Table 1 Notation used for mathematics, image space, measurement space, and physics
From: Algorithms for joint activity–attenuation estimation from positron emission tomography scatter
Symbol | Description |
|---|---|
\(\underline {\boldsymbol {T}}, \boldsymbol {M}, \vec {v}, \vec {1}_{[\cdot ]}\) | Rank-3 tensor, matrix, vector, vector composed of ones |
⊗; \(\odot, \oslash, \overset {\circ }{\text {exp}}\) | Outer product; element-wise multiplication, division, exp, |
\(\vec {\lambda }, \vec {\mu }, \vec {\rho }\) | Spatial distribution of activity, linear attenuation coefficient, electron density |
j, e, s, t | Indices of all, emitting, scattering, transmitting voxels |
\(\vec {y}, \vec {\bar {y}}\); \(\vec {z}, \vec {\bar {z}}\) | Measured/simulated, expected scatter (y) or trues (z) data |
ds,dn | Detectors of scattered, nonscattered photons of a single-scatter coincidence |
E, θ | Energy of scattered photon, associated scattering angle |
l | =(ds,dn), index of an LOR |
i | =(ds,dn,E), index of an SOR |
(i,s) | =(ds,dn,E,s), index of a broken LOR |
Nd,NE,Nl,Ni | Numbers of detectors, energy bins, LORs, SORs |
\(\boldsymbol {A}, \boldsymbol {A}_{\omega }, \boldsymbol {\tilde {A}}\) | SOR system matrices: (Aω, function of parameters;\({~}^{\mathrm {a}} \boldsymbol {\tilde {A}}\) with attenuation) |
\(\boldsymbol {U}, \boldsymbol {\tilde {U}}_{\rho }\) | LOR system matrices (as above) |
\(\underline {\boldsymbol {B}}, b^{{i}}_{s,e}\) | Probability that radiation emitted in e is detected along the broken LOR (i,s), disregarding attenuation, per unit electron density in s, per unit activity in e |
\(\underline {\boldsymbol {{L}}}, l^{{i}}_{s,t}\) | Effective intersection length of photon path along the broken LOR (i,s) with the transmitting (or attenuating) voxel t, taking photon energy into account |
\(\underline {\boldsymbol {{K}}}, k^{{i}}_{s,t}\) | =ls,ti·μt/ρt, effective intersection length times LAC–electron-density ratio |
\(\mathcal {L}_{y}(\bullet)\) | Poisson log-likelihood (LL) given the measured data \(\vec {y}\) |
\(\mathcal {S}(\bullet, \vec {x}^{\,\text {true}})\) | Normalized mean squared error (NMSE) with respect to a reference \(\vec {x}^{\text {true}}\) |