Bi-planar 2D-to-3D Registration in Fourier Domain for Stereoscopic X-Ray Motion Tracking

In this paper we present a new method to track bone movements in stereoscopic X-ray image series of the knee joint. The method is based on two different X-ray image sets: a rotational series of acquisitions of the still subject knee that will allow the tomographic reconstruction of the three-dimensional volume (model), and a stereoscopic image series of orthogonal projections as the subject performs movements. Tracking the movements of bones throughout the stereoscopic image series means to determine, for each frame, the best pose of every moving element (bone) previously identified in the 3D reconstructed model. The quality of a pose is reflected in the similarity between its simulated projections and the actual radiographs. We use direct Fourier reconstruction to approximate the three-dimensional volume of the knee joint. Then, to avoid the expensive computation of digitally rendered radiographs (DRR) for pose recovery, we reformulate the tracking problem in the Fourier domain. Under the hypothesis of parallel X-ray beams, we use the central-slice-projection theorem to replace the heavy 2D-to-3D registration of projections in the signal domain by efficient slice-to-volume registration in the Fourier domain. Focusing on rotational movements, the translation-relevant phase information can be discarded and we only consider scalar Fourier amplitudes. The core of our motion tracking algorithm can be implemented as a classical frame-wise slice-to-volume registration task. Preliminary results on both synthetic and real images confirm the validity of our approach.