Topological X-rays and MRIs

Let K be a compact subset of the interior of the unit disk D in the plane and suppose one can't see through the boundary of D and identify K. However, assume that one can take 'topological X-rays' of D which measure the 'density' of K along the lines of the X-rays. By taking these X-rays from all directions, a 'topological MRI' is generated for the set K. Is K uniquely determined by its MRI?