Zernike矩及Opencv实现

<p style="color: rgb(51, 51, 51);">Zernike在1934年引入了一组定义在单位圆<img alt="" src="http://hi.csdn.net/attachment/201104/19/0_1303212176MEUM.gif" /> 上的复值函数集{<img alt="" src="http://hi.csdn.net/attachment/201104/19/0_1303212176qJnN.gif" /> }，{<img alt="" src="http://hi.csdn.net/attachment/201104/19/0_1303212180i63a.gif" /> }具有完备性和正交性，使得它可以表示定义在单位圆盘内的任何平方可积函数。其定义为：</p><p style="color: rgb(51, 51, 51);"><img alt="" src="http://hi.csdn.net/attachment/201104/19/0_1303212181Ikn3.gif" /></p><p style="color: rgb(51, 51, 51);"><img alt="" src="http://hi.csdn.net/attachment/201104/19/0_13032121819tHh.gif" /> 表示原点到点<img alt="" src="http://hi.csdn.net/attachment/201104/19/0_1303212181Ss1x.gif" /> 的矢量长度；<img alt="" src="http://hi.csdn.net/attachment/201104/19/0_1303212181Q024.gif" /> 表示矢量<img alt="" src="http://hi.csdn.net/attachment/201104/19/0_1303212181wawz.gif" /> 与<img alt="" src="http://hi.csdn.net/attachment/201104/19/0_13032121825rVE.gif" /> 轴逆时针方向的夹角。</p><p style="color: rgb(51, 51, 51);"><img alt="" src="http://hi.csdn.net/attachment/201104/19/0_1303212182gal6.gif" /> 是实值径向多项式：</p><p style="color: rgb(51, 51, 51);"><img alt="" src="http://hi.csdn.net/attachment/201104/19/0_1303212182CDDl.gif" /></p><p style="color: rgb(51, 51, 51);">称为Zernike多项式。</p><p style="color: rgb(51, 51, 51);">Zernike多项式满足正交性：</p><p style="color: rgb(51, 51, 51);"><img alt="" src="http://hi.csdn.net/attachment/201104/19/0_1303212182g6Xx.gif" /></p><p style="color: rgb(51, 51, 51);">其中</p><p style="color: rgb(51, 51, 51);"><img alt="" src="http://hi.csdn.net/attachment/201104/19/0_1303212182HLJ6.gif" /> 为克罗内克符号，<img alt="" src="http://hi.csdn.net/attachment/201104/19/0_1303212183MkZo.gif" /> <img alt="" src="http://hi.csdn.net/attachment/201104/19/0_1303212183u72x.gif" /></p><p style="color: rgb(51, 51, 51);"><img alt="" src="http://hi.csdn.net/attachment/201104/19/0_1303212183ZQgo.gif" /> 是<img alt="" src="http://hi.csdn.net/attachment/201104/19/0_1303212183rfRA.gif" /> 的共轭多项式。</p><p style="color: rgb(51, 51, 51);">由于Zernike多项式的正交完备性，所以在单位圆内的任何图像<img alt="" src="http://hi.csdn.net/attachment/201104/19/0_13032121838SKy.gif" /> 都可以唯一的用下面式子来展开：</p><p style="color: rgb(51, 51, 51);"><img alt="" src="http://hi.csdn.net/attachment/201104/19/0_130321218433GG.gif" /></p><p style="color: rgb(51, 51, 51);">式子中<img alt="" src="http://hi.csdn.net/attachment/201104/19/0_1303212184ETxH.gif" /> 就是Zernike矩，其定义为：</p><p style="color: rgb(51, 51, 51);"><img alt="" src="http://hi.csdn.net/attachment/201104/19/0_1303212184G23M.gif" /></p><p style="color: rgb(51, 51, 51);">注意式子中<img alt="" src="http://hi.csdn.net/attachment/201104/19/0_1303212184gpvZ.gif" /> 和<img alt="" src="http://hi.csdn.net/attachment/201104/19/0_1303212184fnNk.gif" /> 采用的是不同的坐标系（<img alt="" src="http://hi.csdn.net/attachment/201104/19/0_1303212185cX3f.gif" /> 采用直角坐标，而<img alt="" src="http://hi.csdn.net/attachment/201104/19/0_1303212185u17M.gif" /> 采用的极坐标系，在计算的时候要进行坐标转换）</p><p style="color: rgb(51, 51, 51);">对于离散的数字图像，可将积分形式改为累加形式：</p><p style="color: rgb(51, 51, 51);"><img