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Purpose: A large number of false positives (FPs) generated by computer-aided detection (CAD) schemes is likely to distract radiologists’ attention and decrease their interpretation efficiency. This study aims to develop projection-based features which characterize true and false positives to increase the specificity while maintaining high sensitivity in detecting colonic polyps. Methods: In this study, two-dimensional projection images are obtained from each initial polyp candidate or volume of interest, and features are extracted from both the gray and color projection images to differentiate FPs from true positives. These projection features were tested to exclude different types of FPs, such as haustral folds, rectal tubes, and residue stool using a database of 325 patient studies (from two different institutions), which includes 556 scans at supine and∕or prone positions with 347 polyps and masses sized from 5 to 60 mm. For comparison, several well-established features were used to generate a baseline reference. The experimental evaluation was conducted for large polyps (≥10 mm) and medium-sized polyps (5–9 mm) separately. Results: For large polyps, the additional usage of the projection features reduces the FP rate from 5.31 to 1.92 per scan at the comparable by-polyp sensitivity level of 93.1%. For medium-sized polyps, the FP rate is reduced from 8.89 to 5.23 at the sensitivity level of 80.6%. The percentages of FP reduction are 63.9% and 41.2% for the large and medium-sized polyps, respectively, without sacrificing detection sensitivity. Conclusions: The results have demonstrated that the new projection features can effectively reduce the FPs and increase the detection specificity without sacrificing the sensitivity. CAD of colonic polyps is supposed to help radiologists to improve their performance in interpreting computed tomographic colonography images.

We confirm that the arguments by Natterer (2001 Inverse Problems 17 113–9) can be extended to the attenuated Radon transform with complex-valued coefficients and readily lead to the range condition derived by Novikov (2002 Ark. Math. 40 145–67). We give a simple example to numerically verify the inversion formula in the case of constant imaginary attenuation. Several applications of using imaginary attenuation are given as well.

Recently, Noo and coworkers discovered an explicit inversion formula for the finite Hilbert transform, which is very important to accurate reconstruction from truncated projections. This letter presents two formulae for the finite inverse Hilbert transform using some elementary complex variable analysis. The new formulae do not contain the constant C and the singular endpoints that exist in the formula in Noo et al (2004 Phys. Med. Biol. 49 3903–23).

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.

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 which annihilates during filtered backprojection (FBP)-type analytical image reconstruction. The other is called range-space noise which propagates into the FBP-type analytically reconstructed image. In particular, the null-space noise can be numerically estimated. The aim of this work is to investigate the relation between the null-space noise and the checkerboard artifacts in the ML-EM image reconstruction from noise projection data. It is expected that removing the null-space noise from the projection data could improve the signal-to-noise ratio of the data and, therefore, reduce the checkerboard artifacts in the ML-EM reconstructed images. The expectation was realized by computer simulation studies with application 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.

This paper investigated an accurate reconstruction method to invert the attenuated Radon transform in non-parallel beam (NPB) geometries. The reconstruction method contains three major steps: (1) performing one-dimensional phase-shift rebinning, (2) implementing nonuniform Hilbert transform, and (3) applying Novikov’s explicit inversion formula. The method seemed to be adaptive to different settings of fan-beam geometry from very long to very short focal lengths without sacrificing reconstruction accuracy. Compared to the conventional bilinear rebinning technique, the presented method showed a better spatial resolution, as measured by modulation transfer function. Numerical experiments demonstrated its computational efficiency and stability to different levels of Poisson noise. Even with complicated geometries such as varying focal-length and asymmetrical fan-beam collimations, the presented method achieved nearly the same reconstruction quality of parallel-beam geometry. This work can facilitate quantitative reconstruction of single photon emission computed tomography for cardiac imaging, which may need NPB collimation geometries and require high computational efficiency.

In this work, we first introduced a reorganized form of the Novikov's inversion formula for the attenuated Radon transform with parallel-beam geometry which utilizes the conventional filters (such as the Shepp-Logan, Hanning, etc.) to simplify image reconstruction procedures. Then we developed an accurate reconstruction method for non-parallel beam geometries. The method contains three major steps: (1) one-dimensional rebinning using phase-shifting technique, (2) performing non-uniform Hilbert transform, and (3) applying the reorganized Novikov's formula. Numerical evaluations demonstrated its computational efficiency and sound stability to different levels of Poisson noise. It also seemed to be very robust to different settings of fan-beam geometry from very long to very short focal lengths without affecting the reconstruction accuracy.

Spatial resolution of SPECT imaging is mainly limited by the geometry of collimator holes. A long collimator hole with a small diameter improves the spatial resolution at the cost of low detection efficiency. With a finite size for the holes, the resolution changes as a function of the distance from the collimator surface. Lewitt et al. (1988) developed an approximate restoration algorithm to correct for the distance-dependent collimator blurring by the Fourier transform of sinogram. Van Elmbt et al. (1993) presented an analytical formula assuming an approximated resolution variation function. In this work, an accurate restoration procedure is investigated for a fan integral projection data based on the collimator hole geometry. The variable collimator response is illustrated as a fan-region integral of the radionuclide distribution, which is different from the conventional line integral, when using the FBP algorithm to reconstruct the image. In a parallel-beam acquisition geometry, if the radionuclide distribution is restricted to a bounded region, the line integral can be accurately computed from the fan integral near the boundary, then by a bootstrap technique, all these line integrals can be obtained. Therefore the variable collimator response can be completely restored theoretically.

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 variety of inversions of exponential Radon transform has been derived based on the circular harmonic transform in Fourier space by several research groups. However, these inversions cannot be directly applied to deal with the reconstruction for fan-beam or variable-focal-length fan-beam collimator geometries in single photon emission computed tomography (SPECT). In this paper, the authors derived a Cormack-type inversion of the exponential Radon transform by employing the circular harmonic transform directly in the projection space and the image space instead of the Fourier space. Thus, a unified reconstruction framework is established for parallel-, fan-, and variable-focal-length fan-beam collimator geometries. Compared to many existing algorithms, the presented one greatly mitigates the difficulty of image reconstruction due to the complicated collimator geometry and significantly reduces the computational burden of the special functions, such as Chebyshev or Bessel functions. By the well-established fast-Fourier transform (FFT), the authors' algorithm is very efficient, as demonstrated by several numerical simulations.

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.