Optimization of Low-Dose Tomography via Binary Sensing Matrices

Abstract

X-ray computed tomography (CT) is one of the most widely used imaging modalities for diagnostic tasks in the clinical application. As X-ray dosage given to the patient has potential to induce undesirable clinical consequences, there is a need for reduction in dosage while maintaining good quality in reconstruction. The present work attempts to address low-dose tomography via an optimization method. In particular, we formulate the reconstruction problem in the form of a matrix system involving a binary matrix. We then recover the image deploying the ideas from the emerging field of compressed sensing (CS). Further, we study empirically the radial and angular sampling parameters that result in a binary matrix possessing sparse recovery parameters. The experimental results show that the performance of the proposed binary matrix with reconstruction using TV minimization by Augmented Lagrangian and ALternating direction ALgorithms (TVAL3) gives comparably better results than Wavelet based Orthogonal Matching Pursuit (WOMP) and the Least Squares solution.

A Appendix: Wavelet Based Orthogonal Matching Pursuit (WOMP)

The conventional form of orthogonal matching pursuit proposed by Troop et al. [12] is a greedy method which builds up the support set of the reconstructed sparse vector iteratively by adding one index to the current support set at each iteration. The input parameters for the conventional OMP algorithm are the measurement matrix (binary matrix) and the measurement vector. Here, we modified the existing OMP algorithm by incorporating the sparsifying transform (i.e. wavelet transform) to further sparsify the binary matrix. We call the modified algorithm as WOMP, which is given below:

Algorithm

Input Parameters: measurement matrix A, wavelet matrix W, measurement vector b, and the error threshold $\epsilon _{0}$ Initialization: Initialize $k=0$, and set

• The initial solution $(Wx)^{0}=0$.

• The initial residual $r^{0}=b-(AW^{T})(Wx)^{0}=b$.

• The initial solution support $S^{0}=Support\{(Wx)^{0}\}=\phi$

Main Iteration: Increment k by 1 and perform the following steps:

• Sweep: Compute the errors $\epsilon (j)=\min _{z_{j}}\Vert (a_{j} w_{j}^{T})z_{j}-r^{k-1}\Vert _{2}^{2}$ for all j using the optimal choice $z_{j}^{*}=(a_{j}^{T} w_{j}^{T})^{T} r^{k-1}/\Vert a_{j}w_{j}^{T}\Vert _{2}^{2}$.

• Update Support: Find a minimizer, $j_{0}$ of $\epsilon (j):\forall j\notin S^{k-1}, \epsilon (j_{0})\leqslant \epsilon (j)$, and update $S^{k}=S^{k-1}\cup \{j_{0}\}$.

• Update Provisional Solution: Compute $(Wx)^{k}$, the minimizer of $\Vert (AW^{T}) (Wx)-b\Vert _{2}^{2}$ subject to $Support\{(Wx)\}=S^{k}$

• Update Residual: Compute $r^{k}=b-(AW^{T})(Wx)^{k}$.

• Stopping Rule: If $\Vert r^{k}\Vert _{2}<\epsilon _{0}$, stop. Otherwise, apply another iteration.

Output: The WOMP solution is $(Wx)^{k}$ obtained after k iterations.