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.
An you got your bezier-spline/catmull-rom-spline


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.

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