Dix inversion constrained by L1-norm optimization

Conclusions and Discussions

We explore three methods to constrain Dix inversion in an $L1$ nature: an improved IRLS method, a hybrid $L1/L2$ method and conjugate direction $L1$ method. The IRLS and hybrid methods are implemented in a non-linear least-squares scheme by adding a diagonal weighting function. Conjugate direction $L1$ method is realized by a weighted median solver.

The IRLS method is improved by the physical explanation of the cutoff number $\sigma$, allowing this numerical parameter to be determined automatically. The hybrid method has a novel plane search scheme based on Taylor's series at each residual. Conjugate direction $L1$ method has an iterative plane search scheme using a weighted median solver. Both of hybrid and the conjugate direction $L1$ are designed to reduce major computational cost by expending more effort in finding a better next step.

In the numerical experiment, we find that the conjugate direction $L1$ method decreases the iteration number for the outer loop significantly. Hence, the value of spending more to find a better next step is proved. The same concept can be applied to the hybrid method as well. We can expect better inversion results and faster convergence by adding iterations to plane search, which has not been demonstrated before.

In the current study of the conjugate direction $L1$ method, we keep only two equations exactly satisfied in the whole system when searching the plane. In future research, we can add as many equations as needed by Gram Schmidt process. Hopefully, this process can lead us to an even better next step.

Dix inversion constrained by L1-norm optimization