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1、湖北工业大学 硕士学位论文 正交矩的高精度算法研究 姓名：罗柱 申请学位级别：硕士 专业：电力电子与电力传动 指导教师：付波 20100601 湖 北 工 业 大 学 硕 士 学 位 论 文 I 摘 要 自 20 世纪 80 年代，正交矩提出以来，立刻引起各国学者的关注与研究，并 被广泛用于图像处理的各种领域。与其它矩相比，正交矩具有如下独特优点：1 具 有反变换，从理论上讲，利用其反变换可以完全重建原始图像；2 具有最小信息 冗余度，使得图像特征提取的工作量大幅度减少。然而，矩的计算精度和算法效 率依然制约着正交矩在模式识别领域的发展与应用。正交矩的离散误差以及传递 误差均与矩运算量具有不可
2、调和的矛盾，而如何妥善解决正交矩算法精度与算法 效率之间矛盾，仍然是正交矩高精度算法研究中的盲点。本文根据正交矩的发展， 将其分为了以 Zernike 矩为代表的连续正交矩和以 Krawtchouk 矩为代表的离散正 交矩两类，并分别研究了这两类正交矩的高精度算法理论，进而探索了高精度正 交矩模型下的快速算法。 本文首先在总结前人工作的基础上，将已有的几种主要算法进行了归类、分 析和评价，并介绍了目前普遍使用的几种算法误差评价体系。连续正交矩的算法 思想主要有边界法、变换法以及迭代法；而离散正交矩则经历了经典迭代法和对 称法两个历程。 在连续正交矩方面，本文提出了一种在笛卡尔坐标系下精确计算
3、Zernike 矩的 三角积分算法。该算法原理为：先将 Zernike 矩表示为 Fourier-Mellin 矩的线性组 合，接着通过几何区域变换将矩形像素点积分转化为三角元的矩积分，然后利用 三角函数的积分性质推导了一种高精度的迭代算法来计算三角元的矩积分，最后 得到了 Zernike 矩的高精度算法。实验证明该算法比已有算法具有更高精度。 在离散正交矩方面，本文讨论了 Krawtchouk 矩在参数 p0.5 时的计算精度问 题，并提出了一种结合对称性的双向递推算法用于精确计算参数 p(0,1)的 Krawtchouk 矩。先利用直线 x=n 与 x+n=N-1 将 x-n 平面划分为四
4、个部分，分别利 用 n 方向的正向迭代和 n 方向的反向迭代计算1Nnxn 和1nxNn 两个部分内的 Krawtchouk 多项式值，接着通过对角线方向的对称性直接得到另外 两个部分的多项式值。在该算法中，最大迭代次数被降低到 N/2 次，保证了多项 式计算值的高度精确性。通过对 400400 的大灰度图像进行重建实验，验证了该 算法的有效性。 关键词：关键词：正交矩，Zernike 矩，三角积分算法，Krawtchouk 矩，双向递推算法 湖 北 工 业 大 学 硕 士 学 位 论 文 II Abstract Since the 80s of 20th century, orthogona
5、l moments have been concerned and studied by many scholars immediately, which are widely utilized in various fields of image processing. In comparison to other moments, orthogonal moments set have several unique advantages as follows: 1 They include inverse transform, which can be used to reconstruc
6、t the primary image completely in theory; 2 They include minimum information redundancy, which largely decreases the workload of image feature extraction. However, computational accuracy and algorithm efficiency of moments still restrict their development and application in the area of pattern recog
7、nition. Discrete errors and propagation errors have irreconcilable conflicts with calculation. Therefore, it becomes a blind spot on high-precision algorithm of orthogonal moments that how to properly resolve the contradiction between precision and efficiency of the algorithm. They should be divided
8、 into two categories: one is continuous orthogonal moments by representative as Zernike moments and the other is discrete orthogonal moments by Krawtchouk moments. In this paper, according to the development of the the orthogonal moments, theories of high-precision arithmetic of the two kinds moment
9、s have been studied respectively, and their fast algorithms under the model of high-precision orthogonal moments have been derived. On the basis of previous work, several key algorithms are classified, analyzed and evaluated and some common valuation systems of algorithm errors are introduced. Algor
10、ithms of continuous moments mainly include boundary method, transform method and iterative method, while those of discrete moments include classic recursive method and symmetric method. For continuous moments, a novel algorithm so called as triangle-integration algorithm is proposed to accurately ca
11、lculate Zernike moments in Cartesian Coordinates. Firstly, Zernike moments with any order and repetition are expressed as a linear combination of the Fourier-Mellin moments. And the moment integration in a pixel domain is re-arranged as a summation of four integrations in four triangle domains respe
12、ctively. Secondly, we convert the classical un-precise integration formula to a set of high-accurate analytic equations in Cartesian Coordinates based on the trigonometric functions. Finally, a set of high efficient computational recursive relations are proposed. Several experiments are designed to
13、verify the performance of the proposed algorithm. For discrete moments, the accuracy of the Krawtchouk moments for the common case of p0 has been discussed and a novel symmetry and bi-recursive algorithm is proposed to accurately calculate the Krawtchouk moments for the case of p(0, 1). Firstly, the
14、 x-n plane is divided into four parts by x=n and x+n=N-1. Then We use the n-ascending recurrence formula to calculate the polynomials in the domain of 1Nnxn and apply the n-descending recurrence relations in the domain of 1nxNn . Finally, with the help of the diagonal symmetry property on x=n, the 湖
15、 北 工 业 大 学 硕 士 学 位 论 文 III Krawtchouk polynomial values of high precision in the whole x-n coordinates are obtained. By limiting the maximum recursion times to N/2, the algorithm ensures that the maximum recursive numerical errors are within an acceptable range. An experiment on a large image of 400
16、400 pixels is designed to demonstrate the performance of the proposed algorithm against the classical method. Keywords: Orthogonal moments， Zernike moments，Triangle-integration algorithm， Krawtchouk moments，Bi-recursive algorithm 学位论文原创性声明和使用授权说明 原创性声明 原创性声明 本人郑重声明：所呈交的学位论文，是本人在导师指导下，独立进行研究工作所取 得的研究成果。除文中已经标明引用的内容外，本论文不包含任何其他个人或集体已经 发表或撰写过的研究成果。对本文的研究做出贡献的个人和集体，均已在文中以明确方 式标明。本声明的法律结果由本人承担。 学位论文作者签名： 日期： 年 月 日 学位论文版权使用授权书 学位论文版权使用授权书 本学位论文作者完全了解学校有关保留、使
