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Image Reconstruction Image reconstruction by using local inverse for full field of view
Abstract
The iterative refinement method (IRM) has been very successfully applied in
many different fields for examples the modern quantum chemical calculation and
CT image reconstruction. It is proved that the refinement method can create an
exact inverse from an approximate inverse with a few iterations. The IRM has
been used in CT image reconstruction to lower the radiation dose. The IRM
utilize the errors between the original measured data and the recalculated data
to correct the reconstructed images. However if it is not smooth inside the
object, there often is an over-correction along the boundary of the organs in
the reconstructed images. The over-correction increase the noises especially on
the edges inside the image. One solution to reduce the above mentioned noises
is using some kind of filters. Filtering the noise before/after/between the
image reconstruction processing. However filtering the noises also means reduce
the resolution of the reconstructed images. The filtered image is often applied
to the image automation for examples image segmentation or image registration
but diagnosis. For diagnosis, doctor would prefer the original images without
filtering process. In the time these authors of this manuscript did the work of
interior image reconstruction with local inverse method, they noticed that the
local inverse method does not only reduced the truncation artifacts but also
reduced the artifacts and noise introduced from filtered back-projection method
without truncation. This discovery lead them to develop the sub-regional
iterative refinement (SIRM) image reconstruction method. The SIRM did good job
to reduce the artifacts and noises in the reconstructed images. The SIRM divide
the image to many small sub-regions. To each small sub-region the principle of
local inverse method is applied.
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For parallel beam geometry the Fourier reconstruction works via the Fourier slice theorem (or central slice theorem, projection slice theorem). For fan beam situation, Fourier slice can be extended to a generalized Fourier slice theorem (GFST) for fan-beam image reconstruction. We have briefly introduced this method in a conference. This paper reintroduces the GFST method for fan beam geometry in details. The GFST method can be described as following: the Fourier plane is filled by adding up the contributions from all fan-beam projections individually; thereby the values in the Fourier plane are directly calculated for Cartesian coordinates such avoiding the interpolation from polar to Cartesian coordinates in the Fourier domain; inverse fast Fourier transform is applied to the image in Fourier plane and leads to a reconstructed image in spacial domain. The reconstructed image is compared between the result of the GFST method and the result from the filtered backprojection (FBP) method. The major differences of the GFST and the FBP methods are: (1) The interpolation process are at different data sets. The interpolation of the GFST method is at projection data. The interpolation of the FBP method is at filtered projection data. (2) The filtering process are done in different places. The filtering process of the GFST is at Fourier domain. The filtering process of the FBP method is the ramp filter which is done at projections. The resolution of ramp filter is variable with different location but the filter in the Fourier domain lead to resolution invariable with location. One advantage of the GFST method over the FBP method is in short scan situation, an exact solution can be obtained with the GFST method, but it can not be obtained with the FBP method. The calculation of both the GFST and the FBP methods are at O(N-3), where N is the number of pixel in one dimension.
The investigation is concerned first with reconstruction of the density function of an object from computed tomography data. The projection (Fourier Slice) theorem is discussed for fan-beam geometry. A rebinning algorithm, based on the slant stacking approach, is introduced in order to transform fan-beam geometry data into its parallel-beam counterpart. The density function is reconstructed using a filtered back-projection algorithm. The sampling criteria for fan beam computed tomography are also derived. The MATLAB package has been utilized to implement these algorithms for the simulation of Shepp–Logan head phantom and implementation of the discrete normal Radon transform based on fan-beam geometry. The reconstruction of transverse cross-sectional micro-structures of fiber reinforced composite material test specimens has also been discussed. In addition, the MATLAB package has been utilized to demonstrate the image reconstruction of the cross-section of a two-phase composite material using parallel beam geometry (Radon transform).
Fast and accurate image reconstruction is the ultimate goal of iterative methods for limited-angle, few-view, interior problems, etc. Recently, a finite-detector-based projection model was proposed for iterative CT reconstructions, which was called area integral model (AIM) and has shown a high spatial resolution but with a high computational complexity. On the other hand, the distance-driven model (DDM) is the state-of-the-art technology to model forward projection and backprojection, which has shown a low computational complexity but relative low spatial resolution than AIM-based method. Inspired by the DDM, here we propose an improved distance-driven model (IDDM), which has a similar computational complexity with the DDM-based method and comparative spatial resolution with the AIM-based method. In an ordered-subset simultaneous algebraic reconstruction technique (OS-SART) framework, the AIM, IDDM and DDM are implemented and evaluated using a sinogram from a phantom experiment on a Discovery CT750 HD scanner. The results show that the computational cost of DDM- and IDDM-based methods is similar, which is 6 to 13~times faster than the AIM-based method assuming the same number of iterations. The spatial resolution of AIM- and IDDM-based method is comparable, which is better than DDM-based method in terms of full-width-of-half-maximum (FWHM).
