Convolution Backprojection Method Research Articles

Improvement of the longitudinal phase space tomography at the J-PARC synchrotrons

The longitudinal phase space tomography, which reconstructs the phase space distribution from the one-dimensional bunch profiles, is used in various accelerators to measure longitudinal beam parameters. At the J-PARC, an implementation of the phase space tomography based on the Convolution Back Projection method (CBP) has been used to measure the momentum spread of the injected beam. The method assumes that the beam distribution rotates without significant deformation during the synchrotron oscillation. Because of the nonlinearity of synchrotron motion with sinusoidal RF voltage, the method can be used only in limited situations such as small amplitude synchrotron oscillation. Algebraic Reconstruction Techniques (ART) in conjunction with particle tracking, which is implemented in the CERN’s tomography code, allows accurate reconstructions even for nonlinear large amplitude synchrotron oscillations. We present the overview of the application of the CERN’s tomography code to the J-PARC synchrotrons. The results of benchmarking are also reported.

Image Reconstruction Methods for Ultrasonic Transmission Mode Tomography in Bubbly Flow Regime

Image reconstruction from projections plays an important role in monitoring flow regimes by ultrasonic transmission mode tomography (UTMT) system. Fast and more accurate methods are necessary in case of on-line process e.g. bubbly flow regimes. In this work, analytical image reconstruction methods such as linear back projection (LBP), filter back projection (FBP) and convolution back projection (CBP) in bubbly flow regime is investigated and found that CBP is superior to other methods. Furthermore, different filters were applied to CBP to investigate the image quality improvement. Among different types of filters for CBP method, Ram-lack outperforms the others for UTMT. The peak signal to noise ratio (PSNR) of reconstructed images in this particular experiment was improved using Ram-lack in noiseless data.

Key parameter estimation for radar rotating object imaging with multi-aspect observations

Rotating object model is commonly used for imaging analysis in high resolution radars such as the inverse synthetic aperture radar (ISAR). For a rotating object, it is known that multi-aspect observations can improve cross-range resolution with the known imaging geometry. For the non-cooperative rotating object with unknown imaging geometry, this paper proposes an integrated scheme to estimate the key parameters, e.g., the rotating velocity and the aspect angle difference between every two observations. Furthermore, convolution back-projection (CBP) method is applied to provide fused imaging result with improved resolution. Also, the accuracy of the ultimate parameter estimation is analyzed, which is strongly related with several important factors like position extraction error of scattering centers and so on. Finally, the results of numerical experiments are provided to demonstrate the effectiveness of the proposed method.

Local volume reconstruction from width-truncated cone-beam projections by convolution backprojection

Cone-beam computed tomography can reproduce a digital volume for an object that can be completely put inside the scan field of view (SFOV). In practice, the detector may not be wide enough to receive the projection, or the incident beam aperture may be confined to part of the object being scanning. Such cases cause width-truncated projections. With a dataset of width-truncated cone-beam projections, we can reconstruct a local volume in the object domain by using a convolution backprojection method. For the width-truncated projections, when the projection width is wider than the kernel length, we find that there exists, at the center of the object domain, a small region that can be tomographically reconstructed just as well as if there had been no truncation. Given the reconstruction kernel length, we propose a simple boundary extension technique to augment the SFOV, i.e., padding the truncated portions beyond the boundary. A continuous extrapolation beyond the boundary can effectively reduce the Gibbs effect, thereby enabling the reconstruction of a subvolume encompassing the SFOV. We demonstrate convolution-backprojection-based local cone-beam tomography by a breast phantom experiment, where the cone-beam projection is truncated by adjusting the x-ray collimator.

Inversion formulas for the spherical Radon transform and the generalized cosine transform

The k-dimensional totally geodesic Radon transform on the unit sphere Sn and the corresponding cosine transform can be regarded as members of the analytic family of intertwining fractional integrals Rαf(ξ)=γn,k(α)∫Snf(x)sind(x,ξ)α+k−ndx,d(x,ξ) being the geodesic distance between x∈Sn and the k-geodesic ξ. We develop a unified approach to the inversion of Rαf for all α⩾0,1⩽k⩽n−1,n⩾2. The cases of smooth f and f∈Lp are considered. A series of new inversion formulas is obtained. The convolution–backprojection method is developed.

