title: A comparison of linear interpolation models for iterative CT reconstruction.
abstract: OBJECTIVE
  Recent reports indicate that model-based iterative reconstruction methods may improve image quality in computed tomography (CT). One difficulty with these methods is the number of options available to implement them, including the selection of the forward projection model and the penalty term. Currently, the literature is fairly scarce in terms of guidance regarding this selection step, whereas these options impact image quality. Here, the authors investigate the merits of three forward projection models that rely on linear interpolation: the distance-driven method, Joseph's method, and the bilinear method. The authors' selection is motivated by three factors: (1) in CT, linear interpolation is often seen as a suitable trade-off between discretization errors and computational cost, (2) the first two methods are popular with manufacturers, and (3) the third method enables assessing the importance of a key assumption in the other methods.
  
  
  METHODS
  One approach to evaluate forward projection models is to inspect their effect on discretized images, as well as the effect of their transpose on data sets, but significance of such studies is unclear since the matrix and its transpose are always jointly used in iterative reconstruction. Another approach is to investigate the models in the context they are used, i.e., together with statistical weights and a penalty term. Unfortunately, this approach requires the selection of a preferred objective function and does not provide clear information on features that are intrinsic to the model. The authors adopted the following two-stage methodology. First, the authors analyze images that progressively include components of the singular value decomposition of the model in a reconstructed image without statistical weights and penalty term. Next, the authors examine the impact of weights and penalty on observed differences.
  
  
  RESULTS
  Image quality metrics were investigated for 16 different fan-beam imaging scenarios that enabled probing various aspects of all models. The metrics include a surrogate for computational cost, as well as bias, noise, and an estimation task, all at matched resolution. The analysis revealed fundamental differences in terms of both bias and noise. Task-based assessment appears to be required to appreciate the differences in noise; the estimation task the authors selected showed that these differences balance out to yield similar performance. Some scenarios highlighted merits for the distance-driven method in terms of bias but with an increase in computational cost. Three combinations of statistical weights and penalty term showed that the observed differences remain the same, but strong edge-preserving penalty can dramatically reduce the magnitude of these differences.
  
  
  CONCLUSIONS
  In many scenarios, Joseph's method seems to offer an interesting compromise between cost and computational effort. The distance-driven method offers the possibility to reduce bias but with an increase in computational cost. The bilinear method indicated that a key assumption in the other two methods is highly robust. Last, strong edge-preserving penalty can act as a compensator for insufficiencies in the forward projection model, bringing all models to similar levels in the most challenging imaging scenarios. Also, the authors find that their evaluation methodology helps appreciating how model, statistical weights, and penalty term interplay together.
authors.affiliation: Pattern Recognition Laboratory, Department of Computer Science, Friedrich-Alexander-Universität, Erlangen-Nürnberg, Martensstr. 3, 91058 Erlangen, Germany; Siemens Healthcare, GmbH 91301, Forchheim, Germany; and Department of Radiology, University of Utah, Salt Lake City, Utah 84108.
authors.forename: Katharina
authors.initials: K
authors.lastname: Hahn
authors.affiliation: Siemens Healthcare, GmbH 91301, Forchheim, Germany.
authors.forename: Harald
authors.initials: H
authors.lastname: Schöndube
authors.affiliation: Siemens Healthcare, GmbH 91301, Forchheim, Germany.
authors.forename: Karl
authors.initials: K
authors.lastname: Stierstorfer
authors.affiliation: Pattern Recognition Laboratory, Department of Computer Science, Friedrich-Alexander-Universität, Erlangen-Nürnberg, Martensstr. 3, 91058 Erlangen, Germany.
authors.forename: Joachim
authors.initials: J
authors.lastname: Hornegger
authors.affiliation: Department of Radiology, University of Utah, Salt Lake City, Utah 84108.
authors.forename: Frédéric
authors.initials: F
authors.lastname: Noo
mesh_terms: D007091:Image Processing, Computer-Assisted / Q000379:methods*; D016014:Linear Models; D008962:Models, Theoretical; D059629:Signal-To-Noise Ratio; D013997:Time Factors; D014057:Tomography, X-Ray Computed