Finite Number Of Projections Research Articles

A Tomographic Reconstruction Method using Coordinate-based Neural Network with Spatial Regularization

Tomographic reconstruction is concerned with computing the cross-sections of an object from a finite number of projections. Many conventional methods represent the cross-sections as images on a regular grid. In this paper, we study a recent coordinate-based neural network for tomographic reconstruction, where the network inputs a spatial coordinate and outputs the attenuation coefficient on the coordinate. This coordinate-based network allows the continuous representation of an object. Based on this network, we propose a spatial regularization term, to obtain a high-quality reconstruction. Experimental results on synthetic data show that the regularization term improves the reconstruction quality significantly, compared to the baseline. We also provide an ablation study for different architecture configurations and hyper-parameters.

Analysis of Digital Breast Tomosynthesis Acquisition Geometries in Sampling Fourier space.

Tomosynthesis acquires projections over a limited angular range and thus samples an incomplete projection set of the object. For a given acquisition geometry, the extent of tomosynthesis sampling can be measured in the frequency domain based on the Fourier Slice Theorem (FST). In this paper we propose a term, "sampling comprehensiveness", to describe how comprehensively an acquisition geometry samples the Fourier domain, and we propose two measurements to assess the sampling comprehensiveness: the volume of the null space and the nearest sampled plane. Four acquisition geometries, conventional (linear), T-shape, bowtie, and circular geometries, were compared on their comprehensiveness. The volume of the null space was estimated as the percentage of voxels subtended by zero slices in the sampled Fourier space. For each voxel in the frequency space, the nearest sampled plane and the distance to that plane were recorded. Among the four, the circular geometry was determined to be the most comprehensive based on the two measurements. We review tomosynthesis sampling with a finite number of projections and discuss how the sampling comprehensiveness should be interpreted. We further suggest that the decision on a system geometry should consider multiple factors including the sampling comprehensiveness, the task to be performed, the thickness of the imaged object, system specifications, and reconstruction algorithm.

Products of Projections, Polar Decompositions and Norms of Differences of Two Projections

Some characterizations of products of two projections on a Hilbert space are generalized to the case of products of a finite number of projections on a Hilbert $$C^*$$ -module. An example is constructed to show that in the Hilbert $$C^*$$ -module case, $$\mathfrak {X}$$ and its subset $$\mathfrak {X}_\bot $$ can be different, where $$\mathfrak {X}$$ denotes the set of all products of two projections on a Hilbert $$C^*$$ -module, and $$\mathfrak {X}_\bot $$ consists of those elements in $$\mathfrak {X}$$ that have the polar decomposition. Some new phenomena are revealed when an operator is taken in $$\mathfrak {X}$$ instead of $$\mathfrak {X}_\bot $$ . Some refinements are made on norms of differences of two projections.

필터보정역투영 CT 영상재구성방법에서 잡음 특성

전산화단층촬영장치의 영상재구성방법으로 필터보정역투영법이 광범위하게 사용되고 있다. 평행빔과 부채살빔의 재구성에 사용되는 투영에 잡음이 포함되었을 때 재구성 된 영상의 잡음을 살펴보았다. 평행빔과 부채살 구조에서 각각 360개, 720개의 투영으로 <TEX>$512{\times}512$</TEX> 크기로 Visual C++을 이용하여 영상재구성하였고, 원본 Shepp-Logan 두부 모형을 매우 잘 복원한다는 것을 확인하였다. 필터보정역투영법의 현실적인 접근(유한한 투영 개수)으로 인해 입력 잡음이 없어도 영상재구성 과정에서 잡음이 발생하였다. 입력 잡음비 0.5% 이하에서 잡음이 빠르게 증가하기 때문에 CT 장치의 잡음 제거 기술 및 영상처리 기법의 개발이 필요할 것이다. The filtered back projection in the image reconstruction algorithms for the clinic computed tomography system has been widely used. Noise of the reconstructed image was examined under the input noise for parallel and fan beam geometries. The reconstruction images of <TEX>$512{\times}512$</TEX> size were carried out under 360 and 720 projection by the Visual C++ for parallel beam and fan beam, respectively, and those agreed with the original Shepp-Logan head phantom very much. Noise was generated because of intrinsic restriction (finite number of projections) for the image reconstruction algorithm, filtered back projection, when no input noise was applied. Because the result noise was rapidly increased under 0.5% input noise ratio, technologies for reducing noise in CT system and image processing is important.

Image Reconstruction From Finite Number of Projections: Method of Transferring Geometry

This paper introduces a novel method of image reconstruction from a finite number of projections by processing the image along parallel rays. The geometry from the image plane is transferred to the Cartesian lattice by means of using the original image's line-integrals to calculate the line-sums of the discrete image to be reconstructed. Such a transformation of geometry allows for the 2D discrete paired transform, whose complete set of functions is defined by directions, to be effectively used in the exact reconstruction of the original image. The model of image reconstruction is described, and both examples and experimental results of implementation of the proposed method are provided for reconstruction on the Cartesian lattice of size 2(r) × 2(r), where r ≥ 2.

