Memory-efficient algorithm for stored projection and backprojection matrix in helical CT

Iterative image reconstruction with polar coordinate discretized system matrix for optical CT radiochromic gel dosimetry

BACKGROUND

Gel dosimeters are a potential tool for measuring the complex dose distributions that characterize modern radiotherapy. A prototype tabletop solid-tank fan-beam optical CT scanner for readout of gel dosimeters was recently developed. This scanner does not have a straight raypath from source to detector, thus images cannot be reconstructed using filtered backprojection (FBP) and iterative techniques are required. Iterative image reconstruction requires a system matrix that describes the geometry of the imaging system. Stored system matrices can become immensely large, making them impractical for storage on a typical desktop computer.

PURPOSE

Here we develop a method to reduce the storage size of optical CT system matrices through use of polar coordinate discretization while accounting for the refraction in optical CT systems.

METHODS

A ray tracing simulator was developed to track the path of light rays as they traverse the different mediums of the optical CT scanner. Cartesian coordinate discretized system matrices (CCDSMs) and polar coordinate discretized system matrices (PCDSMs) were generated by discretizing the reconstruction area of the optical CT scanner into a Cartesian pixel grid and a polar coordinate pixel grid, respectively. The length of each ray through each pixel was calculated and used to populate the system matrices. To ensure equal weighting during iterative reconstruction, the radial rings of PCDSMs were asymmetrically spaced such that the area of each polar pixel was constant. Two clinical phantoms and several synthetic phantoms were produced and used to evaluate the reconstruction techniques under known conditions. Reconstructed images were analyzed in terms of spatial resolution, signal-to-noise ratio (SNR), contrast-to-noise ratio (CNR), signal nonuniformity (SNU), and Gamma map pass percentage.

RESULTS

A storage size reduction of 99.72% was found when comparing a PCDSM to a CCDSM with the same total number of pixels. Images reconstructed with a PCDSM were found to have superior SNR, CNR, SNU, and Gamma (1 mm, 1%) pass percentage compared to those reconstructed with a CCDSM. Increasing spatial resolution in the radial direction with increasing radial distance was found in both PCDSM and CCDSM reconstructions due to the outer regions refracting light more severely. Images reconstructed with a PCDSM showed a decrease in spatial resolution in the azimuthal directions as radial distance increases, due to the widening of the polar pixels. However, this can be mitigated with only a slight increase in storage size by increasing the number of projections. A loss of spatial resolution in the radial direction within 5 mm radially from center was found when reconstructing with a PCDSM, due to the large innermost pixels. However, this was remedied by increasing the number of radial rings within the PCDSM, yielding radial spatial resolution on par with images reconstructed with a CCDSM and a storage size reduction of 99.26%.

CONCLUSIONS

Discretizing the image pixel elements in polar coordinates achieved a system matrix storage size reduction of 99.26% with only minimal reduction in the image quality.

Impact of the non-negativity constraint in model-based iterative reconstruction from CT data

PURPOSE

Model-based iterative reconstruction is a promising approach to achieve dose reduction without affecting image quality in diagnostic x-ray computed tomography (CT). In the problem formulation, it is common to enforce non-negative values to accommodate the physical non-negativity of x-ray attenuation. Using this a priori information is believed to be beneficial in terms of image quality and convergence speed. However, enforcing non-negativity imposes limitations on the problem formulation and the choice of optimization algorithm. For these reasons, it is critical to understand the value of the non-negativity constraint. In this work, we present an investigation that sheds light on the impact of this constraint.

METHODS

We primarily focus our investigation on the examination of properties of the converged solution. To avoid any possibly confounding bias, the reconstructions are all performed using a provably converging algorithm started from a zero volume. To keep the computational cost manageable, an axial CT scanning geometry with narrow collimation is employed. The investigation is divided into five experimental studies that challenge the non-negativity constraint in various ways, including noise, beam hardening, parametric choices, truncation, and photon starvation. These studies are complemented by a sixth one that examines the effect of using ordered subsets to obtain a satisfactory approximate result within 50 iterations. All studies are based on real data, which come from three phantom scans and one clinical patient scan. The reconstructions with and without the non-negativity constraint are compared in terms of image similarity and convergence speed. In select cases, the image similarity evaluation is augmented with quantitative image quality metrics such as the noise power spectrum and closeness to a known ground truth.

