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This paper presents a new type of filtered backprojection (FBP) algorithm for fan-beam full- and partial-scans. The filtering is shift-invariant with respect to the angular variable. The backprojection does not include position-dependent weights through the Hilbert transform and the one-dimensional transformation between the fan- and parallel-beam coordinates. The strong symmetry of the filtered projections directly leads to an exact reconstruction for partial data. The use of the Hilbert transform avoids the approximation introduced by the nonuniform cutoff frequency required in the ramp filter-based FBP algorithm. Variance analysis indicates that the algorithm might lead to a better uniformity of resolution and noise in the reconstructed image. Numerical simulations are provided to evaluate the algorithm with noise-free and noisy projections. Our simulation results indicate that the algorithm does have better stability over the ramp-filter-based FBP and circular harmonic reconstruction algorithms. This may help improve the image quality for in place computed tomography scanners with single-row detectors.

Fan-beam collimators with spatially varying focal lengths are used in single photon emission computed tomography (SPECT) to improve the imaging sensitivity and to reduce the reconstruction artifacts resulting from the truncation of projection data. An angular representation of the detector coordinates for the projection data is adopted to investigate several aspects of the reconstruction problem for this type of collimation. A rebinning reconstruction algorithm is derived. We prove a conjecture posed by Zeng and Gullberg (1994) and obtain a simplified form of their backprojection filtering algorithm. Some computer simulations are presented to investigate the performance of the rebinning algorithm.

Recently derivative-backporjection image reconstruction algorithms have received significant attention in X-ray CT and in nuclear medicine imaging because of the ability to reconstruct truncated projection data. Specifically for uniformly attenuated SPECT data, algorithms using the derivative-backprojection method have been developed. This paper extends the development to that of a derivative-backprojection algorithm for non-uniformly attenuated SPECT data. It turns out that the image reconstruction algorithm can be reduced to that similar to the situation of a uniform attenuator. This has potential application for special situations in cardiac and brain SPECT. One advantage of this newly developed algorithm is its potential application in local tomography with truncated SPECT data.

In recent years the development of cone-beam reconstruction algorithms has been an active research area in x-ray computed tomography (CT), and significant progress has been made in the advancement of algorithms. Theoretically exact and computationally efficient analytical algorithms can be found in the literature. However, in single photon emission computed tomography (SPECT), published cone-beam reconstruction algorithms are either approximate or involve iterative methods. The SPECT reconstruction problem is more complicated due to degradations in the imaging detection process, one of which is the effect of attenuation of gamma ray photons. Attenuation should be compensated for to obtain quantitative results. In this paper, an analytical reconstruction algorithm for uniformly attenuated cone-beam projection data is presented for SPECT imaging. The algorithm adopts the DBH method, a procedure consisting of differentiation and backprojection followed by a finite inverse cosh-weighted Hilbert transform. The significance of the proposed approach is that a selected region of interest can be reconstructed even with a detector with a reduced field of view. The algorithm is designed for a general trajectory. However, to validate the algorithm, a numerical study was performed using a helical trajectory. The implementation is efficient and the simulation result is promising.

The expectation maximization (EM) algorithm for the maximum likelihood (ML) image reconstruction criterion generates severe checkerboard artifacts in the presence of noise. A classical remedy is to impose an a priori constraint for a penalized ML or maximum a posteriori probability solution. The penalty reduces the checkerboard artifacts and also introduces uncertainty because a priori information is usually unknown in clinic. Recent theoretical investigation reveals that the noise can be divided into two components: one is called null-space noise and the other is range-space noise. The null-space noise can be numerically estimated using filtered backprojection (FBP) algorithm. By the FBP algorithm, the null-space noise annihilates in the reconstruction while the range-space noise propagates into the reconstructed image. The aim of this work is to investigate the relation between the null-space noise and the checkerboard artifacts in the ML-EM reconstruction from noisy projection data. Our study suggests that removing the null-space noise from the projection data could improve the signal-to-noise ratio of the projection data and, therefore, reduce the checkerboard artifacts in the ML-EM reconstructed images. This study reveals an in-depth understanding of the different noise propagations in analytical and iterative image reconstructions, which may be useful to single photon emission computed tomography, where the noise has been a major factor for image degradation. The reduction of the ML-EM checkerboard artifacts by removing the null-space noise avoids the uncertainty of using a priori penalty.

It is well known that in fan-beam X-ray computed tomography (CT) projection data acquired over 2pi contain redundant information and that accurate and stable reconstruction can be obtained from less than 2pi data. In single photon emission computed tomography (SPECT), since the photon absorption is distance-dependent, the conjugate projections are not identical even in the absence of noise. However, projection data over 2pi are still redundant for a noiseless image reconstruction except that the redundancy is not as apparent as in X-ray CT. We study the redundant information in the fan-beam SPECT projection data in this work and developed a reconstruction algorithm only requiring data measured over a short scan. Experiment with computer-simulated data is performed to establish the validity and efficiency of the solution. Besides the explicit inversion formula, the most significant property of the algorithm is that truncations are allowed in the projection data along the detector at each view angle, which makes the reconstruction of region of interest (ROI) feasible in some particular geometries.

The estimation of rigid motion parameters has been investigated using Fourier transform and iterative techniques for over two decades. But the numerical efficiency remains a challenge. Especially for 3D case, there is no analytical algorithm. In this work, we propose to use the moment covariance matrix to estimate the orthogonal transform (i.e. rigid motion in real world). This algorithm does not depend on the dimension of the problem and is an analytical method. The numerical implementation is easy and efficient. The test results show that the proposed algorithm is accurate and well tolerent to noise variation.

A filtered backprojection (FBP) reconstruction algorithm for attenuated fan-beam projections has been derived based on Novikov’s inversion formula. The derivation uses a common transformation between parallel-beam and fan-beam coordinates. The filtering is shift-invariant. Numerical evaluation of the FBP algorithm is presented as well. As a special application, we also present a shift-invariant FBP algorithm for fan-beam SPECT reconstruction with uniform attenuation compensation. Several other fan-beam reconstruction algorithms are also discussed. In the attenuation-free case, our algorithm reduces to the conventional fan-beam FBP reconstruction algorithm.