Increasingly, three-dimensional (3-D) imaging technologies are used in medical diagnosis, for therapy planning, and during interventional procedures. We describe the possibilities of fast 3-D-reconstruction of high-contrast objects with high spatial resolution from only a small series of two-dimensional (2-D) planar radiographs. The special problems arising from the intended use of an open, mechanically unstable C-arm system are discussed. For the description of the irregular sampling geometry, homogeneous coordinates are used thoroughly. The well-known Feldkamp algorithm is modified to incorporate corresponding projection matrices without any decomposition into intrinsic and extrinsic parameters. Some approximations to speed up the whole reconstruction procedure and the tradeoff between image quality and computation time are also considered. Using standard hardware the reconstruction of a 256(3) cube is now possible within a few minutes, a time that is acceptable during interventions. Examples for cranial vessel imaging from some clinical test installations will be shown as well as promising results for bone imaging with a laboratory C-arm system.

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This paper addresses reprojection of three-dimensional (3-D) reconstructions obtained from cone-beam scans using a C-arm imaging equipment assisted by a pose-determining system. The emphasis is on reprojecting without decomposing the estimated projection matrix (P-matrix) associated with a pose. Both voxel- and ray-driven methods are considered. The voxel-driven reprojector follows the algorithm for backprojection using a P-matrix. The ray-driven reprojector is derived by extracting from the P-matrix the equation of the line joining a detector-pixel and the X-ray source position. This reprojector can be modified to a ray-driven backprojector. When the geometry is specified explicitly in terms of the physical parameters of the imaging system, the projection matrices can be constructed. The resulting "projection-matrix method" is advantageous, especially when the scanning trajectory is irregular. The algorithms presented are useful in iterative methods of image reconstruction and enhancement procedures, apart from their well-known role in visualization and volume rendering. Reprojections of 3-D patient data compare favorably with the original X-ray projections obtained from a prototype C-arm system. The algorithms for reprojection can be modified to compute perspective maximum intensity projection.

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.

We present an efficient implementation of an approximate cone-beam image reconstruction algorithm for application in tomography, which accounts for scanner mechanical misalignment. The implementation is based on the algorithm proposed by Feldkamp et al. and is directed at circular scan paths. The algorithm has been developed for the purpose of reconstructing volume data from projections acquired in an experimental x-ray micro-tomography (microCT) scanner. To mathematically model misalignment we use matrix notation with homogeneous coordinates to describe the scanner geometry, its misalignment, and the acquisition process. For convenience analysis is carried out for x-ray CT scanners, but it is applicable to any tomographic modality, where two-dimensional projection acquisition in cone beam geometry takes place, e.g., single photon emission computerized tomography. We derive an algorithm assuming misalignment errors to be small enough to weight and filter original projections and to embed compensation for misalignment in the backprojection. We verify the algorithm on simulations of virtual phantoms and scans of a physical multidisk (Defrise) phantom.