RESULTS

For cases with moderate inconsistencies in the data, associated with noise and bone-induced beam hardening, our results show that the non-negativity constraint offers little benefit. By varying the regularization parameters in one of the studies, we observed that sufficient edge-preserving regularization tends to dilute the value of the constraint. For cases with strong data inconsistencies, the results are mixed: the constraint can be both beneficial and deleterious; in either case, however, the difference between using the constraint or not is small relative to the overall level of error in the image. The results with ordered subsets are encouraging in that they show similar observations. In terms of convergence speed, we only observed one major effect, in the study with data truncation; this effect favored the use of the constraint, but had no impact on our ability to obtain the converged solution without constraint.

CONCLUSIONS

Our results did not highlight the non-negativity constraint as being strongly beneficial for diagnostic CT imaging. Altogether, we thus conclude that in some imaging scenarios, the non-negativity constraint could be disregarded to simplify the optimization problem or to adopt other forward projection models that require complex optimization machinery to be used together with non-negativity.

Accelerating iterative coordinate descent using a stored system matrix

PURPOSE

The computational burden associated with model-based iterative reconstruction (MBIR) is still a practical limitation. Iterative coordinate descent (ICD) is an optimization approach for MBIR that has sometimes been thought to be incompatible with modern computing architectures, especially graphics processing units (GPUs). The purpose of this work is to accelerate the previously released open-source FreeCT_ICD to include GPU acceleration and to demonstrate computational performance with ICD that is comparable with simultaneous update approaches.

METHODS

FreeCT_ICD uses a stored system matrix (SSM), which precalculates the forward projector in the form of a sparse matrix and then reconstructs on a rotating coordinate grid to exploit helical symmetry. In our GPU ICD implementation, we shuffle the sinogram memory ordering such that data access in the sinogram coalesce into fewer transactions. We also update N_S voxels in the xy-plane simultaneously to improve occupancy. Conventional ICD updates voxels sequentially (N_S  = 1). Using N_S  > 1 eliminates existing convergence guarantees. Convergence behavior in a clinical dataset was therefore studied empirically.

RESULTS

On a pediatric dataset with sinogram size of 736 × 16 × 13860 reconstructed to a matrix size of 512 × 512 × 128, our code requires about 20 s per iteration on a single GPU compared to 2300 s per iteration for a 6-core CPU using FreeCT_ICD. After 400 iterations, the proposed and reference codes converge within 2 HU RMS difference (RMSD). Using a wFBP initialization, convergence within 10 HU RMSD is achieved within 4 min. Convergence is similar with N_S values between 1 and 256, and N_S  = 16 was sufficient to achieve maximum performance. Divergence was not observed until N_S  > 1024.

CONCLUSIONS

With appropriate modifications, ICD may be able to achieve computational performance competitive with simultaneous update algorithms currently used for MBIR.

Technical Note: FreeCT_ICD: An open-source implementation of a model-based iterative reconstruction method using coordinate descent optimization for CT imaging investigations

PURPOSE

To facilitate investigations into the impacts of acquisition and reconstruction parameters on quantitative imaging, radiomics and CAD using CT imaging, we previously released an open-source implementation of a conventional weighted filtered backprojection reconstruction called FreeCT_wFBP. Our purpose was to extend that work by providing an open-source implementation of a model-based iterative reconstruction method using coordinate descent optimization, called FreeCT_ICD.

