Cone-beam Projections Research Articles

Improved two-dimensional rebinning of helical cone-beam computerized tomography datausing John’s equation

This paper deals with three-dimensional (3D) image reconstruction for helical computerizedtomography with multi-row detectors. We describe a method to improve the accuracy ofthe rebinning algorithms, which separate 3D reconstruction into independent 2Dreconstructions for a set of oblique slices. Each oblique slice is reconstructed from anestimate of its 2D Radon transform obtained from the cone-beam projections acquiredwhile the x-ray source moves along a segment of the helix. Because this helixsegment is not contained within the oblique slice, the estimated 2D Radon transformis approximate. In theory, exact rebinning could be achieved by solving John’spartial differential equation to virtually move the x-ray source within the obliqueslice. In contrast with previous work by Patch (2002 IEEE Trans. Med. Imaging 21801–13), we do not attempt to solve John’s equation exactly. Instead, we use John’sequation to compute a first order correction to the rebinning algorithm. Testswith simulated data demonstrate a significant improvement of image quality,obtained with a negligible increase of the computation time and of the sensitivity tonoise.

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.

Analytical image reconstruction of cone-beam projections from limited-angle Compton camera data

Since the integration of gamma-rays on a cone is measured with Compton cameras, some sort of image reconstruction method is necessary. Parra (2000) developed an analytical reconstruction algorithm based on a spherical harmonics expansion of projection data that covers the entire scattering-angle range. The measurable scattering angle range is limited due to the electrical noise of the detector and to the finite detector configuration. In this article, a reconstruction algorithm from the projection of a limited scattering angle range is presented. The algorithm was based on an inversion of the summed backprojection. The variance of the reconstructed image of a uniform source was calculated and found proportional to the third power of the number of expansion terms. Thus, window functions were introduced to suppress noise at the cost of spatial resolution. The dependency of the spatial resolution on the number of expansion terms was analyzed. From this, the dependency of the variance on the spatial resolution was found to be variance/spl prop/(spatial resolution)/sup -3/.

General surface reconstruction for cone-beam multislice spiral computed tomography.

A new family of cone-beam reconstruction algorithm, the General Surface Reconstruction (GSR), is proposed and formulated in this paper for multislice spiral computed tomography (CT) reconstructions. It provides a general framework to allow the reconstruction of planar or nonplanar surfaces on a set of rebinned short-scan parallel beam projection data. An iterative surface formation method is proposed as an example to show the possibility to form nonplanar reconstruction surfaces to minimize the adverse effect between the collected cone-beam projection data and the reconstruction surfaces. The improvement in accuracy of the nonplanar surfaces over planar surfaces in the two-dimensional approximate cone-beam reconstructions is mathematically proved and demonstrated using numerical simulations. The proposed GSR algorithm is evaluated by the computer simulation of cone-beam spiral scanning geometry and various mathematical phantoms. The results demonstrate that the GSR algorithm generates much better image quality compared to conventional multislice reconstruction algorithms. For a table speed up to 100 mm per rotation, GSR demonstrates good image quality for both the low-contrast ball phantom and thorax phantom. All other performance parameters are comparable to the single-slice 180 degrees LI (linear interpolation) algorithm, which is considered the "gold standard." GSR also achieves high computing efficiency and good temporal resolution, making it an attractive alternative for the reconstruction of next generation multislice spiral CT data.

