Geometric Transformations via Matrix Multiplications Using Hardware/Software Co-design

Abstract

The standard methods of transformations of a geometric object in n-dimensional space are often expressed in the form of an n x n matrix multiplication by the an n x 1column vector, where the n x n matrix and the vector represent the transformation and the point in the homogeneous coordinate system respectively. This enable us to represent a series of transformations in terms of a single composite transformation in the resulting product matrix of each sequent transformation through the matrix maultiplications, where each individual matrix may be a translation, rotation, or scaling, or the cobinations of all the above. Therefore the matrix multiplications play an important role in such operation. In this paper, we first study the computational complexicity of matrix multiplications. Then we employ the hardware/software codesign on the matrix multiplications during their intensive computationally processes. In our codesign, we exploit the highly parallel nature of matrix multiplications, which cannot be exploited in our purely software implementation[4]. The hardware part of our codesign system is responsible for performing the arithmetic operations. This includes the matrix multiplier and adder, which perform concurrent multiplication and addition operations of matrix multiplication. Our matrix multiplier and adder are modeled in VHDL and runs on an ARC-PCI FPGA board