存1Gb)上成功计算了未知量为25万的电大尺寸目标散射.