Over the past two decades, rapid system and hardware development of x-ray computed tomography (CT) technologies has been accompanied by equally exciting advances in image reconstruction algorithms. The algorithmic development can generally be classified into three major areas: analytical reconstruction, model-based iterative reconstruction, and application-specific reconstruction. Given the limited scope of this chapter, it is nearly impossible to cover every important development in this field; it is equally difficult to provide sufficient breadth and depth on each selected topic. As a compromise, we have decided, for a selected few topics, to provide sufficient high-level technical descriptions and to discuss their advantages and applications.
The reconstruction problem in in-line X-ray Phase-Contrast Tomography is usually approached by solving two independent linearized sub-problems: phase retrieval and tomographic reconstruction. Both problems are often ill-posed and require the use of regularization techniques that lead to artifacts in the reconstructed image. We present a novel reconstruction approach that solves two coupled linear problems algebraically. Our approach is based on the assumption that the frequency space of the tomogram can be divided into bands that are accurately recovered and bands that are undefined by the observations. This results in an underdetermined linear system of equations. We investigate how this system can be solved using three different algebraic reconstruction algorithms based on Total Variation minimization. These algorithms are compared using both simulated and experimental data. Our results demonstrate that in many cases the proposed algebraic algorithms yield a significantly improved accuracy over the conventional L2-regularized closed-form solution. This work demonstrates that algebraic algorithms may become an important tool in applications where the acquisition time and the delivered radiation dose must be minimized.
Purposes:
Interior tomography problem can be solved using the so-called differentiated backprojection-projection onto convex sets (DBP-POCS) method, which requires a priori knowledge within a small area interior to the region of interest (ROI) to be imaged. In theory, the small area wherein the a priori knowledge is required can be in any shape, but most of the existing implementations carry out the Hilbert filtering either horizontally or vertically, leading to a vertical or horizontal strip that may be across a large area in the object. In this work, we implement a practical DBP-POCS method with radial Hilbert filtering and thus the small area with the a priori knowledge can be roughly round (e.g., a sinus or ventricles among other anatomic cavities in human or animal body). We also conduct an experimental evaluation to verify the performance of this practical implementation.
Methods:
We specifically re-derive the reconstruction formula in the DBP-POCS fashion with radial Hilbert filtering to assure that only a small round area with the a priori knowledge be needed (namely radial DBP-POCS method henceforth). The performance of the practical DBP-POCS method with radial Hilbert filtering and a priori knowledge in a small round area is evaluated with projection data of the standard and modified Shepp-Logan phantoms simulated by computer, followed by a verification using real projection data acquired by a computed tomography (CT) scanner.
Results:
The preliminary performance study shows that, if a priori knowledge in a small round area is available, the radial DBP-POCS method can solve the interior tomography problem in a more practical way at high accuracy.
Conclusions:
In comparison to the implementations of DBP-POCS method demanding the a priori knowledge in horizontal or vertical strip, the radial DBP-POCS method requires the a priori knowledge within a small round area only. Such a relaxed requirement on the availability of a priori knowledge can be readily met in practice, because a variety of small round areas (e.g., air-filled sinuses or fluid-filled ventricles among other anatomic cavities) exist in human or animal body. Therefore, the radial DBP-POCS method with a priori knowledge in a small round area is more feasible in clinical and preclinical practice.
A data sufficiency condition for 2D or 3D region-of-interest (ROI) reconstruction from a limited family of line integrals has recently been introduced using the relation between the backprojection of a derivative of the data and the Hilbert transform of the image along certain segments of lines covering the ROI. This paper generalizes this sufficiency condition by showing that unique and stable reconstruction can be achieved from an even more restricted family of data sets, or, conversely, that even larger ROIs can be reconstructed from a given data set. The condition is derived by analysing the inversion of the truncated Hilbert transform, here defined as the problem of recovering a function of one real variable from the knowledge of its Hilbert transform along a segment which only partially covers the support of the function but has at least one end point outside that support. A proof of uniqueness and a stability estimate are given for this problem. Numerical simulations of a 2D thorax phantom are presented to illustrate the new data sufficiency condition and the good stability of the ROI reconstruction in the presence of noise.
We report a reconstruction method, called a back-projection filtered (BPF) algorithm, for fan beam differential phase contrast computed tomography (DPC-CT) with equidistant geometrical configuration. This work comprises a numerical study of the algorithm and its experimental verification with a three-grating interferometer and an x-ray tube source. The numerical simulation and experimental results demonstrate that the proposed method can deal with several classes of truncated datasets. It could be of interest in future medical phase contrast imaging applications.
