  We show that it is possible to construct a well-defined effective field
theory incorporating string winding modes without using strong constraint in
double field theory. We show that X-ray (Radon) transform on a torus is
well-suited for describing weakly constrained double fields, and any weakly
constrained fields are represented as a sum of strongly constrained fields.
Using inverse X- ray transform we define a novel binary operation which is
compatible with the level matching constraint. Based on this formalism, we
construct a consistent gauge transform and gauge invariant action without using
strong constraint. We then discuss the relation of our result to the closed
string field theory. Our construction suggests that there exists an effective
field theory description for massless sector of closed string field theory on a
torus in an associative truncation.
