Table 1.

Investigated image-based approaches for the determination of relative electron density and/or effective atomic number from DECT. The mutual mathematical equivalence of the approaches is manifest via the respective parameter transformations (last column).

ID	Reference	$\widehat{n}$ eq.	Zeff eq.	Fix water calibration	$m$	Parameter transformations
BRO	Brooks (1977) [21]	(4), (6), (10)	(4), (6), (10)	Yes	3.1	$Q_1\equiv Z_{\text{eff,w}}^m{B}_{\text{h}}/A_{\text{h}}$, $Q_2\equiv Z_{\text{eff,w}}^m{B}_{\text{l}}/A_{\text{l}}$
HEI	Heismann et al. (2003) [7]	(16)a	(17)	No	3.0	$\beta \equiv {\stackrel{\sim }{A}}_{\text{l}}\equiv {\stackrel{\sim }{A}}_{\text{h}}$b, $g_1\equiv {\stackrel{\sim }{B}}_{\text{l}}$, $g_2\equiv {\stackrel{\sim }{B}}_{\text{h}}$
SAI1	Saito (2012) [9]	(1), (2)	–	Yes	–	$\alpha \equiv B_{\text{h}}$
LAN	Landry et al. (2013) [10]	–	(9)	No	3.3	$A_{80\text{kVp}}\equiv A_{\text{l}}$, $A_{140\text{kVp}}\equiv A_{\text{h}}$, $C_{80\text{kVp}}\equiv B_{\text{l}}$, $C_{140\text{kVp}}\equiv B_{\text{h}}$
HUE	Hünemohr et al. (2014) [11]	(14)	(19)	Yes	3.1	$c_{\text{e}}\equiv -B_{\text{h}}$, $d_{\text{e}}\equiv A_{\text{h}}$
SAI2	Saito and Sagara (2017) [16]	(1), (2)	(8)	Yes	3.3	$\alpha \equiv B_{\text{h}}$, ${\gamma }_{\text{L}}\equiv \frac{A_{\text{l}}}{B_{\text{l}}{Z}_{\text{eff,w}}^m}+1$

^aMass density replaced by relative electron density.

^bThe equality ${\stackrel{\sim }{A}}_{\text{l}}\equiv {\stackrel{\sim }{A}}_{\text{h}}$is an additional assumption in HEI (see Section 4).