﻿ Gaussian-Hermite矩旋转不变矩的构建
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Gaussian-Hermite矩旋转不变矩的构建

Construction of invariants of Gaussian-Hermite moments
Zhang Chaoxin, Xi Ping, Hu Bifu
School of Mechanical Engineering and Automation, Beijing University of Aeronautics and Astronautics, Beijing 100191, China
Abstract:Moments and functions of moment are widely used in image processing and pattern recognition due to their strong ability to represent features of an image. However, the researches on Gaussian-Hermite moments with which own orthogonal properties are still relatively less currently. Gaussian-Hermite moments were deeply studied, and correspondently, Polar-Gaussian-Hermite moments were proposed in polar coordinate. The method to compute the functions of both moments using raising and lowering operators were presented. Finally, the rotation invariants of Gaussian-Hermite moments were derived based on the Polar-Gaussian-Hermite moments and an independent and complete set of the invariants were given. The given experiment validates the correctness and good digital stability of the proposed invariants.
Key words: Gaussian-Hermite moments     Polar-Gaussian-Hermite moments     rotation invariants     pattern recognition     image processing

1 GH矩 1.1 GH矩基函数

 图 1 GH矩的前5阶矩的基函数Fig. 1 Basic functions under order 5 of GH moments
1.2 GH矩定义

1.3 用升降算符计算GH矩的基

1.4 离散形式

GH矩是定义在连续区间(-∞,∞)上的.GH矩中存在尺度因子σ,在计算矩之前应先设定好大小.对于不同大小的图像,为了更方便设定尺度因子,应把图像坐标转换到一个固定的区间里.因此,这里根据通常的使用习惯选择区间[-1,1],即对于一个定义在区间[0≤i,j≤K-1]上的数据图像I(i,j),图像坐标首先通过下面的式子进行转换:

2 PGH矩 2.1 PGH矩的基函数

 图 2 PGH矩前5阶矩基函数Fig. 2 Basic functions under order 5 of PGH moments
2.2 PGH矩定义

2.3 GH矩与PGH矩的关系

3 GH矩的旋转不变矩

3.1 旋转不变的推导

3.2 GHM旋转不变矩的独立完备集

2,3阶矩:

4阶矩:

5阶矩:

6阶矩:

4 实验证明

 图 3 医学灰度图像及其旋转版本Fig. 3 Medical image and its rotational versions

 角度/(°) ψ1 ψ2 ψ3 ψ4 ψ5 ψ6 ψ7 ψ8 ψ9 0 13.3810 3.1951 4.0437 1.1853 4.6365 3.9185 14.0222 1.7622 -0.3699 20 13.3818 3.1947 4.0416 1.1862 4.6340 3.9162 14.0179 1.7641 -0.3687 70 13.3816 3.1968 4.0453 1.1890 4.6345 3.9282 14.0184 1.7659 -0.3683 110 13.3818 3.1947 4.0416 1.1862 4.6340 3.9162 14.0179 1.7641 -0.3687 170 13.3825 3.1940 4.0415 1.1896 4.6287 3.9259 14.0180 1.7630 -0.3696 200 13.3818 3.1947 4.0416 1.1862 4.6340 3.9162 14.0179 1.7641 -0.3687 230 13.3821 3.1955 4.0412 1.1903 4.6317 3.9329 14.0193 1.7642 -0.3672 310 13.3812 3.1979 4.0462 1.1843 4.6390 3.9185 14.0187 1.7660 -0.3728 偏差/% 0.0026 0.0303 0.0410 0.1560 0.0423 0.1418 0.0070 0.0492 0.3120 角度/(°) ψ10 ψ11 ψ12 ψ13 ψ14 ψ15 ψ16 ψ17 ψ18 0 -1.7296 -1.4225 7.8066 4.8990 -0.9706 1.3113 1.2079 -26.2475 -42.1389 20 -1.7265 -1.4251 7.8041 4.8980 -0.9704 1.3112 1.2028 -26.2279 -42.0656 70 -1.7279 -1.4292 7.8020 4.8990 -0.9697 1.3069 1.2054 -26.2292 -42.1714 110 -1.7265 -1.4251 7.8041 4.8980 -0.9704 1.3112 1.2028 -26.2279 -42.0656 170 -1.7261 -1.4266 7.8003 4.8965 -0.9690 1.3110 1.2068 -26.1806 -42.1137 200 -1.7265 -1.4251 7.8041 4.8980 -0.9704 1.3112 1.2028 -26.2279 -42.0656 230 -1.7235 -1.4314 7.8068 4.9001 -0.9680 1.3102 1.2039 -26.1296 -42.1269 310 -1.7275 -1.4221 7.8089 4.9019 -0.9714 1.3115 1.2096 -26.2772 -42.2547 偏差/% 0.0688 0.1673 0.0274 0.0243 0.0828 0.0774 0.1801 0.1208 0.1131

5 结 束 语

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#### 文章信息

Zhang Chaoxin, Xi Ping, Hu Bifu
Gaussian-Hermite矩旋转不变矩的构建
Construction of invariants of Gaussian-Hermite moments

Journal of Beijing University of Aeronautics and Astronsutics, 2014, 40(11): 1602-1608.
http://dx.doi.org/10.13700/j.bh.1001-5965.2013.0677