opencv中的图像矩（空间矩，中心矩，归一化中心矩，Hu矩）

拼搏现实的明天。 2022-06-02 10:25 253阅读 0赞

**1. [ opencv计算图像矩的API][opencv_API]**

**C++:**Moments moments (InputArray **array**, bool **binaryImage**=false )

**Python:**cv2. moments (array \[, binaryImage \] ) → retval

**C:**void cvMoments (const CvArr\* **arr**, CvMoments\* **moments**, int **binary**=0 )

<table style="margin:20px -.5em 10px; border:0px; color:rgb(0,0,0); font-family:'Open Sans','Lucida Grande','Lucida Sans Unicode',Geneva,Verdana,sans-serif; font-size:14px; line-height:18.200000762939453px; text-align:justify"> 
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  <tr> 
   <th style="text-align:left; padding:1px 8px 1px 5px; border:0px!important"> Parameters:</th> 
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    <ul style="margin:0px; padding-left:1em"> 
     <li style="text-align:justify; line-height:18.200000762939453px; margin:0px"><strong>array</strong>&nbsp;– Raster image (single-channel, 8-bit or floating-point 2D array) or an array (1xN or Nx1&nbsp;) of 2D points(<span>Point</span>&nbsp; &nbsp;or&nbsp;<span>Point2f</span>&nbsp;).</li> 
     <li style="text-align:justify; line-height:18.200000762939453px; margin:0px"><strong>binaryImage</strong>&nbsp;– If it is true, all non-zero image pixels are treated as 1’s. The parameter is used for images only.</li> 
     <li style="text-align:justify; line-height:18.200000762939453px; margin:0px"><strong>moments</strong>&nbsp;– Output moments.</li> 
    </ul> </td> 
  </tr> 
 </tbody> 
</table>

class Moments
    {
       
    public:
        Moments();
        Moments(double m00, double m10, double m01, double m20, double m11,
                double m02, double m30, double m21, double m12, double m03 );
        Moments( const CvMoments& moments );
        operator CvMoments() const;
    
        // spatial moments空间矩
        double  m00, m10, m01, m20, m11, m02, m30, m21, m12, m03;
        // central moments中心矩
        double  mu20, mu11, mu02, mu30, mu21, mu12, mu03;
        // central normalized moments归一化的中心矩
        double  nu20, nu11, nu02, nu30, nu21, nu12, nu03;
    }

Calculates seven Hu invariants.

**C++:**void HuMoments (const Moments& **m**, OutputArray **hu** )

**C++:**void HuMoments (const Moments& **moments**, double **hu**\[7\] )

**Python:**cv2. HuMoments (m \[, hu \] ) → hu

**C:**void cvGetHuMoments (CvMoments\* **moments**, CvHuMoments\* **hu\_moments** )

<table style="margin:20px -.5em 10px; border:0px; color:rgb(0,0,0); font-family:'Open Sans','Lucida Grande','Lucida Sans Unicode',Geneva,Verdana,sans-serif; font-size:14px; line-height:18.200000762939453px; text-align:justify"> 
 <tbody> 
  <tr> 
   <th style="text-align:left; padding:1px 8px 1px 5px; border:0px!important"> Parameters:</th> 
   <td style="padding:1px 8px 1px 5px; text-align:left; border:0px!important"> 
    <ul style="margin:0px; padding-left:1em"> 
     <li style="text-align:justify; line-height:18.200000762939453px; margin:0px"><strong>moments</strong>&nbsp;– Input moments computed with&nbsp;<a href="https://docs.opencv.org/3.0-beta/modules/imgproc/doc/structural_analysis_and_shape_descriptors.html#Moments%20moments%28InputArray%20array,%20bool%20binaryImage%29" title="Moments moments(InputArray array, bool binaryImage)" style="color:rgb(40,120,162)" rel="nofollow"><span>moments()</span></a>&nbsp;.</li> 
     <li style="text-align:justify; line-height:18.200000762939453px; margin:0px"><strong>hu</strong>&nbsp;– Output Hu invariants.</li> 
    </ul> </td> 
  </tr> 
 </tbody> 
</table>

