Opencv--图像处理之一阶和二阶偏导数

 1. 一阶差分：

2. 二阶偏导数的推导和近似：

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：

4. 同理：

5. 进而可推导：

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;
}

转载自：https://blog.csdn.net/xiaofengsheng/article/details/6023368