Oblique reconstructions in tomosynthesis. I. Linear systems theory

Purpose: By convention, slices in a tomosynthesis reconstruction are created on planes parallel to the detector. It has not yet been demonstrated that slices can be generated along oblique directions through the same volume, analogous to multiplanar reconstructions in computed tomography (CT). The purpose of this work is to give a proof-of-principle justification for oblique reconstructions in tomosynthesis, which acquires projection images over a smaller angular range than CT. Methods: To investigate the visibility of individual frequencies in an oblique reconstruction, a theoretical framework is developed in which the reconstruction of a sinusoidal input is calculated. The test frequency is pitched at an angle in a 2D parallel-beam acquisition geometry. Reconstructions are evaluated along the pitch of the object. The modulation transfer function (MTF) is calculated from the relative signal at various test frequencies. The MTF determines whether modulation is within detectable limits in oblique reconstructions. In the previous linear systems (LS) model [B. Zhao and W. Zhao, “Three-dimensional linear system analysis for breast tomosynthesis,” Med. Phys. 35(12), 5219–5232 (2008)], the MTF was calculated only in reconstructed slices parallel to the detector. This paper generalizes the MTF calculation to reconstructed slices at all possible pitches. Unlike the previous LS model, this paper also analyzes the effect of object thickness on the MTF. A second test object that is considered is a rod whose long axis is pitched similar to the sinusoidal input. The rod is used to assess whether the length of an object can be correctly estimated in oblique reconstructions. Results: To simulate the conventional display of the reconstruction, slices are first created along a 0° pitch. This direction is perpendicular to the rays of the central projection. The authors show that the input frequency of a pitched sinusoidal object cannot be determined using these slices. By changing the pitch of the slice to match the object, it is shown that the input frequency is properly resolved. To prove that modulation is preserved in pitched slices, the MTF is also calculated. Modulation is within detectable limits over a broad range of pitches if the object is thin, but is detectable over a narrower range of pitches if the object is thick. Turning next to the second test object, it is shown that the length of a pitched rod can be correctly determined in oblique reconstructions. Concordant with the behavior of the MTF, the length estimate is accurate over a broad range of pitches if the object is thin, but is correct over a narrower range of pitches if the object is thick. Conclusions: This work justifies the feasibility of oblique reconstructions in tomosynthesis. It is demonstrated that pitched test objects are most easily visualized with oblique reconstructions instead of conventional reconstructions. In order to achieve high image quality over a broad range of pitches, the object must be thin. By analyzing the effect of reconstruction pitch and object thickness on image quality, this paper generalizes the previous LS model for tomosynthesis.