Stability results for uniquely determined sets from two directions in discrete tomography

In this paper we prove several new stability results for the reconstruction of binary images from two projections. We consider an original image that is uniquely determined by its projections and possible reconstructions from slightly different projections. We show that for a given difference in the projections, the reconstruction can only be disjoint from the original image if the size of the image is not too large. We also prove an upper bound for the size of the image given the error in the projections and the size of the intersection between the image and the reconstruction.     

A multi-scale image reconstruction algorithm for electrical capacitance tomography

Electrical capacitance tomography (ECT) is considered as a promising process tomography (PT) technology, and its successful applications depend mainly on the precision and speed of the image reconstruction algorithms. In this paper, based on the wavelet multi-scale analysis method, an efficient image reconstruction algorithm is presented. The original inverse problem is decomposed into a sequence of inverse problems, which are solved successively from the largest scale to the smallest scale. At different scales, the inverse problem is solved by a generalized regularized total least squares (TLS) method, which is developed using a combinational minimax estimation method and an extended stabilizing functional, until the solution of the original inverse problem is found. The homotopy algorithm is employed to solve the objective functional. The proposed algorithm is tested by the noise-free capacitance data and the noise-contaminated capacitance data, and excellent numerical performances and satisfactory results are observed. In the cases considered in this paper, the reconstruction results show remarkable improvement in the accuracy. The spatial resolution of the reconstructed images by the proposed algorithm is enhanced and the artifacts in the reconstructed images can be eliminated effectively. As a result, a promising algorithm is introduced for ECT image reconstruction.     

Exact reconstruction in photoacoustic tomography with circular integrating detectors II: Spherical geometry

Illuminating tissue with pulsed electromagnetic waves generates acoustic waves inside an object which can be measured and converted into a three‐dimensional (3d) image. This text is concerned with a two‐step reconstruction method where the acoustic pressure is measured with circular integrating detectors. In the first step, reconstruction formulas for some kind of projection of the source distribution are derived; in the second step, an inversion formula for a circular radon transform on the sphere is developed. Copyright © 2010 John Wiley & Sons, Ltd.     

Positron Emission Tomography and Random Coefficients Regression

We further explore the relation between random coefficients regression (RCR) and computerized tomography. Recently, Beran et al. (1996, Ann. Statist., 24, 2569–2592) explored this connection to derive an estimation method for the non-parametric RCR problem which is closely related to image reconstruction methods in X-ray computerized tomography. In this paper we emphasize the close connection of the RCR problem with positron emission tomography (PET). Specifically, we show that the RCR problem can be viewed as an idealized (continuous) version of a PET experiment, by demonstrating that the nonparametric likelihood of the RCR problem is equivalent to that of a specific PET experiment. Consequently, methods independently developed for either of the two problems can be adapted from one problem to the other. To demonstrate the close relation between the two problems we use the estimation method of Beran, Feuerverger and Hall for image reconstruction in PET.     

Generic iterative subset algorithms for discrete tomography

Discrete tomography deals with the reconstruction of images from their projections where the images are assumed to contain only a small number of grey values. In particular, there is a strong focus on the reconstruction of binary images (binary tomography). A variety of binary tomography problems have been considered in the literature, each using different projection models or additional constraints. In this paper, we propose a generic iterative reconstruction algorithm that can be used for many different binary reconstruction problems. In every iteration, a subproblem is solved based on at most two of the available projections. Each of the subproblems can be solved efficiently using network flow methods. We report experimental results for various reconstruction problems. Our results demonstrate that the algorithm is capable of reconstructing complex objects from a small number of projections.     

Analysis on the strip-based projection model for discrete tomography

Discrete tomography deals with image reconstruction of an object with finitely many gray levels (such as two). Different approaches are used to model the raw detector reading. The most popular models are line projection with a lattice of points and strip projection with a lattice of pixels/cells. The line-based projection model fits some applications but involves a major approximation since the x-ray beams of finite widths are simplified as line integrals. The strip-based projection model formulates projection equations according to the fractional areas of the intersection of each strip-shaped beam and the rectangular grid of an image to be reconstructed, so is more realistic in some applications. In this paper, we characterize the strip-based projection model and establish an equivalence between the system matrices generated by the strip-based and line-based projection models.     

On the use of graphs in discrete tomography

In this tutorial paper, we consider the basic image reconstruction problem which stems from discrete tomography. We derive a graph theoretical model and we explore some variations and extensions of this model. This allows us to establish connections with scheduling and timetabling applications. The complexity status of these problems is studied and we exhibit some polynomially solvable cases. We show how various classical techniques of operations research like matching, 2-SAT, network flows are applied to derive some of these results.      

