Convolution Backprojection Research Articles

A statistical correction method for minimization of systemic artefact in a continuous-rotate X-ray based industrial CT system

The use of a linear detector array (LDA) in X-ray computed tomography (CT) imaging is well established. Generally the CT system using an LDA operates in a fan-beam configuration. In a non-medical tomography set-up, the X-ray source and detectors are stationary and the object is rotated for scanning. Equi-spaced angular projections over a complete 360° object rotation are required for CT image reconstruction using standard Convolution Back Projection (CBP) algorithm as applied to the fan-beam scanning geometry. If there is a lack of timing synchronization between the LDA data acquisition system and the rotary motion in a continuous-rotate system, the incremental angle for the acquired projections may not be known exactly. The error may cause artefacts and blurring in the CT image. The present paper describes a possible way of numerically tuning the acquired transmission data matrix for minimization of such artefacts. It is based on finding out statistically similar projections, which represent 0° and 360° angular views. The correction method has been developed to avoid modification in detector hardware and interface devices. The experimental CT system for industrial applications has been developed by making use of an independent scintillator-based linear detector array, commonly used for on-line radiography of low-density specimens. The proposed data pre-processing algorithm with some experimental results as well as limitations of the correction method is discussed.

Fourier transform moiré tomography for high-sensitivity mapping asymmetric 3-D temperature field

Fourier transform evaluation for moiré deflectogram is proposed to automatically map the temperature field. The moiré deflectogram is generated by conventional deflectometer and is analyzed by means of Fourier transform algorithm. The convolution backprojection algorithm is used for the optical tomography. Asymmetric 3-D gas temperature distribution for a given layer is reconstructed.

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.

Image Reconstruction from Incomplete Projection Data using Combined ART- CBP Algorithm

The present work is concerned with reconstructing cross-sections of composite structures when complete projection data is not available from the experiments. A novel approach combining the algebraic reconstruction technique (ART) and convolution back projection (CBP) algorithm has been developed for reconstruction with such data. The numerical results obtained reveal that a combination of ART and CBP is able to reconstruct cross-sections from incomplete data to an extent that is meaningful for non-destructive evaluation.

A mixed approach to artifacts minimization in a continuous-rotate X-ray based tomographic imaging system using linear detector array

Adaptation of a general-purpose digital radiographic scintillator-based linear detector array (LDA) in a continuous-rotate third generation X-ray industrial tomographic imaging system and the resulting artifacts are discussed. Sources of error in the projection data are identified and a mixed approach, involving neighborhood averaging and row-wise application of a low-pass filter for the minimization of artifacts in the reconstructed images using weighted convolution back projection algorithm is illustrated. It is observed that the recently launched LDA can easily be adapted for industrial tomographic imaging of low-density specimen in a cost effective way by incorporating the suggested artifact removal technique in the given configuration.

The measurement of 3-D asymmetric temperature field by using real time laser interferometric tomography

A real time nondestructive temperature measurement technique based on laser holographic interference tomography technique is presented. An He–Ne laser is used as light source, and a CCD video camera is used to grab the interferogram. This laser holographic tomography technique is applied to the measurement of the temperature fields generated by two heated rods. Since data error is inevitable in engineering measurement, it is necessary to study the reconstruction techniques for reconstructing the temperature field. Three techniques including convolution back projection (CBP), algebra reconstruction technique (ART) and simultaneous iterative reconstruction technique (SIRT) are studied. Based on the reconstruction techniques and experimental situation, ART is used to reconstruct the asymmetric temperature fields. The thermocouples are used to measure the temperatures of the two heated rods. Comparing the reconstructed result with the measured temperature value, a satisfactory result is obtained.

CBP and ART image reconstruction algorithms on media and DSP processors

Computed Tomography (CT) image reconstruction algorithms such as convolution back-projection (CBP) and algebraic reconstruction technique (ART) are highly compute-intensive for today's single processor systems. In this work, we investigate the suitability of TM-1000 media processor and Analog Device's ADSP 21160 as a compute engine for the execution of image reconstruction algorithms. Philips Trimedia TM-1000, a very large instruction word (VLIW) processor, is a high performance media processor optimized for real-time processing of audio, video, graphifcs, and communication data streams. TM-1000 has a high performance digital signal processor (DSP) core, supported by multiple functional units. The DSP core and the functional blocks operate in parallel, driven by a mix of RISC, multimedia, SIMD-type DSP and floating point instructions. A typical DSP such as Analog Device's ADSP 21160 is based on super harvard architecture (SHARC) and is an optimized processor for digital signal processing applications. It has two sets of computation units. Each computation unit comprises of three functional blocks namely, arithmetic and logic unit (ALU), multiplier and shifter unit. ADSP 21160 supports single instruction multiple data (SIMD) computation model to handle dual computation units. Both sets of computation units operate concurrently. We compare the performance of TM-1000 media processor and ADSP 21160 DSP processor to execute the image reconstruction algorithms by comparing the execution time of CBP and ART algorithms on them. The image reconstruction algorithms normally break down to a repetitive multiply-accumulate operation (MAC). All DSP processors support single-cycle MAC and zero-overhead loop instructions. The media processors normally do not support single-cycle MAC instruction and zero-overhead loop functionality. However, media processors are equipped with multiple functional units that perform multiple operations in a single instruction time. A DSP processor is expected to execute image reconstruction algorithms much faster than a multimedia processor. However, the experimental results show that the execution time on DSP and media processor are more or less same when 16-bit representation is used for data. When floating point data is used for implementation, DSP processor has an edge. ADSP 21160 gives same execution time for both floating point and 16-bit fixed point data. But, the execution time almost doubles when data is in floating point format on the media processor, compared to a 16-bit implementation. This can be attributed to the fact that TM-1000 processes two sets of operands in a single instruction time when data is in 16-bit format. The executable code for ADSP 21160 was generated from an optimized assembly language program whereas the executable code for TM-1000 was generated from an optimized ‘C’ with a few custom operations.