Po-topic III-02: Three-dimensional cone-beam reconstruction for the micro computed tomography breast imager

The breast biopsy is a final examination for classifying breast cancer. However, the specimen in breast biopsy can only be viewed in 2D mode, so the diagnostic accuracy of the breast biopsy sometimes is limited. The 3D cone-beam micro computed tomography (microCT) breast imager produces tomographic slices of the specimen. Those images can be reformatted for 3D display, and then 3D computer-aided diagnosis can be performed in conjunction with clinical diagnosis for classifying breast cancer. The objective of this study is to implement and evaluate a 3D cone-beam reconstruction algorithm for the microCT breast imager. We obtained the cone-beam microCT projections of the specimen from electron bombarded charged coupled device (EBCCD). The 3D reconstruction from 2D projections, which are acquired by EBCCD based microCT, are then done by using a modified FDK algorithm. We also propose an alternative approach based on a cone-beam to parallel-beam rebinning step, a corresponding rebinning step into a rectangular virtual detector plane and a filtered backprojection. Based on this idea, we reconstructed cone-beam projections by two different algorithms, the parallel-beam reconstruction and the fan-beam reconstruction. The regions of interest in the reconstructed images were obtained from a uniform cubic phantom. In the parallel-beam reconstruction, the contrast (C) is 1.72, the mean (M) is 0.7239, the standard deviation (SD) is 0.1176, and the coefficient of variation (CV) is 0.1731. In the fan-beam method, the C is 1.99, the M is 0.6987, the SD is 0.1176, and the CV is 0.1683. We found that the C and CV of the images reconstructed by the fan-beam reconstruction are better than those of images reconstructed by the parallel-beam reconstruction. We used the Student’s t-tests with confidence interval of 95% for contrast and CV to compare these two methods, and we found that there is a meaningful difference between the two kinds of algorithms. Thus, we expect the images reconstructed by the 3D cone-beam reconstruction will also be better than those of the fan-beam reconstruction.

Flat panel detector‐based cone beam computed tomography with a circle‐plus‐two‐arcs data acquisition orbit: Preliminary phantom study

Cone beam computed tomography (CBCT) has been investigated in the past two decades due to its potential advantages over a fan beam CT. These advantages include (a) great improvement in data acquisition efficiency, spatial resolution, and spatial resolution uniformity, (b) substantially better utilization of x-ray photons generated by the x-ray tube compared to a fan beam CT, and (c) significant advancement in clinical three-dimensional (3D) CT applications. However, most studies of CBCT in the past are focused on cone beam data acquisition theories and reconstruction algorithms. The recent development of x-ray flat panel detectors (FPD) has made CBCT imaging feasible and practical. This paper reports a newly built flat panel detector-based CBCT prototype scanner and presents the results of the preliminary evaluation of the prototype through a phantom study. The prototype consisted of an x-ray tube, a flat panel detector, a GE 8800 CT gantry, a patient table and a computer system. The prototype was constructed by modifying a GE 8800 CT gantry such that both a single-circle cone beam acquisition orbit and a circle-plus-two-arcs orbit can be achieved. With a circle-plus-two-arcs orbit, a complete set of cone beam projection data can be obtained, consisting of a set of circle projections and a set of arc projections. Using the prototype scanner, the set of circle projections were acquired by rotating the x-ray tube and the FPD together on the gantry, and the set of arc projections were obtained by tilting the gantry while the x-ray tube and detector were at the 12 and 6 o'clock positions, respectively. A filtered backprojection exact cone beam reconstruction algorithm based on a circle-plus-two-arcs orbit was used for cone beam reconstruction from both the circle and arc projections. The system was first characterized in terms of the linearity and dynamic range of the detector. Then the uniformity, spatial resolution and low contrast resolution were assessed using different phantoms mainly in the central plane of the cone beam reconstruction. Finally, the reconstruction accuracy of using the circle-plus-two-arcs orbit and its related filtered backprojection cone beam volume CT reconstruction algorithm was evaluated with a specially designed disk phantom. The results obtained using the new cone beam acquisition orbit and the related reconstruction algorithm were compared to those obtained using a single-circle cone beam geometry and Feldkamp's algorithm in terms of reconstruction accuracy. The results of the study demonstrate that the circle-plus-two-arcs cone beam orbit is achievable in practice. Also, the reconstruction accuracy of cone beam reconstruction is significantly improved with the circle-plus-two-arcs orbit and its related exact CB-FPB algorithm, as compared to using a single circle cone beam orbit and Feldkamp's algorithm.

