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Interconversion Between Truncated Cartesian and Polar Expansions of Images

机译:截断的笛卡尔和图像的极扩展之间的互转换

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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.
机译:在本文中,我们提出了一种算法,当从二维实值函数(映射:$ {BBR} ^ 2 mapsto {BBR} $)表示采样时,笛卡尔坐标和极坐标之间的数据无损转换作为一种截断的扩展。我们将Laguerre函数和傅立叶基础用于极坐标表示。 Hermite函数用于笛卡尔坐标表示。截断展开的有限数量的系数指定每个坐标系中的函数。我们得出两个坐标系的系数之间的关系。基于这种关系,我们提出了两个坐标系之间无损转换的算法。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。重新采样的数据用于计算新的系数集,以避免与直接转换系数相关的数值不稳定。为了将我们的算法应用于离散图像数据,我们提出了一种将截断表达式最佳地拟合到给定图像的方法。我们还量化了此过滤过程可能产生的误差。最后将该算法用于求解极笛卡尔插值问题。

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