Analytical derivation of the point spread function for pinhole collimators

The point spread function (PSF) of a pinhole collimator plays an important role in determining the resolution and characterizing the sensitivity of the accepted photons from a given point in the image space. The focus of this paper is to derive an analytical expression for the PSF of two different types of focusing pinhole collimators that are based on (1) right-circular double cones and (2) oblique-circular double cones. Conventionally, focusing pinhole collimators used in multi-pinhole SPECT were designed using right-circular double cones, as they were easier to fabricate. In this work, a novel focusing collimator consisting of oblique-circular double cones was designed and its properties were studied in detail with respect to right-circular double-cone based collimators. The main advantage of determining the PSF is the fact that they can be used to accurately model the PSF during the reconstruction, thereby improving the resolution of the reconstructed image. The PSF of the focusing collimators based on oblique-circular cones were found to be almost shift invariant for low and medium energy photons (below 200 keV). This property is very advantageous as algorithms such as slice-by-slice reconstruction can be used for resolution recovery thereby drastically reducing the reconstruction time. However, the PSF of focusing oblique-circular double cones (FOCDC) for higher energy photons were found to be asymmetric and hence need to be modelled more accurately during the reconstruction. On the other hand, the PSF for the right-circular cone based collimators were found to be asymmetric for all energy levels. However, due to the smaller acceptance angle used, the number of penetration photons was found to be far less than that observed for oblique-circular cones. This results in a smaller PSF making right-circular cone based collimators preferable for high-resolution small animal imaging, especially where very small pinhole diameters are used. The analytically derived PSF for both collimators were validated using a ray-tracing based Monte Carlo approach and found to agree well with a mean square error of less than 1%.