**2. 矩的计算及特性**

2.1空间矩

空间矩的实质为面积或者质量。可以通过一阶矩计算质心/重心。

![\\texttt\{m\} \_\{ji\}= \\sum \_\{x,y\}  \\left ( \\texttt\{array\} (x,y)  \\cdot x^j  \\cdot y^i \\right )][texttt_m_ _ji_ _sum _x_y_ _left _ _texttt_array_ _x_y_ _cdot x_j _cdot y_i _right]

重心：![\\bar\{x\} = \\frac\{\\texttt\{m\}\_\{10\}\}\{\\texttt\{m\}\_\{00\}\} , \\; \\bar\{y\} = \\frac\{\\texttt\{m\}\_\{01\}\}\{\\texttt\{m\}\_\{00\}\}][bar_x_ _ _frac_texttt_m_10_texttt_m_00_ _ _ _bar_y_ _ _frac_texttt_m_01_texttt_m_00]

2.2中心矩

中心矩体现的是图像强度的最大和最小方向（中心矩可以构建图像的协方差矩阵），其只具有平移不变性，所以用中心矩做匹配效果不会很好。

![\\texttt\{mu\} \_\{ji\}= \\sum \_\{x,y\}  \\left ( \\texttt\{array\} (x,y)  \\cdot (x -  \\bar\{x\} )^j  \\cdot (y -  \\bar\{y\} )^i \\right )][texttt_mu_ _ji_ _sum _x_y_ _left _ _texttt_array_ _x_y_ _cdot _x - _bar_x_ _j _cdot _y - _bar_y_ _i _right]

2.3归一化的中心矩

归一化后具有尺度不变性。

![\\texttt\{nu\} \_\{ji\}= \\frac\{\\texttt\{mu\}\_\{ji\}\}\{\\texttt\{m\}\_\{00\}^\{(i+j)/2+1\}\} .][texttt_nu_ _ji_ _frac_texttt_mu_ji_texttt_m_00_i_j_2_1_ .]

2.4Hu矩Hu矩由于具有尺度、旋转、平移不变性，可以用来做匹配。![\\begin\{array\}\{l\} hu\[0\]= \\eta \_\{20\}+ \\eta \_\{02\} \\\\ hu\[1\]=( \\eta \_\{20\}- \\eta \_\{02\})^\{2\}+4 \\eta \_\{11\}^\{2\} \\\\ hu\[2\]=( \\eta \_\{30\}-3 \\eta \_\{12\})^\{2\}+ (3 \\eta \_\{21\}- \\eta \_\{03\})^\{2\} \\\\ hu\[3\]=( \\eta \_\{30\}+ \\eta \_\{12\})^\{2\}+ ( \\eta \_\{21\}+ \\eta \_\{03\})^\{2\} \\\\ hu\[4\]=( \\eta \_\{30\}-3 \\eta \_\{12\})( \\eta \_\{30\}+ \\eta \_\{12\})\[( \\eta \_\{30\}+ \\eta \_\{12\})^\{2\}-3( \\eta \_\{21\}+ \\eta \_\{03\})^\{2\}\]+(3 \\eta \_\{21\}- \\eta \_\{03\})( \\eta \_\{21\}+ \\eta \_\{03\})\[3( \\eta \_\{30\}+ \\eta \_\{12\})^\{2\}-( \\eta \_\{21\}+ \\eta \_\{03\})^\{2\}\] \\\\ hu\[5\]=( \\eta \_\{20\}- \\eta \_\{02\})\[( \\eta \_\{30\}+ \\eta \_\{12\})^\{2\}- ( \\eta \_\{21\}+ \\eta \_\{03\})^\{2\}\]+4 \\eta \_\{11\}( \\eta \_\{30\}+ \\eta \_\{12\})( \\eta \_\{21\}+ \\eta \_\{03\}) \\\\ hu\[6\]=(3 \\eta \_\{21\}- \\eta \_\{03\})( \\eta \_\{21\}+ \\eta \_\{03\})\[3( \\eta \_\{30\}+ \\eta \_\{12\})^\{2\}-( \\eta \_\{21\}+ \\eta \_\{03\})^\{2\}\]-( \\eta \_\{30\}-3 \\eta \_\{12\})( \\eta \_\{21\}+ \\eta \_\{03\})\[3( \\eta \_\{30\}+ \\eta \_\{12\})^\{2\}-( \\eta \_\{21\}+ \\eta \_\{03\})^\{2\}\] \\\\ \\end\{array\}][begin_array_l_ hu_0_ _eta _20_ _eta _02_ _ hu_1_ _eta _20_- _eta _02_2_4 _eta _11_2_ _ hu_2_ _eta _30_-3 _eta _12_2_ _3 _eta _21_- _eta _03_2_ _ hu_3_ _eta _30_ _eta _12_2_ _ _eta _21_ _eta _03_2_ _ hu_4_ _eta _30_-3 _eta _12_ _eta _30_ _eta _12_ _eta _30_ _eta _12_2_-3_ _eta _21_ _eta _03_2_3 _eta _21_- _eta _03_ _eta _21_ _eta _03_3_ _eta _30_ _eta _12_2_-_ _eta _21_ _eta _03_2_ _ hu_5_ _eta _20_- _eta _02_ _eta _30_ _eta _12_2_- _ _eta _21_ _eta _03_2_4 _eta _11_ _eta _30_ _eta _12_ _eta _21_ _eta _03_ _ hu_6_3 _eta _21_- _eta _03_ _eta _21_ _eta _03_3_ _eta _30_ _eta _12_2_-_ _eta _21_ _eta _03_2_-_ _eta _30_-3 _eta _12_ _eta _21_ _eta _03_3_ _eta _30_ _eta _12_2_-_ _eta _21_ _eta _03_2_ _ _end_array]

