Computer and Information Science and Engineering Department, University of Florida, Gainesville, FL 32611, USA.
IEEE Trans Med Imaging. 2012 Aug;31(8):1532-41. doi: 10.1109/TMI.2012.2191417. Epub 2012 Mar 20.
B-splines are attractive basis functions for the continuous-domain representation of biomedical images and volumes. In this paper, we prove that the extended family of box splines are closed under the Radon transform and derive explicit formulae for their transforms. Our results are general; they cover all known brands of compactly-supported box splines (tensor-product B-splines, separable or not) in any number of dimensions. The proposed box spline approach extends to non-Cartesian lattices used for discretizing the image space. In particular, we prove that the 2-D Radon transform of an N-direction box spline is generally a (nonuniform) polynomial spline of degree N-1. The proposed framework allows for a proper discretization of a variety of tomographic reconstruction problems in a box spline basis. It is of relevance for imaging modalities such as X-ray computed tomography and cryo-electron microscopy. We provide experimental results that demonstrate the practical advantages of the box spline formulation for improving the quality and efficiency of tomographic reconstruction algorithms.
B 样条是生物医学图像和体数据的连续域表示的有吸引力的基函数。在本文中，我们证明了扩展的盒样条族在 Radon 变换下是封闭的，并推导出了它们的变换的显式公式。我们的结果是通用的；它们涵盖了所有已知品牌的紧支撑盒样条（张量积 B 样条，可分离或不可分离）在任何维数。所提出的盒样条方法扩展到用于离散图像空间的非笛卡尔格。特别是，我们证明了 N 方向盒样条的 2-D Radon 变换通常是 N-1 阶的（非均匀）多项式样条。所提出的框架允许在盒样条基中对各种层析重建问题进行适当的离散化。它与 X 射线计算机断层扫描和冷冻电子显微镜等成像方式有关。我们提供了实验结果，证明了盒样条公式在提高层析重建算法的质量和效率方面的实际优势。
