Analytical back projection

To produce an image analytical algorithms generate a back projection utilising Comp- ton kinematics, in the case of this work the back projection is a cone described by the Equation 2.3. A two dimensional approach is taken within the analytical method utilised for this work. This method projects s series of two dimensional conics onto a plane in z, generating an image on an intensity map. This two dimensional method removes the need to generate a full cone, reducing the computational time required. The trigonometry used to build the cone is shown in Figure 3.5

Figure 3.5: A simple case showing a gamma-ray Compton scattering from the front edge of the scatterer. Knowing the scattering angleθthe half radiusR of the cone can be calculated for a given distanceZ using trigonometry.

The conics are traced onto an intensity map. As the number increases the conics overlap, producing the images seen in Figure 3.6 shows this for 10 (a), 100 (b) and 1000 (c) cones.

(a) (b) (c)

Figure 3.6: Examples of images created with 10 (a), 100 (b) and 1000 (c) conics. As the number of cones increases the source position is located more accurately, with the source positioned at the point of most overlap. These images are taken from the same GAMOS simulated data set. The black box on the image indicates the position of the absorber detector in relative space.

clearer with the area of maximum overlap indicating the source position, this is true until around 10,000 cones when additional cones do not make a significant difference. A measure of the image quality can be found by taking cuts through each axis at the maximum point in the intensity distribution, the FWHM of the image can then be found in both directions to quantify the image quality, examples can be seen in Figure 3.7. A quadratic background fit is performed and the peak is then fitted using a Lorentzian function based on this background, providing the FWHM measurement. The maximum point also provides an estimate of the x and y position of the source, a fit can be used to locate the source more accurately. The image quality can also be presented as an angular resolution, this removes the dependence upon source to detector distance.

(a) (b)

Figure 3.7: Examples of the cuts taken through the X (a) and Y (b) axes taken from the GAMOS simulated data set. The Lorentzian fit is shown for each overlaid in green with the quadratic background in blue. This fit is used to provide the location of the source from the mean value and also the FWHM of the image, taken from the standard deviation of the fitted curve.

Imaging different z slices across a range will make the image come in and out of focus. The best FWHM value should be at the z slice in which the source is located. To find the distance to the source the FWHM of the reconstructed image is calculated for a number of different z slices and then plotted against the slice number. This plot will show a parabolic function as shown in Figure 3.8, where the minima gives the approximate location of the source. This method is not accurate however it does provide a guide, this will be investigated further in a later chapter.

Analytical methods in general utilise less computational power than iterative meth- ods allowing the images to be produced faster and online imaging is also possible. Analytical images are of a poor quality when compared to iterative images, this is due to the simple back projection retaining a number of artifacts from conics which do not add to the image.

Figure 3.8: Plot of the FWHM found in X against the Z slice. The source position in Z is estimated by the minima of the curve.

Methods used

In this work the detectors used are both double sided strip detectors. The simplest events to reconstruct are made up of single voxel interactions within the scatterer and absorber so only events which have energy in one strip from each side of each detector are recorded. These events are called fold 1,1,1,1.

The interaction is assumed to have occurred at the centre of the rectangular voxel, in X, Y and Z. Within our detectors this leads to an error of ±half of the strip pitch in X and Y and a ± half of the thickness error in Z when raw segmentation is used. The Z slice is defined as zero at the back edge of the absorber and increases in units of millimetres as shown in Figure 3.9.

Figure 3.9: Schematic diagram showing the zero of Z for imaging purposes as being the back edge of the absorber crystal.

To generate an image the interaction point is taken to be the centre of a square voxel with a size equal to the pitch strip and a thickeness equal to that of the detector. This will introduce a slight error when applied to the circular silicon detector used for this project however the number of interactions that occur in the affected edge voxels means that error is negligible from this. The use of the centre point of a voxel produces

an error on the position of interaction of a half strip pitch (2.5mm in this case) and a half of the thickness (4mm and 10mm in the case of the two detectors used here). The images presented represent the 2π view of the detector.

A Lorentzian fit is used to provide both the position in X and Y of the source, using the mean fit values along the cuts taken where the bin with the maximum number of conic intersections is and the image FWHM. The fit is placed on a quadratic back- ground allowing an improved measurement of the FWHM to be made, as not all of the conic overlaps present on the distribution add to the final image. Conics are rejected which will not fit onto the intensity map however others will be present due to random coincidences and also coincident events which do not fully deposit their energy.

The imaging code provided uses a series of gates to select data, these are selected on energy and also scattering angle. The energy gate will be used to place a gate around the photopeak seen within an addback spectra, the image will therefore only be created from events which have an addback energy within this selection. The angle gate is used to select data which scatters within a given selection of angles, this is especially useful at lower gamma ray energies as the increased likelihood of backscattered events can cause the source location to become lost in the image generated. All the images produced will have the detector system centred on (430,430), the use of this offset is required by the imaging code.

A variable bin size is used for the 2D image intensity map. This bin size, called the image compression, can be optimised by the user to match the available statistics, provide a good quality of image visually, allow the fitting program to fit the data properly and also the image size if known. As the image compression is increased the associated error on the source position and the image FWHM will also increase, the likelihood of a conic crossing a bin will also be increased so the statistics will not be scaled by the compression value. Examples of this can be seen in Figure 3.10. In these examples a 3mm and 5mm compression have been used, showing that the fit is better with the 5mm compression, but the image is visually more diffuse while the FWHM is actually better at 3mm. The user must decide on the compression to be used, the value used for each image created will be stated. The compression of the image and the error in the fit to measure the FWHM and position provide an estimation of the error in the final image. Half of the compression value will be added in quadrature to the error associated with the fit to provide an estimate of the error on the reconstructed image position and FWHM.