Multicriterion cross-entropy minimization approach to positron emission tomographic imaging.

A multicriterion cross-entropy minimization approach to positron emission tomographic (PET) imaging is described. An unexplored multicriterion cross-entropy optimization algorithm based on weighted-sum scalarization is used to solve this problem. The efficacy of the algorithm is compared with that of the single-criterion optimization algorithm and the convolution backprojection method for image reconstruction from computer-generated projection data and Siemens PET scanner data. The algorithms described have been implemented on a PIII/686 microcomputer.

A Fast and Accurate Multilevel Inversion of the Radon Transform

A number of imaging technologies reconstruct an image function from its Radon projection using the convolution backprojection method. The convolution is an O(N2 log N) algorithm, where the image consists of N X N pixels, while the backprojection is an O(N3 ) algorithm, thus constituting the major computational burden of the convolution backprojection method. An O(N2 log N) multilevel backprojection method is presented here. When implemented with a Fourier-domain postprocessing technique, also presented here, the resulting image quality is similar or superior to the image quality of the classical backprojection technique.

Inverse radon transform with one-dimensional wavelet transform

In this paper, the wavelet inverse formula of Radon transform is obtained with one-dimensional wavelet. The convolution back-projection method of Radon transform is derived from this inverse formula. An asymptotic relation between wavelet inverse formula of Radon transform and convolution-back projection algorithm of Radon transform in 2 dimensions is established.

Fast direct Fourier methods, based on one- and two-pass coordinate transformations, yield accurate reconstructions of x-ray CT clinical images

The conversion from polar to Cartesian coordinates can be carried out with two-pass algorithms. The paper describes two different methods based on concentric square frames and octagonal frames and their results, obtained with accurate interpolations based on the `moving window Shannon reconstruction' (MWSR). The embedding of these algorithms in direct Fourier methods (DFMs) of tomographic reconstruction is discussed. With respect to one-pass methods and to the use of octagonal frames, the square frame method makes it possible to carry out the first pass, a radial resampling, in the direct space, before computing 1D Fourier transforms (FTs) of projections. Reconstructions of clinical images from the raw data of a third-generation x-ray tomograph are presented and compared with those obtained with one-pass DFMs and with the convolution back-projection method (CBPM) performed by the instrument. The simple algorithm using square frames yields results in complete agreement with other DFM protocols and the CBPM. On a general-purpose computer, the execution of DFM protocols based on one-pass and two-pass coordinate transformations is 35 to 55 times faster than the CBPM and make the algorithms attractive for modern instrumentation.

Multilevel algebraic reconstruction technique for x-ray computed tomography

The performance of an iterative image reconstruction technique, the Algebraic Reconstruction Technique (ART), was improved to be basically a one-iteration noniterative technique producing even better image quality for x-rayCT. In this thesis, the factors affecting the performance of ART were first analyzed. Then a novel projection access order, the multilevel scheme (MLS) for ART was proposed. MLS is exactly the sequence of 1D FFT if the number of projections is a power of 2. Experimental tests using real CT data demonstrated that the new MLS-ART technique outperforms the conventional Convolution Backprojection (CBP) method, by producing higher spatial resolution when the number of projections is taken satisfying the sampling criterion and smaller noise when the number of projections is relatively small. A computer simulation study which matches the real CT dosage and noise conditions further quantifies that MLS produces a larger MTF when the number of projections is taken above half of that required by the sampling criterion and a larger SNR when the number of projections is taken below that half. It also improves the performance of ART itself, in both the computational speed (by more than 10 times) and the physical image quality (both the high and low contrast detectabilities). This work provides a thorough physical comparison among different CTreconstruction techniques and it confirms that MLS ART will find applications in reconstructions of different CT models, especially in situations where the projection data are limited.

