Implicitly defined PCCT material decomposition estimator and learned physics-informed neural proxy

Photon counting detector-based CT systems provide spectral count measurements, enabling material decomposition for quantitative imaging. Maximum-likelihood estimation (MLE) for material decomposition offers asymptotically unbiased and efficient (minimum variance) results but is usually solved iteratively, making the entire process computationally expensive and time-consuming. Conversely, representative empirical methods relying on grid calibration aim to construct a direct measurement-decomposition conversion, which can be fast but may suffer from bias or noise amplification. In this work, we show that the iterative MLE method implicitly defines the functional mapping from measurements to estimates, i.e. material decomposition results, and the corresponding mean and noise yield analytical approximation forms from the Implicit Function Theorem. From this perspective, we demonstrate that it is possible to distill knowledge from the implicit function defined by the iterative MLE, i.e., finding the explicit proxy, by leveraging universal approximators such as neural networks and the derivative-aware Sobolev Learning paradigm. We show that the proposed method, namely proxy MD, is both computationally efficient (providing >3000 times speedup in numerical simulation) and approaches the performance of iterative MLE. Thus, it outperforms conventional empirical methods and enables high-quality real-time quantitative spectral imaging.