A common wish in non-destructive testing is to inves-tigate a large object with a small interesting detail inside. Due to practical circumstances, the projec-tions may sometimes be truncated. According to the theory on tomography, it is then impossible to recon-struct the object. However, sometimes it is possible to receive an approximate result. It turns out that the key-point is how to implement the ramp-filter. The quality of the result depends on the object itself. We show one good experiment on real data, linear cone-beam tomography for logs. We also show ex-periments on the Shepp-Logan phantom, well-known from medical CT, and discuss the varying reconstruc-tion quality.
The discrete Radon transform (DRT) was defined by Abervuch et al. as an analog of the continuous Radon transform for discrete data. Both the DRT and its inverse are computable in O ( n 2log n ) operations for images of size n × n . In this paper, we demonstrate the applicability of the inverse DRT for the reconstruction of a 2-D object from its continuous projections. The DRT and its inverse are shown to model accurately the continuum as the number of samples increases. Numerical results for the reconstruction from parallel projections are presented. We also show that the inverse DRT can be used for reconstruction from fan-beam projections with equispaced detectors.
A convolution-backprojection formula is deduced for direct reconstruction of a three-dimensional density function from a set of two-dimensional projections. The formula is approximate but has useful properties, including errors that are relatively small in many practical instances and a form that leads to convenient computation. It reduces to the standard fan-beam formula in the plane that is perpendicular to the axis of rotation and contains the point source. The algorithm is applied to a mathematical phantom as an example of its performance.
In computer tomography (CT), truncated projections are produced due to scanning large objects with a detector that is limited in width. Applying filtered back-projection(FBP) method directly to truncated projections, the reconstructed image will contain truncation artifacts - bright rings on the boundary of region of interest (ROI). Extrapolation algorithms can be used to reduce the truncation artifacts; however extrapolations are usually double the length of the projection data; resulting in an increased calculation time. This paper introduces mixed extrapolation, which is a combination of exponential and quadratic extrapolation. It is proven that doubling the length of the projection data for the mixed extrapolation can be avoided. The projections were extrapolated according to the boundary values and their derivatives. The algorithm achieves equivalence to the extrapolation approach with negligible increased calculation time. Supplementary functions are introduced in order to simplify the calculations. These functions can be calculated prior to extrapolation process, hence the calculation time is significantly reduced. The calculation times are compared between fast extrapolation introduced in this paper and normal extrapolation with doubling the length of projection data.
Despite major advances in x-ray sources, detector arrays, gantry mechanical design and especially computer performance, one component of computed tomography (CT) scanners has remained virtually constant for the past 25 years-the reconstruction algorithm. Fundamental advances have been made in the solution of inverse problems, especially tomographic reconstruction, but these works have not been translated into clinical and related practice. The reasons are not obvious and seldom discussed. This review seeks to examine the reasons for this discrepancy and provides recommendations on how it can be resolved. We take the example of field of compressive sensing (CS), summarizing this new area of research from the eyes of practical medical physicists and explaining the disconnection between theoretical and application-oriented research. Using a few issues specific to CT, which engineers have addressed in very specific ways, we try to distill the mathematical problem underlying each of these issues with the hope of demonstrating that there are interesting mathematical problems of general importance that can result from in depth analysis of specific issues. We then sketch some unconventional CT-imaging designs that have the potential to impact on CT applications, if the link between applied mathematicians and engineers/physicists were stronger. Finally, we close with some observations on how the link could be strengthened. There is, we believe, an important opportunity to rapidly improve the performance of CT and related tomographic imaging techniques by addressing these issues.
Non-circular scanning geometries such as helix or circular sinusoid have been used or proposed for cone-beam computed tomography (CBCT), because they provide sufficient data for numerically stable and exact image reconstruction within the scanned volume. Analytic algorithms have been developed for image reconstruction from cone-beam data acquired with a full-scan circular sinusoidal trajectory. In this work, we propose an innovative imaging approach in which a reduced-scan circular sinusoidal trajectory is used for acquiring data sufficient for exact 3D image reconstruction. A filtered backprojection (FBP) algorithm based on Pack-Noo's reconstruction formula is applied for image reconstruction in reduced-scan circular sinusoidal scans. We have conducted numerical studies to demonstrate the reduced-scan approach and to validate the FBP reconstruction algorithm in the proposed approach. The proposed scanning method can contribute to increasing the throughput of a scanner, while improving the image quality compared to a conventional circular scan.
A new integral transform pair has been introduced for the reconstruction of slices from fan beam projection data. This integral transform plays the same role as the Hankel transform (or Fourier Bessel transform) in the parallel beam case and includes the Hankel transform as a special case. The relations between the new integral transform and the Hankel transform pair are derived and the implementation of the algorithm is described. Memory space requirements and speed of the transform are discussed and the advantage of precalculations are mentioned. The transform pair is especially useful in case the two-dimensional object slice to be reconstructed has only few low frequency components in the azimuthal direction. Furthermore the method is capable of treating the case where no analytic a priori information is to be taken into account. In this case projections into the measurement directions are performed and the differences between these projection data and the measured data are reconstructed in order to improve the object data. Alternating transformations between projection and object space are efficient and fast. Finally some results of simulations including the influence of noise are shown and possible applications of the method are discussed.
