Hessian Identifier

The problem of identifying notable points that can be easily recognized between two images has initially been addressed by shifting the focus towards detecting corner points in the image, thereby discarding those portions of the image that lack texture or contain only edges.

The Hessian operator (Hessian detector) (Bea78), based on the Hessian matrix derived from the Taylor series expansion around the point to be described, seeks those regions of the image that exhibit strong derivatives in orthogonal directions. This algorithm is based on the analysis of the matrix of second derivatives, known as the Hessian.

$$\mathbf{H}(\mathbf{x}, \sigma)=\begin{bmatrix} I_{xx}(\mat... ...thbf{x}, \sigma) & I_{yy}(\mathbf{x}, \sigma)\\ \end{bmatrix}$$ (5.1)

The algorithm computes the second derivatives of the image $I_{xx}$, $I_{xy}$, $I_{yy}$ for each point in the image and identifies the points where the determinant of the Hessian

$$\det \left(\mathbf{H}(\mathbf{x}, \sigma) \right) = I_{xx}(... ...a) I_{yy}(\mathbf{x}, \sigma) - I_{xy}^{2}(\mathbf{x}, \sigma)$$ (5.2)

reaches its maximum. This search is typically performed on the image of the Hessian determinant, to which a Non-Maxima Suppression is applied over a window $3 \times 3$. The maxima of the Hessian determinant response are usually located at corners and in areas of the image with strong texture. The use of the Hessian determinant makes this algorithm invariant to rotation.

In practical applications, the original image is never used; instead, a low-pass filtered version is obtained through a Gaussian filter.