Scanning of logs with linear cone-beam tomography

This paper reports an ongoing feasibility study in industrial X-ray tomography. Full volume scanning of logs may have a large potential in terms of increased sales value. On the other hand, there is no technology commercially available today which can meet the requirements for speed. In conventional medical 3D-tomography, the source-detector system rotates around a slowly translating patient. We do not believe such an arrangement is able to reach our speed requirements. Instead, we have been simulating an arrangement with two fixed source-detector systems rotated 90° relative to each other. Each system consists of an X-ray source and a 2D detector. The logs are to be translated through this arrangement at relatively high speed (2–3 m/s) lengthwise on a conveyor belt, while cone-beam projections are acquired by each of the source-detector systems. The 2D-detectors are composed of a number of 1D-detectors side by side. For cost reasons we will try to limit the number of rows to nine, forming a 2D-array of 256×9 detector elements. Due to the unconventional scanning geometry, we have developed a new reconstruction algorithm, which also takes special care to avoid many artifacts due to sparse and/or missing data. Experiments (still simulated), performed on voxelized CT data, indicate that the knots are reconstructed with sufficient accuracy to allow for quantitative optimization. Due to the missing data, however, heartwood can barely be distinguished from sapwood, but his fact does not seem to compromise the subsequent knot detection. Note that this paper only describes the reconstruction part of the simulated experiments. To be complete, the feasibility study should also include segmentation, which is outside the scope of this paper.

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.

Modelling tomographic cone-beam projection data froma polyhedral phantom

Analytical phantoms are used to generate projectiondata for testing reconstruction accuracy in computed axialtomography. A circular source locus (equivalent to rotatingspecimen with a fixed source) provides insufficient data for`exact' reconstruction in cone-beam transmission tomography, thusphantom data are useful for studying the consequent errors andalso for investigating alternative scanning loci andreconstruction techniques. We present an algorithm that cancompute phantom cone-beam projection data from a phantomcomprising geometrically defined polyhedra. Each polyhedron isdefined as a set of polygons enclosing a volume of fixed linearattenuation coefficient. The algorithm works by projecting eachpolygon in turn onto the modelled detector array, whichaccumulates the product of source to polygon intersection distance(for the rays intersecting each detector element), linearattenuation coefficient and sign of projected polygon area(indicating whether rays enter or exit the polyhedron at thisface). The phantom data are rotated according to the projectionangle, whilst the source location and detector plane remain fixed. Polyhedra can be of simple geometric form, or complexsurfaces derived from 3D images of real specimens. This algorithmis illustrated using a phantom comprising 989 238 polygons,representing an iso-surface generated from a microtomographicreconstruction of a piece of walrus tusk.

An accurate iterative reconstruction algorithm for sparse objects: application to 3D blood vessel reconstruction from a limited number of projections

Based on the duality of nonlinear programming, this paper proposes an accurate row-action type iterative algorithm which is appropriate to reconstruct sparse objects from a limited number of projections. The cost function we use is the Lp norm with p ≈ 1.1. This norm allows us to pick up a sparse solution from a set of feasible solutions to the measurement equation. Furthermore, since it is both strictly convex and differentiable, we can use the duality of nonlinear programming to construct a row-action type iterative algorithm to find a solution. We also impose the bound constraint on pixel values to pick up a better solution. We demonstrate that this method works well in three-dimensional blood vessel reconstruction from a limited number of cone beam projections.

Analysis of an exact inversion algorithm for spiral cone-beam CT

In this paper we continue studying a theoretically exact filtered backprojection inversion formula for cone beam spiral CT proposed earlier by the author. Our results show that if the phantom f is constant along the axial direction, the formula is equivalent to the 2D Radon transform inversion. Also, the inversion formula remains exact as spiral pitch goes to zero and in the limit becomes again the 2D Radon transform inversion formula. Finally, we show that according to the formula the processed cone beam projections should be backprojected using both the inverse distance squared law and the inverse distance law.

