Null functions in three-dimensional imaging of alpha and beta particles

Abstract

Null functions of an imaging system are functions in the object space that give exactly zero data. Hence, they represent the intrinsic limitations of the imaging system. Null functions exist in all digital imaging systems, because these systems map continuous objects to discrete data. However, the emergence of detectors that measure continuous data, e.g. particle-processing (PP) detectors, has the potential to eliminate null functions. PP detectors process signals produced by each particle and estimate particle attributes, which include two position coordinates and three components of momentum, as continuous variables. We consider Charged-Particle Emission Tomography (CPET), which relies on data collected by a PP detector to reconstruct the 3D distribution of a radioisotope that emits alpha or beta particles, and show empirically that the null functions are significantly reduced for alpha particles if ≥3 attributes are measured or for beta particles with five attributes measured.

Fisher information analysis of list-mode SPECT emission data for joint estimation of activity and attenuation distribution

“…Of most importance, LM data yielded a lower standard deviation for the attenuation coefficient in comparison to even when up to four energy bins were considered. These results are in agreement with other recently conducted studies [31,32,36,46,47], which have all shown that binning of LM attributes leads to loss of information.…”

“…To obtain the elements of the FIM for a given value of the activity and attenuation coefficient, the quantities obtained in equations ( 54)-(57) must be averaged with respect to the observed LM data at that value of the activity and attenuation coefficient. Before averaging, note that where in the second step, we use the expressions from equation (40) and equation (47), while in the third step, the mixture-model definition (equation ( 7)) is used.…”

Towards continuous-to-continuous 3D imaging in the real world

“…On the other hand, a CC operator such as L may have a null space, but for the setup of section 3 and if K 3, the null space is not demanded by dimensionality considerations. Thus, it is possible that some-conceivably all-of the estimability problems inherent in the world of conventional digital imaging may disappear (Ding et al 2017b). Notice that replacing the operator H with L does not, in general, ensure that an inverse problem that was ill-posed (Natterer 1986, Bertero andBoccacci 1998) in the CD case becomes well-posed when a CC operator is used.…”

“…Although this paper has been limited to imaging with gamma-ray photons, the same methodology can be applied to imaging with charged particles, such as alpha and beta particles. Some work (Ding et al 2017b) has already been done in this direction, and it was shown that estimating additional parameters (such as direction and/or particle's residual energy) besides positions, does provide enormous benefits in terms of reduced null space.…”

Is there a role for image science in the brave new world of artificial intelligence?

“…In simulations of tomographic imaging with alpha or beta particles, where we can also estimate direction of travel of each particle, we have demonstrated that the null functions nearly disappear if we have 4-6 attributes per event. 30,31 3 Computational Methods for Image Science To the unitiated, concepts from image science may seem arcane, even bewildering. How do we actually compute the performance of an ideal observer or the null functions of a particular imaging system?…”