A study of errors due to reconstruction filters in computed tomography

In the summation convolution backprojection method of image reconstruction in computed tomography, the final image accuracy depends on the convolution filter used. Filters are designed to attenuate high spatial frequencies when noisy projection data are used. This paper explores the differences between the images reconstructed using a range of filters, and compares the results with the case of the ramp filter that provides the “best” image for ideal, noise-free, projection data. It is shown that systematic errors between these images and the best image exist, and that these errors are related to the second differential of the reconstruction filter with respect to spatial frequency. This error determination may be used to correct computed tomography images that have been reconstructed using inappropriate filters, and this theory is tested using noise-free projection data from two computer simulated images. It is shown that the corrected images are far closer to the original images.

A Study of Errors due to Reconstruction Filters in Computed Tomography

Abstract In the summation convolution backprojection method of image reconstruction in computed tomography, the final image accuracy depends on the convolution filter used. Filters are designed to attenuate high spatial frequencies when noisy projection data are used. This paper explores the differences between the images reconstructed using a range of filters, and compares the results with the case of the ramp filter that provides the “best” image for ideal, noise-free, projection data. It is shown that systematic errors between these images and the best image exist, and that these errors are related to the second differential of the reconstruction filter with respect to spatial frequency. This error determination may be used to correct computed tomography images that have been reconstructed using inappropriate filters, and this theory is tested using noise-free projection data from two computer simulated images. It is shown that the corrected images are far closer to the original images.

An investigation of the theoretical error in tomographic images

The non-destructive technique of computed tomography (CT) has been successfully used for imaging cross-sections of a variety of specimens using scanners of differing designs and spatial resolutions. The image reconstruction algorithm used in this study is the convolution backprojection method (CBP) with a variety of different filter functions. The reconstructed images show the effect of different filters as expected with decreasing high spatial frequency content as the smoothing is increased. This decrease in fine detail is quantified in terms of an error value, and this is compared with the predictions of an error formula. Reasonable agreement between the experimental results using a variety of specimens and this error formula is found for a number of filters.

Investigation of smoothing techniques for positron emission tomography imaging

Recently, many techniques have been introduced to improve on the reconstruction and enhancement of biomedical images, in particular, positron emission tomography. Often these techniques either perform poorly in the presence of noise or are computationally expensive. Nonintensive computational techniques are presented that aim to reduce noise while preserving edges working in conjunction with the filtered convolution back-projection method of reconstruction. First we present an adaptive windowing technique that bases window size on the count value. Then a maximum likelihood technique is developed. Finally, a method of image enhancement based on directional operators and template matching attempts to diminish noise while preserving edges. These techniques are applied both in the image domain and in the sinogram domain and then evaluated using a similarity measure against the original Shepp-Logan phantom.

Design of Optimal Statistical Filter for the Discrete Convolution Backprojection Method

SYNOPTIC ABSTRACTA statistically optimal convolving function (filter) is derived for discrete implementation of the convolution backprojection method. The method of derivation simultaneously takes into account (i) “the process” generating the object images, (ii) the data collection geometry, (iii) the discretization scheme used, (iv) various interpolations used, and (v) the data noise. The proposed filter minimizes the expected weighted sum of squared errors in pixel reconstructions. The performance of this optimal filter, in reconstructing discretized images by convolution backprojection method, is compared with that of some of the commonly used filters.

Sinogram recovery with the method of convex projections for limited-data reconstruction in computed tomography

Image reconstruction from incomplete projection data is strongly required in widespread applications of computed tomography. This problem can be formulated as a sinogram–recovery problem. The sinogram–recovery problem is to find a complete sinogram that is compatible with the Helgason–Ludwig consistency condition, the measured incomplete sinogram, and other a priori knowledge about the sinogram in question. The direct use of the Helgason–Ludwig consistency condition considerably reduces computational requirements and the accumulation of digital-processing errors over the conventional iterative reconstruction–reprojection method. Most research for solving the sinogram–recovery problem is based on directly inverting systems of linear equations associated with the Helgason–Ludwig consistency condition. However, these noniterative techniques cannot be applied to various different types of limited-data situations in a unified way. Moreover, nonlinear a priori constraints such as the nonnegativity and the amplitude limit are not easily incorporated. We solve the sinogram–recovery problem by using an iterative signal-recovery technique known as the method of projection onto convex sets. Once an estimation of the complete sinogram is obtained, the conventional convolution–backprojection method can be utilized to reconstruct an image. The performance of the proposed method is investigated both with numerical phantoms and with actual x-ray data.

