A novel reconstruction method based on basis function decomposition for snapshot CAXRDT system

<i>Objective.</i>X-ray diffraction (XRD) is a non-destructive technique capable of obtaining molecular structural information of materials and achieving higher sensitivity than transmission tomography (CT) for substances with similar densities. It has great potential in medical and security applications, such as rapid breast cancer screening, calculi composition analysis, and detection of drugs and explosives. Among various XRD tomography (XRDT) systems, snapshot coded aperture XRDT (SCA-XRDT) achieves the fastest scanning speed, making it well-suited for practical medical imaging and security inspection. However, SCA-XRDT suffers from poor data condition and an ill-posed reconstruction problem, leading to significant challenges in accurate image reconstruction. In this work, we explore the inherent characteristics of XRD patterns and incorporate a novel and effective prior accordingly into an iterative reconstruction algorithm, thereby improving the reconstruction performance.<i>Approach.</i>By analyzing the key physical factors that shape XRD patterns, we represent XRD patterns as a linear combination of basis functions, and validate the feasibility and generality of this representation using experimental data. Building upon this, we propose a novel basis-function-decomposition reconstruction (BFD-Recon) method that incorporates the basis function representation as a prior into a model-based SCA-XRDT reconstruction framework. This method transforms the optimization target from entire XRD patterns to parameters of basis functions. We further impose smoothness and sparsity constraints on the parameters to restrict the solution space. We employ the Split Bregman algorithm to iteratively solve the optimization problem. Both simulation and experimental results demonstrate the effectiveness of the proposed BFD-Recon method.<i>Main-results.</i>Compared with a conventional MBIR method for XRDT reconstruction, the proposed BFD-Recon method results in more accurate reconstruction of XRD patterns, especially the sharp peaks that closely match the ground truth. It substantially suppresses the noise and the impact of background signals on the reconstructed XRD patterns. Since the proposed basis function decomposition and the prior align well with the characteristics of XRD patterns, its value is well manifested along the spectral dimension of the reconstructed images. It also reduces blur along the x-ray path in the spatial dimension. Quantitatively, BFD-Recon increases the correlation coefficients between the reconstructed and ground-truth XRD patterns by up to 10% and the average PSNR by 20%.<i>Significance.</i>Through theoretical analysis and experiments, we propose a basis function decomposition method for XRD patterns and demonstrate its effectiveness and general applicability. Incorporating the basis-function-decomposition into the model-based iterative reconstruction can significantly enhance the XRDT reconstruction performance. The method provides prior information on XRD patterns and reduces the number of unknowns by at least one order of magnitude by transforming the optimization target to basis function parameters, which effectively alleviates the ill-posedness of the reconstruction problem.