Base of target detection hessian matrix---haisen matrices

Is the sea race (sea color) matrix, search on the internet has.

In mathematics, a sea-color matrix is a square matrix of second-order partial derivatives of an independent variable as a real-valued function of a vector.

Hessian matrices are second-order partial derivative matrices of multidimensional variable functions, H (I,J) =d^2 (f)/(d (XI) d (XJ))



1. Definitions of extreme values (maxima or minima)

There is a function defined on the area D  RN y=f (x) =f (x1,..., xn). For an inner point of region D x0= (X10,..., xn0), if there is a neighbor ud of x0, it makes

             F (x) ≤f (x0)     x∈u

     The x0 is called the Maximal point of f (X), and F (x0) is called the maximum value of f (x).

     Conversely, as

             F (x) ≥f (x0)     x∈u

     The x0 is called the minimum of f (x), and F (x0) is called the minimum of f (x).

2. Sea Race (Hessian) matrix

  Set function y=f (x) =f (x1,..., xn) in a neighborhood of Point x0= (X10,..., xn0) where all second derivative derivatives are contiguous, then the following Matrix H is f (x) in the x0 point of the sea-race matrix.

Obviously the sea-race matrix is symmetrical, so that all its characteristic roots are real numbers.

3. Extremum

The necessary conditions for existence

If x0 is the extreme point of f (x), if present, then

Further, the second derivative is contiguous in one neighborhood, and H is the X0 matrix at the point.

(1) X0 is the minimum point of f (x) h≥0, that is, the characteristic roots of H are non-negative.

(2) X0 is the maximal point of f (x) h≤0, that is, the characteristic root of H is non-positive.

If there is a x0 point, it is said that x0 is the critical point of f (X), and F (x0) is the critical value.

4. Sufficient conditions for the existence of extreme values

Set F (x) in a neighborhood of x0 in which all second derivative derivatives are contiguous, and x0 is the critical point (i.e.) of f (x) and H is f (x) at the x0 point of the sea-race matrix, then

(1) h>0, i.e. H is a positive definite matrix x0 is the minimum of f (x).

(2) h<0, that is, H is a negative-definite matrix x0 is the maximal point of f (x).

(3) The characteristic root of H has a positive negative x0 not an extreme point of f (x).

(4) In the remaining case, it is not possible to determine whether x0 is or is not an extreme point of f (x).

5. Sufficient conditions for the existence of a two-dollar function extremum

As a special case of 4. The sufficient conditions for the existence of the extremum of two-yuan function are observed.

Set Z=f (x, y) in a neighborhood (x0,y0) in which all second derivative derivatives are contiguous, and,

Remember.

Well, the sea-race matrix.

(1) If a>0,deth=ac-b2>0, then H positive definite, thus (x0,y0) is f (x, y) of the minimum point.

(2) If a<0,deth=ac-b2>0, then H is negative, thus (X0,Y0) is the maximum point of f (x, y).

(3) If deth=ac-b2<0, the characteristic root of H has positive negative, thus (X0,Y0) is not the extreme point of f (x, y).

(4) If deth=ac-b2=0, it is not possible to determine whether (x0,y0) is the extreme point of f (x, y).

6. Conditional extremum

The function y=f (x) =f (x1,..., xn) x∈drn (1),

Under Constraints: QK (x) =qk (x1,..., xn) =0,k=1,..., m,m<n (2),

The extremum, called the conditional extremum problem.

Here, it is assumed that the rank of the Jacobian matrix is m everywhere in D, that is to say that the M constraints are independent.

Direct Surrogate method

The M variables are solved directly from the constraint (2), substituting into (1), and the problem is converted to the direct extremum problem of n-m variable function.

Lagrange (Lagrange) Multiplier method

Introducing Lagrange Functions:

(3)

Where Λ1,...,λm is called Lagrange multiplier, it is the undetermined constant.

The conditional extremum problem (1) and (2) can be converted to the direct extremum problem of Lagrange function (3).

(1) If the x0 is a condition extreme point of (1) and (2), then x0 satisfies the equations

The point of satisfying the above-mentioned equations is called the critical point of the conditional extremum problem. It is obvious that the extremum point is a point, and the threshold must not be extreme.

(2) If X0 is the critical point, HL is the Lagrangian function L at X0 Point of the sea-race matrix, you can be given in 4 of the extremum exists sufficient conditions, by HL positive definite, negative or indefinite, judging x0 is a minimum, maximum point or not extreme point.
Http://zhidao.baidu.com/link?url=p1cPMKHMIGidZRYfTDDP5RwTW9sAe0xPk4Y-DQR03htxWCNFElxq1Ql809b17ROi8GKZctHnReZadk_xw5Qpwa
Comparison of http://blog.csdn.net/memray/article/details/9174705 Jacobian and Haisen matrices
http://zh.wikipedia.org/wiki/Haisen Matrix Wiki Encyclopedia

Base of target detection hessian matrix---haisen matrices