Hessian matrix, how to combine Ixx & Iyy together?

Question

"Before extracting the lines, you need to detect potential points on them. Apply a Gaussian filter first and use the Sobel filters as derivative operators. Threshold the determinant of the Hessian and then apply non-maximum suppression in 3 × 3 neighborhoods. Ignore pixels for which any of the filters falls even partially out of the image boundaries."

I understand to gaussian an image first to eliminate noise, then take twice with Sobel_x and Sobel_y, respectively, which became Ixx and Iyy in Hessian that would show horizontal line and vertical line in image.But how am I suppose to get Ixxyy? But how could I combine these two image together to make Ixxyy as the right bottom in Hessian matrix?

Answer 1

The two off-diagonal elements of the Hessian matrix are d^2/dxdy. That is, they are the first derivative along y applied to the first derivative along x.

If the top-left element is obtained by Sobel_x( Sobel_x( image )), and the bottom-right element is Sobel_y( Sobel_y( image )), then the two other elements are both Sobel_y( Sobel_x( image )) or, equivalently, Sobel_x( Sobel_y( image )) (note that these two should be identical).

Do take into account that negative values are important here, and you should thus be careful to compute the Sobel filter in a way that preserves those negative values—don't store them in an unsigned integer array!