The paper describes a new accurate two-dimensional (2D) image reconstruction method consisting of two steps. In the first step, the backprojected image is formed after taking the derivative of the parallel projection data. In the second step, a Hilbert filtering is applied along certain lines in the differentiated backprojection (DBP) image. Formulae for performing the DBP step in fanbeam geometry are also presented. The advantage of this two-step Hilbert transform approach is that in certain situations, regions of interest (ROIs) can be reconstructed from truncated projection data. Simulation results are presented that illustrate very similar reconstructed image quality using the new method compared to standard filtered backprojection, and that show the capability to correctly handle truncated projections. In particular, a simulation is presented of a wide patient whose projections are truncated laterally yet for which highly accurate ROI reconstruction is obtained.
This paper is about calibration of cone-beam (CB) scanners for both x-ray computed tomography and single-photon emission computed tomography. Scanner calibration refers here to the estimation of a set of parameters which fully describe the geometry of data acquisition. Such parameters are needed for the tomographic reconstruction step. The discussion is limited to the usual case where the cone vertex and planar detector move along a circular path relative to the object. It is also assumed that the detector does not have spatial distortions. We propose a new method which requires a small set of measurements of a simple calibration object consisting of two spherical objects, that can be considered as 'point' objects. This object traces two ellipses on the detector and from the parametric description of these ellipses, the calibration geometry can be determined analytically using explicit formulae. The method is robust and easy to implement. However, it is not fully general as it is assumed that the detector is parallel to the rotation axis of the scanner. Implementation details are given for an experimental x-ray CB scanner.
A data sufficiency condition for 2D or 3D region-of-interest (ROI) reconstruction from a limited family of line integrals has recently been introduced using the relation between the backprojection of a derivative of the data and the Hilbert transform of the image along certain segments of lines covering the ROI. This paper generalizes this sufficiency condition by showing that unique and stable reconstruction can be achieved from an even more restricted family of data sets, or, conversely, that even larger ROIs can be reconstructed from a given data set. The condition is derived by analysing the inversion of the truncated Hilbert transform, here defined as the problem of recovering a function of one real variable from the knowledge of its Hilbert transform along a segment which only partially covers the support of the function but has at least one end point outside that support. A proof of uniqueness and a stability estimate are given for this problem. Numerical simulations of a 2D thorax phantom are presented to illustrate the new data sufficiency condition and the good stability of the ROI reconstruction in the presence of noise.
This work is concerned with 2D image reconstruction from fan-beam projections. It is shown that exact and stable reconstruction of a given region-of-interest in the object does not require all lines passing through the object to be measured. Complete (non-truncated) fan-beam projections provide sufficient information for reconstruction when 'every line passing through the region-of-interest intersects the vertex path in a non-tangential way'. The practical implications of this condition are discussed and a new filtered-backprojection algorithm is derived for reconstruction. Experiments with computer-simulated data are performed to support the mathematical results.
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