Image Reconstruction Formula Research Articles

A new weighting scheme for arc based circle cone-beam CT reconstruction.

In this paper, we present an arc based fan-beam computed tomography (CT) reconstruction algorithm by applying Katsevich's helical CT image reconstruction formula to 2D fan-beam CT scanning data. Specifically, we propose a new weighting function to deal with the redundant data. Our weighting function ϖ(x_,λ) is an average of two characteristic functions, where each characteristic function indicates whether the projection data of the scanning angle contributes to the intensity of the pixel x_. In fact, for every pixel x_, our method uses the projection data of two scanning angle intervals to reconstruct its intensity, where one interval contains the starting angle and another contains the end angle. Each interval corresponds to a characteristic function. By extending the fan-beam algorithm to the circle cone-beam geometry, we also obtain a new circle cone-beam CT reconstruction algorithm. To verify the effectiveness of our method, the simulated experiments are performed for 2D fan-beam geometry with straight line detectors and 3D circle cone-beam geometry with flat-plan detectors, where the simulated sinograms are generated by the open-source software "ASTRA toolbox." We compare our method with the other existing algorithms. Our experimental results show that our new method yields the lowest root-mean-square-error (RMSE) and the highest structural-similarity (SSIM) for both reconstructed 2D and 3D fan-beam CT images.

Compensation of shear waves in photoacoustic tomography with layered acoustic media

An image reconstruction formula is presented for photoacoustic computed tomography that accounts for conversion between longitudinal and shear waves in a planar-layered acoustic medium. We assume the optical absorber that produces the photoacoustic wave field is embedded in a single fluid layer and any elastic solid layers present are separated by one or more fluid layers. The measurement aperture is assumed to be planar. Computer simulation studies are conducted to demonstrate and investigate the proposed reconstruction formula.

Image reconstruction in photoacoustic tomography involving layered acoustic media

Photoacoustic tomography (PAT), also known as thermoacoustic or optoacoustic tomography, is a rapidly emerging biomedical imaging technique that combines optical image contrast with ultrasound detection principles. Most existing reconstruction algorithms for PAT assume the object of interest possesses homogeneous acoustic properties. The images produced by such algorithms can contain significant distortions and artifacts when the object's acoustic properties are spatially variant. In this work, we establish an image reconstruction formula for PAT applications in which a planar detection surface is employed and the to-be-imaged optical absorber is embedded in a known planar layered acoustic medium. The reconstruction formula is exact in a mathematical sense and accounts for multiple acoustic reflections between the layers of the medium. Computer-simulation studies are conducted to demonstrate and investigate the proposed method.

GPR Migration Imaging Algorithm Based on NUFFT

Stolt migration is one of the useful GPR imaging algorithms, which has the ad- vantage of fast implementation speed. The fast Fourier transform of integral formula in image reconstruction is realized by coordinate transformation and done through interpolation. Inter- polation determines the efiectiveness of imaging directly. In the practical applications, zero-flll interpolation in time-domain is used to improve accuracy, but which will increase the compu- tational complexity. In this paper, non-uniform fast Fourier transform (NUFFT) algorithm is introduced to realize the fast computing of integral formula in image reconstruction directly. The simulation results show that this method achieve good results. Stolt migration is one of the useful GPR imaging algorithms which has the advantage of fast implementation speed. The date, which is uniformly distribute in frequency-wavenumber domain, is transformed from frequency-wavenumber domain to wavenumber-wavenumber domain. Then using interpolation method to make sure that the date is uniformly distribute in wavenumber- wavenumber domain. So it will be possibility to use the FFT, which lead e-ciencies, to calculate the date. That is the key method of Stolt migration method. Usually, the Linear interpolation and sinc interpolation (2,3) is used. Interpolation determines the efiectiveness of imaging directly, in the practical applications, zero-flll interpolation in time domain is used to improve accuracy but which will increase the computational complexity. This paper analyzed the root causes of why could not use FFT to calculate the formula for image reconstruction before stolt interpolation. Then introduced the algorithms of computing Fourier transform with unequally spaced data in any of the domain (time or frequency) have been proposed. These algorithms are known as Non-Uniform FFT (NUFFT), NUFFT was flrst proposed by Dutt and Rokhlin in 1993 (4), Beylkin; Nguyen (5) and Liu; Fessler (6) and Sutton (7) are made great contributions to its development. In this paper, NUFFT is introduced to realize the fast computing of formula in image reconstruc- tion directly. Both the stolt interpolation and the flnal range inverse FFT have been substituted by a single NUFFT. This paper describes the detail application of the NUFFT's to the stolt migration algorithm. Compared the new method with the old ones in calculation e-ciency and image result. The performance is illustrated with numerical simulations. The simulation of B-scan data of a two-layered medium is generated by the Gprmax2.0 software. The simulation results show that this method reduce the complexity of the interpolation while ensure the operational speed. at the same time, it use least-square to approximate the function so the energy of the image result is more concentrated. This method achieves good results.

