X-ray computed tomography (CT) is an important technique for noninvasive clinical diagnosis and nondestructive testing. In many applications a number of image processing steps are needed before the image features are available. One of these processing steps is image segmentation, which generates the edge and the structural features of the regions of interest. The conventional flow is to first reconstruct images and then apply image segmentation methods on reconstructed images. In contrast, an emerging technique obtains the tomographic image and segmentation simultaneously, which is especially useful in the case of limited data. An iterative method for simultaneous reconstruction and segmentation (SRS) with Mumford-Shah model has been proposed, which not only regularizes the ill-posed tomographic reconstruction problem, but also produces the image segmentation at the same time. The Mumford-Shah model is both mathematically and computationally challenging. In this paper, we propose an asynchronous ray-parallel algorithm of the SRS method and accelerate it using field-programmable gate array (FPGA) devices, which drastically improves the energy efficiency. Experimental results show that the FPGA implementation achieves a 1:2× speedup with an energy efficiency as great as 58×, over the GPU implementation.
While classic CT theory targets exact reconstruction of a whole cross-section or an entire object, practical applications often focus on a region of interest (ROI). The long-standing interior problem is well known that an internal ROI cannot be exactly reconstruct only from truncated projection data associated with x-rays through the ROI. Although lambda tomography was developed to target gradient-like features of an internal ROI for the interior problem, it has not been well accepted in the biomedical community. On the other hand, approximate local reconstruction methods are subject to biases and artifacts. Recently, the interior problem is re-visited with appropriate prior knowledge, delivering practical results. First, the interior problem can be exactly and stably solved if a sub-region in an ROI is known. Thereafter, the sub-region knowledge can be replaced by certain rather weak constraints. For local reconstruction, a candidate image can be represented as the sum of the truth and an ambiguity component. Very surprisingly, the ROI image is prove to be the unique minimizer of the total variation (TV) or high order total variation (HOT) functional subject to the measurement, if the ROI is piece-wise constant or polynomial. Interior tomography algorithms based on HOT minimization have been developed for x-ray CT, and then extended for interior SPECT and interior differential phasecontrast tomography, respectively. In this paper, we will summarize the main theoretical and algorithmic results.
X-ray imaging is of paramount importance for clinical and pre-clinical applications but it is fundamentally restricted by the attenuation-based contrast mechanism, which has remained essentially the same since Roentgen's discovery a century ago. Recently, based on the Talbot effect, groundbreaking work was reported using 1D gratings for x-ray phase-contrast imaging with a hospital-grade x-ray tube instead of a synchrotron or micro-focused source. In this paper, we
report an extension of our earlier 2D-grating-based work to the case of Gaussian beams. This 2D-grating-based approach has the potential to reduce the imaging time, increase the spatial coherence, and enhance the accuracy and robustness compared to current 1D-grating-based phase-contrast imaging techniques.
Diffuse Optical Tomography (DOT) is a functional medical imaging modality which can determine the spatial optical parameters' distributions inside a medium. The forward model of DOT is described by the diffusion approximation of radiative transform equation (RTE) while the DOT is to recover optical parameters of a medium from the boundary measurements induced by external near-infared (NIR) light. In this paper, we propose a mathematic model of DOT and then give a novel iterative reconstruction method of the proposed model. The new iterative reconstruction method is based on the assumption that the measurement noise is Poissonian while previous iterative reconstruction methods are mostly base on the assumption that the measurement noise is Gaussian, and are of least-squares type. The proposed algorithm is a variant of the well-known EM algorithm. It can also be used to deal with the incomplete boundary measurements. The performance of the reconstruction algorithm including spatial resolution and contrast are investigated with 2-dimensional numerical experiments.
Differential Phase Contrast Imaging (DPCI) has the potential to vastly increase soft tissue contrast. DPCI requires spatial and temporal coherence as generated by a synchrotron or a micro-focus
X-ray source; however, recent research demonstrates DPCI can be implemented using a conventional X-ray source with three transmission gratings (Pfeiffer et al., Nature 2006). This paper describes the optimization of the essential system parameters (system size, delivered dose, spatial resolution) of this implementation from a theoretical perspective. The optimization of these parameters is an essential step in practical application of DPCI. We conclude that the minimum size of the system is approximately 700 mm, the minimum resolution is 100 um, and the dose is 1/1000 that of conventional absorption CT.
The bioluminescence tomography (BLT) emerges as a novel molecular imaging technology for small animal studies. BLT is an ill-posed inverse source problem subject to Cauchy data of the diffusion model. Several algorithms were developed to regularize this problem. Although those algorithms produce encouraging results, they theoretically require the completely measured data on the external surface. In practice, the observed data is often incomplete due to physical limitations. The BLT problem in this situation is similar to limited angle X-ray CT although the imaging model is more complicated with BLT. In this work, we formulate a mathematical model for BLT from partial data and generalize our previous results on the solution uniqueness to the partial data case. Also, we extend our previous methods to handle incomplete data. The first method is a variant of the well-known EM algorithm. The second one is based on the Landweber scheme. Both methods allow incorporation of knowledge-based constraints. Numerical and physical phantoms are used to evaluate and validate the proposed algorithms.
