Converting a b-spline curve to a catmull rom spline

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Hello everyone, I have a b spline that is represented by 30 control points and I want to convert it to equivalant catmull rom spline with the same number of control points. What is the best way that I can achieve this. I know I must invert the Catmull rom spline basis matrix and multiply it with the bezier basis matrix but is there a way to transform all the control points in one sweep? The basis matrix would be 4 X 4, and my control point vector has 30 elements. So, how is this supposed to work? Cheers, xarg [Edited by - xargon123 on July 6, 2008 3:35:42 AM]

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Hello,

A bit more detail:
It seems it should be as follows (found it after googling):

T x B1 x G1 = T X B2 x G2

G1 = B1'x B2 x G2

B1 = Basis matrix for spline 1.
B2 = Basis matrix for spline 2.
G1 = control points for spline 1.
G2 = control points for spline 2.

So, do I transform each of the conrol points according to this equation. So, if I have 100 control points, I should loop over them and transform each of them separately according to this equation? Also, what if my control points are 2D, should I pad them with 2 0s to make them a 4D point considering that the multiplying matrix would be 4 x 4?

Many thanks.

Cheers,
xarg

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A Bezier curve with 30 control points is described by a 29th-order polynomial, and cannot be described (except in ridiculously degenerate cases) by a cubic spline such as a Catmull-Rom spline. The best you'd be able to do would be approximating it. Are you sure you actually have a Bezier curve, rather than a B-spline?

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Sorry! Yes, it is a b-spline, not a bezier!

xarg

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Catmull-Rom splines are just C1 continuous bezier curves, hence bezier-splines(not the same as bspline!!)

If you have a bspline, you can convert it to a bezier-spline by inserting the knots repeatedly to get multiplicity n for inner knots and on the borders you need to clamp the bspline to get endpoint interpolation.

If n is your degree and the knot vector is [t0,...tn+m]
you need to insert tn and tm up to multiplicity n+1
(called clamping)

Use the algorithm of Boehm to insert knots.
Note(detail of the boehm algorithm):
Inserting tm is valid in this case, because the linear weight for the right side of the last control point is zero.

Your knot vector should look like this(n=3):
[t0,...,t2,tn,tn,tn,tn,tn+1,tn+1,tn+1,...,tm,tm,tm,tm,tm+1,..,tm+n]

You can remove the last and first n control points & knots, since these are not part of the curve.

Note: for a parameter u < tn or u > tm the sum of the bspline basis functions is <1.0, thus these segments can be removed, since it is stated, that the basis functions need to be a convex combination (sum = 1.0)

You will have a knot vector like this now(n=3):
[00001112223333]
each interval [0,1][1,2][2,3] is now a degree 3 bezier curve.

I hope that helps, I'll check this thread if you have further questions.

P.S.: "The Nurbs Book" is a pretty interesting read for this kind of work.

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Hello,

Thanks for your reply. I guess I started the post wrongly and confused everyone.

So, what I have is a set of uniform data points and I have fitted a b-spline to it in the least square sense. Now, for rendering I want to use a Catmull Rom spline as they have the nice property that the spline passes through the control points.

I had googled and found pages like this that described this simple matrix inversion and multiplication with the other spline basis function to transform the control points:

One example is here:
http://groups.csail.mit.edu/graphics/classes/6.837/F04/assignments/assignment8/>

http://www.cs.virginia.edu/~gfx/Courses/2002/Intro.fall.02/Lectures/lecture23.ppt (Slide 32, for example).

So, I thought it would be something simple as they have described. Unfortunately, I do not quite understand the solution that you prescribed but will try and figure it out.

Please do post any suggestions you have. I would be really grateful as I am struggling quite a bit with this! My whole weekend has vanished!

Cheers,
xarg

[Edited by - xargon123 on July 6, 2008 6:03:40 AM]

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if you have a knot vector, pick a knot and insert it up to multiplicity n, then the curves will go through through the corresponding control point. So the segment defined by an interval is just a bezier curve of the same degree as your bspline.

Lookup "bsplines multiple knots"

Here is a graphical illustration of the influence of knots insertions, look at picture e) the spike is 1.0, thus the curve goes through the control point.

http://www.cs.mtu.edu/~shene/COURSES/cs3621/NOTES/spline/B-spline/bspline-property.html

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