Abstract
This paper presents the derivation of the Epipolar Consistency Conditions (ECC) between two X-ray images from the Beer-Lambert law of X-ray attenuation and the Epipolar Geometry of two pinhole cameras, using Grangeat's theorem. We motivate the use of Oriented Projective Geometry to express redundant line integrals in projection images and define a consistency metric, which can be used, for instance, to estimate patient motion directly from a set of X-ray images. We describe in detail the mathematical tools to implement an algorithm to compute the Epipolar Consistency Metric and investigate its properties with detailed random studies on both artificial and real FD-CT data. A set of six reference projections of the CT scan of a fish were used to evaluate accuracy and precision of compensating for random disturbances of the ground truth projection matrix using an optimization of the consistency metric. In addition, we use three X-ray images of a pumpkin to prove applicability to real data. We conclude, that the metric might have potential in applications related to the estimation of projection geometry. By expression of redundancy between two arbitrary projection views, we in fact support any device or acquisition trajectory which uses a cone-beam geometry. We discuss certain geometric situations, where the ECC provide the ability to correct 3D motion, without the need for 3D reconstruction.
Highlights
Consistency conditions have been known for decades, yet only few practical applications have been demonstrated
We have presented a consistency-based algorithm which can recover 2D and 3D transformations of an unknown object with a set of reference X-ray projections
Epipolar consistency assumes that the object is fully visible on the detector
Summary
Computed tomography (CT) reconstructs a three-dimensional (3D) object from its two-dimensional (2D) projection images. Projection images can be from either a transmission-only (e.g. X-ray) or emission-only modalities (e.g., single-photon emission CT in nuclear medicine). This section begins by introducing CT reconstruction and discussing how the reconstruction process is prone to various artifacts in the reconstructed images. The concept of data redundancy of multiple projection images of the same object is introduced which leads to mathematical consistency conditions. The main topic of this thesis is presented as the estimation of motion using a certain type of consistency conditions. The author’s contribution to this field of research is briefly listed, and the contents of each chapter of this thesis are briefly summarized
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