Image reconstruction for fan-beam differential phase contrast computed tomography

Recently, x-ray differential phase contrast computed tomography (DPC-CT) has been experimentally implemented using a conventional tube combined with gratings. Images were reconstructed using a parallel-beam reconstruction formula. However, parallel-beam reconstruction formulae are not applicable when the parallel-beam approximation fails. In this paper, we present a new image reconstruction formula for fan-beam DPC-CT. There are several novel features of the new image reconstruction formula: (i) when the scanning angular range of data acquisition is larger than π + γm (γm is the full fan angle), the entire field of view can be exactly reconstructed; (ii) when the scanning angular range is smaller than π + γm, a local region of interest (ROI) can be exactly reconstructed; (iii) it enables an exact reconstruction for a local ROI when the projection data are truncated at some view angles; (iv) it enlarges the imaging field of view when the detector is asymmetrically placed. In this last case, the data are truncated from every view angle. Numerical simulations have been conducted to validate the new reconstruction formula.

A local region of interest image reconstruction via filtered backprojection for fan-beam differential phase-contrast computed tomography

Recently, x-ray differential phase contrast computed tomography (DPC-CT) has been experimentally implemented using a conventional source combined with several gratings. Images were reconstructed using a parallel-beam reconstruction formula. However, parallel-beam reconstruction formulae are not directly applicable for a large image object where the parallel-beam approximation fails. In this note, we present a new image reconstruction formula for fan-beam DPC-CT. There are two major features in this algorithm: (1) it enables the reconstruction of a local region of interest (ROI) using data acquired from an angular interval shorter than 180°+ fan angle and (2) it still preserves the filtered backprojection structure. Numerical simulations have been conducted to validate the image reconstruction algorithm.

TH‐D‐L100J‐02: Exact Image Reconstruction in Reverse Helical Cone‐Beam CT for Radiation Therapy

Purpose: To propose an exact image reconstruction algorithm based on Pack‐Noo formula for a reverse helical cone‐beam CT (CBCT) which is potentially a useful scanning geometry for image‐acquisition in image‐guided radiation therapy. Method and Materials: Last year, we proposed a novel scanning geometry named reverse helical trajectory for a CBCT imager mounted on a linear accelerator (LINAC) treatment system, and applied a chord‐based backprojection filtration (BPF) algorithm for exact image reconstruction. It was shown that there is a middle gap in the reconstructed image which is due to lack of chords passing through the gap. The gap was removed when we used an additional long line segment connecting the end points of the revere helices. However, larger patient dose and longer scan time due to additional line scan were pointed out as drawbacks of the proposed method. In this work, we applied Pack‐Noo formula, which does not depend on the existence of chords for reconstructing the exact image, to propose an exact image reconstruction algorithm for a reverse helical trajectory. Cone‐beam data requirement for the proposed algorithm was exploited to address the concept of reduced cone‐beam scan. A numerical study using 3D Shepp‐Logan phantom was performed. Results: We showed the proposed algorithm can reconstruct the image without any gap in the middle from the cone‐beam data acquired by a reverse helical scan. Additional line scan, which is necessary for chord‐based BPF algorithms to remove the middle gap, is not needed for exact image reconstruction with the proposed algorithm. Conclusion: A new algorithm based on Pack‐Noo formula for exact image reconstruction in reverse helical cone‐beam CT was proposed. Without any additional line scan which is necessary for chord‐based BPF algorithms to reconstruct the complete image, the proposed algorithm successfully reconstructed the image.

