## 1 Introduction  
Consider a solid material sample composed of similar crystals with a uniform distribution of directions. There are three times as many crystallographic planes in each crystal as there are Miller indices. The sample if subject to a narrow neutron beam with a known distribution of kinetic energy. The neutrons behave as waves with the energy inversely proportional to the wavelength, and this wavelength is a similar magnitude to the crystallographic planes [1, 2]. Parallel crystallographic planes with \(d\) separation scatter an incident wave if the wavelength is a multiple of \(2d\sin\theta\). If we detect only the transmitted neutrons, the scattered neutrons will be lost. If we look the number of transmitted neutrons as a function of wavelength, then at the wavelength \(2d\) the transmission leaps dramatically. This jump is labelled as the Bragg edges. If the material is subjected to a \(\varepsilon\) linear elastic strain, the separation between planes with a normal \(\mathbf{n}\) vector is in proportion to the strain component \(\mathbf{n}\cdot\varepsilon\cdot\mathbf{n}\). For more information and additional sources, see [3, 4, 5]. If the strain were consistent along the neutron path, the Bragg edge would be shifted. Within ideal conditions, a linear wavelength function multiplied by the unit step function can model the Bragg edges. [PERSON] et al. [6] had previously explored whether strain could be reconstructed from the average of each ray shifts of a Bragg edge. [PERSON] and [PERSON] [7] noted that the transverse ray transformation is feasible for X-ray strain measurements. Given the longitudinal ray transformation (lrt) data plane by plane, an attempt could be made to recover the strain from the ray measurements in that plane using appropriate additional a priori information [8]. The Lame system could be represented by finite-element or finite difference methods provided that the Lame coefficients are known. A linear system could be developed, and the least square solution determined together with a discretization of the lrt and prior knowledge. This method has been adopted by [PERSON] et al. [5] for axisymmetric objects, [PERSON] et al. [9] for synthetic data, and [PERSON] et al. [10] for a generic case experimentally. We consider here the measurement of the Bragg edges based on the calculation of the propagation spectrum using flight time or energy-resolved technologies. We note that other approaches exist [3; 11; 12; 13]. The major contribution of neutron transmission strain tomography is the optimal use of the incident beam flux to reduce the data collection time. Recent developments in detector technology allow the acquisition of high-resolution strain images with Bragg neutron transmission [14]. The sample is placed on a rotating plate between the source and a micro-channel plate detector (MCP) [15; 16]. This detector is pixelated with a spatial resolution of up to \(512\times 512\) pixels of up to \(55\upmu\)m. This imaging raises the possibility of strain tomography, and efforts have been made over the past few years to solve the resulting tensor reconstruction challenge--special cases such as axial symmetry revolved around reconstruction techniques [5; 17]. A well-known fact about this inverse problems is that it is not well-posed. Therefore it is not possible to reconstruct the strain without the inclusion of additional conditions or without easing assumptions [10]. The question of how to better execute these additional criteria was discussed and various approaches were used in the literature [5; 9]. Recently, reconstruction has been accomplished by incorporating either equilibrium or compatibility conditions. The main distinction between these approaches is the set of basis functions representing the strain field [6; 10]. Recently, considerable attention has been given in the science and engineering community to the concept of using meshfree methods as a modern tool for numerical solution for partial differential equations.