Emulating Optically Inspired Massively Parallel non-Boolean Operators on FPGA Andr´ as Kiss, Zolt´ an Nagy, P´ eter Szolgay and Tam´ as Roska Computer and Automation Research Institute Hungarian Academy of Sciences and Dept. Information Technology P´ azm´ any P´ eter Catholic University Budapest, Hungary Gy¨ orgy Csaba and Xiaobo Sharon Hu and Wolfgang Porod Dept. of Electrical Engineering University of Notre Dame, Notre Dame, IN, USA Abstract—In this paper, we demonstrate two optically inspired massively parallel non-Boolean operators running on FPGA. One of the algorithm is based on the Paraxial Helmholtz Equation: which describes the beam propagation through different media with different refractive indices, and the other is based on the concepts of optical computing: quasi-optical wave equations are solved numerically, using FPGA-accelerated hardware. The second algorithm describes a holographic pattern-matching al- gorithm. Both of the two FPGA-based implementations are very well parallelizable, consequently they are also be amenable to mega-core architectures. I. I NTRODUCTION There are a number of CNN templates that are known to support the propagation of waves and there are several algorithms that are known to exploit wave phenomena for computation [1][2]. Non-linear autowaves were extensively studied. However, the behavior of these waves is complex, untreatable with analytical tools and therefore, it is challenging to design computing systems utilizing such non-linear waves. This motivates our study for the CNN-emulation of a linear wave equation, the Paraxial Helmholtz Equation (PHE). This is a well known partial differential equation (PDE) in optics and can be used to simulate propagation of light through a system of lenses, filters and diffraction gratings (see Fig. 1). Since an optical system may be used for linear image processing tasks, solution of the PHE can be a basis of PDE-based image processing algorithms. The second example we point out is a special-purpose Beam propaga*on k k 1 k 2 k 3 k 5 x y z →t k 4 Fig. 1. Beam propagation along the z axis in different media. computing algorithm, which solves the two-dimensional wave equation, which can, in fact serve as a computational engine of a pattern-matching algorithm. Pattern matching itself is a basis of widely used image processing / recognition algorithms. Since the solution of the wave equation is well paralellizable, the pattern-matching algorithm itself will be possible to dis- tribute to a large number of computing cores. II. PROPOSED ARCHITECTURE OF THE 3D LIGHT PROPAGATION SIMULATOR Modern FPGAs, like the Zynq on the ZedBoard [3] (introduced in 2012), usually contain hard-processor cores, like the Cortex A9 ARM processors. It is appropriate to utilise the power of the processor, therefore an efficient hardware/software co-design is proposed for simulating light propagation. The description of the operators, such as lenses, can be car- ried out by the ARM processor. The resulting implementation, the lens-medium parameter and the distance between the lens and the focal point (which can be computed from the refraction of the light paths) can be stored on the on-chip memory. By putting these types of operators next to each other various algorithms can be simulated. The PHE can be solved by the arithmetic unit, which can be optimised according to the discretised governing equations. By simulating a light propagation in a medium the arithmetic unit should compute the governing equations in several stages according to the distance of the focal point. These stages should be computed sequentially, because the output of one stage is going to be the input of a subsequent stage. III. BASICS HOLOGRAPHIC COMPUTING The principle of pattern matching is shown in Fig. 2a). A two-dimensional medium is assumed, which can support the propagation of waves. For now we refer to the waves in an abstract sense, i.e. an excitation that shows interference phenomenon. The pattern recognition algorithm works by simply using different wave source distributions (test patterns) in the bottom 978-1-4799-6007-1/14/$31.00 c  2014 IEEE