Development and evaluation of an exact fan‐beam reconstruction algorithm using an equal weighting scheme via locally compensated filtered backprojection (LCFBP)

A novel exact fan-beam image reconstruction formula is presented and validated using both phantom data and clinical data. This algorithm takes the form of the standard ramp filtered backprojection (FBP) algorithm plus local compensation terms. This algorithm will be referred to as a locally compensated filtered backprojection (LCFBP). An equal weighting scheme is utilized in this algorithm in order to properly account for redundantly measured projection data. The algorithm has the desirable property of maintaining a mathematically exact result for: the full scan mode (2pi), the short scan mode (pi+ full fan angle), and the supershort scan mode [less than (pi+ full fan angle)]. Another desirable feature of this algorithm is that it is derivative-free. This feature is beneficial in preserving the spatial resolution of the reconstructed images. The third feature is that an equal weighting scheme has been utilized in the algorithm, thus the new algorithm has better noise properties than the standard filtered backprojection image reconstruction with a smooth weighting function. Both phantom data and clinical data were utilized to validate the algorithm and demonstrate the superior noise properties of the new algorithm.

Theory and algorithms for image reconstruction on chords and within regions of interest

We introduce a formula for image reconstruction on a chord of a general source trajectory. We subsequently develop three algorithms for exact image reconstruction on a chord from data acquired with the general trajectory. Interestingly, two of the developed algorithms can accommodate data containing transverse truncations. The widely used helical trajectory and other trajectories discussed in literature can be interpreted as special cases of the general trajectory, and the developed theory and algorithms are thus directly applicable to reconstructing images exactly from data acquired with these trajectories. For instance, chords on a helical trajectory are equivalent to the n-PI-line segments. In this situation, the proposed algorithms become the algorithms that we proposed previously for image reconstruction on PI-line segments. We have performed preliminary numerical studies, which include the study on image reconstruction on chords of two-circle trajectory, which is nonsmooth, and on n-PI lines of a helical trajectory, which is smooth. Quantitative results of these studies verify and demonstrate the proposed theory and algorithms.

A general exact reconstruction for cone-beam CT via backprojection-filtration

In this paper, we prove a generalized backprojection-filtration formula for exact cone-beam image reconstruction with an arbitrary scanning locus. Our proof is independent of the shape of the scanning locus, as long as the object is contained in a region where there is a chord through any interior point. As special cases, this generalized formula can be applied with cone-beam scanning along nonstandard spiral and saddle curves, as well as in an n-PI window setting. The algorithmic implementation and numerical results are described to support the correctness of our general claim.

Filtered backprojection formula for exact image reconstruction from cone‐beam data along a general scanning curve

Recently, Katsevich proved a filtered backprojection formula for exact image reconstruction from cone-beam data along a helical scanning locus, which is an important breakthrough since 1991 when the spiral cone-beam scanning mode was proposed. In this paper, we prove a generalized Katsevich's formula for exact image reconstruction from cone-beam data collected along a rather flexible curve. We will also give a general condition on filtering directions. Based on this condition, we suggest a natural choice of filtering directions, which is more convenient than Katsevich's choice and can be applied to general scanning curves. In the derivation, we use analytical techniques instead of geometric arguments. As a result, we do not need the uniqueness of the PI lines. In fact, our formula can be used to reconstruct images on any chord as long as a scanning curve runs from one endpoint of the chord to the other endpoint. This can be considered as a generalization of Orlov's classical theorem. Specifically, our formula can be applied to (i) nonstandard spirals of variable radii and pitches (with PI- or n-PI-windows), and (ii) saddlelike curves.

