N. Navab, A. Bani-Hashemi, M. Nadar, K. Wiesent, P. Durlak, T. Brunner, K. Barth, R. Graumann 3D Reconstruction from Projection Matrices in a C-Arm Based 3D-Angiography System Proceedings of Medical Image Computing and Computer-Assisted Intervention (MICCAI 1998), Cambridge, MA, October 1998 (bib)
	3D reconstruction of arterial vessels from planar radiographs obtained at several angles around the object has gained increasing interest. The motivating application has been interventional angiography. In order to obtain a three-dimensional reconstruction from a C-arm mounted X-Ray Image Intensifier (XRII) traditionally the trajectory of the source and the detector system is characterized and the pixel size is estimated. The main use of the imaging geometry characterization is to provide a correct 3D-2D mapping between the 3D voxels to be reconstructed and the 2D pixels on the radiographic images. We propose using projection matrices directly in a voxel driven backprojection for the reconstruction as opposed to that of computing all the geometrical parameters, including the imaging parameters. We discuss the simplicity of the entire calibration-reconstruction process, and the fact that it makes the computation of the pixel size, source to detector distance, and other explicit imaging parameters unnecessary. A usual step in the reconstruction is sinogram weighting, in which the projections containing corresponding data from opposing directions have to be weighted before they are filtered and backprojected into the object space. The rotation angle of the C-arm is used in the sinogram weighting. This means that the C-arm motion parameters must be computed from projection matrices. The numerical instability associated with the decomposition of the projection matrices into intrinsic and extrinsic parameters is discussed in the context. The paper then describes our method of computing motion parameters without matrix decomposition. Examples of the calibration results and the associated volume reconstruction are also shown.
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