  We show that a significant quantum gain corresponding to squeezed or
over-squeezed spin states can be obtained in multiparameter estimation by
measuring the Hadamard coefficients of a 1D or 2D signal. The physical platform
we consider consists of twolevel atoms in an optical lattice in a squeezed-Mott
configuration, or more generally by correlated spins distributed in spatially
separated modes. Our protocol requires the possibility to locally flip the
spins, but relies on collective measurements. We give examples of applications
to scalar or vector field mapping and compressed sensing.