Image reconstruction on PI-lines by use of filtered backprojection in helical cone-beam CT

Recently, we have derived a general formula for image reconstruction from helical cone-beam projections. Based upon this formula, we have also developed an exact algorithm for image reconstruction on PI-line segments from minimum data within the Tam–Danielsson window. This previous algorithm can be referred to as a backprojection-filtration algorithm because it reconstructs an image by first backprojection of the data derivatives and then filtration of the backprojections on PI-line segments. In this work, we propose an alternative algorithm, which reconstructs an image by first filtering the modified data along the cone-beam projections of the PI-lines onto the detector plane and then backprojecting the filtered data onto PI-line segments. Therefore, we refer to this alternative algorithm as the filtered-backprojection algorithm. A preliminary computer-simulation study was performed for validating and demonstrating this new algorithm. Furthermore, we derive a practically useful expression to accurately compute the derivative of the data function for image reconstruction. The proposed filtered-backprojection algorithm can reconstruct the image within any selected ROI inside the helix and thus can handle naturally the long object problem and the super-short scan problem. It can also be generalized to reconstruct images from data acquired with other scanning configurations such as the helical scan with a varying pitch.

Calibration of a synthetic aperture radiometer

Calibration algorithms for a synthetic aperture microwave radiometer are presented. The calibration is geared to Earth remote sensing applications and is demonstrated on an airborne prototype thinned array imager. Two approaches to the system calibration are presented. The first utilizes commonly available reference brightness temperature scenes, such as open water, and the second utilizes data collected on the antenna range. Both algorithms yield spatial response information which is cast in matrix form and inverted to obtain the image reconstruction formula. Experimental results are examined, and errors in some reconstructed images are linked to the present prototype antenna design. Algorithms for improving the synthesized antenna pattern sidelobe performance are also presented. In one solution, the pattern efficiency is optimized by minimizing the pattern outside a defined beam. In another solution, the patterns are matched to a desired model pattern by the method of least squared errors. Both techniques offer an attractive alternative to aperture weighting.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Cone beam tomography with circular, elliptical and spiral orbits

A general formula for image reconstruction from cone beam data is described. Applying this formula to various cone beam geometries results in a class of filtered backprojection algorithms. This formula is known to lead to exact reconstructions in cases in which the cone vertices form certain unbounded curves. An example of such a curve is an infinite straight line. In the case where the curve is a circle, this formula leads to the well-known Feldkamp algorithm, for which the reconstructions are only approximations to the true image. The authors apply this general formula to the cases where the curve is an ellipse and a sprial, and new algorithms are derived. The properties of these algorithms are investigated through studies of the point spread function and reconstructions of computer generated phantom data.

Fast Magnetic Resonance Imaging with Simultaneously Oscillating and Rotating Fiell Gradients

We present an approach to magnetic resonance imaging employing a magnetic field gradient that rotates 180 degrees in the image plane while the gradient magnitude oscillates rapidly during the rotation. A single free induction decay recorded during this rotation contains all the information needed to reconstruct a two-dimensional image. In effect, each sinusoidal oscillation of the gradient provides information corresponding to one projection in more conventional Fourier-projection approaches. Since the data acquisition can be achieved in a period less than T2, the method offers the potential of great speed, which is limited only by the gradient modulation frequency. An explicit image reconstruction formula is derived that gives, when evaluated, a reconstruction of the magnetization equal to the true magnetization convolved with a space-invariant point spread function. This point spread function is derived and characterizes the resolving power and sidelobe response of the technique. Moreover, we derive a similar reconstruction formula which is valid when known inhomogeneities in the static field H0 and T2 are present. Finally, we show how the general approach can be extended to three dimensions.

A New Approach to the Fan Beam Geometry Image Reconstruction for a CT Scanner

A new derivation of the fan beam geometry image reconstruction formula for CT scanners has been made in this paper. Moreover there are two points discussed here in order to develop fast algorithms based upon this formula. First is to establish a principle of filterered back-projection algorithm for fan beam geometry which enables us to use FFT algorithm. Second is to develop a fast algorithm for fan back-projection process. With these algorithms, the fan beam image reconstruction time to compute can be made comparable with the filtered back-projection algorithm for the parallel beam geometry image reconstruction.