An iterative algorithm, based on recent work in compressive sensing, is developed for volume image reconstruction from a circular cone-beam scan. The algorithm minimizes the total variation (TV) of the image subject to the constraint that the estimated projection data is within a specified tolerance of the available data and that the values of the volume image are non-negative. The constraints are enforced by the use of projection onto convex sets (POCS) and the TV objective is minimized by steepest descent with an adaptive step-size. The algorithm is referred to as adaptive-steepest-descent-POCS (ASD-POCS). It appears to be robust against cone-beam artifacts, and may be particularly useful when the angular range is limited or when the angular sampling rate is low. The ASD-POCS algorithm is tested with the Defrise disk and jaw computerized phantoms. Some comparisons are performed with the POCS and expectation-maximization (EM) algorithms. Although the algorithm is presented in the context of circular cone-beam image reconstruction, it can also be applied to scanning geometries involving other x-ray source trajectories.

/pdf/image-reconstruction-in-circular-cone-beam-computed-4wndbkpv6w.pdf

Image reconstruction in circular cone-beam computed tomography by constrained, total-variation minimization

This paper describes a statistical image reconstruction method for X-ray computed tomography (CT) that is based on a physical model that accounts for the polyenergetic X-ray source spectrum and the measurement nonlinearities caused by energy-dependent attenuation. We assume that the object consists of a given number of nonoverlapping materials, such as soft tissue and bone. The attenuation coefficient of each voxel is the product of its unknown density and a known energy-dependent mass attenuation coefficient. We formulate a penalized-likelihood function for this polyenergetic model and develop an ordered-subsets iterative algorithm for estimating the unknown densities in each voxel. The algorithm monotonically decreases the cost function at each iteration when one subset is used. Applying this method to simulated X-ray CT measurements of objects containing both bone and soft tissue yields images with significantly reduced beam hardening artifacts.

/pdf/statistical-image-reconstruction-for-polyenergetic-x-ray-5dfg3tu602.pdf

Statistical image reconstruction for polyenergetic X-ray computed tomography

Iterative reconstruction methods in X-ray CT

The ordered subsets EM (OSEM) algorithm has enjoyed considerable interest for emission image reconstruction due to its acceleration of the original EM algorithm and ease of programming. The transmission EM reconstruction algorithm converges very slowly and is not used in practice. In this paper, we introduce a simultaneous update algorithm called separable paraboloidal surrogates (SPS) that converges much faster than the transmission EM algorithm. Furthermore, unlike the 'convex algorithm' for transmission tomography, the proposed algorithm is monotonic even with nonzero background counts. We demonstrate that the ordered subsets principle can also be applied to the new SPS algorithm for transmission tomography to accelerate 'convergence', albeit with similar sacrifice of global convergence properties as for OSEM. We implemented and evaluated this ordered subsets transmission (OSTR) algorithm. The results indicate that the OSTR algorithm speeds up the increase in the objective function by roughly the number of subsets in the early iterates when compared to the ordinary SPS algorithm. We compute mean square errors and segmentation errors for different methods and show that OSTR is superior to OSEM applied to the logarithm of the transmission data. However, penalized-likelihood reconstructions yield the best quality images among all other methods tested.

/pdf/ordered-subsets-algorithms-for-transmission-tomography-1u6synb0lw.pdf

Ordered subsets algorithms for transmission tomography.

The problem of forming cross-sectional or tomographic images of the attenuation characteristics of objects arises in a variety of contexts, including medical x-ray computed tomography (CT) and nondestructive evaluation of objects in industrial inspection. In the context of emission imaging, such as positron emission tomography (PET) [1, 2], single photon emission computed tomography (SPECT) [3], and related methods used in the assay of containers of radioactive waste [4], it is useful to be able to form "attenuation maps," tomographic images of attenuation coefficients, from which one can compute attenuation correction factors for use in emission image reconstruction. One can measure the attenuating characteristics of an object by transmitting a collection of photons through the object along various paths or "rays" and observing the fraction that pass unabsorbed. From measurements collected over a large set of rays, one can reconstruct tomographic images of the object. Such image reconstruction is the subject of this chapter.

In all the above applications, the number of photons one can measure in a transmission scan is limited. In medical x-ray CT, source strength, patient motion, and absorbed dose considerations limit the total x-ray exposure. Implanted objects such as pacemakers also significantly reduce transmissivity and cause severe artifacts [5]. In industrial applications, source strength limitations, combined with the very large attenuation coefficients of metallic objects, often result in a small fraction of photons passing to the detector unabsorbed. In PET and SPECT imaging, the transmission scan only determines a "nuisance" parameter of secondary interest relative to the object's emission properties, so one would like to minimize the transmission scan duration. All the above considerations lead to "low-count" transmission scans. This chapter discusses algorithms for reconstructing attenuation images from low-count transmission scans. In this context, we define low-count to mean that the mean number of photons per ray is small enough that traditional filtered-backproject on (FBP) images, or even methods based on the Gaussian approximation to the distribution of the Poisson measurements (or logarithm thereof), are inadequate. We focus the presentation in the context of PET and SPECT transmission scans, but the methods are generally applicable to all low-count transmission studies. See [6] for an excellent survey of statistical approaches for the emission reconstruction problem.

Statistical Image Reconstruction Methods for Transmission Tomography

An iterative algorithm is presented for accelerated reconstruction of cone beam transmission CT data (CBCT). CBCT supplies an attenuation map for SPECT attenuation compensation and anatomical correlation. Iterative algorithms are necessary to reduce truncation artifacts and 3D reconstruction artifacts. An existing transmission maximum-likelihood algorithm (TRML) is accurate but the reconstruction time is too long. The new algorithm is a modified EM algorithm, based on ordered subsets (OSEM). OSEM was evaluated in comparison to TRML using a thorax phantom and a 3D Defrise phantom. A wide range of image measures were evaluated, including spatial resolution, noise, log likelihood, region quantification, truncation artifact removal, and 3D artifact removal. For appropriate subset size, OSEM produced essentially the same image as TRML, but required only one-tenth as many iterations. Thus, adequate images were available in two to four iterations (20-30 min on a SPARC 2 workstation). Further, OSEM still approximately maximizes likelihood: divergence occurs only for very high (and clinically irrelevant) iterations. Ordered subsets are likely to be useful in other geometries (fan and parallel) and for emission CT as well. Therefore, with ordered subsets, high-quality iterative reconstruction is now available in clinically practical reconstructions times.

https://iopscience.iop.org/article/10.1088/0031-9155/40/7/006/pdf

R Narayanaswamy

Papers

Transmission maximum-likelihood reconstruction with ordered subsets for cone beam CT.