Bone-tissue decomposition of a single X-ray image via solving a Laplace equation

Because bones are often enveloped by soft tissues, their visibility in X-ray images is compromised, resulting in a lack of clarity. Addressing this challenge, our article introduces an innovative approach to virtually decompose an X-ray image into distinct components: one representing soft tissues and the other, the bone structure. To achieve this separation, we have formulated a novel mathematical model. With proper assumptions, the model is reduced to a standard Laplace equation, which has fast numerical solvers. Our method has two important properties. First, the bone image derived from this process is theoretically guaranteed to have enhanced contrast relative to the original, thereby accentuating the visibility of bony details. Second, our method is computationally fast. Our method can process a $2\text{,}044\times 1\text{,}514$ resolution image within 0.35 s on a laptop (8.8 million pixels per second). Our methodology has been validated through a series of numerical experiments, demonstrating its efficacy and efficiency. With such performance, this technique holds promise for a broad spectrum of X-ray imaging applications, including but not limited to clinical diagnostics, surgical planning, pattern recognition, and advanced deep learning applications.