. Author manuscript; available in PMC: 2009 Dec 31.

Published in final edited form as: Proc IEEE Int Symp Biomed Imaging. 2008 May 14;5:911–914. doi: 10.1109/ISBI.2008.4541145

Table 1.

Proposed basis functions

P(r) basis	Corresponding S(q)/S₀ basis

$r_{1}^{4}$	$(12 - 48 q_{1}^{2} + 16 q_{1}^{4}) e^{- q^{T} q}$
$r_{2}^{4}$	$(12 - 48 q_{2}^{2} + 16 q_{2}^{4}) e^{- q^{T} q}$
$r_{3}^{4}$	$(12 - 48 q_{3}^{2} + 16 q_{3}^{4}) e^{- q^{T} q}$
$r_{1}^{2} r_{2}^{2}$	$(- 2 + 4 q_{1}^{2}) (- 2 + 4 q_{2}^{2}) e^{- q^{T} q}$
$r_{2}^{2} r_{2}^{3}$	$(- 2 + 4 q_{2}^{2}) (- 2 + 4 q_{3}^{2}) e^{- q^{T} q}$
$r_{1}^{2} r_{2}^{3}$	$(- 2 + 4 q_{1}^{2}) (- 2 + 4 q_{3}^{2}) e^{- q^{T} q}$
$r_{1}^{2} r_{2} r_{3}$	$4 q_{2} q_{3} (- 2 + 4 q_{1}^{2}) e^{- q^{T} q}$
$r_{1} r_{2}^{2} r_{3}$	$4 q_{1} q_{3} (- 2 + 4 q_{2}^{2}) e^{- q^{T} q}$
$r_{1} r_{2} r_{2}^{3}$	$4 q_{1} q_{2} (- 2 + 4 q_{3}^{2}) e^{- q^{T} q}$
$r_{1}^{3} r_{2}$	$- 2 q_{2} (12 q_{1} - 8 q_{1}^{3}) e^{- q^{T} q}$
$r_{1}^{3} r_{3}$	$- 2 q_{3} (12 q_{1} - 8 q_{1}^{3}) e^{- q^{T} q}$
$r_{1} r_{2}^{3}$	$- 2 q_{1} (12 q_{2} - 8 q_{2}^{3}) e^{- q^{T} q}$
$r_{2}^{3} r_{3}$	$- 2 q_{3} (12 q_{2} - 8 q_{2}^{3}) e^{- q^{T} q}$
$r_{1} r_{3}^{3}$	$- 2 q_{1} (12 q_{3} - 8 q_{3}^{3}) e^{- q^{T} q}$
$r_{2} r_{3}^{3}$	$- 2 q_{2} (12 q_{3} - 8 q_{3}^{3}) e^{- q^{T} q}$

$r_{1}^{i} r_{2}^{j} r_{3}^{k}$	${(\partial ∕ \partial q_{1})}^{i} {(\partial ∕ \partial q_{2})}^{j} {(\partial ∕ \partial q_{3})}^{k} B (q)$