Rees algebras and almost linearly presented ideals

Consider a grade 2 perfect ideal I in R=k[x1,⋯,xd] which is generated by forms of the same degree. Assume that the presentation matrix φ is almost linear, that is, all but the last column of φ consist of entries which are linear. For such ideals, we find explicit forms of the defining ideal of the Rees algebra R(I). We also introduce the notion of iterated Jacobian duals.