Arbitrary mode-n convolution of physical tensors with applications in optics

Abstract

Multidimension functions frequently appear in Fourier Optics. For example, in a polychromatic optical system, one could have three spatial dimensions for space, one dimension for time, and one dimension for wavelength. Such systems would require multi-dimension mathematical models that could be efficiently analyzed or solved using Tensor Analysis. In Fourier Optics, convolution describes light propagation in multidimensional linear shift-invariant media and image formation in multidimensional linear shift-invariant imaging systems. In Tensor Analysis, many tensor operations, including convolution, are well defined in the literature. However, in this literature, tensor convolution is defined under three limiting assumptions 1) tensors to be convolved must have the same size; 2) tensors are expected to be convolved along all its dimensions; 3) tensors to be convolved should represent the same physical variables on each of its dimensions. In practice, one could possibly seek convolution along a specific subset of physical variables, which would not be well defined by this standard definition of tensor convolution. In this thesis, to overcome these limitations inherent in the definition of tensor convolution, we defined arbitrary mode-n convolution that allows convolution of different size tensors along a specific subset of their physical variables. We then applied our novel arbitrary mode-n convolution method to simulate three simple multidimensional Fourier Optics problems, i.e., free space propagation, diffraction by an aperture, and imaging using a thin lens. We simulated these problems using 1) full-sized tensors; 2) Tensor Tucker Decomposition; and 3) Tensor Train Decomposition. Our numerical results demonstrated that the Tensor Train approach is most efficient in terms of accuracy, storage requirement, and computation time.