The radon transform of an image $f(x,y)$ can be written as:
\begin{equation} p(\alpha,s)=\int_{-\infty}^{\infty}f(x(z),y(z))dz \\ = \int_{-\infty}^{\infty}f(z\sin\alpha+s\cos \alpha , -z\cos \alpha + s \sin \alpha)dz \end{equation}
The discrete matrix version of this equation can be written as \begin{equation} R \vec{f}=\vec{p} \end{equation}
Where $R$ is a matrix, $\vec{f}$ is the image vector and $\vec{p}$ is the radon transform vector. I'm seeking to find the continuous analogue to $RR^T\vec{f}$.
Maybe irrelevant context: I'm creating an quadratic optimization algorithm where I need to quickly calculate the minimum of $\frac{1}{2}x^TAx+b^Tx$ subject to $x\geq 0$ where \begin{equation} A=\frac{1}{2}\begin{bmatrix} RR^T & -RR^T\\ -RR^T & RR^T \end{bmatrix} \end{equation}
\begin{equation} b=\begin{bmatrix} R\vec{f}\\ 1-R\vec{f} \end{bmatrix} \end{equation}