[Insight-developers] itkPoint from a math perspective

[Luis Ibanez]

On Thu, 19 Jul 2001, George Stetten wrote:
> Is this what you mean?
>
> Since gradients are "intensity per unit distance," changing units from feet to inches makes the
> gradient get smaller (in numbers), while it makes distance get larger (in numbers).
>

Yes,

That's a very clear way to interpret it,
at least for gradients of functions.

a gradient of 12 graylevels / feet is
a gradient of  1 graylevel  / inch

so applying a factor of 1/12 to the basic
measure unit (the coordinate system), you
get the same factor (1/12) in gradient
components. (so gradients are "Covariant")

On the other hand, distances (the components
of vectors) got multiplied by 12 when measured
in inches as opposed to feet. (so distance
vectors are "Contravariant").

A non-isotropic scale, for example, will use
feet to measure distances in X, while using
inches to measure distances in Y.

That will make distance vectors to rotate in
one direction while gradient vector will rotate
in the other.



BTW, this nice explanation suggest that
     we should "volunteer" George for
     writting some of the tutorials   :-)


Luis