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We propose a physically motivated parametrization for the unpolarized generalized parton distributions, H and E, valid at both zero and non-zero values of the skewness variable, \zeta. Our approach follows a previous detailed study of the \zeta=0 case where H and E were determined using constraints from simultaneous fits of the experimental data on both the nucleon elastic form factors and the deep inelastic structure functions in the non singlet sector. Additional constraints at \zeta \neq 0 are provided by lattice calculations of the higher moments of generalized parton distributions. We illustrate a method for extracting generalized parton distributions from lattice moments based on a reconstruction using sets of orthogonal polynomials. The inclusion in our fit of data on Deeply Virtual Compton Scattering is also discussed. Our method provides a step towards a model independent extraction of generalized distributions from the data. It also provides an alternative to double distributions based phenomenological models in that we are able to satisfy the polynomiality condition by construction, using a combination of experimental data and lattice, without resorting to any specific mathematical construct., Comment: 29 pages, 8 figures; added references, changed text in several places