We explore two characteristic features of x-ray computed tomography inversion formulas in two and three dimensions that are dependent on π-lines. In such formulas the data from a given source position contribute only to the reconstruction of f(x) for x in a certain region, called the region of back projection. The second characteristic is a certain small artifact in the reconstruction called a comet tail artifact. We propose that the comet tail artifact is closely related to the boundary of the region of back projection and make this relationship precise, developing a general theory of the region of back projection, its boundary, and the location of the artifact in helical and fan-beam tomography. This theory is applied to a number of specific examples and confirmed by numerical experiments. Furthermore it is demonstrated that a strong comet tail artifact appears in numerical reconstructions from misaligned fan-beam data. A numerical method for using the artifact to find the correct alignment is suggested.
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We present a novel approximate image reconstruction technique for
helical cone-beam CT, called the Advanced Multiple Plane Reconstruction
(AMPR). The method is an extension of the ASSR algorithm presented in
Medical Physics vol. 27, no. 4, 2000 by Kachelriess et al. In the ASSR,
the pitch is fixed to a certain value and dose usage is not optimum.
These limitations have been overcome in the AMPR algorithm by
reconstructing several image planes from any given half scan range of
projection angles. The image planes are tilted in two orientations so as
to optimally use the data available on the detector. After
reconstruction of several sets of tilted images, a subsequent
interpolation step reformats the oblique image planes to a set of voxels
sampled on a cartesian grid. Using our novel approach on a scanner with
16 slices, we can achieve image quality superior to what is currently a
standard for four-slice scanners. Dose usage in the order of 95% for all
pitch values can be achieved. We present simulations of
semi-antropomorphic phantoms using a standard CT scanner geometry and a
16 slice design.
In various applications of computed tomography (CT), it is common that the reconstructed object is over the field of view (FOV) or we may intend to sue a FOV which only covers the region of interest (ROI) for the sake of reducing radiation dose. These kinds of imaging situations often lead to interior reconstruction problems which are difficult cases in the reconstruction field of CT, due to the truncated projection data at every view angle. In this paper, an interior reconstruction method is developed based on a rotation-translation (RT) scanning model. The method is implemented by first scanning the reconstructed region, and then scanning a small region outside the support of the reconstructed object after translating the rotation centre. The differentiated backprojection (DBP) images of the reconstruction region and the small region outside the object can be respectively obtained from the two-time scanning data without data rebinning process. At last, the projection onto convex sets (POCS) algorithm is applied to reconstruct the interior region. Numerical simulations are conducted to validate the proposed reconstruction method.
To reconstruct images from truncated projections collected along a spiral trajectory, the most common approach is to smoothly extrapolate each projection to suppress truncation artifacts. Based on the work of Dennerlein, a local reconstruction algorithm is developed for spiral cone beam computed tomography (CT), which does not need explicit projection extrapolation. This algorithm is a filtered-backprojection (FBP) format and contains three major steps: cosine weight, filtration, and backprojection. While the cosine weight and backprojection steps are the same as the classical Feldkamp - Davis - Kress (FDK) scheme, the filtering step contains two steps: Laplace filtering and Radon transform-based filtering. Numerical simulation results show that the proposed algorithm has a better performance than FDK algorithm for region-of-interest (ROI) reconstruction of spiral cone-beam CT.
ALGOL procedures for the iterative refinement of an approximation to a generalized inverse of a matrix and some results of numerical tests with ill-conditioned and rank-deficient matrices are given.
In this paper we study three-dimensional cone beam local tomography. We analyze the local tomography function fΛc, which was proposed earlier in [A.K. Louis and P. Maass, IEEE Trans. Medical Imaging, 12 (1993), pp. 764-769]. Let f be an unknown density distribution inside an object being scanned. We find a relationship between the wave fronts of fΛc and f and compute the principal symbol of the operator which maps f into fΛc. Our results prove the fact, which was first noted in Louis and Maass, that one can recover most of the singularities of f knowing fΛc. It is shown that these are precisely the singularities of f that are visible from the data. A simple and efficient algorithm for finding values of jumps of f knowing local cone beam data is proposed. The nature of artifacts inherent in cone beam local tomography is studied.
