Feature Detection And Scale Space
Feature Detection at a given scale in scale-space
Consider scale-space representation of image \(I(x, y)\):
\[L(x, y, \sigma) = G(x, y, \sigma) * I(x, y),\]where \(\sigma \in \mathbb{R}_+\) is the scale parameter, and \(G(x, y, \sigma)\) is the two-dimensional variable-scale Gaussian kernel written as
\[{G(x, y, \sigma) = \frac{1}{2\pi \sigma^2} e^{ -(x^2 + y^2) / 2 \sigma ^ 2}}.\]The 2-jet at a given image point and a given scale contains the partial derivatives
\[(L_x, L_y, L_{xx}, L_{xy}, L_{yy}).\]From the five components in the 2-jet, four differential invariants can be constructed, which are invariant to local rotations; the gradient magnitude \(\|\nabla L\|\), the Laplacian \(\nabla ^2 L\), the determinant of the Hessian \(det\mathcal{H}L\) and the rescaled level curve curvature \(\tilde{\mathbb{k}}(L)\):
- \(\|\nabla L\|^2 = L_x^2 + L_y^2\),
- \(\nabla ^2 L = L_{xx} + L_{yy}\),
- \(det\mathcal{H}L = L_{xx}L_{yy}-L_{xy}^2\),
- \(\tilde{\mathcal{k}}(L)=L_x^2L_{yy},+L_y^2L_{xx}-2L_xL_yL_{xy}\).
These operators can detect different types of local structures:
- A single-scale blob detector can be expressed from the minima and the maxima of the Laplacian response \(\nabla^2L\).
- An affine covariant blob detector or a saddle detector can be expressed from the maxima and the minima of the determinant of the Hessian \(det \mathcal{H}L\).
- A straightforward and affine covariant corner detector can be expressed from the maxima and minima of the rescaled level curve curvature \(\tilde{\mathbb{k}}(L)\).
Some conclusions drawn by SURF:
- Hessian-based detectors are more stable and repeatable than Harris-based counterparts.
- Using the determinant of the Hessian matrix rather than its trace (the Laplacian) seems advantageous, as it fires less elongated, ill-localised structures.
Reference
Lindeberg, Tony. “Scale-space.”
The impressive graphs of Gaussian derivative functions
