We propose and validate a framework for the inverse identification of the constitutive parameter fields of functionally graded hyperelastic materials. We demonstrate estimation of relative parameter fields for all parameters of the Neo-Hookean and other isotropic hyperelastic solids upto a multiplicative constant only using local displacement field data (not full-field displacement data) without the need for boundary forces. Our proposed framework can also achieve exact parameter field identification if boundary force/forces are provided, as the elastography problem can be solved only upto a multiplicative constant in the absence of boundary forces. We achieve elastography of 2-D (thin membranes) and 3-D geometries while using many orders of magnitude lesser parameters and compute time compared to past works that use neural networks for elastography. We achieve this by formulating an objective function that satisfies balance of linear momentum and smoothness regularization of the discovered field. Furthermore we benchmark the proposed framework on various geometries and hyperelastic models using experimentally realistic 2D-DIC or DVC displacement data perturbed by various levels of correlated and uncorrelated noise. We also demonstrate that the proposed framework can accurately characterize homogeneous materials and materials with distinct heterogeneities, thereby offering significant characterization improvements over our previously developed Hetero-EUCLID framework. We believe that the proposed framework will be effective for biomedical contexts that involve hyperelastic materials with continuously varying properties, and for rapidly characterizing composite materials.