Purpose: Cone-beam CT (CBCT) plays an important role in image guided radiation therapy (IGRT). However, the large radiation dose from serial CBCT scans in most IGRT procedures raises a clinical concern, especially for pediatric patients who are essentially excluded from receiving IGRT for this reason. The goal of this work is to develop a fast GPU-based algorithm to reconstruct CBCT from undersampled and noisy projection data so as to lower the imaging dose. Methods: The CBCT is reconstructed by minimizing an energy functional consisting of a data fidelity term and a total variation regularization term. We developed a GPU-friendly version of the forward-backward splitting algorithm to solve this model. A multi-grid technique is also employed. Results: It is found that 20~40 x-ray projections are sufficient to reconstruct images with satisfactory quality for IGRT. The reconstruction time ranges from 77 to 130 sec on a NVIDIA Tesla C1060 GPU card, depending on the number of projections used, which is estimated about 100 times faster than similar iterative reconstruction approaches. Moreover, phantom studies indicate that our algorithm enables the CBCT to be reconstructed under a scanning protocol with as low as 0.1 mAs/projection. Comparing with currently widely used full-fan head and neck scanning protocol of ~360 projections with 0.4 mAs/projection, it is estimated that an overall 36~72 times dose reduction has been achieved in our fast CBCT reconstruction algorithm. Conclusions: This work indicates that the developed GPU-based CBCT reconstruction algorithm is capable of lowering imaging dose considerably. The high computation efficiency in this algorithm makes the iterative CBCT reconstruction approach applicable in real clinical environments.
An iterative refinement technique for the sparse approximate inverse factor Z of a symmetric matrix S, where ZZT≈S−1 is proposed. Its efficiency is demonstrated for the approximate inverse square root and Cholesky factorization of the overlap matrix S in the generalized eigenvalue problem occurring in linear scaling electronic structure theory using a localized, nonorthogonal basis set. A version of the refinement method useful for local perturbations of the overlap matrix is also proposed. In addition to its applications in electronic structure theory, the scheme may be useful for iterative refinement of preconditioners.
The problem of using a divergent fan beam convolution reconstruction algorithm in conjunction with a minimal complete (180° plus the fan angle) data set is reviewed. It is shown that by proper weighting of the initial data set, image quality essentially equivalent to the quality of reconstructions from 360° data sets is obtained. The constraints on the weights are that the sum of the two weights corresponding to the same line‐integral must equal one, in regions of no data the weights must equal zero, and the weights themselves as well as the gradient of the weights must be continuous over the full 360°. After weighting the initial data with weights that satisfy these constraints, image reconstruction can be conveniently achieved by using the standard (hardwired if available) convolver and backprojector of the specific scanner.
Application of the total variation (TV) minimization algorithm is applied to image reconstruction in few-view fan-beam computed tomography (CT). In particular, the impact of noise is investigated for different levels of tolerance on the data error. The artifacts due to noise are seen to vary dramatically in spatial frequency as the data error tolerance changes.
C-arm cone-beam CT could replace preoperative multi-detector CT scans in the cardiac interventional setting. However, cardiac gating results in view angle undersampling and the small size of the detector results in projection data truncation. These problems are incompatible with conventional tomographic reconstruction algorithms. In this paper, the prior image constrained compressed sensing (PICCS) reconstruction method was adapted to solve these issues. The performance of the proposed method was compared to that of FDK, FDK with extrapolated projection data (E-FDK), and total variation-based compressed sensing. A canine projection dataset acquired using a clinical C-arm imaging system supplied realistic cardiac motion and anatomy for this evaluation. Three different levels of truncation were simulated. The relative root mean squared error and the universal image quality index were used to quantify the reconstruction accuracy. Three main conclusions were reached. (1) The adapted version of the PICCS algorithm offered the highest image quality and reconstruction accuracy. (2) No meaningful variation in performance was observed when the amount of truncation was changed. (3) This study showed evidence that accurate interior tomography with an undersampled acquisition is possible for realistic objects if a prior image with minimal artifacts is available.
This work discusses the iterative refinement for finding the numerical inverse of a given non-singular matrix. The algorithm and convergence analysis of the iterative procedures are presented. Numerical examples are given to show the effectiveness of the iterative refinement.
Neural networks have some applications in computerized tomography, in particular to reconstruct an image from projections. The presented paper describes a new practical approach to the reconstruction problem using a Hopfield-type neural network. The methodology of this reconstruction algorithm resembles a transformation formula—the so-called ρ-filtered layergram method. The method proposed in this work is adapted for discrete fan beam projections, already used in practice. Performed computer simulations show that the neural network reconstruction algorithm designed to work in this way outperforms conventional methods in obtained image quality, and in perspective of hardware implementation in the speed of the reconstruction process.
Proposed is a theoretically exact formula for inversion of data obtained by a spiral computed tomography (CT) 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 backprojected in order to form an image. Another property is that the formula solves the so-called long object problem. Limitations of the algorithm are discussed. Results of numerical experiments are presented.