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.

Image reconstruction of simulated specimens using convolution back projection

This paper reports about the reconshuction of cross-sections of composite structures. The convolution back projection (CBP) algor~thm hns been used to capture the attenuation field over the specimen. Five different test cases have been taken up for evaluation. These cases represent varying degrees of complexity. In addition, the role of filters on the nature of the reconstruction errors has also been discussed. Numerical results obtained in the study reveal that CBP algorithm is a useful tool for qualitative as well as quantitative assessment of composite regions encountered in engineering applications.

Subfringe integration method for automatic analysis of moire deflection tomography

A new method of fringe analysis by subfringe integration al- gorithm (SIA) is proposed to analyze moire deflectograms and to map the phase object, in which the fringe phase is retrieved and phase object message is obtained automatically. Optical tomography is applied to the moire deflectometry for the measurement of asymmetric phase object. The convolution backprojection algorithm is used to obtain the 3-D tem- perature field. Theoretical analysis, experimental result and simulation calculations are presented. © 2000 Society of Photo-Optical Instrumentation Engi- neers. (S0091-3286(00)01310-6)

Region of Interest Reconstruction of Tomographic Images using Filter Banks

A new technique for ROI image reconstruction from projections using filter banks is reported in this paper. It is known that the convolution back projection operator, used for reconstruction of images from 1D-projections, has a non-local filter which requires global projection data. In X-ray tomography, the exposure time of the object to harmful radiation is thereby increased. Recently, it has been proved that this filtering operation can be done on a chosen wavelet function instead of the projections. This localizes the filter and leads to ROI image reconstruction with reduced projections. This concept was extended using 2D and 1D MRA filter banks, in which the filters are combined with the non-local CBP filter to get short length filters. In this paper it is proved that the presence of unwanted information in all previous algorithms is due to aliasing and a new technique is proposed for reconstruction without any aliasing. The proposed scheme is implemented using Shepp-Logan phantom head image and the performance is compared.

On intermediate view estimation in computed tomography

In many applications in computed tomography, practical limitations in data acquisition restrict the number of projections (views). The use of the standard convolution backprojection algorithm for reconstruction from an inadequate number of projections results in view aliasing artifacts. One approach to alleviating the effects of such artifacts consists of artificially increasing the number of views, by estimating a set of intermediate views. Two possible methods of estimating the intermediate views are interpolation and reprojection. In this paper, a study of the two is considered. Based on the merits and demerits of the two methods, a combination of the two methods is investigated. Specifically, a reconstruction from the available sinogram augmented by intermediate view reprojections, and the projections interpolated from the original views and the reprojections, provide an additional improvement with respect to view aliasing artifacts. The advantage of computing reprojections over smaller regions of interest is discussed. When the number of available projections is reasonably high but not adequate to produce an artifact-free reconstruction, estimating the intermediate views by interpolation provides an improvement without much additional degradation, at minimal computational cost.

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.

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.

Polar sampling in k-space: reconstruction effects.

Magnetic resonance images are most commonly computed by taking the inverse Fourier transform of the k-space data. This transformation can potentially create artifacts in the image, depending on the reconstruction algorithm used. For equally spaced radial and azimuthal k-space polar sampling, both gridding and convolution backprojection are applicable. However, these algorithms potentially can yield different resolution, signal-to-noise ratio, and aliasing characteristics in the reconstructed image. Here, these effects are analyzed and their tradeoffs are discussed. It is shown that, provided the modulation transfer function and the signal-to-noise ratio are considered together, these algorithms perform similarly. In contrast, their aliasing behavior is different, since their respective point spread functions (PSF) differ. In gridding, the PSF is composed of the mainlobe and ringlobes that lead to aliasing. Conversely, there are no ringlobes in the convolution backprojection PSF, thus radial aliasing effects are minimized. Also, a hybrid gridding and convolution backprojection reconstruction is presented for radially nonequidistant k-space polar sampling.