基于字典学习方法的CT不完全投影图像重建算法

对于不完全投影角度的重建研究是CT图像重建中一个重要的问题.将压缩感知中字典学习的方法与CT重建算法ART迭代算法相结合.字典学习方法中字典更新采用K-SVD(K-奇异值分解)算法,稀疏编码采用OMP(正交匹配追踪)算法.最后通过对标准Head头部模型进行仿真实验,验证了字典学习方法在CT图像重建中对于提高图像的重建质量和提高信噪比的可行性与有效性.另外还研究了字典学习中图像块大小和滑动距离对重建图像的影响     

Wavelet Filtering for Filtered Backprojection in Computed Tomography

In this paper, we present a characterization of MRA biorthogonal wavelet filters with full frequency supports. Based on this characterization, it is established that wavelet ramp filters are biorthogonal wavelets if the original wavelets are sufficiently regular. An efficient subband coding algorithm is developed for wavelet filtering in filtered backprojection, which is the most popular method in computed tomography (CT). Computer simulation suggests that this wavelet filtering process is a useful tool for improving image quality and reducing computational time in local CT reconstruction.     

An improved exact filtered backprojection algorithm for spiral computed tomography

Proposed is a theoretically exact formula for inversion of data obtained by a spiral computed tomography scan with a two-dimensional detector array. The detector array is supposed to be of limited extent in the axial direction. The main property of the formula is that it can be implemented in a truly filtered backprojection fashion. First, one performs shift-invariant filtering of a derivative of the cone beam projections, and, second, the result is back-projected in order to form an image. Compared with an earlier reconstruction algorithm proposed by the author, the new one is two times faster, requires a smaller detector array, and does not impose restrictions on how big the patient is inside the gantry. Results of numerical experiments are presented.     

A Subpixel Image Restoration Algorithm

Abstract

In statistical image reconstruction, data are often recorded on a regular grid of squares, known as pixels, and the reconstructed image is defined on the same pixel grid. Thus, the reconstruction of a continuous planar image is piecewise constant on pixels, and boundaries in the image consist of horizontal and vertical edges lying between pixels. This approximation to the true boundary can result in a loss of information that may be quite noticeable for small objects, only a few pixels in size. Increasing the resolution of the sensor may not be a practical alternative. If some prior assumptions are made about the true image, however, reconstruction to a greater accuracy than that of the recording sensor's pixel grid is possible. We adopt a Bayesian approach, incorporating prior information about the true image in a stochastic model that attaches higher probability to images with shorter total edge length. In reconstructions, pixels may be of a single color or split between two colors. The model is illustrated using both real and simulated data.     

Image reconstruction model for limited-angle CT based on prior image induced relative total variation

Limited-angle computed tomography (CT) reconstruction has a great potential to reduce X-ray radiation dose or scanning time. Suppressing shading artifacts is challenging, but of great practical significance in limited-angle CT. Traditional methods based on total variation (TV) cannot effectively remove the shading artifacts, prior image constrained compressed sensing (PICCS) is a promising method, but is sensitive to the quality of the prior image. In micro-CT, a prior image reconstructed by filtered back-projection (FBP) may contain high-level noise. An image reconstructed by PICCS tends to inherit both structures and noise of the prior image. In this study, to suppress noise and shading artifacts, we propose a new limited-angle CT reconstruction model called prior image induced relative total variation (piiRTV), that uses the structure information of a prior image to guide limited-angle CT reconstruction. The proposed piiRTV is compared to TV and PICCS. Numerical simulations and experiments on real CT projections demonstrate the effectiveness of piiRTV in suppression of noise and shading artifacts. In addition, the proposed piiRTV is more robust to the prior image quality than PICCS.     

Approximate inverse meets local tomography

Local or lambda tomography reconstructs Λƒ which has the same discontinuities as the searched‐for density distribution ƒ. Computing Λƒ, however, requires only local tomographic measurements. Local tomography is usually implemented by a filtered backprojection algorithm (FBA). In the present article we design reconstruction filters for the FBA such that Λ^2m+1ƒ will be reconstructed for a given m∈ℕ₀. Moreover, we prove convergence and convergence rates for the FBA as the discretization step size goes to zero. To this end we express the FBA in the framework of approximate inverse. Based on our analysis we further propose a scheme which yields a proper scaling of the reconstruction filters. Numerical experiments illustrate the analytic results. Copyright © 2000 John Wiley & Sons, Ltd.     

