Let us introduce some basic notations. $C$ is defined as an operator from $X\u2254L2(R2\xd7R+)$ to $Y\u2254L2(R2\xd7[0,\pi 2])$, and the respective inner products of those spaces are defined by Display Formula
$(f1,f2)X=\u222bR\u222bR2f1(x,y,z)f2(x,y,z)dxdydz,(g1,g2)Y=\u222bR2\u222b0\pi /2g1(\zeta ,\xi ,\omega )g2(\zeta ,\xi ,\omega )d\omega d\zeta d\xi .$(22)
The adjoint transform $C\u2020$ of $C$, closely related to the backprojection operator, is defined as the transform from $Y$ to $X$ that verifies Display Formula$(g,Cf)Y=(C\u2020g,f)X.$(23)
In order to derive an expression for $C\u2020$, we start from the left side of the last identity and introduce the definition of $C$ given in Eq. (3) writing Display Formula$(g,Cf)Y=\u222bR2\u222b0\pi /2g(\zeta ,\xi ,\omega )Cf(\zeta ,\xi ,\omega )d\omega d\zeta d\xi =\u222bR2\u222b0\pi /2g(\zeta ,\xi ,\omega )=\u222b0\u221e\u222b02\pi 1rf(\zeta +r\u2009sin\u2009\omega \u2009cos\u2009\psi ,\xi +r\u2009sin\u2009\omega \u2009sin\u2009\psi ,r\u2009cos\u2009\omega )d\psi drd\omega d\zeta d\xi ,$(24)
and apply a change of variables in the form $x=\zeta +tz\u2009cos\u2009\psi $, $y=\xi +tz\u2009sin\u2009\psi $, and Fubini’s theorem to have Display Formula$(g,Cf)Y=\u222bR\u222bR21zf(x,y,z)\u222b0\pi /2\u222b02\pi g(x\u2212z\u2009tan\u2009\omega \u2009cos\u2009\psi ,y\u2212z\u2009tan\u2009\omega \u2009sin\u2009\psi ,\omega )d\psi d\omega dxdydz.$(25)