[opencv_API]: https://docs.opencv.org/3.0-beta/modules/imgproc/doc/structural_analysis_and_shape_descriptors.html
[texttt_m_ _ji_ _sum _x_y_ _left _ _texttt_array_ _x_y_ _cdot x_j _cdot y_i _right]: https://docs.opencv.org/3.0-beta/_images/math/d9bd5f5326c5ed2f2ddf8b674f776e80edaa528b.png
[bar_x_ _ _frac_texttt_m_10_texttt_m_00_ _ _ _bar_y_ _ _frac_texttt_m_01_texttt_m_00]: https://docs.opencv.org/3.0-beta/_images/math/0af8534226c23a39387269570965e276a70d80a8.png
[texttt_mu_ _ji_ _sum _x_y_ _left _ _texttt_array_ _x_y_ _cdot _x - _bar_x_ _j _cdot _y - _bar_y_ _i _right]: https://docs.opencv.org/3.0-beta/_images/math/b2d04bb91f234f0f3a14bb2c760bd3462d4132d7.png
[texttt_nu_ _ji_ _frac_texttt_mu_ji_texttt_m_00_i_j_2_1_ .]: https://docs.opencv.org/3.0-beta/_images/math/3570e87610d702dd1a18c9b1d0bfeedfc5c340a8.png
[begin_array_l_ hu_0_ _eta _20_ _eta _02_ _ hu_1_ _eta _20_- _eta _02_2_4 _eta _11_2_ _ hu_2_ _eta _30_-3 _eta _12_2_ _3 _eta _21_- _eta _03_2_ _ hu_3_ _eta _30_ _eta _12_2_ _ _eta _21_ _eta _03_2_ _ hu_4_ _eta _30_-3 _eta _12_ _eta _30_ _eta _12_ _eta _30_ _eta _12_2_-3_ _eta _21_ _eta _03_2_3 _eta _21_- _eta _03_ _eta _21_ _eta _03_3_ _eta _30_ _eta _12_2_-_ _eta _21_ _eta _03_2_ _ hu_5_ _eta _20_- _eta _02_ _eta _30_ _eta _12_2_- _ _eta _21_ _eta _03_2_4 _eta _11_ _eta _30_ _eta _12_ _eta _21_ _eta _03_ _ hu_6_3 _eta _21_- _eta _03_ _eta _21_ _eta _03_3_ _eta _30_ _eta _12_2_-_ _eta _21_ _eta _03_2_-_ _eta _30_-3 _eta _12_ _eta _21_ _eta _03_3_ _eta _30_ _eta _12_2_-_ _eta _21_ _eta _03_2_ _ _end_array]: https://docs.opencv.org/3.0-beta/_images/math/6f3e6a1dd05317afa7a9317d0d0e87bc3dacf7fd.png