Lewitt R M
Department of Radiology, University of Pennsylvania, Philadelphia 19104-6021.
J Opt Soc Am A. 1990 Oct;7(10):1834-46. doi: 10.1364/josaa.7.001834.
Inverse problems that require the solution of integral equations are inherent in a number of indirect imaging applications, such as computerized tomography. Numerical solutions based on discretization of the mathematical model of the imaging process, or on discretization of analytic formulas for iterative inversion of the integral equations, require a discrete representation of an underlying continuous image. This paper describes discrete image representations, in n-dimensional space, that are constructed by the superposition of shifted copies of a rotationally symmetric basis function. The basis function is constructed using a generalization of the Kaiser-Bessel window function of digital signal processing. The generalization of the window function involves going from one dimension to a rotationally symmetric function in n dimensions and going from the zero-order modified Bessel function of the standard window to a function involving the modified Bessel function of order m. Three methods are given for the construction, in n-dimensional space, of basis functions having a specified (finite) number of continuous derivatives, and formulas are derived for the Fourier transform, the x-ray transform, the gradient, and the Laplacian of these basis functions. Properties of the new image representations using these basis functions are discussed, primarily in the context of two-dimensional and three-dimensional image reconstruction from line-integral data by iterative inversion of the x-ray transform. Potential applications to three-dimensional image display are also mentioned.
需要求解积分方程的反问题在许多间接成像应用中是固有的，比如计算机断层扫描。基于成像过程数学模型离散化或基于积分方程迭代反演解析公式离散化的数值解，需要对潜在的连续图像进行离散表示。本文描述了在n维空间中通过旋转对称基函数的平移副本叠加构建的离散图像表示。基函数是使用数字信号处理中的凯泽 - 贝塞尔窗函数的推广来构建的。窗函数的推广涉及从一维到n维的旋转对称函数，以及从标准窗的零阶修正贝塞尔函数到涉及m阶修正贝塞尔函数的函数。给出了三种在n维空间中构建具有指定（有限）连续导数数量的基函数的方法，并推导了这些基函数的傅里叶变换、x射线变换、梯度和拉普拉斯算子的公式。主要在通过x射线变换的迭代反演从线积分数据进行二维和三维图像重建的背景下，讨论了使用这些基函数的新图像表示的性质。还提到了对三维图像显示的潜在应用。
