  Let $X$ and $X^*$ denote a restricted ray transform along curves and a
corresponding backprojection operator, respectively. Theoretical analysis of
reconstruction from the data $Xf$ is usually based on a study of the
composition $X^* D X$, where $D$ is some local operator (usually a derivative).
If $X^*$ is chosen appropriately, then $X^* D X$ is a Fourier Integral Operator
(FIO) with singular symbol. The singularity of the symbol leads to the
appearance of artifacts (added singularities) that can be as strong as the
original (or, useful) singularities. By choosing $D$ in a special way one can
reduce the strength of added singularities, but it is impossible to get rid of
them completely.
  In the paper we follow a similar approach, but make two changes. First, we
replace $D$ with a nonlocal operator $\tilde D$ that integrates $Xf$ along a
curve in the data space. The result $\tilde D Xf$ resembles the generalized
Radon transform $R$ of $f$. The function $\tilde D Xf$ is defined on pairs
$(x_0,\Theta)\in U\times S^2$, where $U\subset\mathbb R^3$ is an open set
containing the support of $f$, and $S^2$ is the unit sphere in $\mathbb R^3$.
Second, we replace $X^*$ with a backprojection operator $R^*$ that integrates
with respect to $\Theta$ over $S^2$. It turns out that if $\tilde D$ and $R^*$
are appropriately selected, then the composition $R^* \tilde D X$ is an
elliptic pseudodifferential operator of order zero with principal symbol 1.
Thus, we obtain an approximate reconstruction formula that recovers all the
singularities correctly and does not produce artifacts. The advantage of our
approach is that by inserting $\tilde D$ we get access to the frequency
variable $\Theta$. In particular, we can incorporate suitable cut-offs in $R^*$
to eliminate bad directions $\Theta$, which lead to added singularities.
