Interconversion between truncated Cartesian and polar expansions of images.
Interconversion between truncated Cartesian and polar expansions of images.
复制标题
图像的截断笛卡尔膨胀和极坐标膨胀之间的相互转换。
DOI：
10.1109/tip.2007.899190
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发表时间：
2007
期刊：
影响因子：
--
通讯作者：
Chirikjian,GregoryS
中科院分区：
文献类型：
--
作者：
Park,Wooram;Chirikjian,GregoryS
本文提出了一种从二维实值函数(映射：${bbr}^2\mapstto{\bbr}$)采集的数据在笛卡尔坐标和极坐标之间进行无损转换的算法。我们使用拉盖尔函数和傅立叶基来表示极坐标。Hermite函数用于笛卡尔坐标表达式。截断展开的有限个系数指定每个坐标系中的函数。我们推导了两个坐标系的系数之间的关系。基于这种关系，我们提出了一种在两个坐标系之间进行无损转换的算法。重采样可用于评估互补坐标系上的截断展开，而无需计算新的系数集。重采样的数据被用来计算新的系数集，以避免与系数的直接转换相关的数值不稳定性。为了将我们的算法应用于离散图像数据，我们提出了一种方法来使截断表达式最适合于给定的图像。我们还量化了这个过滤过程可能产生的误差。最后，将该算法应用于极-笛卡尔插补问题。
In this paper, we propose an algorithm for lossless conversion of data between Cartesian and polar coordinates, when the data is sampled from a 2-D real-valued function (a mapping: ${\BBR}^2 \mapsto {\BBR}$) expressed as a particular kind of truncated expansion. We use Laguerre functions and the Fourier basis for the polar coordinate expression. Hermite functions are used for the Cartesian coordinate expression. A finite number of coefficients for the truncated expansion specifies the function in each coordinate system. We derive the relationship between the coefficients for the two coordinate systems. Based on this relationship, we propose an algorithm for lossless conversion between the two coordinate systems. Resampling can be used to evaluate a truncated expansion on the complementary coordinate system without computing a new set of coefficients. The resampled data is used to compute the new set of coefficients to avoid the numerical instability associated with direct conversion of the coefficients. In order to apply our algorithm to discrete image data, we propose a method to optimally fit a truncated expression to a given image. We also quantify the error that this filtering process can produce. Finally the algorithm is applied to solve the polar-Cartesian interpolation problem.