In a previous study, we proposed a helical scanning configuration with triple X-ray sources symmetrically positioned
and established its reconstruction algorithm. Although symmetrically positioned sources are convenient in practice,
artifacts can be produced in a reconstructed image if the physical sources are not equally spaced. In this work, we
develop an exact backprojection filtration (BPF) type algorithm for the configuration with unequally spaced triple
sources to reduce the artifacts. Similar to the Tam-Danielsson window, we define the minimum detection window as the
region bounded by the most adjacent turns of two helices. The sum of the heights of the three consequent minimum
detection windows is equal to that of the traditional Tam-Danielsson window for a single source. Furthermore, we prove
that inter-helix PI-lines satisfy the existence and uniqueness properties (i.e., through any point inside the triple helices,
there exists one and only one inter-helix PI-line for any pair of helices). The proposed algorithm is of the
backprojection-filtration (BPF) type and can be implemented in three steps: 1) differentiation of the cone-beam
projection from each source; 2) weighted backprojection of the derivates on the inter-helix PI-arcs; 3) inverse Hilbert
transformation along one of the three inter-helix PI-lines. Numerical simulations with 3D Shepp-Logan phantoms are
performed to validate the algorithm. We also demonstrate that artifacts are generated when the algorithm for the
symmetric configuration is applied to the unequally spaced helices setting.
Multiple source helical cone-beam scanning is a promising technique for dynamic volumetric CT/micro-CT. In the previous studies, we had proposed a helical cone-beam scanning mode with triple x-ray source and detector assemblies that are symmetrically arranged, and proved the property of minimum detection windows under this configuration. Moreover, we had established an exact backprojection filtration (BFP) reconstruction algorithm for this setting. In this paper, we perform simulation studies for this reconstruction algorithm with 3D Shepp-Logan and Defrise phantoms. The implementation of the BFP algorithm in the planar detector geometry consists of three steps. First, the cone-beam projection from each of the three sources is differentiated respectively. Second, the derivates on the three inter-helix PI-arcs are summed up with weights to form the backprojection. Third, inverse Hilbert transformations are performed along each of the three inter-helix PI-lines. The reconstructed images validate the proposed algorithm. Furthermore, this work can be generalized to the case of multiple source helical cone-beam CT.
Motivated by bioluminescent imaging needs for studies on gene therapy and other applications in the mouse models, a bioluminescence tomography (BLT) system is being developed by our group. While the forward imaging model is described by the diffusion approximation, BLT is the inverse problem to recover an internal bioluminescent source distribution subject to Cauchy data for the diffusion equation. This inverse source problem is ill-posed and does not yield the unique solution in the general case. The
uniqueness problem under practical constraints was recently studied by our group. It was found that all the inverse source solutions can be expressed as the unique minimal energy source solution plus a nonradiating source. We demonstrate that the minimal energy source solution is not physically favorable for bioluminescence tomography, although the minimal energy constraint is utilized in other applications. To find a physically meaningful unique solution, adequate prior knowledge must be utilized. Here we propose two iterative approaches in this work. The first one is a variant of the well-known EM algorithm. The second one is based on the Landweber scheme. Either of the methods is suitable for incorporating
knowledge-based constraints. We discuss several issues related to the implementation of these methods, including the initial guess and stopping criteria. Also, we report our numerical simulation results to demonstrate the feasibility of bioluminescence tomography.
In this paper, we propose a helical cone-beam scanning configuration of triple symmetrically located X-ray sources, and study minimum detection windows to extend the traditional Tam-Danielsson window for exact image reconstruction. For three longitudinally displaced scanning helices of the same radius and a source location on any helix, the corresponding minimum detection window is bounded by the most adjacent turns respectively selected from the other two helices. The height of our proposed minimum detector window is only 1/3 of that in the single helix case. Associated with proposed minimum detection windows, we define the inter-helix PI-line and establish its existence and uniqueness property: through any point inside the triple helices, there exists one and only one inter-helix PI-line for any pair of the helices. Furthermore, we prove that cone-beam projection data from such a triple-source helical scan are sufficient for exact image reconstruction. Although there are certain redundancies among those projection data, the redundant part cannot be removed by shrinking the detector window without violating the data sufficiency condition. Those results are important components for development of exact or quasi-exact image reconstruction algorithms in the case of triple-source helical cone-beam scanning in the future.
The Landweber method provides a framework to formulate iterative
algorithms for image reconstruction problems with large, sparse and
unstructured system matrices. In a previous study, the authors established the convergence conditions for a general Landweber scheme in both simultaneous and block-iterative [or ordered-subset
(OS)] formats with either consistent or inconsistent data, without constraints. Constrained iterative algorithms provide a mechanism
for incorporating prior knowledge such nonnegativity, bounds, finite
spatial or spectral supports, <i>etc</i>. Hence, they have been widely used in practice. Although the simultaneous constrained (or projected) Landweber scheme was well studied, the convergence of the
constrained block-iterative Landweber scheme is unknown. Block-iterative schemes are recently intensively studied theoretically and applied widely. In this paper, we report convergence conditions of a constrained block-iterative Landweber scheme. Prior knowledge is represented as convex sets in which an image of interest must stay. The constrained block-iterative Landweber scheme is constructed by alternatively performing a projection onto convex sets (POCS) and a conventional block-iterative Landweber iteration. The POCS method has been used before for constrained image reconstruction to satisfy both imaging equations and convex constraints. Our approach is different from the conventional application of the POCS method in that we use Landweber iteration for the imaging equations and perform POCS only for the convex constraints. While the conventionally applied POCS method requires Moore-Penrose inverses of matrix blocks, our constrained block-iterative method only takes transposes of such matrix blocks, and improves the computational complexity greatly.