[Insight-users] BSplineDeformableTransform parameters?

[Ramón Casero Cañas]

Luis Ibanez wrote:
> 
> 3) When we refer to Cubic BSplines, we are talking about the largest order
> of
>     the polynomials used for the interpolation.  In this case, = 3.


Hi Luis,

Maybe the following note is appropriate.

"Polynomial order" and "polynomial degree" are used interchangeably, 
although WoframMathWorld notes [0]:

<QUOTE>
Polynomial Order: The highest order power in a univariate polynomial is 
known as its order (or, more properly, its polynomial degree). [...] It 
is preferable to use the word "degree" for the highest exponent in a 
polynomial, since [...]
</QUOTE>

But I think that in B-splines it actually makes a difference, as the 
following nomenclature is common:

	* "degree" is used for the maximum degree of the polynomial in
	each piece,

	* "order = degree + 1" is the number of coefficients.

Thus, isn't it quite standard to say that a cubic spline has degree 3 
and order 4?

For example, see Wikipedia [1], Matlab Spline Toolbox [2], Rogers1990 
(p. 305) [3], Dubuc1999 [4], or more explicitly in Bankman2000 [5]:

<QUOTE>
The cubic B-spline is a piecewise polynomial function of degree 3. It 
does not correspond to what is generally understood as cubic 
convolution, the latter being a Key's function made of piecewise 
polynomials of degree 3 (like the cubic B-spline) and of maximal order 3 
(contrary to the cubic B-spline, for which the order is 4).
</QUOTE>


However, what you mean above is that in ITK, e.g. 
itk::BSplineDeformableTransform< TScalarType, NDimensions, VSplineOrder 
 > [6], "order" is used in the sense of "degree", isn't it?




[0] http://mathworld.wolfram.com/PolynomialOrder.html

[1] http://en.wikipedia.org/wiki/Spline_(mathematics)

[2] 
http://www.mathworks.com/access/helpdesk/help/toolbox/splines/index.html?/access/helpdesk/help/toolbox/splines/&http://www.mathworks.com/access/pagenotfound.html

[3] D.F. rogers, "Mathematical Elements for Computer Graphics", 
McGraw-Hill, New York, 1990.

[4] S. Dubuc and G. Deslauriers, "Spline functions and the theory of 
wavelets", AMS Bookstore, 1999. 
http://books.google.com/books?id=Lo3WD-ByHwIC&pg=PA90&dq=spline+degree+order

[5] I.N. Bankman, "Handbook of medical imaging", Academic Press, 2000. 
http://books.google.com/books?id=nSFeCrYDylMC&pg=RA2-PA397&dq=spline+degree+order

[6] 
http://www.itk.org/Doxygen/html/classitk_1_1BSplineDeformableTransform.html



Cheers,

Ramon.

-- 
Ramón Casero Cañas, DPhil

Computational Biology, Computing Laboratory
University of Oxford
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