Biomedical Imaging

$={\displaystyle \sum _{m=0}^{K-1}{\displaystyle \underset{0}{\overset{\text{π}}{\int }}\text{d}\varphi w_m(\varphi )}\frac{1}{K}{\displaystyle \sum _{k=1}^K{U}_m(\underset{\_}{x}\cdot {\underset{\_}{n}}_{{\varphi }_{\kappa }})}{U}_m\left(\cos ({\varphi }_k-\varphi )\right)}(4.127)$

$=\frac{1}{K}{\displaystyle \sum _{k=1}^K{\displaystyle \sum _{m=0}^{K-1}{U}_m(\underset{\_}{x}\cdot {\underset{\_}{n}}_{{\varphi }_{\kappa }})}\underset{=\frac{m+1}{2}{c}_m({\varphi }_k)}{\underbrace{{\displaystyle \underset{0}{\overset{\text{π}}{\int }}\text{d}\varphi w_m(\varphi )}{U}_m\left(\cos ({\varphi }_k-\varphi )\right)}}}.(4.128)$

In this way, we have been able to discretize the angle, but it seems that we still need the full – i.e., continuous – information (f)(ϕk , s) for each angle ϕk. On the other hand, we also know that the RT equals a polynomial of at most order K − 1 in s, up to a prefactor of $\sqrt{1-s^2}$see equation (4.116). Therefore, the right hand side of

$c_m({\varphi }_k)=\frac{2}{\text{π}}{\displaystyle \underset{-1}{\overset{1}{\int }}\text{ds}\text{(}\mathcal{R}f\text{)}\text{(}{\varphi }_k},s)U_m(s),(4,129)$

is an integral over a polynomial of at most order 2(K − 1), multiplied ...