We introduce three generalizations of homotopy equivalence in digital images, to allow us to express whether a bounded and an unbounded digital image with standard adjacencies are similar with respect to homotopy. We show that these three generalizations are not equivalent to ordinary homotopy equivalence, and give several examples. We show that, like homotopy equivalence, our three generalizations imply isomorphism of fundamental groups, and are preserved under wedges and Cartesian products.