Let $X$ be a random vector taking values in $\mathbb R^2$ with probability density $p(x) = p_1(x_1)p_2(x_2)$, i.e. the components of $X$ are independent.

Let $V$ be an open set in $\mathbb S^1$, the $1$-sphere. Now, given Radon projections of $X$ along $v \in V$, i.e. the densities of the random variables $\langle v,X \rangle$, can we reconstruct $p_1$ and $p_2$?

My comments:

We have for the characteristic function \begin{align*} \varphi_{\langle v,X \rangle}(t) = \varphi_{X_1}(s v_1)\varphi_{X_2}(s v_2). \end{align*} The left-hand sideis given for specific $v \in V$ and is the product of the characteristic functions of the marginal densities. The question is, if $v \in V$ and $V$ open in $\mathbb S^1$ suffices to determine the factors (i.e. the marginal densities).