METHODS

Model-based iterative reconstruction offers the potential for substantial radiation dose reduction, but can impose substantial computational processing and storage requirements. FreeCT_ICD is an open-source implementation of a model-based iterative reconstruction method that provides a reasonable tradeoff between these requirements. This was accomplished by adapting a previously proposed method that allows the system matrix to be stored with a reasonable memory requirement. The method amounts to describing the attenuation coefficient using rotating slices that follow the helical geometry. In the initially proposed version, the rotating slices are themselves described using blobs. We have replaced this description by a unique model that relies on trilinear interpolation together with the principles of Joseph's method. This model offers an improvement in memory requirement while still allowing highly accurate reconstruction for conventional CT geometries. The system matrix is stored column-wise and combined with an iterative coordinate descent (ICD) optimization. The result is FreeCT_ICD, which is a reconstruction program developed on the Linux platform using C++ libraries and released under the open-source GNU GPL v2.0 license. The software is capable of reconstructing raw projection data of helical CT scans. In this work, the software has been described and evaluated by reconstructing datasets exported from a clinical scanner which consisted of an ACR accreditation phantom dataset and a clinical pediatric thoracic scan.

RESULTS

For the ACR phantom, image quality was comparable to clinical reconstructions as well as reconstructions using open-source FreeCT_wFBP software. The pediatric thoracic scan also yielded acceptable results. In addition, we did not observe any deleterious impact in image quality associated with the utilization of rotating slices. These evaluations also demonstrated reasonable tradeoffs in storage requirements and computational demands.

CONCLUSION

FreeCT_ICD is an open-source implementation of a model-based iterative reconstruction method that extends the capabilities of previously released open-source reconstruction software and provides the ability to perform vendor-independent reconstructions of clinically acquired raw projection data. This implementation represents a reasonable tradeoff between storage and computational requirements and has demonstrated acceptable image quality in both simulated and clinical image datasets.

System matrix computation vs storage on GPU: A comparative study in cone beam CT

PURPOSE

Iterative reconstruction algorithms in computed tomography (CT) require a fast method for computing the intersection distances between the trajectories of photons and the object, also called ray tracing or system matrix computation. This work focused on the thin-ray model is aimed at comparing different system matrix handling strategies using graphical processing units (GPUs).

METHODS

In this work, the system matrix is modeled by thin rays intersecting a regular grid of box-shaped voxels, known to be an accurate representation of the forward projection operator in CT. However, an uncompressed system matrix exceeds the random access memory (RAM) capacities of typical computers by one order of magnitude or more. Considering the RAM limitations of GPU hardware, several system matrix handling methods were compared: full storage of a compressed system matrix, on-the-fly computation of its coefficients, and partial storage of the system matrix with partial on-the-fly computation. These methods were tested on geometries mimicking a cone beam CT (CBCT) acquisition of a human head. Execution times of three routines of interest were compared: forward projection, backprojection, and ordered-subsets convex (OSC) iteration.

RESULTS

A fully stored system matrix yielded the shortest backprojection and OSC iteration times, with a 1.52× acceleration for OSC when compared to the on-the-fly approach. Nevertheless, the maximum problem size was bound by the available GPU RAM and geometrical symmetries. On-the-fly coefficient computation did not require symmetries and was shown to be the fastest for forward projection. It also offered reasonable execution times of about 176.4 ms per view per OSC iteration for a detector of 512 × 448 pixels and a volume of 384³ voxels, using commodity GPU hardware. Partial system matrix storage has shown a performance similar to the on-the-fly approach, while still relying on symmetries.

CONCLUSION

Partial system matrix storage was shown to yield the lowest relative performance. On-the-fly ray tracing was shown to be the most flexible method, yielding reasonable execution times. A fully stored system matrix allowed for the lowest backprojection and OSC iteration times and may be of interest for certain performance-oriented applications.

A review of GPU-based medical image reconstruction

Tomographic image reconstruction is a computationally demanding task, even more so when advanced models are used to describe a more complete and accurate picture of the image formation process. Such advanced modeling and reconstruction algorithms can lead to better images, often with less dose, but at the price of long calculation times that are hardly compatible with clinical workflows. Fortunately, reconstruction tasks can often be executed advantageously on Graphics Processing Units (GPUs), which are exploited as massively parallel computational engines. This review paper focuses on recent developments made in GPU-based medical image reconstruction, from a CT, PET, SPECT, MRI and US perspective. Strategies and approaches to get the most out of GPUs in image reconstruction are presented as well as innovative applications arising from an increased computing capacity. The future of GPU-based image reconstruction is also envisioned, based on current trends in high-performance computing.