Corner Detection - The Multi-scale Harris Operator

The Multi-scale Harris Operator

The computation of the second moment matrix (sometimes also referred to as the structure tensor) in the Harris operator, requires the computation of image derivatives in the image domain as well as the summation of non-linear combinations of these derivatives over local neighbourhoods. Since the computation of derivatives usually involves a stage of scale-space smoothing, an operational definition of the Harris operator requires two scale parameters: (i) a local scale for smoothing prior to the computation of image derivatives, and (ii) an integration scale for accumulating the non-linear operations on derivative operators into an integrated image descriptor.

With denoting the original image intensity, let denote the scale space representation of obtained by convolution with a Gaussian kernel

with local scale parameter :

and let and denote the partial derivatives of . Moreover, introduce a Gaussian window function with integration scale parameter . Then, the multi-scale second-moment matrix can be defined as

\mu(x, y; t, s) = \int_{\xi = -\infty}^{\infty} \int_{\eta = -\infty}^{\infty} \begin{bmatrix} L_x^2(x-\xi, y-\eta; t) & L_x(x-\xi, y-\eta; t) \, L_y(x-\xi, y-\eta; t) \\ L_x(x-\xi, y-\eta; t) \, L_y(x-\xi, y-\eta; t) & L_y^2(x-\xi, y-\eta; t) \end{bmatrix} g(\xi, \eta; s) \, d\xi \, d\eta.

Then, we can compute eigenvalues of in a similar way as the eigenvalues of and define the multi-scale Harris corner measure as

.

Concerning the choice of the local scale parameter and the integration scale parameter, these scale parameters are usually coupled by a relative integration scale parameter such that, where is usually chosen in the interval . Thus, we can compute the multi-scale Harris corner measure at any scale in scale-space to obtain a multi-scale corner detector, which responds to corner structures of varying sizes in the image domain.

In practice, this multi-scale corner detector is often complemented by a scale selection step, where the scale-normalized Laplacian operator

is computed at every scale in scale-space and scale adapted corner points with automatic scale selection (the "Harris-Laplace operator") are computed from the points that are simultaneously:

  • spatial maxima of the multi-scale corner measure
  • local maxima or minima over scales of the scale-normalized Laplacian operator :

Read more about this topic:  Corner Detection

Famous quotes containing the word harris:

    “Mother came to us destitute. She brings a child into the world, takes one look at him and promptly dies. Without leaving so much as a forwarding name and address.”
    —Vernon Harris (c. 1910)