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The Hessian matrix plays an important role in Morse theory and catastrophe theory, because its kernel and eigenvalues allow classification of the critical points. [2] [3] [4] The determinant of the Hessian matrix, when evaluated at a critical point of a function, is equal to the Gaussian curvature of the function considered as a manifold. The ...
The following test can be applied at any critical point a for which the Hessian matrix is invertible: If the Hessian is positive definite (equivalently, has all eigenvalues positive) at a, then f attains a local minimum at a. If the Hessian is negative definite (equivalently, has all eigenvalues negative) at a, then f attains a local maximum at a.
A detailed analysis of the selection properties of the determinant of the Hessian operator and other closely scale-space interest point detectors is given in (Lindeberg 2013a) [1] showing that the determinant of the Hessian operator has better scale selection properties under affine image transformations than the Laplacian operator.
The Hessian affine region detector is a feature detector used in the fields of computer vision and image analysis.Like other feature detectors, the Hessian affine detector is typically used as a preprocessing step to algorithms that rely on identifiable, characteristic interest points.
The determinant of the Hessian matrix is used as a measure of local change around the point and points are chosen where this determinant is maximal. In contrast to the Hessian-Laplacian detector by Mikolajczyk and Schmid, SURF also uses the determinant of the Hessian for selecting the scale, as is also done by Lindeberg.
Then the Gaussian curvature of the surface at p is the determinant of the Hessian matrix of f (being the product of the eigenvalues of the Hessian). (Recall that the Hessian is the 2×2 matrix of second derivatives.) This definition allows one immediately to grasp the distinction between a cup/cap versus a saddle point.
The graph colouring techniques explore sparsity patterns of the Hessian matrix and cheap Hessian vector products to obtain the entire matrix. Thus these techniques are suited for large, sparse matrices. The general strategy of any such colouring technique is as follows. Obtain the global sparsity pattern of
where | g | is the absolute value of the determinant of the matrix of scalar coefficients of the metric tensor . These are useful when dealing with divergences and Laplacians (see below). The covariant derivative of a vector field with components is given by: