Kim H J, Zeeberg B R, Reba R C
Department of Radiology, George Washington University, Washington, DC.
Phys Med Biol. 1993 Jul;38(7):881-95.
For the reconstruction of a series of computer simulations of statistically-independent noisy realizations of projection data, the total error of the ith reconstructed voxel in the rth realization, Er,i, is composed of the statistical error, Sr,i, and the (deterministic) inaccuracy in the presence of noise, Di+. Di+ is composed of the (deterministic) inaccuracy in the absence of noise, Di-, and the (deterministic) additional inaccuracy in the presence of noise, Di delta. E(Er,i), the theoretical expected value of Er,i, is given by E(Er,i) = E(Di+) + E(Sr,i). Similarly, E(Di+) = E(Di-) + E(Di delta). The corresponding theoretical variances are given by sigma 2(Er,i) = sigma 2(Di+)+2C(Di+, Sr,i)+ sigma 2(Sr,i) and sigma 2(Di+) = sigma 2(Di-)+2C(Di-, Di delta)+ sigma 2(Di delta), where C(.,.) is the covariance. We have utilized these relationships to evaluate three reconstruction algorithms: standard filtered back projection (FBP), an iterative reconstruction algorithm (IRA), and a version of the IRA which incorporates a linear transformation (TIRA). For simulated brain images in which the projection data (500,000 events detected) were degraded as the result of convolution of the true radioactivity distribution with a realistic distance-dependent detector response function, for FBP the major contribution to both E(Er,i) and sigma 2(Er,i) was Di-. For the IRA and TIRA, the major contributions to E(Er,i) were Di- and Di delta, and the major contribution to sigma 2(Er,i) was Sr,i, although in some cases Di delta was also a contributing factor. Furthermore, the errors due to sigma 2(Er,i) (that is, [sigma 2(Er,i)]0.5) were more severe than those due to E(Er,i). We conclude that, in contrast to FBP, the effects of statistical noise are an important limiting factor for the IRA and TIRA, and that the future development of tomographic devices with higher sensitivity would expand the quantitative potential of the IRA and TIRA.
