Progress made over the last few years in the hardware for magnetic induction tomography makes it possible to detect signal changes for low conductivity, low contrast variations in the target. However, image reconstruction of the data obtained presents a challenge especially when internal features within the target must be imaged. This paper focuses on the image reconstruction problem. A method of computing the forward problem with a lower computational cost than a model which needs to compute all the fields' values every time is presented. The method has been tested for its accuracy. Furthermore, the method can be used to produce a coefficient matrix which relates the contribution of each voxel to the total signal. This linear description of the problem forms the basis for image reconstruction. The images produced by direct linear reconstruction using the proposed matrix have been tested against maps produced by the dot product of the electric fields.