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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 \(\vec {x}^{\text {true}}\)
  1. aThese parameters are assumed to be constant within one iteration, but can be updated between iterations