Continuous-time image reconstruction for binary tomography

Binary tomography is the process of reconstructing a binary image from a finite number of projections. We present a novel method for solving binary tomographic inverse problems using a continuous-time image reconstruction (CIR) system described by nonlinear differential equations based on the minimization of a double Kullback–Leibler divergence. We prove theoretically that the divergence measure monotonically decreases in time. Moreover, we demonstrate numerically that the quality of the reconstructed images of the nonlinear CIR system is better than those from an iterative reconstruction method.

Deciding the dimension of effective dimension reduction space for functional and high-dimensional data

In this paper, we consider regression models with a Hilbert-space-valued predictor and a scalar response, where the response depends on the predictor only through a finite number of projections. The linear subspace spanned by these projections is called the effective dimension reduction (EDR) space. To determine the dimensionality of the EDR space, we focus on the leading principal component scores of the predictor, and propose two sequential χ2 testing procedures under the assumption that the predictor has an elliptically contoured distribution. We further extend these procedures and introduce a test that simultaneously takes into account a large number of principal component scores. The proposed procedures are supported by theory, validated by simulation studies, and illustrated by a real-data example. Our methods and theory are applicable to functional data and high-dimensional multivariate data.

SU-FF-T-632: Rotational Smooth for Arc Therapy Dose Calculation

Purpose: In rotational radiotherapy, such as TomoTherapy® or VMAT, the gantry rotates during radiationdelivery. While the gantry rotation is continuous, dose can only be calculated for a finite number of projections due to the limitation of computing power. Ideally, using more finely sampled projections approximates better the continuous delivery. We proposed and demonstrated a method that effectively produces the result of supersampling in projections without increasing memory and calculation complexity. Method: We start with regular convolution/superposition dose calculation to get a dose distribution. Each leaf opens for a certain amount of angle in each projection. Then we calculate rotations of the dose distribution up to the mean angle of all such angles. The mean of rotations of the dose distribution is the proposed rotational smooth of the dose distribution. Results: We compared dose calculation using 51 projections as in standard TomoTherapySM planning with and without rotation smooth against supersampling (459) of projections for a 2D disc of radius 20 cm with a tumor of radius 5 cm at the center. The result of dose calculation for 51 projections without rotation smooth showed star‐shaped artifact at distance far away from the center, while the result with rotation smooth is much closer to the super‐sampled case. Conclusions: Simulation demonstrated the effectiveness of the proposed approach. Rotational smooth improves geometric accuracy for arc therapy dose calculation. This technique may be used for rotational delivery modalities, so that the geometric correction can be applied without increasing the complexity to reduce the artifact due to finite sampling in rotation angles.

Decentralized Diagnosis of Stochastic Discrete Event Systems

We investigate the decentralized diagnosis of stochastic discrete event systems (SDESs) by using multiple local stochastic diagnosers, each possessing its own sensors to deal with different information. We formalize the notions of decentralized diagnosis for SDESs by defining the concept of codiagnosability for stochastic automata, in which any communication among the local stochastic diagnosers or to any coordinators is not involved. These notions are weaker than the corresponding notions of decentralized diagnosis of classical DESs. A stochastic system being codiagnosable means that a fault can be detected by at least one local stochastic diagnoser within a finite delay. We construct a codiagnoser from a given stochastic system with a finite number of projections whose each diagnosis component uses the complete model of the system. We also deal with a number of basic properties of the codiagnoser. In particular, a necessary and sufficient condition of the codiagnosability for SDESs is presented, which generalizes the corresponding results of centralized diagnosis for SDESs. Also, we give a computing method in detail to check the codiagnosability of SDESs. As an application of our results, some examples are described.

Ridge functions, natural pixels and minimal norm reconstruction

In this work reconstruction from a finite number of projections using ridge functions within the framework of parallel beam geometry is considered. Theoretically, ridge functions are continuous solutions of a system of integral equations and their sum results in the minimal norm solution of the reconstruction problem. In practice, discretized versions of ridge functions are considered to obtain reconstruction from real data. Discrete projection data are modeled with the help of natural pixels. Analytical inversion formulas based both on ridge functions and natural pixels are derived. Results of numerical experiments are discussed.

Reconstruction of elongated structures using ridge functions and natural pixels

In this work, reconstruction of elongated structures from a finite number of projections using ridge functions within the framework of parallel beam geometry is considered. Discretized versions of ridge functions, or natural pixels are used to obtain reconstruction from sampled data. Inversion formulae for both ray- and strip-based projection data are derived. Possibilities of detecting the direction of elongated structures from few projections and detecting the defects in regular textures are discussed.

What information about a wave function is given by “measuring” it with the help of the tomographic method of recovering the Wigner distribution?

Mathematical problems connected with “measuring” a wave function by the tomographic method of recovering the Wigner distribution are considered. Inequalities showing what information is contained in a finite number of projections of the Wigner distribution are obtained. Bibliography: 20 titles.