Complexity results for reconstructing binary images with disjoint components from horizontal and vertical projections

The problem of reconstructing a special class of binary images from their horizontal and vertical projections is considered. We present a general framework for analyzing the worst case complexity of this task if the image consists of more than one pairwise disjoint component. Applying the presented technique we analyze the complexity of reconstructing canonical hv-convex binary images. We also present parameterized complexity results on general and so-called glued hv-convex images. Moreover, we study how our results are related to the reconstruction of permutation matrices from four projections.     

由投影重建图象的样条拟合法

曾获得诺贝尔医学奖的CT(计算机断层扫描)技术,其应用范围已从医学扩展到地质勘探、工业无损探伤等许多工程和科技领域.它的理论基础就是由投影重建图象这样一个数学问题,即已知某函数(在屏幕上表现为图象)沿低维流形(二维情形为直线或曲线,三维情形为平面或曲面)的积分值,反求函数(图象)本身.拉当早在1 916年就提出并研究了这一问题,一般称为拉当变换及其反演.目前已有很多实用算法,如代数重建法,卷积反投影法等田.而CT技术应用范围不断扩大,又促进了算法的研究     

电磁层析成像(EMT):正问题的理论分析

基于感应原理的电磁层析成像(EMT)是一种新颖的工业过程成像技术。通过解析求解特定的两组分流成像正问题,导出了二维磁场分布的精确表达式,定量分析了传感器内部的物理机制和EMT技术的可检测性极限,建立了场灵敏度的直接数学表达式和灵敏度图。在一定程度上,为传感器的定量设计,EMT技术术的可检测性分析和两组分过程的图像重建提供了数理基础。     

Constraining by a family of strictly nonexpansive idempotent functions with applications in image reconstruction

When solving an image reconstruction problem a previous knowledge concerning the original image may lead to various constraining strategies. A convergence result has been previously proved for a constrained version of the Kaczmarz projection algorithm with a single strictly nonexpansive idempotent function with a closed image. In this paper we consider a more general projection based iterative method and a family of such constraining functions with some additional hypotheses in order to better use the a priori information for every approximation calculated. We present a particular family of box-constraining functions which satisfies our assumptions and we design an adaptive algorithm that uses an iteration-dependent family of constraining functions for some numerical experiments of image reconstruction on Tomographic Particle Image Velocimetry.     

基于PICCS的非局部TGV能谱CT重建算法

能谱CT将宽谱划分为窄谱,导致通道内光子数目明显减少,加大了噪声影响,故从噪声投影中重建出高质量图像是能谱CT的一个研究热点.传统全变分(total variational,TV)容易造成重建图像中出现块状伪影等问题,总广义全变分(total generalized variation,TGV)算法可以逼近任意阶函数,再结合非局部均值算法的思想,同时考虑到不同能谱通道下重建图像的相关性,将高质量全能谱重建图像作为先验图像指导能谱CT重建,提出了基于先验图像约束压缩感知(prior image constrained compressed sensing,PICCS)的非局部TGV重建算法.实验结果表明,所提算法在抑制噪声的同时能够有效复原图像细节及边缘信息,且收敛速度快.     

Detection of Subsurface Bubbles with Discrete Electromagnetic Geotomography

In electromagnetic geotomography, the inversion is very ill-posed due to the limita- tion of an angular range of projections, and consequently the reconstructed image is different than the true image. Nevertheless, the reconstruction can be much im- proved if we are interested only in a piece of information on the interior volume of the examined object, e.g. detection of some anomalies (bubbles). We assume that the image can be represented by binary values, hence: Discrete Electromagnetic Geotomography (DEG). In this paper, we discuss two discrete image reconstruction algorithms. The first one uses the D.C. programming to find the integer solution to the penalized least-squares problem. The penalty term is based on the Gibbs distribution. In the second one, the mean field annealing is applied to maximize the Gibbs-Boltzmann distribution associated with the penalized least-squares function. The numerical results are presented for noise-free and noisy data.     

Family constraining of iterative algorithms

In constraining iterative processes, the algorithmic operator of the iterative process is pre-multiplied by a constraining operator at each iterative step. This enables the constrained algorithm, besides solving the original problem, also to find a solution that incorporates some prior knowledge about the solution. This approach has been useful in image restoration and other image processing situations when a single constraining operator was used. In the field of image reconstruction from projections a priori information about the original image, such as smoothness or that it belongs to a certain closed convex set, may be used to improve the reconstruction quality. We study here constraining of iterative processes by a family of operators rather than by a single operator.