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The key idea discussed in this paper is to reconstruct an image from overlapped projections so that the data acquisition process can be shortened while the image quality remains essentially uncompromised. To perform image reconstruction from overlapped projections, the conventional reconstruction approach (e.g., filtered backprojection (FBP) algorithms) cannot be directly used because of two problems. First, overlapped projections represent an imaging system in terms of summed exponentials, which cannot be transformed into a linear form. Second, the overlapped measurement carries less information than the traditional line integrals. To meet these challenges, we propose a compressive sensing-(CS-) based iterative algorithm for reconstruction from overlapped data. This algorithm starts with a good initial guess, relies on adaptive linearization, and minimizes the total variation (TV). Then, we demonstrated the feasibility of this algorithm in numerical tests.

The popular CT scheme takes projection data from an X-ray source being scanned along a trajectory and reconstructs an image from these data that are essentially line integrals through an object. In real-world applications, higher temporal resolution has been constantly pursued, such as for dynamic medical CT, micro-, and nano-CT. The multisource scanning mode is well known to improve temporal resolution but the data acquisition and field of view are seriously restricted to avoid overlapped projections, such as in the case of the classic dynamic spatial reconstructor (DSR). As shown in Figure

Imaging geometry for collection of overlapped projections from two

In the overlapped projection geometry, two (or more) sources, A and B, emit X-rays simultaneously through an object to be reconstructed from various orientations. As a result, the resultant X-ray projections are overlapped onto the same detector array. The overlapped projections use the same detector array at the same time but complicate the imaging model. To perform image reconstruction from overlapped projections, the conventional reconstruction approach (e.g., filtered backprojection (FBP) algorithms) cannot be directly used because of the two problems. First, overlapped projections represent an imaging system in terms of summed exponentials, which cannot be transformed into a linear form, since the X-ray intensity through an object follows an exponential decaying function. Second, overlapped measurement carries less information than the traditional line integrals, due to the additional uncertainty from mixing two ray sums, leading to an underdetermined imaging system.

Compressive sensing (CS) is a new technique being rapidly developed over the past years [

An image

Now, the key issue is how to reconstruct an image from overlapped projection data. To alleviate the underdetermined measurement due to the overlapped nature of projection data, the compressed sensing (CS) principles are employed in our reconstruction process. To utilize the sparsity of an underlying image, it is first transformed into a gradient counterpart, and then the L-1 norm of the gradient, which is known as the total variation (TV), is minimized, subject to the overlapped projection data. The entire reconstruction process can therefore be casted into a constrained nonlinear optimization problem:

Clearly, there are various ways to solve the above constrained TV minimization problem. In the CS field, a projection onto convex sets (POCS) and gradient descent search approach has been successfully used to solve this type of MRI and CT imaging problems [

However, in the case of overlapped projections from two sources, the constraint equations, which are the sums of two exponentials, cannot be transformed into a linear form. Therefore, a different approach is needed. Our solution is to make a good initial guess, such as a low-resolution CT image first. This blurry image will serve as a starting point, and the difference between this initial reference and the actual image will be iteratively updated, and at the same time the current guess will be also updated. Since the difference is assumed to be small, we can perform a Taylor series expansion to linearize the imaging system by omitting high-order terms. Then, we can apply the POCS-gradient algorithm on this linearly approximated system iteratively.

Mathematically, let us denote

To demonstrate the feasibility of our proposed algorithm for image reconstruction from overlapped projections, we developed a program in MATLAB, and implemented the traditional algebraic reconstruction technique (ART) for comparison. A Modified 2D Shepp-Logan phantom (Table

Parameters used in the numerical tests.

ART: Single Source | IROP: Two Source | ART: Single Source | IROP: Two Source | ART: Single Source | IROP: Two Source | |

15 | 150 | 30 | 150 | NA | 150 | |

50 | 500 | 100 | 500 | NA | 500 | |

NA | 3 | NA | 3 | NA | 3 | |

50 | 50 | 50 | 50 | NA | 50 | |

NA | 5 | NA | 5 | NA | 5 |

Parameters of the 2D modified Shepp-Logan phantom.

Axis length ( | Center ( | Angle ( | Density |
---|---|---|---|

We performed both ART and IROP reconstructions under these conditions, with blurry and constant initial guesses. Representative results are in Figures

Reconstructed images of the Shepp-Logan phantom in the first test. (a) A reconstruction using ART, (b) a reconstruction using IROP, and (c) the profiles along the central vertical line of the phantom, where the dotted and solid lines are for the phantom and the IROP reconstruction, respectively (the display window: [0,0.5]).

Reconstructed images of the Shepp-Logan phantom in the second test. (a) A reconstruction using ART, (b) a reconstruction using IROP, and (c) the profiles along the central vertical line of the phantom, where the dotted and solid lines are for the phantom and the IROP reconstruction, respectively (the display window: [0,0.5]).

Reconstructed images of the Shepp-Logan phantom in the third test. (a) A reconstruction using IROP with a constant initial guess, and (b) the profiles along the central vertical line of the phantom, where the dotted and solid lines are for the phantom and the IROP reconstruction, respectively (the display window: [0,0.5]).

To investigate the convergence of IROP, we first introduce an evaluation metric

Convergence plots for (a) Test 1, (b) Test 2, and (c) Test 3, with 3 linearization steps and 50 iterations after each linearization.

Convergence plots for (a) Test 1, (b) Test 2, and (c) Test 3, with 10 linearization steps and 3 iterations after each linearization step.

The primary advantage of the IROP scheme is to improve the data acquisition speed. In one exemplary application, we can assume that the two sources are fairly close so that the detector collimation can work effectively for both the sources. If a good number of sources are used, scattering effects could be a concern. In that case, scattering correction may be needed using hardware (such as some degree of multiplexing) and/or software (such as model- or image-based compensation) methods [

In Algorithm

The implementation of Algorithm

A theoretical analysis on the convergence of the IROP scheme has not been performed yet but we hypothesize that the global convergence can be established if a guess is appropriately chosen, as numerically shown in the preceding section. Actually, the IROP problem is much better posed than many well-known inverse problems such as diffuse optical tomography (DOT) [

Since the IROP scheme mixes line integrals pairwise, the IROP problem may lead to an underdetermined system of measurement equations, especially when the number of samples is not sufficiently large for ultrafast imaging performance. To address this issue, we have implemented the CS principles in Algorithm

It is emphasized that our IROP approach can be extended to multiple other imaging scenarios. For example, in transmission ultrasound imaging, we can use multiple ultrasound sources and a single array of detectors (transducers). This may be also related to the area of signal unmixing. The common task would be to unravel an underlying signal or image from mixed measures. There seem good research opportunities along this direction.

In conclusion, we have proposed the idea to perform image reconstruction from overlapped projection data and formulated a CS-based iterative algorithm for this new imaging problem. Our IROP algorithm starts with a good initial guess, relies on adaptive linearization, and minimizes the TV. Also, we have demonstrated the feasibility of this algorithm in numerical simulation. Further research is being performed to characterize and improve our IROP approach.

This work was partially supported by NIH/NIBIB Grants (EB002667, EB004287, and EB007288). The authors thank Dr. Jun Zhao for computational support.