title: Modelling the penumbra in Computed Tomography1.
abstract: In computed tomography (CT), the spot geometry is one of the main sources of error in CT images. Since X-rays do not arise from a point source, artefacts are produced. In particular there is a penumbra effect, leading to poorly defined edges within a reconstructed volume. Penumbra models can be simulated given a fixed spot geometry and the known experimental setup.
  
  
  
  This paper proposes to use a penumbra model, derived from Beer's law, both to confirm spot geometry from penumbra data, and to quantify blurring in the image.
  
  
  
  Two models for the spot geometry are considered; one consists of a single Gaussian spot, the other is a mixture model consisting of a Gaussian spot together with a larger uniform spot.
  
  
  
  The model consisting of a single Gaussian spot has a poor fit at the boundary. The mixture model (which adds a larger uniform spot) exhibits a much improved fit. The parameters corresponding to the uniform spot are similar across all powers, and further experiments suggest that the uniform spot produces only soft X-rays of relatively low-energy.
  
  
  
  Thus, the precision of radiographs can be estimated from the penumbra effect in the image. The use of a thin copper filter reduces the size of the effective penumbra.
authors.affiliation: Statistics Department, University of Warwick, Warwick, UK.
authors.forename: Audrey
authors.initials: A
authors.lastname: Kueh
authors.affiliation: Warwick Manufacturing Group, University of Warwick, Warwick, UK.
authors.forename: Jason M
authors.initials: JM
authors.lastname: Warnett
authors.affiliation: Warwick Manufacturing Group, University of Warwick, Warwick, UK.
authors.forename: Gregory J
authors.initials: GJ
authors.lastname: Gibbons
authors.affiliation: Statistics Department, University of Warwick, Warwick, UK.
authors.forename: Julia
authors.initials: J
authors.lastname: Brettschneider
authors.affiliation: Statistics Department, University of Warwick, Warwick, UK.|Warwick Manufacturing Group, University of Warwick, Warwick, UK.
authors.forename: Thomas E
authors.initials: TE
authors.lastname: Nichols
authors.affiliation: Warwick Manufacturing Group, University of Warwick, Warwick, UK.
authors.forename: Mark A
authors.initials: MA
authors.lastname: Williams
authors.affiliation: Statistics Department, University of Warwick, Warwick, UK.
authors.forename: Wilfrid S
authors.initials: WS
authors.lastname: Kendall
mesh_terms: D007091:Image Processing, Computer-Assisted / Q000379:methods*; D008962:Models, Theoretical; D017711:Nonlinear Dynamics; D014057:Tomography, X-Ray Computed / Q000379:methods*
keywords: Computed tomography; focal spot; nonlinear least squares; penumbra; secondary radiation