are widely used in image processing. Gilmore–Perelomov coherent states The construction of coherent states using group representations described above is not sufficient. Already it cannot yield the CCS, since these are not indexed by the elements of the Heisenberg group, but rather by points of the quotient of the latter by its center, that quotient being precisely . The key observation is that the center of the Heisenberg group leaves the vacuum vector invariant, up to a phase. Generalizing this idea, Gilmore and Perelomov consider a locally compact group and a unitary irreducible representation of on the Hilbert space , not necessarily square integrable. Fix a vector in , of unit norm, and denote by the subgroup of consisting of all elements that leave it invariant up to a phase, that is, where is a real-valued function of . Let be the left coset space and an arbitrary element in . Choosing a coset representative , for each coset , we define the vectors The dependence of these vectors on the specific choice of the coset representative is only through a phase. Indeed, if instead of , we took a different representative for the same coset , then since for some , we would have . Hence, quantum mechanically, both and represent the same physical state and in particular, the projection operator depends only on the coset. Vectors defined in this way are called Gilmore–Perelomov coherent states. Since is assumed to be irreducible, the set of all these vectors as runs through is dense in . In this definition of generalized coherent states, no resolution of the identity is postulated. However, if carries an invariant measure, under the natural action of , and if the formal operator defined as is bounded, then it is necessarily a multiple of the identity and a resolution of the identity is again retrieved. Gilmore–Perelomov coherent states have been generalized to quantum groups, but for this we refer to the literature. Further generalization: Coherent states on coset spaces The Perelomov construction can be used to define coherent states for any locally compact group. On the other hand, particularly in case of failure of the Gilmore–Perelomov construction, there exist other constructions of generalized coherent states, using group representations, which generalize the notion of square integrability to homogeneous spaces of the group. Briefly, in this approach one starts with a unitary irreducible representation and attempts to find