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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.

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.

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.