Truncated projections can arise from a detector with a limited field of view (LFOV). Truncation artifacts can be reduced through extrapolation methods; however the reconstructed images with extrapolation are often over-corrected or under-corrected. Recently, an iterative reconstruction-reprojection algorithm was developed, which incorporated extrapolation with iterative algorithm. It gave the possibility to better reduce truncation artifacts compared to using the extrapolation method alone. This article builds a theoretic foundation for the above iterative reconstruction-reprojection algorithm. The theoretic foundation is suitable to parallel-beam, fan-beam and cone-beam computed tomography(CT). Two assumptions are summarized from the CT system. Then, a truncation-artifact-free solution for the problem of LFOV is derived from these assumptions. The solution contains a "local inverse" of matrix. The local inverse of a matrix is defined using the sub-matrix and its general inverse. The solution can be approximately implemented as the iterative reconstruction-reprojection algorithm which is just the algorithm mentioned above.
This article considers the problem of reconstructing cone-beam computed tomography (CBCT) images from a set of undersampled and potentially noisy projection measurements.
The authors cast the reconstruction as a compressed sensing problem based on l1 norm minimization constrained by statistically weighted least-squares of CBCT projection data. For accurate modeling, the noise characteristics of the CBCT projection data are used to determine the relative importance of each projection measurement. To solve the compressed sensing problem, the authors employ a method minimizing total-variation norm, satisfying a prespecified level of measurement consistency using a first-order method developed by Nesterov.
The method converges fast to the optimal solution without excessive memory requirement, thanks to the method of iterative forward and back-projections. The performance of the proposed algorithm is demonstrated through a series of digital and experimental phantom studies. It is found a that high quality CBCT image can be reconstructed from undersampled and potentially noisy projection data by using the proposed method. Both sparse sampling and decreasing x-ray tube current (i.e., noisy projection data) lead to the reduction of radiation dose in CBCT imaging.
It is demonstrated that compressed sensing outperforms the traditional algorithm when dealing with sparse, and potentially noisy, CBCT projection views.
We clarify that the derivative-free fan-beam reconstruction formula [IEEE Trans. Image Process. 2, 543-547, 1993] only allows exact reconstruction of an object for a circular trajectory or at the origin of the coordinate system for a radially symmetric noncircular trajectory.
The exterior computed tomography (CT) problem is one kind of truncation problem. It is very ill-posed, so that accurate reconstruction of the attenuation function is hardly possible from real data. Based on projection onto convex sets (POCS) algorithm, total variation minimization (TVM) methods, and C-V model, we develop and investigate a new iterative reconstruction algorithm, which is referred to as subregion-averaged-TVM-POCS (SA-TVM-POCS). Numerical simulations are presented to illustrate the efficiency of the algorithm. The results of this paper can be easily applied to other x-ray CT reconstruction problems.
Cone-beam CT (CBCT) plays an important role in image guided radiation therapy (IGRT). However, the large radiation dose from serial CBCT scans in most IGRT procedures raises a clinical concern, especially for pediatric patients who are essentially excluded from receiving IGRT for this reason. The goal of this work is to develop a fast GPU-based algorithm to reconstruct CBCT from undersampled and noisy projection data so as to lower the imaging dose.
The CBCT is reconstructed by minimizing an energy functional consisting of a data fidelity term and a total variation regularization term. The authors developed a GPU-friendly version of the forward-backward splitting algorithm to solve this model. A multigrid technique is also employed.
It is found that 20-40 x-ray projections are sufficient to reconstruct images with satisfactory quality for IGRT. The reconstruction time ranges from 77 to 130 s on an NVIDIA Tesla C1060 (NVIDIA, Santa Clara, CA) GPU card, depending on the number of projections used, which is estimated about 100 times faster than similar iterative reconstruction approaches. Moreover, phantom studies indicate that the algorithm enables the CBCT to be reconstructed under a scanning protocol with as low as 0.1 mA s/projection. Comparing with currently widely used full-fan head and neck scanning protocol of approximately 360 projections with 0.4 mA s/projection, it is estimated that an overall 36-72 times dose reduction has been achieved in our fast CBCT reconstruction algorithm.
This work indicates that the developed GPU-based CBCT reconstruction algorithm is capable of lowering imaging dose considerably. The high computation efficiency in this algorithm makes the iterative CBCT reconstruction approach applicable in real clinical environments.
Analytic simulation in computed tomography(CT) generates projection data for evaluating and improving CT image reconstruction algorithms and has played an important role in the research and development of x-ray CT. The simulation is desired to be as realistic as possible while the computation needs to be efficient and accurate. Early primitive equation-based phantoms such as Shepp-Logan and FORBILD use only boxes, cylinders, and quadrics to simulate body parts. The superquadrics have been used in Computer Graphics since 1980's, and in CT since 1990's. While their more complex shapes make them more realistic in simulation, the difficulties in solving their equations increase dramatically, which restricts their use, especially in ray-tracing and x-ray transform. Zhu et al. developed an algorithm for ray-intersecting the superellipsoid and used it to build a thorax phantom. No algorithms for ray-intersecting the supertoroid, however, are known up to now to the best of our knowledge. In this paper we propose such an algorithm and use it in computation of x-ray transform. The algorithm was tested by a phantom consisting of the top portion of vertebrae and a few ribs. Cone-beam data were produced from the phantom, and then used to reconstruct the phantom.