Filtering point spread function in backprojection cone-beam CT and its applications in long object imaging

In backprojection cone-beam CT the cone-beam projection images are first filtered, then 3D backprojected into the object space. In this paper the point spread function (PSF) for the filtering operation is studied. For the cases where the normalization matrix is a constant, ie all integration planes intersect the scan path the same number of times, the derivation of the PSF is extended to the general case of limited angular range for the Radon line integrals. It is found that the 2D component of the PSF can be reduced to the form of space-variant 1D Hilbert transforms. The application of the PSF to a number of aspects in long object imaging will be discussed.

Cone-beam iterative reconstruction of a segment of a long object

This study was performed to investigate the iterative reconstruction of a segment of a long object using cone-beam projections acquired with a practical-sized detector. This reconstruction problem is commonly called the cone-beam long object problem. The cone-beam focal-point trajectories studied were either helical or helical with planar arcs. We assume that the long object is confined in a tall supporting cylinder and that the cone-beam detector is wide enough that the projections are not truncated transversely. If the detector is not wide enough, we use two asymmetric cone-beam detectors, each covering at least one-half of the field of view in the transverse direction. The detector size in the axial direction is not large enough to cover the segment. The detector size is used to determine the maximal helical pitch. A data sufficiency condition is established to design a focal-point trajectory. The trajectory then defines a convex hull. Projection rays that touch the region outside the convex hull and inside the supporting cylinder are discarded in the iterative algorithm. Computer simulations and experimental data verify that the proposed data acquisition and discarding strategy result in accurate segment reconstruction.

Theoretically Exact Filtered Backprojection-Type Inversion Algorithm for Spiral CT

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.

Cone-beam image reconstruction using spherical harmonics

Image reconstruction from cone-beam projections is required for both x-raycomputed tomography (CT) and single photon emission computed tomography(SPECT). Grangeat's algorithm accurately performs cone-beam reconstructionprovided that Tuy's data sufficiency condition is satisfied and projectionsare complete. The algorithm consists of three stages:(a) Forming weighted plane integrals by calculating the line integralson the cone-beam detector, and obtaining the first derivative of the planeintegrals (3D Radon transform) by taking the derivative of the weighted planeintegrals.(b) Rebinning the data and calculating the second derivative withrespect to the normal to the plane.(c) Reconstructing the image using the 3D Radon backprojection.A new method for implementing the first stage of Grangeat'salgorithm was developed using spherical harmonics. The method assumes that thedetector is large enough to image the whole object without truncation.Computer simulations show that if the trajectory of the cone vertex satisfiesTuy's data sufficiency condition, the proposed algorithm provides an exactreconstruction.

Image quality in extended arc filtered digital tomosynthesis.

To study image quality in filtered digital tomosynthesis (FDTS) tomograms as a function of their reconstruction arc, using isocentrically acquired, fluoroscopic projection data. Both digital tomosynthesis (DTS) and cone beam CT (CBCT) reconstruction algorithms are based on backprojection and use cone beam projection data as input. Under limited angle conditions, CBCT is reduced to FDTS, where only a subset of projection data are used for reconstruction. The effect of the reconstruction arc on the spatial resolution, slice thickness, contrast sensitivity, shape distortion and artifacts, was also experimentally studied. The investigation was performed using both simulated and actual fluoroscopic images. Image quality in terms of spatial resolution, slice thickness, shape distortion and artifacts, improved with increasing reconstruction arc and was optimized at 180 degrees, while contrast continued to improve as the arc was increased to 360 degrees. However, DTS was determined to be the technique of choice when reconstruction arcs of less than 40 degrees were used. Consequently, FDTS may be successfully implemented in applications involving extended arc reconstructions, in the range between 40 degrees delimiting the DTS domain and 360 degrees corresponding to CBCT.

