【作者】 梁俊；

【导师】 罗立民； 舒华忠；

【作者基本信息】 东南大学 ， 生物医学工程， 2006， 硕士

【摘要】 矩函数在图像分析中有着广泛的应用。目前图像矩主要包括以下几种类型:几何矩,复数矩,旋转矩,连续正交矩和离散正交矩。正交矩具有以下特点:(1)矩变换是可逆的,从而可以对变换前的信息进行恢复;(2)正交矩的各阶矩是彼此独立的,具有最小冗余信息。连续正交矩的优点非常显著,它有非常简单的反变换形式,解决了用矩进行图像重建这一原本困难的问题。但是连续正交矩必须对积分进行近似。离散正交矩由于不需要积分的近似化,值的动态范围变化较小,不需要坐标空间的转换而具有比连续矩更好的应用前景。以离散正交多项式为基本集的离散正交矩是本文的主要研究内容,并用它们来进行图像重构,以验证它们的图像表征能力。一种新的以离散正交Hahn多项式为变换核的离散正交矩--Hahn矩被介绍。同时为了提高Hahn矩的计算精度,我们推导出它的加权正则化形式,并推导了该多项式的递推关系式,并利用x方向的对称性来减少Hahn矩计算中的误差,从而使被重建的图像具有更高的准确性。我们还介绍了Racah矩和Dual-Hahn矩。构建他们的Racah多项式和Dual-Hahn多项式是一类在非均匀网格上正交的离散正交多项式。为了提高Racah矩和Dual-Hahn矩的计算精度,我们推导出它们的规则化形式,同时推导了这两种多项式的递推关系式,并对图像进行了重建,结果证明了该方法的有效性。最后,本文还对这些用于构建离散正交矩的多项式进行总结。更多还原

【Abstract】 A new discrete orthogonal moment—Hahn moment based on the discrete classical Hahn polynomials is introduced. In order to ensure numerical stability, the Hahn polynomials are normalized, thus creating a set of weighted orthonormal Hahn polynomials, to define the so-called Hahn moments. The recurrence relations of the Hahn polynomials are derived. These relations, combining with the symmetry property, are used to evaluate the Hahn polynomial values. Such a strategy permits us to reduce the accumulation of numerical errors in the computation of high order polynomial values so that the original image can be accurately reconstructed.1 The image reconstruction ability using Hahn moment is compared with that using Tchebichef moment, and the experimental results demonstrate the effectiveness of the proposed method.A novel set of discrete orthogonal moments based on these polynomials which are built on the nonuniform lattices are introduced--Racah polynomials and Dual Hahn polynomials. In order to ensure numerical stability, the Racah polynomials are normalized, thus creating a set of weighted orthonormal Racah polynomials, to define the so-called Racah moments--the same to Dual Hahn polynomials. The recurrence relations of the both polynomials are derived. And the experiments about the reconstruction of the original images have been done. The experimental results demonstrate the effectiveness of the proposed method. At last the series of discrete orthogonal polynomials used to be the base sets of the discrete orthogonal moment have been discussed.更多还原