To present the theory for image reconstruction of a high-pitch, high-temporal-resolution spiral scan mode for dual-source CT (DSCT) and evaluate its image quality and dose.
With the use of two x-ray sources and two data acquisition systems, spiral CT exams having a nominal temporal resolution per image of up to one-quarter of the gantry rotation time can be acquired using pitch values up to 3.2. The scan field of view (SFOV) for this mode, however, is limited to the SFOV of the second detector as a maximum, depending on the pitch. Spatial and low contrast resolution, image uniformity and noise, CT number accuracy and linearity, and radiation dose were assessed using the ACR CT accreditation phantom, a 30 cm diameter cylindrical water phantom or a 32 cm diameter cylindrical PMMA CTDI phantom. Slice sensitivity profiles (SSPs) were measured for different nominal slice thicknesses, and an anthropomorphic phantom was used to assess image artifacts. Results were compared between single-source scans at pitch = 1.0 and dual-source scans at pitch = 3.2. In addition, image quality and temporal resolution of an ECG-triggered version of the DSCT high-pitch spiral scan mode were evaluated with a moving coronary artery phantom, and radiation dose was assessed in comparison with other existing cardiac scan techniques.
No significant differences in quantitative measures of image quality were found between single-source scans at pitch = 1.0 and dual-source scans at pitch = 3.2 for spatial and low contrast resolution, CT number accuracy and linearity, SSPs, image uniformity, and noise. The pitch value (1.6 pitch 3.2) had only a minor impact on radiation dose and image noise when the effective tube current time product (mA s/pitch) was kept constant. However, while not severe, artifacts were found to be more prevalent for the dual-source pitch = 3.2 scan mode when structures varied markedly along the z axis, particularly for head scans. Images of the moving coronary artery phantom acquired with the ECG-triggered high-pitch scan mode were visually free from motion artifacts at heart rates of 60 and 70 bpm. However, image quality started to deteriorate for higher heart rates. At equivalent image quality, the ECG-triggered high-pitch scan mode demonstrated lower radiation dose than other cardiac scan techniques on the same DSCT equipment (25% and 60% dose reduction compared to ECG-triggered sequential step-and-shoot and ECG-gated spiral with x-ray pulsing).
A high-pitch (up to pitch = 3.2), high-temporal-resolution (up to 75 ms) dual-source CT scan mode produced equivalent image quality relative to single-source scans using a more typical pitch value (pitch = 1.0). The resultant reduction in the overall acquisition time may offer clinical advantage for cardiovascular, trauma, and pediatric CT applications. In addition, ECG-triggered high-pitch scanning may be useful as an alternative to ECG-triggered sequential scanning for patients with low to moderate heart rates up to 70 bpm, with the potential to scan the heart within one heart beat at reduced radiation dose.
This paper presents a novel data sufficiency condition that unique and stable ROI reconstruction can be achieved from a more flexible family of data sets. To the interior problem, it allows the ROI (Region-of-interest) can be reconstructed from the line integrals passing through this ROI and a small region B located anywhere as long as the image is known on B. Especially, ROI reconstruction can be achieved without any other a priori knowledge when region B is placed outside the object support. We also develop a general reconstruction algorithm with the DBP-POCS (Differentiated backprojection-projection onto convex sets) method. Finally, both numerical and real experiments were done to illustrate the new data sufficiency condition and the good stability of the ROI reconstruction algorithm.
While conventional wisdom is that the interior problem does not have a unique solution, by analytic continuation we recently showed that the interior problem can be uniquely and stably solved if we have a known sub-region inside a region of interest (ROI). However, such a known sub-region is not always readily available, and it is even impossible to find in some cases. Based on compressed sensing theory, here we prove that if an object under reconstruction is essentially piecewise constant, a local ROI can be exactly and stably reconstructed via the total variation minimization. Because many objects in computed tomography (CT) applications can be approximately modeled as piecewise constant, our approach is practically useful and suggests a new research direction for interior tomography. To illustrate the merits of our finding, we develop an iterative interior reconstruction algorithm that minimizes the total variation of a reconstructed image and evaluate the performance in numerical simulation.
Hybrid methods have been known for a long time as very efficient algorithms for attenuation correction in single-photon emission computed tomography, but only recently have efforts been made to formulate them with more rigorous mathematics. This has allowed us to explain their efficiency in terms of approximate inversion, and to establish a convergence condition. The present study focuses on the convergence problem and emphasizes the question of symmetry. Hybrid method operators are not symmetrical; therefore the convergence condition is not easily verified. New schemes based on a modified conjugate gradient method are presented. Convergence is proved and performances are shown to be at least as good as the standard hybrid schemes on perfect and noisy simulated data.