IMAGE QUALITY IN EXTENDED ARC FILTERED DIGITAL TOMOSYNTHESIS

AbstractPurpose: To study image quality in filtered digital tomosynthesis (FDTS) tomograms as a function of their reconstruction arc, using isocentrically acquired, fluoroscopic projection data.Material and Methods: Both digital tomosynthesis (DTS) and cone beam CT (CBCT) reconstruction algorithms are based on backprojection and use cone beam projection data as input. Under limited angle conditions, CBCT is reduced to FDTS, where only a subset of projection data are used for reconstruction. The effect of the reconstruction arc on the spatial resolution, slice thickness, contrast sensitivity, shape distortion and artifacts, was also experimentally studied. The investigation was performed using both simulated and actual fluoroscopic images.Results and Conclusion: Image quality in terms of spatial resolution, slice thickness, shape distortion and artifacts, improved with increasing reconstruction arc and was optimized at 180°, while contrast continued to improve as the arc was increased to 360°. However, DTS...

Cone-beam reconstruction by backprojection and filtering.

A new analytical method for tomographic image reconstruction from cone-beam projections acquired on the source orbits lying on a cylinder is presented. By application of a weighted cone-beam backprojection, the reconstruction problem is reduced to an image-restoration problem characterized by a shift-variant point-spread function that is given analytically. Assuming that the source is relatively far from the imaged object, a formula for an approximate shift-invariant inverse filter is derived; the filter is presented in the Fourier domain. Results of numerical experiments with circular and helical orbits are considered.

Reconstruction of cone-beam projections from Compton scattered data

The problem of reconstructing a 3D source distribution from Compton scattered data can be separated into two tasks. First, the angular distribution of line projections at different observation points within the detector volume are reconstructed. Then, reconstruction techniques are applied to the resulting cone-beam projections to synthesize the 3D source distribution. This paper describes an analytic solution for the first, yet unsolved, task. Building on the convolution theorem in spherical coordinates, a back-projection and inverse filtering technique in terms of spherical harmonics is formulated, The rotation invariance of the point response of the back-projection in spherical coordinates is proved; and the corresponding inverse filter function is derived. The resulting filtered back-projection algorithm then consists of a summation over all detected events of fixed and known event response functions. Measurement errors, which for Compton scatter detectors are typically different for each detected event, can easily be accounted for in the proposed algorithm. The computational cost of the algorithm is O(NT/sup 2/), where N is the number of detected events and /spl pi//T is the desired angular resolution.

A solution to the long-object problem in helical cone-beam tomography

This paper presents a new algorithm for the long-object problem in helical cone-beam (CB) computerized tomography (CT). This problem consists in reconstructing a region-of-interest (ROI) bounded by two given transaxial slices, using axially truncated CB projections corresponding to a helix segment long enough to cover the ROI, but not long enough to cover the whole axial extent of the object. The new algorithm is based on a previously published method, referred to as CB-FBP (Kudo et al 1998 Phys. Med. Biol. 43 2885-909), which is suitable for quasi-exact reconstruction when the helix extends well beyond the support of the object. We first show that the CB-FBP algorithm simplifies dramatically, and furthermore constitutes a solution to the long-object problem, when the object under study has line integrals which vanish along all PI-lines. (A PI-line is a line which connects two points of the helix separated by less than one pitch.) Exploiting a geometric property of the helix, we then show how the image can be expressed as the sum of two images, where the first image can be reconstructed from the measured CB projections by a simple backprojection procedure, and the second image has zero PI-line integrals and hence can be reconstructed using the simplified CB-FBP algorithm. The resulting method is a quasi-exact solution to the long-object problem, called the ZB method. We present its implementation and illustrate its performance using simulated CB data of the 3D Shepp phantom and of a more challenging head-like phantom.