Hessian Matrix

In mathematics, the Hessian matrix (or simply the Hessian) is the square matrix of second-order partial derivatives of a function; that is, it describes the local curvature of a function of many variables. The second derivative can be applied at a non-degenerate critical point x. If the Hessian is positive definite at x, then f attains a local minimum at x. If the Hessian is negative definite at x, then f attains a local maximum at x. If the Hessian has both positive and negative eigenvalues then x is a saddle point for f (this is true even if x is degenerate). Otherwise the test is inconclusive. In view of what has just been said, the second derivative test for functions of one and two variables is simple. In one variable, the Hessian contains just one second derivative; if it is positive then x is a local minimum, if it is negative then x is a local maximum; if it is zero then the test is inconclusive. In two variables, the discriminant can be used, because the determinant is the product of the eigenvalues. If it is positive then the eigenvalues are both positive, or both negative. If it is negative then the two eigenvalues have different signs. If it is zero, then the second derivative test is inconclusive(wikipdia).

consider a continuous function of two variables such that the value of the function at (x,y) is given by f(x,y). The Hessian matrix, H, is the matrix of partial derivates of the function f.



The determinant of this matrix, known as the discriminant, is calculated by:


The value of the discriminant is used to classify the maxima and minima of the function by the second order derivative test. Since the determinant is the product of eigenvalues of the Hessian we can classify the points based on the sign of the result. If the determinant is negative then the eigenvalues have different signs and hence the point is not a local extremum; if it is positive then either both eigenvalues are positive or both are negative and in either case the point is classified as an extremum.

NOTE:

We can calculate the derivatives by convolution with an appropriate kernel. In the case of SURF the second order scale normalised Gaussian is the chosen filter as it allows for analysis over scales as well as space. We can construct kernels for the Gaussian derivatives in x, y and combined xy direction such that we calculate the four entries of the Hessian matrix. Use of the Gaussian allows us to vary the amount of smoothing during the convolution stage so that the determinant is calculated at different scales. Furthermore, since the Gaussian is an isotropic (i.e. circularly symmetric) function, convolution with the kernel allows for rotation invariance.

Lowe found performance increase in approximating the Laplacian of Gaussians by a difference of Gaussians. In a similiar manner, Bay proposed an approximation to the Laplacian of Gaussians by using box filter representations of the respective kernels.