Mathematics for computer tomography

Computerized tomography requires not only fast computers, but also analysis of mathematical models and construction of numerical algorithms. Classical mathematical theory is combined with modern numerical analysis to form the basis for efficient implementation on fast computers.The solution of the inverse problem of finding the image from given x-ray projections is theoretically obtained by the inverse Radon transform. Since only a finite number of projections are available, some approximation must be found, and this leads to a discrete counterpart of the continuous problem. There are three major classes of numerical solution methods: the Algebraic Reconstruction Method, the Filtered Back projection Method and the Direct Fourier Method. Much research is devoted to making the methods faster and more robust. The first one was used for the original tomography machine, the second one is used on almost all current machines in use. The third one has great potential for the future, since almost all computation is done by using the fast discrete Fourier transform.We shall describe the basic mathematical problem in computer tomography and the computational methods mentioned above for solving it. In particular we shall emphasize the special difficulties that are built into the problem. However, this is not a review article. Instead, it is intended to describe the influence of modern numerical methods on a fundamental problem of great significance for the society. We shall also indicate how computerized tomography has initiated new important research in central fields of numerical analysis, that can be used for problems in many other applications.

Statistical limitations in transaxial tomography

A general analysis is presented of the limitations on transaxial tomographic imaging due to the quantum statistics of the radiation source. An idealized model is used in which reconstruction errors due to divergence of the X-ray beam, the finite number of projections, and the finite detector size are neglected. The results, therefore, represent an upper bound to the performance attainable with this method. An operator formalism is introduced to describe the various linear reconstruction algorithms, all of which are shown to be statistically equivalent. Expressions are derived for the mean signal, rms noise, resolution, and required radiation dose for a given signal-to-noise ratio. The results are valid for any object and any linear reconstruction algorithm.

Recovering multidimensional signals from their projections

Some algorithms are presented for the reconstruction of multidimensional signals from their projections. The reconstruction problem is broken into two distinct steps; first samples of the Fourier transform of the unknown signal are computed from a series of digitized projections, then the unknown is estimated from these samples of its Fourier transform. Reconstructions are considered from several sets of samples in Fourier space. In the process a particular set of samples, the concentric squares raster of samples, is developed, the reconstructions from which are often superior to those from the more traditional polar raster of samples. Furthermore for an important class of unknowns exact reconstructions can be performed from a concentric squares raster from a finite number of projections. For this class of unknowns a single projection is sufficient. A detailed treatment of the one-projection reconstruction problem is presented and the difficulties associated with its solutions are explored.

Image reconstruction from finite numbers of projections

Several results are obtained appertaining to the reconstruction of a two- dimensional image from a finite number of projections. Several schemes are considered for interpolating between the given data. When a trigonometrical Fourier series is used for angular interpolation then one finds, firstly, a consistency condition whereby a posteriori estimates can be made of the errors in the given data, and secondly, a basic image which contains only that information common to all physically permissible interpolation schemes. This basic image is necessarily free of misleading artefacts but it is computationally slow. Several computationally rapid interpolation schemes (based on the fast Fourier transform algorithm) are found to give good quality images, provided the given number of projections is sufficient to resolve the major details of the true image. A computational example is presented showing that a contrived image can be accurately reconstructed from a single projection.

The reconstruction of a three-dimensional structure from projections and its application to electron microscopy. II. Direct methods.

Under certain conditions it is possible to obtain the approximate projected density of a specimen from an electron micrograph, and by taking electron micrographs in different directions a number of different projections of a specimen can be determined. Several methods have been proposed for reconstructing the three-dimensional density distribution of an object from a series of projections whose planes are equally spaced in angle about an axis. One of these methods uses the Fourier transforms of the projections, while other methods operate directly on the projection data. The limitations of the direct method of ‘back-projection’ are discussed, and it is shown how a valid reconstruction may be achieved by back-projection from modified projected densities. This direct method is shown to be equivalent to the Fourier method and also to certain methods of reconstruction developed for the solution of analogous problems in other fields. An expression is given for the resolution of the reconstruction obtainable from a finite number of projections. Because of its computational simplicity this method is very suitable for reconstructing the density of objects with rotational or helical symmetry.

Resolution in reconstructing objects from electron micrographs

Some basic problems associated with the reconstruction of objects from electron micrographs and X-ray photographs are investigated. A precise definition of the problem is given. It is proved that interesting images cannot be uniquely described by any finite number of projections, and thus any reconstruction technique can only hope to give an approximation to the original. The root-mean-square distance is discussed as a measure of the approximation, and it is shown to be sensitive to changes in fine detail. Various suggested ways of measuring resolution are discussed and precise definitions for the resolution of reconstruction techniques are suggested both with and without the use of Fourier transforms. A theoretical explanation is given for the success, from the point of view of resolution, of the algebraic reconstruction techniques using only few projections from a limited range of angles.