图像处理中的一阶偏导数和二阶偏导数

男娘i 2022-07-16 16:44 174阅读 0赞

1. 一阶差分：

![0_1290233319aNcF.gif][]

2. 二阶偏导数的推导和近似：

![0_1290233325LMGR.gif][]

3. 上式以点（i+1，j）为中心，用i代换i+1可得以（i,j）为中心的二阶偏导数则有：

![0_12902333308RB1.gif][]

4. 同理：

![0_12902333356g42.gif][]

5. 进而可推导：

![0_1290233339EzVx.gif][]

6. 这样我们就可以很好的运用其他的一阶偏导的定义，如SIFT特征OpenCV实现版本中的一阶以及二阶偏导：

/\* Computes the partial derivatives in x, y, and scale of a pixel in the DoG scale space pyramid. @param dog\_pyr DoG scale space pyramid @param octv pixel's octave in dog\_pyr @param intvl pixel's interval in octv @param r pixel's image row @param c pixel's image col @return Returns the vector of partial derivatives for pixel I \{ dI/dx, dI/dy, dI/ds \}^T as a CvMat\* \*/ static CvMat\* deriv\_3D( IplImage\*\*\* dog\_pyr, int octv, int intvl, int r, int c ) \{ CvMat\* dI; double dx, dy, ds; dx = ( pixval32f( dog\_pyr\[octv\]\[intvl\], r, c+1 ) - pixval32f( dog\_pyr\[octv\]\[intvl\], r, c-1 ) ) / 2.0; dy = ( pixval32f( dog\_pyr\[octv\]\[intvl\], r+1, c ) - pixval32f( dog\_pyr\[octv\]\[intvl\], r-1, c ) ) / 2.0; ds = ( pixval32f( dog\_pyr\[octv\]\[intvl+1\], r, c ) - pixval32f( dog\_pyr\[octv\]\[intvl-1\], r, c ) ) / 2.0; dI = cvCreateMat( 3, 1, CV\_64FC1 ); cvmSet( dI, 0, 0, dx ); cvmSet( dI, 1, 0, dy ); cvmSet( dI, 2, 0, ds ); return dI; \} /\* Computes the 3D Hessian matrix for a pixel in the DoG scale space pyramid. @param dog\_pyr DoG scale space pyramid @param octv pixel's octave in dog\_pyr @param intvl pixel's interval in octv @param r pixel's image row @param c pixel's image col @return Returns the Hessian matrix (below) for pixel I as a CvMat\* / Ixx Ixy Ixs / <BR> | Ixy Iyy Iys | <BR> / Ixs Iys Iss / \*/ static CvMat\* hessian\_3D( IplImage\*\*\* dog\_pyr, int octv, int intvl, int r, int c ) \{ CvMat\* H; double v, dxx, dyy, dss, dxy, dxs, dys; v = pixval32f( dog\_pyr\[octv\]\[intvl\], r, c ); dxx = ( pixval32f( dog\_pyr\[octv\]\[intvl\], r, c+1 ) + pixval32f( dog\_pyr\[octv\]\[intvl\], r, c-1 ) - 2 \* v ); dyy = ( pixval32f( dog\_pyr\[octv\]\[intvl\], r+1, c ) + pixval32f( dog\_pyr\[octv\]\[intvl\], r-1, c ) - 2 \* v ); dss = ( pixval32f( dog\_pyr\[octv\]\[intvl+1\], r, c ) + pixval32f( dog\_pyr\[octv\]\[intvl-1\], r, c ) - 2 \* v ); dxy = ( pixval32f( dog\_pyr\[octv\]\[intvl\], r+1, c+1 ) - pixval32f( dog\_pyr\[octv\]\[intvl\], r+1, c-1 ) - pixval32f( dog\_pyr\[octv\]\[intvl\], r-1, c+1 ) + pixval32f( dog\_pyr\[octv\]\[intvl\], r-1, c-1 ) ) / 4.0; dxs = ( pixval32f( dog\_pyr\[octv\]\[intvl+1\], r, c+1 ) - pixval32f( dog\_pyr\[octv\]\[intvl+1\], r, c-1 ) - pixval32f( dog\_pyr\[octv\]\[intvl-1\], r, c+1 ) + pixval32f( dog\_pyr\[octv\]\[intvl-1\], r, c-1 ) ) / 4.0; dys = ( pixval32f( dog\_pyr\[octv\]\[intvl+1\], r+1, c ) - pixval32f( dog\_pyr\[octv\]\[intvl+1\], r-1, c ) - pixval32f( dog\_pyr\[octv\]\[intvl-1\], r+1, c ) + pixval32f( dog\_pyr\[octv\]\[intvl-1\], r-1, c ) ) / 4.0; H = cvCreateMat( 3, 3, CV\_64FC1 ); cvmSet( H, 0, 0, dxx ); cvmSet( H, 0, 1, dxy ); cvmSet( H, 0, 2, dxs ); cvmSet( H, 1, 0, dxy ); cvmSet( H, 1, 1, dyy ); cvmSet( H, 1, 2, dys ); cvmSet( H, 2, 0, dxs ); cvmSet( H, 2, 1, dys ); cvmSet( H, 2, 2, dss ); return H; \}

参考：

（1）[http://hi.baidu.com/shareshow/blog/item/34abdf544725cf54d109069b.html][http_hi.baidu.com_shareshow_blog_item_34abdf544725cf54d109069b.html]

（2）SIFT的OpenCV实现

[0_1290233319aNcF.gif]: /images/20220709/d0ddd719f6a9457ab841bbd319540a09.png
[0_1290233325LMGR.gif]: /images/20220709/66c06091f1064cbca675b663db8f4339.png
[0_12902333308RB1.gif]: /images/20220709/5cc0dce3338a4253aa96896bcfd38c02.png
[0_12902333356g42.gif]: /images/20220709/f3aa48754ce34977903e29c219577a13.png
[0_1290233339EzVx.gif]: /images/20220709/81adad9ae45c4efca8eea95908987d75.png
[http_hi.baidu.com_shareshow_blog_item_34abdf544725cf54d109069b.html]: http://hi.baidu.com/shareshow/blog/item/34abdf544725cf54d109069b.html