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The main objective of the current work is to determine meshless methods using the radial basis function (<span style="font-variant: small-caps;">rbf</span>) approach to estimate the elastic strain field from energy-resolved neutron imaging. To this end, we first discretize the longitudinal ray transformation with <span style="font-variant: small-caps;">rbf</span> methods to give us an unconstrained optimization problem. This discretization is then transformed into a constrained optimization problem by adding equilibrium conditions to ensure uniqueness. The efficiency and accuracy of this approach are investigated for the situation of <span style="font-variant: small-caps;">2d</span> plane stress. In addition, comparisons are made between the results obtained with <span style="font-variant: small-caps;">rbf</span> collocation, finite-element (<span style="font-variant: small-caps;">fem</span>) and analytical solution methods for test problems. The method is then applied to experimentally measured continuous and discontinuous strain fields using steel samples for an offset ring-and-plug and crushed ring, respectively.