Information‐theoretic discrepancy based iterative reconstructions (IDIR) for polychromatic x‐ray tomography

Purpose: X‐ray photons generated from a typical x‐ray source for clinical applications exhibit a broad range of wavelengths, and the interactions between individual particles and biological substances depend on particles' energy levels. Most existing reconstruction methods for transmission tomography, however, neglect this polychromatic nature of measurements and rely on the monochromatic approximation. In this study, we developed a new family of iterative methods that incorporates the exact polychromatic model into tomographic image recovery, which improves the accuracy and quality of reconstruction.Methods: The generalized information‐theoretic discrepancy (GID) was employed as a new metric for quantifying the distance between the measured and synthetic data. By using special features of the GID, the objective function for polychromatic reconstruction which contains a double integral over the wavelength and the trajectory of incident x‐rays was simplified to a paraboloidal form without using the monochromatic approximation. More specifically, the original GID was replaced with a surrogate function with two auxiliary, energy‐dependent variables. Subsequently, the alternating minimization technique was applied to solve the double minimization problem. Based on the optimization transfer principle, the objective function was further simplified to the paraboloidal equation, which leads to a closed‐form update formula. Numerical experiments on the beam‐hardening correction and material‐selective reconstruction were conducted to compare and assess the performance of conventional methods and the proposed algorithms.Results: The authors found that the GID determines the distance between its two arguments in a flexible manner. In this study, three groups of GIDs with distinct data representations were considered. The authors demonstrated that one type of GIDs that comprises “raw” data can be viewed as an extension of existing statistical reconstructions; under a particular condition, the GID is equivalent to the Poisson log‐likelihood function. The newly proposed GIDs of the other two categories consist of log‐transformed measurements, which have the advantage of imposing linearized penalties over multiple discrepancies. For all proposed variants of the GID, the aforementioned strategy was used to obtain a closed‐form update equation. Even though it is based on the exact polychromatic model, the derived algorithm bears a structural resemblance to conventional methods based on the monochromatic approximation. The authors named the proposed approach as information‐theoretic discrepancy based iterative reconstructions (IDIR). In numerical experiments, IDIR with raw data converged faster than previously known statistical reconstruction methods. IDIR with log‐transformed data exhibited superior reconstruction quality and faster convergence speed compared with conventional methods and their variants.Conclusions: The authors' new framework for tomographic reconstruction allows iterative inversion of the polychromatic data model. The primary departure from the traditional iterative reconstruction was the employment of the GID as a new metric for quantifying the inconsistency between the measured and synthetic data. The proposed methods outperformed not only conventional methods based on the monochromatic approximation but also those based on the polychromatic model. The authors have observed that the GID is a very flexible means to design an objective function for iterative reconstructions. Hence, the authors expect that the proposed IDIR framework will also be applicable to other challenging tasks.