Q: Second derivative test for three-dimensional function
Cueball
I have been writing an algorithm that will detect peaks in an image, and have been using the second derivative test to localize possible
maxima candidates in a set of points (x,y) each of which with an intensity value F.
Now I would like to do the same for a three-dimensional data sequence, containing points (x,y,z) with an intensity value F. Looking at
http://mathworld.wolfram.com/SecondDerivativeTest.html, I would assume the discriminant can be used here as well, just pushed up another
dimension. But I wanted to make sure. Does anyone know what the generalized second derivative test look like?
TIA,
-Mark
Karl
if you have a point where the gradient of a function f is equal to the
zero vector, you can check if there is a local maximum by finding out if
the Hessian there is negative definite, i.e. all its eigenvalues are
negative. if this is true, there is a local maximum at the point.
Ciao
Karl
David Ziskind
<7ipsn013ultq9blia...@4ax.com> about the availability of
a test for identifying relative extrema for an R^3 -> R function.
This is a traditional topic in (advanced) calculus. Some texts go into
the n dimensional case (here n is 3), others stop with the two
dimensional situation.
The general idea is to determine if the Hessian matrix is positive
definite (or negative definite). The "Hessian matrix corresponding to
function F" (Hessian(F)) is defined to be: the matrix whose term in row
i, column j is the j th partial derivative of the i th partial
derivative of F.
The following formulae hold (F is assumed to be twice continuously
differentiable):
1. If [ Hessian(F) at a is negative definite
and each (first partial of F at a) = 0]
then F has a relative maximum at a.
2. A matrix M is negative definite
IFF(def) [ M is real, symmetric
and For all x~=0, Sum{i: j: M_ij*x_i*x_j < 0}]
3. If M is a real, symmetric matrix
then [M is negative definite IFF all the eigenvalues of M are
negative].
[Note that this test is ideal for the situation where M is a
Hessian (hence, real and symmetric).]
In addition to the preceding core formulae, there is also "Sylvester's
Criterion" for determining negative definiteness (see below).
Due to the amount of territory that has to be traveled in order to
obtain all of the results in this note, I will mention that I am
drawing from material in the references listed -- I have not re-derived
these results.
Notes:
1. More information can be found in the following references:
i) Korn, G., & Korn, T., “Mathematical Handbook for Scientists and
Engineers”, Dover, 2000, ISBN 0-486-41147-8;
ii) Bronshtein., I & Semendyayev, I., "Handbook of Mathematics", Van
Nostrand Reinhold, New York, 1985, ISBN 0-442-21171-6; and
iii) Widder, D., "Advanced Calculus", Dover, 1989, ISBN 0-486-66103-2.
2. Sylvester's Criterion, which may be computationally easiest, is
given in i), and (especially) iii).
3. For the situations where Hessian(F) is "negative semi-definite", or
"indefinite"; see references i) or ii).
In the OP, the language seems to indicate that the function F being
considered, is defined by discretized data. The above rules represent
the idealized situation where one has analytic formulae for the
derivatives of F. To transfer this to the realm where the derivatives
are given by numerical data will create some additional complexity.
David Ziskind
zis...@ntplx.net