Modelling the penumbra in Computed Tomography1

Abstract

Background: 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.

Objective: 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.

Methods: 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.

Results: 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.

Conclusions: 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.

Keywords: Computed tomography; focal spot; nonlinear least squares; penumbra; secondary radiation.

Fig.1

The above plots show the varying intensity of rows in the spot model, which is introduced in Section 2, with two different spot sizes. The difference between the two is most pronounced in the penumbra, the blurring at the edge of the object image. When the spot size is small, the penumbra consists of gradient discontinuities; when the spot size is large, the penumbra is almost linear.

Fig.2

Horizontal cross-section of the experimental setup. A possible X-ray path from the source at point S to the detector at point K is shown.

Fig.3

Tails from a single Gaussian spot (top) and the mixture of a Gaussian spot and the uniform spot (bottom). The heavier tails from the mixture can only be seen by magnifying the tail.

Fig.4

A sample radiograph, examined in depth in Sections 4.1 and 4.2. The dotted boxes to either side of the central cylinder profile indicate areas on the radiographs which are used to judge goodness of fit.

Fig.5

Absolute residuals of the single Gaussian spot model, as indicated by Equation (2). Absolute residuals are plotted so that larger residuals stand out. The residuals in close proximity to the cylinder found in columns 870-910 and 1040-1080 are high in absolute value compared to the rest of the image, thus the fit is poor.

Fig.6

Raw data 5 to 75 pixels from the estimated boundaries of the cylinder in the whole image are adjusted for distance from the cylinder, and plotted here. The intensities flatten out at the mean grey level of air approximately 40 pixels from the boundary. This contradicts the single Gaussian model, which predicts that this would happen within 10 pixels from the boundary.

Fig.7

Absolute residuals of the mixture model. The residuals resemble white noise, so the model is adequate at explaining the data.

Fig.8

Properties of the model uniform spot when plotted against power. Note that they are broadly similar across all powers.

Fig.9

Plot of D_g against power. The curve shows the expected unsharpness, as detailed in Equation 5.

Fig.10

Absolute residuals of the single Gaussian spot model of the image. The residuals do not resemble those of white noise, thus the fit is inadequate.

Fig.11

Absolute residuals of the single Gaussian spot model of the filtered image.