Bi-planar 2D-to-3D registration in Fourier domain for stereoscopic x-ray motion tracking

Abstract

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 allows 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 theoretical 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 develop a corollary to the 3-dimensional central-slice theorem and reformulate the tracking problem in the Fourier domain. Under the hypothesis of parallel X-ray beams, the heavy 2D-to-3D registration of projections in the signal domain is replaced by efficient slice-tovolume 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. Results on both synthetic and real images confirm the validity of our approach.

A 2D 3D registration with low dose radiographic system for in vivo kinematic studies

“…In our application, we use the Fourier slice theorem to make the 2D 3D registration [19], [20]. This theorem states that the Fourier Transform (F T ) of a linear projection in a direction is equal to the central slice of the Fourier transform of the object in the same direction.…”

Robust initialization of 2D-3D image registration using the projection-slice theorem and phase correlation

“…Fourierbased methods 48,49 and phase correlation 50,51 have been well explored for 2D-2D and 3D-3D registration. Preliminary results on using the projection-slice theorem were presented in Van der Bom et al 52 and a similar method was proposed later on by Zosso et al 53 It should be noted that, due to reasons discussed in this paper, our method was developed to provide an approximation of the registration parameters and complete convergence is left to a 2D-3D registration method. The feasibility of the approach was tested in experiments on both in vivo and ex vivo data.…”