GPU-accelerated iterative reconstruction from Compton scattered data using a matched pair of conic projector and backprojector

https://doi.org/10.1016/j.cmpb.2016.04.012Get rights and content

Highlights

  • Acceleration of iterative Compton camera reconstruction using graphics processing unit.

  • Exactly matched pair of conic projector and backprojector.

  • Significantly reduced reconstruction time without loss of accuracy.

  • Improvement of reconstruction accuracy in advanced iterative methods.

Abstract

Background and Objective

Iterative reconstruction from Compton scattered data is known to be computationally more challenging than that from conventional line-projection based emission data in that the gamma rays that undergo Compton scattering are modeled as conic projections rather than line projections. In conventional tomographic reconstruction, to parallelize the projection and backprojection operations using the graphics processing unit (GPU), approximated methods that use an unmatched pair of ray-tracing forward projector and voxel-driven backprojector have been widely used. In this work, we propose a new GPU-accelerated method for Compton camera reconstruction which is more accurate by using exactly matched pair of projector and backprojector.

Methods

To calculate conic forward projection, we first sample the cone surface into conic rays and accumulate the intersecting chord lengths of the conic rays passing through voxels using a fast ray-tracing method (RTM). For conic backprojection, to obtain the true adjoint of the conic forward projection, while retaining the computational efficiency of the GPU, we use a voxel-driven RTM which is essentially the same as the standard RTM used for the conic forward projector.

Results

Our simulation results show that, while the new method is about 3 times slower than the approximated method, it is still about 16 times faster than the CPU-based method without any loss of accuracy.

Conclusions

The net conclusion is that our proposed method is guaranteed to retain the reconstruction accuracy regardless of the number of iterations by providing a perfectly matched projector–backprojector pair, which makes iterative reconstruction methods for Compton imaging faster and more accurate.

Introduction

Compton imaging is a three-dimensional (3D) emission imaging technique to visualize the gamma-rays that undergo Compton scattering, whose applications are in a variety of fields such as nuclear medicine, gamma-ray astronomy, and security [1], [2], [3], [4]. A Compton camera uses two types of detectors, a scatterer and an absorber. The valid events are recorded when the photons that reach the scatterer are Compton scattered and detected by the absorber in coincidence with the events in the scatterer (See Fig. 1). The scattering angle of a photon is measured using the energies of the electron at rest and the scattered photon [5]. In this case, the incident direction of the emitted photon on the scatterer can be computed within a conic surface of ambiguity. Therefore, the projection and backprojection operations involved in a reconstruction algorithm for Compton imaging must be performed by the conic surface integral rather than the line integral used for conventional tomographic reconstruction. Due to computational complexity of such operations, reconstruction from Compton scattered data using an iterative method has been of a challenging problem.

Recently, with the successful development of block-iterative methods for emission tomography, fast iterative reconstruction methods for Compton imaging have also been proposed [6], [7], [8], [9] and further accelerated by using the GPU [6], [10]. In fact, there have been substantial developments of accelerating the computational speed of projection and backprojection in iterative reconstruction for both transmission and emission tomography by using the GPU [11]. (An overview of existing methods for Compton camera reconstruction can be found in [12], [13].)

Since the size of the system matrix for a typical Compton camera is intractably large, the conventional central processing unit (CPU)-based methods to calculate conic projection and backprojection, which include a caching scheme that pre-calculates elements of the system matrix and repeatedly use them in every iteration, are almost impractical. To parallelize such time-consuming operations, one can consider utilizing the GPU which has attracted great interest in iterative reconstruction for conventional computed tomography. However, unlike the conic projection operation which can be easily parallelized with a ray-tracing [14] forward projector, the conic backprojection operation is challenging. In [6], to develop GPU-accelerated methods that can rapidly perform conic backprojection on the fly, two different approximated schemes were proposed, where none of the two schemes was matched with the ray-tracing forward projector. Although the approximated methods significantly reduced the computation time, they had an unfortunate effect of causing a visually noticeable error in the reconstruction due to the mismatch between the projector and backprojector when compared with the result obtained by the exact calculation using a matched projector–backprojector pair implemented with the CPU. Moreover, the error gradually became larger as the number of iterations increased.

In this work, to avoid the approximation error, which could be significant when a large number of iterations are required, we propose a new GPU-accelerated ray-tracing method (RTM) for both projection and backprojection which does not use any approximations for parallelizing the operations. Since our method is exact, the result is as accurate as those obtained from the non-accelerated method.

The present work is inspired by our work for parallelizing a matched pair of ray-tracing projector and backprojector for iterative cone-beam CT (CBCT) reconstruction [15] where the matched pair clearly provides better reconstruction accuracy than the unmatched pair. One can easily expect that the improvement of the reconstruction accuracy by using a matched projector–backprojector pair will be more apparent when it is applied to low-resolution reconstruction such as the Compton camera reconstruction considered in this work. While the basic idea proposed in [15] is directly applicable to any conventional tomography reconstruction that involves line integral-based projection and backprojection operations, its application to a Compton camera is uniquely challenging because Compton camera reconstruction involves surface integral-based projection and backprojection operations that are considerably more complicated than the simple line integral-based operations.

In this work, to minimize computations for conic backprojection while keeping their accuracy, we propose a new scheme to properly indicate the possible cones that can pass through a voxel. Then the intersecting area of a cone with the voxel is calculated by summing all of the intersecting chord lengths in the voxel. As the procedure to calculate the intersecting area in this case is identical to that for forward projection, the resulting conic projector–backprojector pair becomes exactly matched.

The remainder of this paper is organized as follows. Section 2 presents an exact and parallelizable method to efficiently perform conic projections and backprojections for Compton camera reconstruction. This section also briefly describes an iterative algorithm for Compton camera reconstruction which is used in our experiments for the performance test of our proposed methods. Section 3 presents our simulation studies to compare the computational performance of the proposed GPU-based method with that of the conventional CPU-based method and also with that of other previous GPU-based methods. Section 4 concludes our work.

Section snippets

Parallelizing conic projection and backprojection

We consider a typical Compton camera geometry consisting of two parallel detector planes as shown in Fig. 1. A valid event in this model is that a photon incident on the camera undergoes Compton scattering in the first detector (the scatterer) and then is completely absorbed in the second detector (the absorber). The quantities measured in this event are: the detected position m(x1,y1,z1) in the scatter, the detected position n(x2,y2,z2) in the absorber, and the scattering angle ω

Simulations and results

To test our proposed method, we modeled a Compton camera system with the six detector pairs placed in the direction of the x-, y- and z-axes as shown in Fig. 4. Each detector pair was placed with a radial offset of 10 cm from the center. The distance between the scatterer and the absorber in each detector pair was 5 cm. Both the scatterer and the absorber were sampled into 16 × 16 discrete detector elements, each of which had the size of (3.125 mm)2. The range of the scattering angle of the

Summary and conclusion

GPU-accelerated methods using unmatched projector–backprojector pairs have been useful for high-performance computing in high-resolution CT reconstruction [11], [19], [23]. However, the use of such approximated methods for Compton camera reconstruction causes visually noticeable artifacts due to its relatively low resolution compared to the conventional CT [6]. Moreover, since Compton camera reconstruction involves surface integral-based projection and backprojection operations rather than line

Acknowledgment

This work was supported by the Vietnam's National Foundation for Science and Technology Development (NAFOSTED) under Grant 102.01-2013.42 for V.-G. Nguyen, and by the National Research Foundation of Korea grant funded by the Korea government (MSIP) under Grant 2014R1A2A2A01002626 for S.-J. Lee.

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