In this paper we report cone-beam CT techniques that permit reconstruction from width-truncated projections. These techniques are variants of Feldkamp's filtered backprojection algorithm and assume quasi-redundancy of ray integrals. Two methods are derived and compared. The first method involves the use of preconvolution weighting of the truncated data. The second technique performs post-convolution weighting preceded by non-zero estimation of the missing information. The algorithms were tested using the three-dimensional Shepp-Logan head phantom. The results indicate that given an appropriate amount of overscan, satisfactory reconstruction can be achieved. These techniques can be used to solve the problem of undersized detectors.
The purpose of this paper is to develop a method of eliminating CT image artifacts generated by objects extending outside the scan field of view, such as obese or inadequately positioned patients. CT projection data are measured only within the scan field of view and thus are abruptly discontinuous at the projection boundaries if the scanned object extends outside the scan field of view. This data discontinuity causes an artifact that consists of a bright peripheral band that obscures objects near the boundary of the scan field of view. An adaptive mathematical extrapolation scheme with low computational expense was applied to reduce the data discontinuity prior to convolution in a filtered backprojection reconstruction. Despite extended projection length, the convolution length was not increased and thus the reconstruction time was not affected. Raw projection data from ten patients whose bodies extended beyond the scan field of view were reconstructed using a conventional method and our extended reconstruction method. Limitations of the algorithm are investigated and extensions for further improvement are discussed. The images reconstructed by conventional filtered backprojection demonstrated peripheral bright-band artifacts near the boundary of the scan field of view. Images reconstructed with our technique were free of such artifacts and clearly showed the anatomy at the periphery of the scan field of view with correct attenuation values. We conclude that bright-band artifacts generated by obese patients whose bodies extend beyond the scan field of view were eliminated with our reconstruction method, which reduces boundary data discontinuity. The algorithm can be generalized to objects with inhomogeneous peripheral density and to true "Region of Interest Reconstruction" from truncated projections.
In computed tomography (CT), the beam hardening effect has been known to be one of the major sources of deterministic error that leads to inaccuracy and artifact in the reconstructed images. Because of the polychromatic nature of the x-ray source used in CT and the energy-dependent attenuation of most materials, Beer's law no longer holds. As a result, errors are present in the acquired line integrals or measurements of the attenuation coefficients of the scanned object. In the past, many studies have been conducted to combat image artifacts induced by beam hardening. In this paper, we present an iterative beam hardening correction approach for cone beam CT. An algorithm that utilizes a tilted parallel beam geometry is developed and subsequently employed to estimate the projection error and obtain an error estimation image, which is then subtracted from the initial reconstruction. A theoretical analysis is performed to investigate the accuracy of our methods. Phantom and animal experiments are conducted to demonstrate the effectiveness of our approach.
In many transmission imaging geometries, the transmitted "beams" of photons overlap on the detector, such that a detector element may record photons that originated in different sources or source locations and thus traversed different paths through the object. Examples include systems based on scanning line sources or on multiple parallel rod sources. The overlap of these beams has been disregarded by both conventional analytical reconstruction methods as well as by previous statistical reconstruction methods. We propose a new algorithm for statistical image reconstruction of attenuation maps that explicitly accounts for overlapping beams in transmission scans. The algorithm is guaranteed to monotonically increase the objective function at each iteration. The availability of this algorithm enables the possibility of deliberately increasing the beam overlap so as to increase count rates. Simulated single photon emission tomography transmission scans based on a multiple line source array demonstrate that the proposed method yields improved resolution/noise tradeoffs relative to "conventional" reconstruction algorithms, both statistical and nonstatistical.
To develop volumetric micro-CT fluoroscopy for small animal imaging, we have proposed a cone-beam system with multiple x-ray sources. In this paper, we extend Parker's single-source half-scan weighting scheme to the case of an odd number of x-ray sources that are equiangularly distributed, and apply it for half-scan Feldkamp-type reconstruction in this unique geometry. In the numerical simulation with the Shepp-Logan phantom, representative images indicate that the proposed half-scan Feldkamp-type algorithm produces temporal resolution significantly superior to that with a single x-ray source cone-beam system.
This paper describes a statistical image reconstruction method for X-ray computed tomography (CT) that is based on a physical model that accounts for the polyenergetic X-ray source spectrum and the measurement nonlinearities caused by energy-dependent attenuation. We assume that the object consists of a given number of nonoverlapping materials, such as soft tissue and bone. The attenuation coefficient of each voxel is the product of its unknown density and a known energy-dependent mass attenuation coefficient. We formulate a penalized-likelihood function for this polyenergetic model and develop an ordered-subsets iterative algorithm for estimating the unknown densities in each voxel. The algorithm monotonically decreases the cost function at each iteration when one subset is used. Applying this method to simulated X-ray CT measurements of objects containing both bone and soft tissue yields images with significantly reduced beam hardening artifacts.