Building upon this scale-adapted second-moment matrix, the "'Harris-Laplace "'detector is a twofold process : applying the Harris corner detector at multiple scales and automatically choosing the " characteristic scale ".
22.
From representing the six special cases, we see a clear advantage of the moment matrix representation, i . e ., it allows a unified representation for seemingly diverse types of knowledge, including linear equations, joint and conditional distributions, and ignorance.
23.
One of these properties is that the square root of the second-moment matrix, M ^ { \ tfrac { 1 } { 2 } } will transform the original anisotropic region into isotropic regions that are related simply through a pure rotation matrix R.
24.
As discussed with the Harris detector, the eigenvalues and eigenvectors of the second-moment matrix, M = \ mu ( \ mathbf { x }, \ Sigma _ I, \ Sigma _ D ) characterize the curvature and shape of the pixel intensities.
25.
The computation of the second moment matrix ( sometimes also referred to as the structure tensor ) A in the Harris operator, requires the computation of image derivatives, and ( ii ) an " integration scale " for accumulating the non-linear operations on derivative operators into an integrated image descriptor.
26.
Furthermore, it was shown that all these differential scale-space interest point detectors defined from the Hessian matrix allow for the detection of a larger number of interest points and better matching performance compared to the Harris and Shi-and-Tomasi operators defined from the structure tensor ( second-moment matrix ).
27.
The matrix \ mathbf X ^ { \ rm T } \ mathbf X is known as the Gramian matrix of \ mathbf X, which possesses several nice properties such as being a positive semi-definite matrix, and the matrix \ mathbf X ^ { \ rm T } \ mathbf y is known as the moment matrix of regressand by regressors.