Development and characterization of a neutron tomography system based on image intensifier/CCD system

A neutron tomography system has been developed using a low cost electronic imaging system which uses commercially available Image intensifier, CCD, Frame grabber. This setup has been used for a series of experiments on neutron tomography at the 400 kW swimming pool type reactor APSARA which has a neutron flux of 10 6 n/cm 2/s at L/ D=90. The main feature of this tomography system is integrated computer controlled imaging and stepper motor control system with online data acquisition, analysis and display of images for each projection angle. The density profile of the projection images and ray sum calculation can be carried out in offline or online manner at any slice and a series of reconstructed image for any slice can be obtained. An algorithm for reconstruction using Convolution Back Projection (CBP) technique has been developed and implemented on a Pentium system. The imaging system is modular in design. The reconstructed image has shown good quality and holes of less than 0.6 mm filled with wax can be revealed in a 60 mm aluminum matrix or 1 mm holes in 40 mm solid brass matrix have been resolved.

Faster tomographic fan-beam back-projection using Cartesian axes pre-projection

The convolution back-projection algorithm is the most common method for reconstructing images from complete sets of fan-beam projections. For each pixel and for every projection, the interception point on the detector array must be determined and a weighted value from the appropriate point on the filtered back projection added. Thus the number of operations required is of order n 2 p, where n is the number of points per projection, and p the number of projections. This can mean a considerable computation time, even with modern, fast computer workstations. The complexity of each pixel operation (weighting and geometric computations) is reduced if the projection is first pre-projected onto one or other of the Cartesian axes. This has been demonstrated to reduce the computational time by a factor of 2, with no loss of accuracy, when compared with a highly optimised implementation of the conventional fan-beam back-projection algorithm.

Direct‐Fourier reconstruction in tomography and synthetic aperture radar

We investigate the use of direct-Fourier (DF) image reconstruction in computed tomography and synthetic aperture radar (SAR). One of our aims is to determine why the convolution-backprojection (CBP) method is favored over DF methods in tomography, while DF methods are virtually always used in SAR. We show that the CBP algorithm is equivalent to DF reconstruction using a Jacobian-weighted two-dimensional periodic sinc-kernel interpolator. This interpolation is not optimal in any sense, which suggests that DF algorithms using optimal interpolators may surpass CBP in image quality. We consider use of two types of DF interpolation: a windowed sinc kernel, and the least-squares optimal Yen interpolator. Simulations show that reconstructions using the Yen interpolator do not possess the expected visual quality, because of regularization needed to preserve numerical stability. Next, we show that with a concentric-squares sampling scheme, DF interpolation can be performed accurately and efficiently, producing imagery that is superior to that obtainable by other algorithms. In the case of SAR, we show that the DF method performs very well with interpolators of low complexity. We also study DF reconstruction in SAR for trapezoidal grids. We conclude that the success of the DF method in SAR imaging is due to the nearly Cartesian shape of the sampling grid. © 1998 John Wiley & Sons, Inc. Int J Imaging Syst Technol, 9, 1–13, 1998

Direct-Fourier reconstruction in tomography and synthetic aperture radar

We investigate the use of direct-Fourier (DF) image reconstruction in computed tomography and synthetic aperture radar (SAR). One of our aims is to determine why the convolution-backprojection (CBP) method is favored over DF methods in tomography, while DF methods are virtually always used in SAR. We show that the CBP algorithm is equivalent to DF reconstruction using a Jacobian-weighted two-dimensional periodic sinc-kernel interpolator. This interpolation is not optimal in any sense, which suggests that DF algorithms using optimal interpolators may surpass CBP in image quality. We consider use of two types of DF interpolation: a windowed sinc kernel, and the least-squares optimal Yen interpolator. Simulations show that reconstructions using the Yen interpolator do not possess the expected visual quality, because of regularization needed to preserve numerical stability. Next, we show that with a concentric-squares sampling scheme, DF interpolation can be performed accurately and efficiently, producing imagery that is superior to that obtainable by other algorithms. In the case of SAR, we show that the DF method performs very well with interpolators of low complexity. We also study DF reconstruction in SAR for trapezoidal grids. We conclude that the success of the DF method in SAR imaging is due to the nearly Cartesian shape of the sampling grid. © 1998 John Wiley & Sons, Inc. Int J Imaging Syst Technol, 9, 1–13, 1998

Hamming Signature of Three Composite Specimens

Abstract Three samples having composite structure are examined by an X-ray tomographic scanner for pixel sizes of 20 to 160 μm. The data are processed by the convolution backprojection algorithm and the resulting tomographic images are analyzed for the Hamming class of filters. The characteristic signature of the three specimens is interpreted with the help of two error theorems.