In this repo, I will present the stochastic optimization algorithms for reconstructing heterogeneous materials using X-ray tomographic projections. As the described in detail below, with stochastic reconstruction, one can accurately reconstructe 3D binary representation of microstructures using limited (around 20 to 40) x-ray tomography projections, which is a significant decrease compared to the traditional reconstruction algorithm like Filtered-backprojection algorithm (FBP), which requires several thousands of projections. Our reconstruction algorithm would save huge space and time collecting and saving x-ray projection data, which will significantly improve the efficiency and application of this technique into 4D material science.
- X-ray tomography
- Parallel-beam geometry
- Cone-beam geometry
- Generating X-ray projections
- 3D reconstruction
- Stochastic optimization
- Simulated annealing
- Reconstruction result
- Multi-modal reconstruction
X-ray tomography microscopy, when properly combined with in situ experiments, is an extremely attractive, non-destructive technique for characterizing microstructure in 3D and 4D. The use of high brilliance and partially coherent synchrotron light allows one to image multi-component materials from the sub-micrometer to nanometer range. X-ray tomography can be conducted in imaging modes based on absorption or phase contrast. The technique can also be used using lab-scale systems. In x-ray tomography, 2D projections are usually obtained at small angular increments. Given a sufficiently large number of such 2D projections, different reconstruction algorithms can be allpied to reconstruction the 3D representation of the target microstructure.
In x-ray tomographic microscopy, an x-ray beam is generated from an x-ray source and then the rays pass through the material sample that is being examined and eventually being captured on a screen, or a detector to generate the tomographic radiographs. As an x-ray passes through the material, some of photons will be scattered and/or absorbed by the materials via different light-matter interactions. Without going into details of such interactions, one can consider that the intensity of the x-ray is attenuated as it passes through the material. Based on the x-ray source, the x-ray tomgoraphy can be categorized as parallel beam geometry or cone beam geometry. For the parallel-beam projection geometry, a set of parallel rays are sent through the material sample from different projection angles and the attenuated intensities of the x-rays passing through the sample are recorded via a detector behind the sample.
Different from the parallel-beam projection geometry, the cone-beam geometry has a point x-ray source, and the x-rays are radically emitted from the source to form a cone pattern. A consequence of such projection geometry is that the spatial density of incident rays is higher at the front side of the material sample which is close to the source and is lower at rear side of the sample. Therefore, each voxel may affect the total attenuation values in multiple detector bins and thus, a detector-bin-based projection method is used to compute the contribution of each voxel to the total attenuation data.
Once we obtain the projections, the next step is to develop the algorithm for 3D reconstruction. In the current project, the reconstruction algorithm is followed the simulated annealing procedure, which considers the reconstruction problem as an inverse optimization problem and allows one to generate virtual microstructures compatible with prescribed structural information and statistics. In particular, we formulate the reconstruction problem as an “energy” minimization problem, where the energy functional 'E' is defined as the square difference of x-ray projections between the target data set 'D' and the corresponding data set 'D*'. 'D' and 'D*' can be considered as target microstructure and the trial microstructure during reconstruction procedure. The assumption is that, with enough projections, we can accurately evolve the trial microstructure to the target microstructure as we reduce the energy to a tolerance level.
Based on wikipedia, simulated annealing is a probabilistic technique for approximating the global optimum of a given function. The name and inspiration come from annealing in metallurgy, a technique involving heating and controlled cooling of a material to increase the size of its crystals and reduce their defects. The notion of cooling implemented in the simulated annealing algorithm is interpreted as a slow decrease in the probability of accepting worse solutions as the solution space is explored (accepting worse solutions is a fundamental property of metaheuristics because it allows for a more extensive search for the optimal solution).
Implementing x-ray tomographic projections and simulated annealing algorithm into C++ code, we were able to reconstruct microstructures in both 2D and 3D. Several heterogeneous structures and composite materials were tested in both parallel beam and conebeam geometry.
Reconstruction of sandstone using both parallel beam and cone beam geometry
Reconstruction of Sn-spheres/Clay composite material
In order to further reduce the projection data, one can adding complementary data as the reconstruction input data, together with the projection data. As the result shows that two-point correlation function (these will be talked in detail in the future repository) is the perfect candidate to do the job.
