Hessian Matrix of the image

What information does a Hessian Matrix of an image provide? Does it provide the information of the stable points? What is Hessian matrix used for?

Hessian matrix describes the 2nd order local image intensity variations around the selected voxel. For the obtained Hessian matrix, eigenvector decomposition extracts an orthonormal coordinate system that is aligned with the second order structure of the image. Having the eigenvalues and knowing the (assumed) model of the structure to be detected and the resulting theoretical behavior of the eigenvalues, the decision can be made if the analyzed voxel belongs to the structure being searched.

The figure below illustrates the correspondence between eigenvalues of the hessian operation on the image and the local features (corner, edge, or flat region).

The Hessian operator is also widely used in 3D images, and it can reflect more local features:

It is widely used in vessel detection in medical images. For more details, please see M.Rudzki et al's Vessel Detection Method Based on Eigenvalues of the Hessian Matrix and its Applicability to Airway Tree Segmentation