ConformalALU: A Conformal Geometric Algebra Coprocessor for Medical Image Processing (original) (raw)

2015, IEEE Transactions on Computers

Medical imaging involves important computational geometric problems, such as image segmentation and analysis, shape approximation, three-dimensional (3D) modeling, and registration of volumetric data. In the last few years, Conformal Geometric Algebra (CGA), based on five-dimensional (5D) Clifford Algebra, is emerging as a new paradigm that offers simple and universal operators for the representation and solution of complex geometric problems. However, the widespread use of CGA has been so far hindered by its high dimensionality and computational complexity. This paper proposes a simplified formulation of the conformal geometric operations (reflections, rotations, translations, and uniform scaling) aimed at a parallel hardware implementation. A specialized coprocessing architecture (ConformalALU) that offers direct hardware support to the new CGA operators, is also presented. The ConformalALU has been prototyped as a complete System-on-Programmable-Chip (SoPC) on the Xilinx ML507 FPGA board, containing a Virtex-5 FPGA device. Experimental results show average speedups of one order of magnitude for CGA rotations, translations, and dilations with respect to the geometric algebra software library Gaigen running on the general-purpose PowerPC processor embedded in the target FPGA device. A suite of medical imaging applications, including segmentation, 3D modeling and registration of medical data, has been used as testbench to evaluate the coprocessor effectiveness. Index Terms-Conformal geometric algebra, five-dimensional clifford algebra, computational geometry, embedded coprocessors, systemson-programmable-chip, FPGA-based prototyping, medical imaging, segmentation, 3D modeling, Volume registration, Growing Neural Gas, marching spheres, iterative closest point (ICP), thin-plate spline robust point matching (TPS-RPM) Ç 1 INTRODUCTION M EDICAL imaging plays an important role in current medical research and clinical practice. Efficient algorithms are required to solve complex geometric problems arising in medical image processing, such as segmentation, shape extraction, three-dimensional (3D) modeling and registration of medical data. A key problem in medical computation is the reconstruction of 3D shapes (of organs, bones, tumors, etc.) from two-dimensional (2D) slices derived from Magnetic Resonance (MR) or Computed Tomography (CT) scans [1], [2], [3], [4]. This is a typical geometric problem that consists in finding a proper surface connecting a set of contour data points. Efficient geometric tools are also required for medical image registration that consists in finding a proper geometrical transformation that aligns different views of the same image taken in different moments or by diverse acquisition modalities [5], [6], [7]. Computational geometry deals with finding solutions to geometric problems that arise in medical imaging as well as in other application domains, such as computer graphics, robotics, computer vision, and