Retrieval of Temperature and Concentration Profiles From Absorption Measurements with Error

Abstract The widely used methods for obtaining the inverse of the Radon transform, the underlying reconstruction problem in the determination of temperature and species concentralion profiles from absorption measurements, have been developed for medical imaging appticaitons and are not well-suited for limited data set problems. The application of a new method, that is especially suited for limited data problems involving smooth distributions, to this retrieval problem has been previously reported. Performance of the new method with error containing absorption data is studied with regard to the stability and accuracy of the retrieved temperature and concentration profiles. Test results demonstrating the superiority of this new method over the convolution backprojection method, a widely used benchmark for the discrete data set inverse Radon transform problem, are presented.

Improved iterative image reconstruction with automatic noise artifact suppression

A method for stabilizing iterative image reconstruction techniques has been developed for improving the image quality of position emission tomography. A damping matrix is introduced, which suppresses noisy correction on a pixel-by-pixel basis, depending on the statistical precision of the iterative correction. The precision is evaluated by comparing a certain number of correction submatrices, each of which is formed from a subset of the projection data. Simulation studies showed that statistical noise is effectively suppressed, while the image of the source object is reconstructed with high resolution, as long as the signal level is higher than the local noise level. In the application to the MLE (maximum likelihood estimator), the minimum RMS error of the image was reduced to 84% for 500 k total counts, and the RMS error increased more slowly with further iterations as compared with the simple MLE. The method was also applied to the FIR (filtered iterative reconstruction) algorithm, and the images were found to be better than those obtained by the convolution backprojection method.

Backprojection by upsampled Fourier series expansion and interpolated FFT

A fast backprojection method through the use of interpolated fast Fourier transform (FFT) is presented. The computerized tomography (CT) reconstruction by the convolution backprojection (CBP) method has produced precise images. However, the backprojection part of the conventional CBP method is not very efficient. The authors propose an alternative approach to interpolating and backprojecting the convolved projections onto the image frame. First, the upsampled Fourier series expansion of the convolved projection is calculated. Then, using a Gaussian function, it is projected by the aliasing-free interpolation of FFT bins onto a rectangular grid in the frequency domain. The total amount of computation in this procedure for a 512x512 image is 1/5 of the conventional backprojection method with linear interpolation. This technique also allows the arbitrary control of the frequency characteristics.

Sinogram recovery with the method of convex projections for limited-data reconstruction in computed tomography

Image reconstruction from incomplete projection data is strongly required in widespread applications of computed tomography. This problem can be formulated as a sinogram–recovery problem. The sinogram–recovery problem is to find a complete sinogram that is compatible with the Helgason–Ludwig consistency condition, the measured incomplete sinogram, and other a priori knowledge about the sinogram in question. The direct use of the Helgason–Ludwig consistency condition considerably reduces computational requirements and the accumulation of digital-processing errors over the conventional iterative reconstruction–reprojection method. Most research for solving the sinogram–recovery problem is based on directly inverting systems of linear equations associated with the Helgason–Ludwig consistency condition. However, these noniterative techniques cannot be applied to various different types of limited-data situations in a unified way. Moreover, nonlinear a priori constraints such as the nonnegativity and the amplitude limit are not easily incorporated. We solve the sinogram–recovery problem by using an iterative signal-recovery technique known as the method of projection onto convex sets. Once an estimation of the complete sinogram is obtained, the conventional convolution–backprojection method can be utilized to reconstruct an image. The performance of the proposed method is investigated both with numerical phantoms and with actual x-ray data.