Likelihood-based bilateral filters for pre-estimated basis sinograms using photon-counting CT

Noise amplification in material decomposition is an issue for exploiting photon-counting computed tomography (PCCT). Regularization techniques and neighborhood filters have been widely used, but degraded spatial resolution and bias are concerns.

This paper proposes likelihood-based bilateral filters that can be applied to pre-estimated basis sinograms to reduce the noise while minimally affecting spatial resolution and accuracy.

The proposed method needs system models (e.g., incident spectrum, detector response) to calculate the likelihood. First, it performs maximum likelihood (ML)-based estimation in the projection domain to obtain basis sinograms. The estimated basis sinograms suffer from severe noise but are asymptotically unbiased without degrading spatial resolution. Then it calculates the neighborhood likelihoods for a given measurement at the center pixel using the neighborhood estimates and designs the weights based on the distance of likelihoods. It is also analyzed in terms of statistical inference, and then two variations of the filter are introduced: one that requires the significance level instead of the empirical hyperparameter. The other is a measurement-based filter, which can be applied when accurate estimates are given without the system models. The proposed methods were validated by analyzing the local property of noise and spatial resolution and the global trend of noise and bias using numerical thorax and abdominal phantoms for a two-material decomposition (water and bone). They were compared to the conventional neighborhood filters and the model-based iterative reconstruction with an edge-preserving penalty applied in the basis images.

The proposed method showed comparable or superior performance for the local and global properties to conventional methods in many cases. The thorax phantom: The full width at half maximum (FWHM) decreased by -2%-31% (-2 indicates that it increased by 2% compared to the best performance from conventional methods), and the global bias was reduced by 2%-19% compared to other methods for similar noise levels (local: 51% of the ML, global: 49%) in the water basis image. The FWHM decreased by 8%-31%, and the global bias was reduced by 9%-44% for similar noise levels (local: 44% of the ML, global: 36%) in the CT image at 65 keV. The abdominal phantom: The FWHM decreased by 10%-32%, and the global bias was reduced by 3%-35% compared to other methods for similar noise levels (local: 66% of the ML, global: 67%) in the water basis image. The FWHM decreased by up to -11%-47%, and the global bias was reduced by 13%-35% for similar noise levels (local: 71% of the ML, global: 70%) in the CT image at 60 keV.

This paper introduced the likelihood-based bilateral filters as a post-processing method applied to the ML-based estimates of basis sinograms. The proposed filters effectively reduced the noise in the basis images and the synthesized monochromatic CT images. It showed the potential of using likelihood-based filters in the projection domain as a substitute for conventional regularization or filtering methods.