[Insight-developers] itkPoint from a math perspective

[George Stetten]

Luis,

I've never been clear what the itkCovariantVector is for.

You say:

itkCovariantVector = itkVector X itkVector (cross product)

What do you use them for and what restrictions apply to their values?

Thanks

George

Luis Ibanez wrote:

> Damion Shelton wrote:
>
> > seedPos.Get_vnl_vector().normalize();
>
> an itkPoint is a position in space.
> Normalization is not defined for points.
> (it doesn't makes sense to normalize the
> components of a position).
>
> In fact, returning a vnl_vector from an
> itkPoint is just a backdoor for type missuse.
>
> Normalization is defined for itkVectors
> which are the relative position of one
> itkPoint with respect to another.
> (in this case the normalized vector
> codes for a direction in space).
>
> If normalization is required in a function,
> maybe the itkPoint is not the expected parameter.
>
> What is this particular function doing ?
>
> >
> > This is a relatively simple operation - for more complex operations (dot
> > products, for instance), I found it neccessary to duplicate Point data in
> > order to avoid horribly messy code.
>
> Dot product is not defined over points in space,
> but over vectors, representing the projection of
> one vector over the other.
>
> Could you post the example of what the function is
> intended to compute ?
>
> The typical experience in computer graphics is that
> these types are often used in the wrong sense, just
> because all look like n-d arrays.
>
> But their difference are quite important if they
> are mapped by a affine transform.
>
> >
> > To summarize: Is there an easy way to interchange
> > Points, CovariantVectors, and vnl_vector's?
>
> yes:
>
> itkVector = itkPoint - itkPoint
> itkPoint  = itkPoint +/- itkVector
>
> itkPoint + itkPoint = Undefined
>
> itkCovariantVector = itkVector X itkVector (cross product)
>
> scalar = itkVector * itkVector (dot product)
>
> Now, all three, itkPoint, itkVector and itkCovariantVector
> have a vnl_vector() method, the only justification for this
> conversion is to use any of the vnl methods that requires
> a vnl_vector as an input. This conversion will imply a hit
> in performance.
>
> > I'm concerned that having to explicity convert back and
> > forth between data types is an impediment to clear code, and possibly a
> > speed handicap as well, since these interconversions would occur many
> > hundreds of thousands of times in my code.
>
> Agree on the performance side,
> but it is important to be careful with the missuse of the types.
> an itkPoint is different from an itkVector, and an itkCovariantVector.
> They are all different mathematical concepts. The possibility of
> converting
> them to vnl_vectors was provided only because vnl doesn't make this
> distintion, and use only vnl_vector in the linear algebra sense as
> opposed of the geometric sense.
>
> In other words, a vnl_vector is like a Nx1 vnl_matrix,
> or like an array in Matlab.
>
> Luis
>
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