alt="" src="http://hi.csdn.net/attachment/201104/19/0_13032121856x5r.gif" /> <img alt="" src="http://hi.csdn.net/attachment/201104/19/0_1303212185AoCr.gif" /></p><p style="color: rgb(51, 51, 51);">我们在计算一副图像的Zernike矩时，必须将图像的中心移到坐标的原点，将图像的像素点映射到单位圆内，由于Zernike矩具有旋转不变性，我们可以将<img alt="" src="http://hi.csdn.net/attachment/201104/19/0_13032121859Muz.gif" /> 作为图像的不变特征，其中图像的低频特征有p值小的<img alt="" src="http://hi.csdn.net/attachment/201104/19/0_1303212186c13v.gif" />提取，高频特征由p值高的<img alt="" src="http://hi.csdn.net/attachment/201104/19/0_130321218645Gm.gif" /> 提取。从上面可以看出，Zernike矩可以构造任意高阶矩。</p><p style="color: rgb(51, 51, 51);">由于Zernike矩只具有旋转不变性，不具有平移和尺度不变性，所以要提前对图像进行归一化，我们采用标准矩的方法来归一化一副图像，标准矩定义为：</p><p style="color: rgb(51, 51, 51);"><img alt="" src="http://hi.csdn.net/attachment/201104/19/0_1303212186NX9l.gif" /> ，</p><p style="color: rgb(51, 51, 51);">由标准矩我们可以得到图像的"重心"，</p><p style="color: rgb(51, 51, 51);"><img alt="" src="http://hi.csdn.net/attachment/201104/19/0_1303212186C40C.gif" /></p><p style="color: rgb(51, 51, 51);"><img alt="" src="http://hi.csdn.net/attachment/201104/19/0_1303212186PfPn.gif" /></p><p style="color: rgb(51, 51, 51);">我们将图像的"重心"移动到单位圆的圆心（即坐标的原点），便解决了平移问题。</p><p style="color: rgb(51, 51, 51);">我们知道<img alt="" src="http://hi.csdn.net/attachment/201104/19/0_1303212187bteB.gif" /> 表征了图像的"面积"，归一图像的尺度无非就是把他们的大小变为一致的，（这里的大小指的是图像目标物的大小，不是整幅图像的大小，"面积"也是目标物的"面积"）。</p><p style="color: rgb(51, 51, 51);">所以，对图像进行变换<img alt="" src="http://hi.csdn.net/attachment/201104/19/0_1303212187lf2L.gif" /> 就可以达到图像尺寸一致的目的。</p><p style="color: rgb(51, 51, 51);">综合上面结果，对图像进行<img alt="" src="http://hi.csdn.net/attachment/201104/19/0_1303212187BCKk.gif" /> 变换，最终图像<img alt="" src="http://hi.csdn.net/attachment/201104/19/0_1303212187X6X8.gif" /> 的Zernike矩就是平移，尺寸和旋转不变的。</p><p style="color: rgb(51, 51, 51);"><![endif]--></p><p style="color: rgb(51, 51, 51);">Zernike 不变矩相比 Hu 不变矩识别效果会好一些，因为他描述了图像更多的细节内容，特别是高阶矩，但是由于Zernike 不变矩计算时间比较长，所以出现了很多快速的算法，大家可以 google 一下。</p><p style="color: rgb(51, 51, 51);">用 Zernike 不变矩来识别手势轮廓，识别率大约在 40%~50% 之间，跟 Hu 不变矩一样， Zernike 不变矩一般用来描述目标物形状占优势的图像，不适合用来描述纹理丰富的图像，对于纹理图像，识别率一般在 20%~30% 左右，很不占优势。</p>

C++代码如下：/*计算一行的像素个数imwidth:图像宽度deep:图像深度（8位灰度图为1，24位彩色图为3）*/#define  bpl(imwidth, deep) ((imwidth*deep*8+31)/32*4)/*获取像素值psrcBmp:图像数据指针nsrcBmpWidth:图像宽度，以像素为单位x,y:像素点deep:图像的位数深度，（1表示8位的灰度图，3表示24位的RGB位图）*/COLORREF J_getpixel( const BYTE *psrcBmp, const int nsrcBmpWidth, const int x, const int y, int deep = 3){if (deep == 3){return RGB(*(psrcBmp + x*3 + y*bpl(nsrcBmpWidth, deep) + 2 ) , *(psrcBmp + x*3 + y*bpl(nsrcBmpWidth, deep) + 1 ) , *(psrcBmp + x*3 + y*bpl(nsrcBmpWidth, deep) +0 ));}else if (deep == 1){return *(psrcBmp + x + y*bpl(nsrcBmpWidth, deep));}}//获取标准矩(只支持8位灰度图)void GetStdMoment(BYTE *psrcBmp ,  int nsrcBmpWidth, int nsrcBmpHeight, double *m){for ( int p = 0 ; p < 2 ; p++ )for ( int q = 0 ; q < 2 ; q++ ){if( p == 1 && q == 1)break;for ( int y = 0 ; y < nsrcBmpHeight ; y++ )for ( int x = 0 ; x < nsrcBmpWidth ; x++ )m[p*2+q] += (pow( (double)x , p ) * pow( (double)y , q ) * J_getpixel(psrcBmp , nsrcBmpWidth , x ,y, 