First-order and second-order partial derivatives in image processing

1. First-order difference:

 

2. derivation and approximation of second-order partial derivatives:

 

3. the above formula is centered on the point (I + 1, J), and the second-order partial derivative that can be centered on (I, j) with I for I + 1 is as follows:

 

4. Likewise:

 

5. Further derivation:

 

6. In this way, we can use other first-order partial guidance definitions, such as sift feature opencv, to implement the first-and second-order partial guidance in the version:

/* <Br/> computes the partial derivatives in X, Y, and scale of a pixel in the dog <br/> Scale Space pyramid. </P> <p> @ Param dog_pyr dog Scale Space pyramid <br/> @ Param octv pixel's Ave ave in dog_pyr <br/> @ Param intvl pixel's interval in octv <br/> @ Param R pixel's image row <br/> @ Param C pixel's image Col </P> <p> @ return returns the vector of partial derivatives pixel I <br/> {di/dx, DI/Dy, DI/DS} ^ t as a cvmat * <br/> */<br/> static cvmat * deriv_3d (iplimage *** dog_pyr, int octv, int intvl, int R, int c) <br/>{< br/> cvmat * di; <br/> double dx, Dy, DS; </P> <p> dx = (pixval32f (dog_pyr [octv] [intvl], R, C + 1) -<br/> pixval32f (dog_pyr [octv] [intvl], R, C-1)/2.0; <br/> DY = (pixval32f (dog_pyr [octv] [intvl], R + 1, C)-<br/> pixval32f (dog_pyr [octv] [intvl], R-1, c)/2.0; <br/> DS = (pixval32f (dog_pyr [octv] [intvl + 1], R, c)-<br/> pixval32f (dog_pyr [octv] [intvl-1], R, c)/2.0; </P> <p> di = cvcreatemat (3, 1, cv_64fc1); <br/> cvmset (Di, 0, 0, dx ); <br/> cvmset (Di, 1, 0, Dy); <br/> cvmset (Di, 2, 0, DS); </P> <p> return di; <br/>}</P> <p>/* <br/> computes the 3D Hessian matrix for a pixel in the dog Scale Space pyramid. </P> <p> @ Param dog_pyr dog Scale Space pyramid <br/> @ Param octv pixel's Ave ave in dog_pyr <br/> @ Param intvl pixel's interval in octv <br/> @ Param R pixel's image row <br/> @ Param C pixel's image Col </P> <p> @ return returns the Hessian matrix (below) for pixel I as a cvmat * </P> <p>/ixx ixy ixs/<br> <br/> | ixy iyy IYS | <br> <br/>/ixs iys iss/<br/> */<br/> static cvmat * hessian_3d (iplimage *** dog_pyr, int octv, int intvl, int R, int c) <br/>{< br/> cvmat * h; <br/> Double V, dxx, dyy, DSS, dxy, dxS, DYS; </P> <p> V = pixval32f (dog_pyr [octv] [intvl], R, C ); <br/> dxx = (pixval32f (dog_pyr [octv] [intvl], R, C + 1) + <br/> pixval32f (dog_pyr [octv] [intvl], R, c-1)-2 * V); <br/> dyy = (pixval32f (dog_pyr [octv] [intvl], R + 1, C) + <br/> pixval32f (dog_pyr [octv] [intvl], R-1, c)-2 * V ); <br/> DSS = (pixval32f (dog_pyr [octv] [intvl + 1], R, c) + <br/> pixval32f (dog_pyr [octv] [intvl-1], R, c)-2 * V); <br/> dxy = (pixval32f (dog_pyr [octv] [intvl], R + 1, C + 1) -<br/> pixval32f (dog_pyr [octv] [intvl], R + 1, C-1)-<br/> pixval32f (dog_pyr [octv] [intvl], R-1, c + 1) + <br/> pixval32f (dog_pyr [octv] [intvl], R-1, C-1)/4.0; <br/> DxS = (pixval32f (dog_pyr [octv] [intvl + 1], R, C + 1) -<br/> pixval32f (dog_pyr [octv] [intvl + 1], R, C-1)-<br/> pixval32f (dog_pyr [octv] [intvl-1], R, c + 1) + <br/> pixval32f (dog_pyr [octv] [intvl-1], R, C-1)/4.0; <br/> DYS = (pixval32f (dog_pyr [octv] [intvl + 1], R + 1, C) -<br/> pixval32f (dog_pyr [octv] [intvl + 1], R-1, c)-<br/> pixval32f (dog_pyr [octv] [intvl-1], R + 1, c) + <br/> pixval32f (dog_pyr [octv] [intvl-1], R-1, c)/4.0; </P> <p> H = cvcreatemat (3, 3, cv_64fc1); <br/> cvmset (H, 0, 0, dxx); <br/> cvmset (H, 0, 1, dxy ); <br/> cvmset (H, 0, 2, DxS); <br/> cvmset (H, 1, 0, dxy); <br/> cvmset (H, 1, 1, dyy); <br/> cvmset (H, 1, 2, Dys); <br/> cvmset (H, 2, 0, DxS ); <br/> cvmset (H, 2, 1, Dys); <br/> cvmset (H, 2, 2, DSS); </P> <p> return h; <br/>}

 

Refer:

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

(2) opencv Implementation of Sift