Hessian affine region detector

The Harris affine detector relies on interest points detected at multiple scales using the Harris corner measure on the second-moment matrix. The Hessian affine also uses a multiple scale iterative algorithm to spatially localize and select scale and affine invariant points. However, at each individual scale, the Hessian affine detector chooses interest points based on the Hessian matrix at that point:

${\displaystyle H(\mathbf {x} )={\begin{bmatrix}L_{xx}(\mathbf {x} )&L_{xy}(\mathbf {x} )\\L_{yx}(\mathbf {x} )&L_{yy}(\mathbf {x} )\\\end{bmatrix}}}$

where ${\displaystyle L_{aa}(\mathbf {x} )}$ is second partial derivative in the ${\displaystyle a}$ direction and ${\displaystyle L_{ab}(\mathbf {x} )}$ is the mixed partial second derivative in the ${\displaystyle a}$ and ${\displaystyle b}$ directions. It's important to note that the derivatives are computed in the current iteration scale and thus are derivatives of an image smoothed by a Gaussian kernel: ${\displaystyle L(\mathbf {x} )=g(\sigma _{I})\otimes I(\mathbf {x} )}$. As discussed in the Harris affine region detector article, the derivatives must be scaled appropriately by a factor related to the Gaussian kernel: ${\displaystyle \sigma _{I}^{2}}$.

At each scale, interest points are those points that simultaneously are local extrema of both the determinant and trace of the Hessian matrix. The trace of Hessian matrix is identical to the Laplacian of Gaussians (LoG):[5]

${\displaystyle {\begin{aligned}DET=\sigma _{I}^{2}(L_{xx}L_{yy}(\mathbf {x} )-L_{xy}^{2}(\mathbf {x} ))\\TR=\sigma _{I}(L_{xx}+L_{yy})\end{aligned}}}$

As discussed in Mikolajczyk et al.(2005), by choosing points that maximize the determinant of the Hessian, this measure penalizes longer structures that have small second derivatives (signal changes) in a single direction.[6] This type of measure is very similar to the measures used in the blob detection schemes proposed by Lindeberg (1998), where either the Laplacian or the determinant of the Hessian were used in blob detection methods with automatic scale selection.

Like the Harris affine algorithm, these interest points based on the Hessian matrix are also spatially localized using an iterative search based on the Laplacian of Gaussians. Predictably, these interest points are called Hessian–Laplace interest points. Furthermore, using these initially detected points, the Hessian affine detector uses an iterative shape adaptation algorithm to compute the local affine transformation for each interest point. The implementation of this algorithm is almost identical to that of the Harris affine detector; however, the above mentioned Hessian measure replaces all instances of the Harris corner measure.