1));}}//阶乘double Factorial( int n ){if( n < 0 )return -1;double m = 1;for(int i = 2 ; i <= n ; i++){m *= i;}return m;}//阶乘数，计算好方便用，提高速度double factorials[11] = {1 , 1 , 2 , 6 , 24 , 120 , 720 , 5040 , 40320 , 362880 , 39916800};//把图像映射到单位圆，获取像素极坐标半径double GetRadii(int nsrcBmpWidth,   int nsrcBmpHeight,   int x0,   int y0,   int x,   int y){double lefttop = sqrt(((double)0 - x0)*(0 - x0) + (0 - y0)*(0 - y0));double righttop = sqrt(((double)nsrcBmpWidth - 1 - x0)*(nsrcBmpWidth - 1 - x0) + (0 - y0)*(0 - y0));double leftbottom = sqrt(((double)0 - x0)*(0 - x0) + (nsrcBmpHeight - 1 - y0)*(nsrcBmpHeight - 1 - y0));double rightbottom = sqrt(((double)nsrcBmpWidth - 1 - x0)*(nsrcBmpWidth - 1 - x0) + (nsrcBmpHeight - 1 - y0)*(nsrcBmpHeight - 1 - y0));double maxRadii = lefttop;maxRadii < righttop ? righttop : maxRadii;maxRadii < leftbottom ? leftbottom : maxRadii;maxRadii < rightbottom ? rightbottom : maxRadii;double Radii = sqrt(((double)x - x0)*(x - x0) + (y - y0)*(y - y0))/maxRadii;if(Radii > 1){Radii = 1;}return Radii;}//把图像映射到单位圆，获取像素极坐标角度double GetAngle(int nsrcBmpWidth,int nsrcBmpHeight,int x,int y){double o;double dia = sqrt((double)nsrcBmpWidth*nsrcBmpWidth + nsrcBmpHeight*nsrcBmpHeight);int x0 = nsrcBmpWidth / 2;int y0 = nsrcBmpHeight / 2;double x_unity = (x - x0)/(dia/2); double y_unity = (y - y0)/(dia/2);if( x_unity == 0 && y_unity >= 0 )o=pi/2;else if( x_unity ==0 && y_unity <0)o=1.5*pi;elseo=atan( y_unity / x_unity );if(o*y<0)    //第三象限o=o+pi;return o;}//Zernike不变矩J_GetZernikeMoment(BYTE *psrcBmp , int nsrcBmpWidth,int nsrcBmpHeight,double *Ze ){double R[count][count] = {0.0};double V[count][count] = {0.0};double M[4] = {0.0};GetStdMoment(psrcBmp , nsrcBmpWidth , nsrcBmpHeight , M);int x0 = (int)(M[2]/M[0]+0.5);int y0 = (int)(M[1]/M[0]+0.5);for(int n = 0 ; n < count ; n++){for (int m = 0 ; m < count ; m++){//优化算法，只计算以下介数if( (n == 1 && m == 0) ||(n == 1 && m == 1) ||(n == 2 && m == 0) ||(n == 2 && m == 1) ||(n == 2 && m == 2) ||(n == 3 && m == 0) ||(n == 3 && m == 1) ||(n == 3 && m == 2) ||(n == 3 && m == 3) ||(n == 4 && m == 0) ||(n == 4 && m == 1) ||(n == 4 && m == 2) ||(n == 4 && m == 3) ||(n == 4 && m == 4)){for(int y = 0 ; y < nsrcBmpHeight ; y++){for (int x = 0 ; x < nsrcBmpWidth ; x++){for(int s = 0 ; (s <= (n - m)/2 ) && n >= m ; s++){R[n][m] += pow( -1.0, s )* ( n - s > 10 ? Factorial( n - s ) : factorials[ n - s ] )* pow( GetRadii( nsrcBmpWidth, nsrcBmpHeight, x0, y0, x, y ), n - 2 * s )/ ( ( s > 10 ? Factorial( s ) : factorials[ s ] )* ( ( n + m ) / 2 - s > 10 ? Factorial( ( n + m ) / 2 - s ) : factorials[ ( n + m ) / 2 - s ] )* ( ( n - m ) / 2 - s > 10 ? Factorial( ( n - m ) / 2 - s ) : factorials[ ( n - m ) / 2 - s ] ) );}Ze[ n * count + m ] += R[ n ][ m ]* J_getpixel( psrcBmp, nsrcBmpWidth, x ,y, 1)* cos( m * GetAngle( nsrcBmpWidth, nsrcBmpHeight, x, y) );//实部V[n][m] += R[ n ][ m ] * J_getpixel( psrcBmp, nsrcBmpWidth, x, y, 1)* sin( m * GetAngle( nsrcBmpWidth, nsrcBmpHeight, x, y ) );//虚部R[n][m] = 0.0;}}*(Ze+n*count + m) = sqrt( (*(Ze+n*count + m))*(*(Ze+n*count + m)) + V[n][m]*V[n][m] )*(n+1)/pi/M[0];}}}}